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A final assignment Report on

“Solution of Non-dimensional Navier-Stokes equations for Flow and Heat transfer by using Scilab CFD codes in LID DRIVEN CAVITY and study of variation of non-dimensional, convergence criteria parameters with different grid structure” Submitted by

S.N.Topannavar [email protected] Cell: +91 9480397798 Sub center: KIT, Kolhapur

Ten Day ISTE Main Workshop on Computational Fluid Dynamics (CFD) Conducted by

Indian Institute of Technology Bombay

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Index Chapter

Content

Page No.

I

Introduction

3-4

II

Literature survey with conclusions

5-13

III

Objectives of the work

14

IV

Physical description of the problem and models for simulations

15-18

V

Mathematical modeling with boundary conditions

19-20

VI

Validation study

21

VII

Results and Discussions

22-91

VIII

Scilab CFD codes used in the problem and algorithm

92-125

IX

Conclusion

126

X

References

127

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-I

Introduction Over the last three decades, the so-called lid-driven cavity flow problem has received considerable attention mainly because of its geometric simplicity, physical abundance, and its close relevance to some fundamental engineering flows. While some fundamental flow phenomena have become clear to us through twodimensional solutions. The recent progress in numerical analyses and computer hardware have made it possible to numerically analyze unsteady flow problems by solving their corresponding Navier-Stokes equations with a large number of grid points within a three-dimensional domain. In a parallel development, a considerable number of experimental studies on this problem have been done since the early 1980s. Due to the relatively inexpensive high speed computers, numerical simulation approach, such as computational fluid dynamics (CFD), is widely adopted for investigating realistic and research problems. Numerical simulation has full control on computing the parameters of problems of different complexities. Therefore, it is able to provide a compromising solution among cost, efficiency and complexity to engineering problems. Although high speed computers and robust numerical techniques have been developed rapidly, the computation of turbulence at high Reynolds number using direct numerical simulation (DNS) is too expensive for practical problems. The large-eddy simulation (LES) is an alternative that demands relatively less computational load. However, it still requires huge amount of computation resources for simulations conducted on sequential computers. The recent advance of supercomputers provides a possibility for conducting these large scale computations. Sequential computer codes could be parallelized directly by compilers but it is unable to fully utilize supercomputers. Therefore, innovative parallel solution techniques are necessary for exploring the power of parallel computing. To facilitate parallel computation the domain is usually divided into several sub-domains according to the structure of the mesh Flow in a lid driven cavity is one of the most widely used benchmark problems to test steady state incompressible fluid dynamics codes. Our interest will be to present this problem as a benchmark for the steady and unsteady state solution. In order to demonstrate the grid independence, code validation and other details like time step, grid size and steadiness criteria; we taken 2D Cartesian (x, y as horizontal and vertical components) square domain of size L x L (1 unit x 1 unit for simplicity of the problem) with bottom, left and right boundaries as solid walls stationary; whereas top wall is like a long conveyor-belt, moving horizontally with a constant velocity (Uo=1unit for the simplicity of the problem). To study the influence of grid sizes, computational time steps on the convergence of the governing equation codes and to catch the oscillations in the contours of different governing parameters; we categorized problem into three models; firstly, a coarse grids i.e. 12 x 12 model secondly, a medium grids i.e. 32 x 32 model and finally, a fine grids i.e. 52 x 52 model. To study the x and y velocity component contours and steam functions in the conservation of mass and momentum; we employed Non-dimensional Navier-Stokes solver with Reynolds number, Prandl number, Grashoff number and Richardson number for the simulation of the problem. And whole study is mainly concentrated on Four different cases like Isothermal fluid flow (all walls of the are at same temperature in all time steps),Non-isothermal forced convection fluid flow ( where Grashoff number is almost negligible because buoyancy induced flow exists), Mixed Convection fluid flow ( where buoyancy as well as inertia induced; the Grashoff number is non zero value) and Natural convection fluid flow ( where inertia is negligible therefore Reynolds not taken into consideration) The study of the stability of two dimensional vortex flows of a viscous fluid is one of the fundamental problems of hydrodynamics, which concerns the problems of control of separation flows. The topological characteristics of a flow can be determined on the basis of solution of the Navier–Stokes equations. However, the number of problems that can be exactly solved in this way is limited; this being so, numerical methods are used in the majority of cases. The problem on the flow of an incompressible viscous fluid in a rectangular cavity with a moving wall is a classical fluid-mechanics problem with closed boundaries. The by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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main structural peculiarities of this flow are characteristic of other separation flows having a more complex geometry; therefore, solution of the problem on the indicated flow is used for testing and comparison of different numerical methods of integrating the Navier–Stokes equations. Comprehensive data on the vortex structure and characteristics of a flow in a rectangular cavity. On the basis of systematization and analysis of the data, steady state convergence criteria 10-4 have been adopted for estimating the quality of the discrete model used. The computation of turbulence at high Reynolds number using direct numerical simulation (DNS) is too expensive for practical problems. The large-eddy simulation (LES) is an alternative that demands relatively less computational load. However, it still requires huge amount of computation resources for simulations conducted on sequential computers. But we facilitated Microsoft Windows XP professional version 2002 operating system, Inter(R) Core(TM) Duo CPU T6670 @ 2.20 GHz processor with small memory of 1.96G, 2.19GHz speed RAM Dell Vastro lap top for large scale computations. Due to time restriction of submission of the report we skipped some fine grid computation of simulations because, it is observed that some simulations will take one and half day computation time. .

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-II

Literature survey with conclusions [1] A numerical study of the vertical flow structure in a confined lid-driven cavity which is defined by a depth-to-width aspect ratio of 1:1 and a span-to-width aspect ratio of 3:1 SAR (Spanwise Aspect Ratio) ( L : B . 1; 2; 3). A simple discretization technique ; third-order QUICK (Leonard 1979) upwind scheme, formulated on the non-uniform basis, to the nonlinear advective fluxes was applied to study carefully examined the computed data that the useful to gain an in-depth knowledge of the complex interactions among secondary eddies, primary eddies, and spiraling span wise motions. Chief of conclusions drawn from this study is to explain how the secondary eddies are intimately coupled with the primary re-circulating flow. Also enlighten in this paper why spiraling vortices inside the upstream secondary eddy tend to destabilize the incompressible flow system and aid development of laminar instabilities. Prior to describing the appearance of TGL (Taylor-GoÈrtler) vortices are studied in detail how eddies of different sizes and attributes are intimately coupled. And same is permitted a systematic approach to understanding the complex interaction among spiraling eddies. The separation surface plotted in this paper furthermore helps to show that fluid flows present in the narrow wavy trough of the separation surface have a higher propensity to develop into TGL vortices. Conclusions: [1] The geometry of the cavity examined is extraordinarily simple; the flow physics in the cavity are nevertheless rich. The physical complexity is attributable to the eddies which are characterized as possessing different sizes and characteristics. Also, how interaction proceeds among the eddies is crucial to the development into laminar instabilities. In the entire flow evolution, the transport mechanism is rooted largely in the spiraling nature of the flow motion established inside the secondary eddies and, of course, in the primary core. According to the finite volume solutions concluded with some important findings from the numerical simulation. The three-dimensional lid-driven cavity flow is manifested by the presence of a spanwise velocity component which arises due to the presence of two vertical end walls. Accompanying the span-wise motion, the flow exhibiting the dominant recirculation flow pattern is prone to spiral. It is interpreted that the presence of USE particles, which are engulfed from regions fairly near the two end walls into the primary core and then spiral monotonically towards the symmetry plane, as being the main cause leading to the flow instability because the two flow streams moving in opposite directions tend to collide with each other at the symmetry plane. This instability causes the surface separating the primary core and the upstream secondary eddy to detach from the upstream side wall. It is this distorted detachment which disrupts the well-balanced force between the centrifugal and pressure-gradient forces established inside the primary re-circulating cell. This paves the way for the onset of Taylor-GoÈrtler vortices. As the end wall is approached, particles in the downstream secondary eddy begin to be engulfed into the primary core and this is followed by suction of particles in the upstream secondary eddy, which is closer to the end wall, into the primary core through the spiral-saddle point. There exists a higher possibility that instabilities will result at spatial locations where the width of the upstream secondary eddy becomes appreciably larger than the width of the downstream secondary eddy. Computational experience from this study reveals that the size of the upstream secondary eddy and the contour lines of zero span-wise velocity at the surface, and the separation surface are closely related.. In the vicinity of the distorted v = 0 contour surface, the sign-switching span-wise velocity induces a free-shear vortex. The pressure field established to support the existence of this vortex further affects the boundary layer of the outward-running spiraling flow in the sense that a wall-shear vortex is formed near the floor of the cavity. This pair of well-established vortices, referred to as Taylor-GoÈrtler vortices, bursts from the spatial location which has the local maximum kinetic energy.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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[3] The work on stabilized finite element formulation proposed by Tezduyar applied to solve steady viscoplastic incompressible flows on unstructured grids. The formulation, originally proposed for Newtonian fluids, allows that equal-order-interpolation velocity-pressure elements are employed, circumventing the Babuska-Brezzi stability condition by introducing two stabilization terms. The first term used is the streamline upwind/Petrov-Galerkin (SUPG) introduced by Brooks and Hughes and the other one is the ressurestabilizing/ Petrov-Galerkin (PSPG) stabilization proposed initially by Hughes for Stokes flows and generalized by Tezduyar to the Navier–Stokes equations. The inexact-Newton methods associated with iterative Krylov solvers have been used to reduce computational efforts related to non-linearities in many problems of computational fluid dynamics, offering a trade-off between accuracy and the amount of computational effort spent per iteration.

A parallel edge-based solution of three dimensional viscoplastic flows governed by the steady Navier– Stokes equations is presented. The governing partial differential equations are discritized using the SUPG (streamline upwind/Petrov-Galerkin)/PSPG (pressure stabilizing/Petrov-Galerkin) stabilized finite element method on unstructured grids. The highly nonlinear algebraic system arising from the convective and material effects is solved by an inexact Newton-Krylov method. The locally linear Newton equations are solved by GMRES with nodal block diagonal pre-conditioner. Matrix-vector products within GMRES are computed edge-by-edge (EDE), diminishing flop counts and memory requirements. A comparison between EDE and element-by-element data structures is presented. The parallel computations were based in a message passing interface standard. Performance tests were carried out in representative three dimensional problems, the sudden expansion for power-law fluids and the flow of Bingham fluids in a lid-driven cavity. Results have shown that edge based schemes requires less CPU time and memory than element based solutions. The SUPG/PSPG finite element formulation with the inexact nonlinear method.

Conclusions: [3] The nonlinear character due to the non-Newtonian viscous and convective terms of the Navier–Stokes equations was treated by an inexact-nonlinear method allowing a good tradeoff between convergence and computational effort. At the beginning of the solution procedure the large linear tolerances produced fast nonlinear steps, and as the solution progresses, the inexact nonlinear method adapts the tolerances to reach the desired accuracy. The linear systems of equations within the nonlinear solution procedure were solved with the nodal block diagonal preconditioned GMRES. An edge-based data structure was introduced and successfully employed to improve the performance of the matrix-vector products within the iterative solver. The results showed that the computing time when using EDE data structure was on the average 2.5 times faster than for those problems using standard EBE. The computations were performed in a message passing parallelism environment presenting good speedup and scalability. [4] A general, efficient, accurate and reliable algorithm developed with emphasis on high Reynolds number flows that still maintains a simple algorithmic structure and which is not hampered by the diffusive time step limit. A new semi-implicit finite element algorithm for time-dependent viscous incompressible flows. The algorithm is of a general type and can handle both low and high Reynolds number flows, although the emphasis is on convection dominated flows. An explicit three-step method is used for the convection term and an implicit trapezoid method for the diffusion term. The consistent mass matrix is only used in the momentum phase of the fractional step algorithm while the lumped mass matrix is used in the pressure phase and in the pressure Poisson equation. An accuracy and stability analysis of the algorithm is provided for the pure convection equation. Two different types of boundary conditions for the end-of-step velocity of the fractional step algorithm have been investigated. Numerical tests for the lid-driven cavity at Re = 1 and Re= 7500 and flow past a circular cylinder at Re =100 are presented to demonstrate the usefulness of the method. Finite element method for predicting time-dependent viscous incompressible flows over a wide range of inertial conditions has been presented. The method is mainly aimed at solving convection dominated flows and employs an explicit three-step algorithm for the convection terms, which gives not only high accuracy by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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but also high efficiency since it allows large Courant numbers. To further improve accuracy for this kind of flows, the consistent mass matrix has also been included. Two variants of the method have been used; one fully explicit scheme with lumped mass matrix and one semi-implicit scheme with the consistent mass matrix in the momentum phase of the fractional step algorithm but with the lumped mass matrix in the pressure phase and in the pressure Poisson equation. The latter of these variants requires an extra system of linear equations to be solved at every time step. This was done in a simple and efficient way by using just a few Jacobi iterations and it was shown that this worked well even for very low Reynolds number flows. Two different kinds of velocity boundary conditions for the end-of-step velocity of the fractional step algorithm have been investigated, one which excludes checker boarding (type 1 B.C.) and one simpler version which does not exclude the checkerboard mode (type 2 b.c.). The type 1 B.C. was found to be slightly more accurate and it was also found to initiate the vortex shedding behind the circular cylinder much earlier than the type 2 b.c.

Conclusions: [4] The simple algorithmic structure and that no extra terms or new higher-order derivatives are needed. In spite of the simplicity, the method is of a general nature and can easily handle complex geometries. Numerical tests show good agreement with other numerical solutions and experimental data and suggest that the proposed method is competitive in terms of both accuracy and efficiency. [5] Fixed point iteration idea employed to linearize the coarse and fine scale sub-problems that arise in the variational multi scale frame work and it lead to a stabilized method for the incompressible Navier–Stokes equations. In the current work we present a consistent linearization of the nonlinear coarse and fine scale sub-problems, and substitution of the fine scales extracted from the fine-scale problem into the coarse-scale variational form leads to the new stabilized method. The solution of the fine-scale or the sub-grid scale problem which is an integral component of the proposed procedure for developing stabilized methods automatically yields an explicit definition of the stabilization operator τ. Another significant contribution of the paper is a numerical technique for evaluating the advection part of the stabilization operator τ that brings in the notion of up-winding in the resulting method. Presented a variational multi-scale-based stabilized formulation for the incompressible Navier–Stokes equations. A novel feature of our method is that fine scales are solved in a direct nonlinear fashion. Consistent linearization of the nonlinear equations in the context of the variational multi scale framework leads to the design of the stabilization terms in the new method A variational multi-scale residual-based stabilized finite element method for the incompressible Navier– Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation furnished by the variational multi-scale framework. A significant feature of the new method is that the fine scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the advection part of the stabilization tensor. The new method circumvents the Babuska–Brezzi condition and yields a stable formulation for high Reynolds number flows. A family of equal-order pressure-velocity elements comprising 4- and 10-node tetrahedral elements and 8- and 27-node hexahedral elements is developed. Convergence rates are reported and accuracy properties of the method are presented via the liddriven cavity flow problem. Presented the strong form and the classical weak form of the incompressible Navier–Stokes equations. Consistent linearization of the nonlinear equations performed in the vartiational multi-scale setting leads to the new multi-scale /stabilized formulation that is developed. The structure of the stabilization tensor and a numerical scheme to evaluate its advection part are presented; a convergence study for a family of 3D tetrahedral and hexahedral elements. An extensive set of numerical simulations of liddriven cavity flows for various Reynolds number are also presented.

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Conclusions: [5] The VMS based stabilized form possesses additional stabilization terms than are present in the classical stabilization methods alone. An important feature of the new method is that a definition of the stabilization operator τ appears naturally via the solution of the fine-scale problem. This stabilization operator is a second order tensor and leads to a full matrix that brings in cross coupling effects in the stabilization terms. A computationally economic scheme is proposed that incorporates up-winding effects in the calculation of the advection part of the stabilization operator τ. Good stability and accuracy properties of the new method are shown for a family of linear and quadratic tetrahedral and hexahedral elements. [6]A scalable numerical model to solve the unsteady incompressible Navier–Stokes equations is developed using the Galerkin finite element method. The coupled equations are decoupled by the fractional-step method and the systems of equations are inverted by the Krylov subspace iterations. The data structure makes use of a domain decomposition of which each processor stores the parameters in its sub-domain, while the linear equations solvers and matrices constructions are parallelized by a data parallel approach. The accuracy of the model is tested by modeling laminar flow inside a two-dimensional square lid-driven cavity for Reynolds numbers at 1,000 as well as three-dimensional turbulent plane and wavy Couette flow and heat transfer at high Reynolds numbers. The parallel performance of the code is assessed by measuring the CPU time taken on an IBM SP2 supercomputer. The speed up factor and parallel efficiency show a satisfactory computational performance. The innovative parallel solution techniques are adopted for exploring the power of parallel computing. Domain decomposition or the Schwarz method that is commonly adopted by CFD analysts. The discretized information is distributed to each processor which is responsible for the calculation in the corresponding sub-domain. The boundary conditions are obtained from the neighboring sub-domains during computations. To facilitate parallel computation the domain is usually divided into several sub-domains according to the structure of the mesh. A semi-implicit second-order accurate fractional-step method is used to decouple unsteady incompressible Navier–Stokes equation. The quasi-minimal residual (QMR) and the conjugate gradient (CG) methods are used to solve the non-symmetric and symmetric systems of equations, respectively. These are non-stationary iterations that involve some constants to be calculated at each iteration. Typically these constants are calculated by either taking products of matrices and vectors, or inner products of the vectors. Hence, the iterations are parallelized once the above two products are able to do so. A data parallel approach is adopted to perform these two parallelizations in the study. Conclusions: [6]A computation model based on equal-order FEM interpolating polynomials is developed for solving both velocity and pressure of the Navier–Stokes equations. The governing equations are decoupled by a four-step fractional method. The spatial domain is solved by the Galerkin FEM while the temporal domain is integrated by the Crank–Nicolson scheme, both of second-order accuracy. The main advantage of the current model is its simplicity in prescribing the boundary conditions for the velocity and pressure formulation. The proposed solution procedure is parallelized for porting on distributed-memory machines. Those expensive computational loads such as data storage, matrices/vectors constructions and linear equations solvers are parallelized by employing either domain decomposition or data parallel approaches. The developed parallel model is implemented on a distributed memory IBM SP2 supercomputer which is a SPMD type model. Improvement on solution accuracy of an equal-order FEM is shown by comparing laminar flow solution inside a two-dimensional square cavity at a Reynolds number of 1000. In addition, the current model is validated by a three-dimensional DNS of fluid turbulence in plane Couette flow at a Reynolds number of 5000. The capability of the current numerical scheme in large-scale scientific computation is further demonstrated through DNS of turbulent Couette flow over wavy surface. The parallel performance of the proposed parallel strategy is tested by analyzing the CPU time taken on an IBM SP2 supercomputer. Two scales, namely the small and large, of computations consisting of millions of elements are performed on different numbers of processors and improved computational performance is obtained by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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upon parallelism. By measuring the speed up factor and the parallel efficiency, the large scale calculation shows better parallel performance and scalability compared with its small scale counterpart. [7]The solution accuracy is compared with the existing interpolation functions such as the discretized Dirac delta function and the reproducing kernel interpolation function. The finite element shape function is easy to implement and it naturally satisfies the reproducing condition. They are interpolated through only one element layer instead of smearing to several elements. A pressure jump is clearly captured at the fluid–solid interface. Two example problems are studied and results are compared with other numerical methods. A convergence test is thoroughly conducted for the independent fluid and solid meshes in a fluid–structure interaction system. The required mesh size ratio between the fluid and solid domains is obtained. The discretized Dirac delta function used in the immersed boundary method and the reproducing particle method used in the immersed finite element method satisfy this condition. Propose and implemented a finite element interpolation function for non-uniform background fluid grid to capture a sharper fluid–structure interface than the reproducing kernel interpolation function used in the immersed finite element method and the Dirac delta function used in the immersed boundary method. The solutions are examined thoroughly and compared with other published results. The convergence test will be performed and a range of allowable mesh size ratios between the fluid and solid domains will be identified. A comprehensive convergence test is performed using this example. We pay special attention to the allowable fluid– solid mesh size ratios that can be used to yield convergent solutions. For a coupled fluid– structure problem, the convergence rate is computed independently with Lagrangian mesh element size and Eulerian grid spacing. Since there is no analytical solution for this problem, the errors of fluid velocity and solid displacement are calculated based on the solution obtained from a finely discretized system. The convergence of the solid displacement is calculated by refining the Lagrangian mesh while keeping the Eulerian mesh fixed at a refined state. Similarly, the convergence of the N-S solver is studied by refining the fluid mesh while keeping the solid mesh at a very fine resolution. Both components are performed with uniform mesh spacings for consistencies. Errors in the fluid velocity and solid displacement are calculated in L2 norms for steady state solutions. Conclusions: [7] The interpolation functions used in the immersed boundary method and the immersed finite element method, i.e. the discretized Dirac delta function and the reproducing kernel function. Proposed a straightforward finite element interpolation function that is capable of producing sharper interface that preserves the accuracy in interface solutions and to be used on unstructured background fluid meshes. The finite element interpolation function naturally satisfies the reproducing condition and it is easy to implement. Comparing to the previously mentioned techniques, the thickness of the interface can be narrowed by approximately 65% when using uniform grids, and can be improved even further when non uniform or unstructured grids are used. Through the example problems, we performed a thorough convergence test and examined the mesh size compatibility requirement for the fluid and solid domains. We found a mesh size ratio of 0.5 is required for the fluid and solid discretization to avoid numerical issues. If the fluid mesh size is less than half of the solid mesh size, then a leaking phenomenon would occur and lead the solutions to diverge. This value is consistent for several mesh resolutions. We also observed a relatively large volume change when the solid comes near a moving fluid boundary that generates large velocity gradient. A volume correction algorithm is imposed to enforce this incompressibility constraint. This correction algorithm can dramatically improve the durability of the incompressibility assumption and enhance the performance of the simulation. In summary, this paper introduces a finite element interpolation function to be used in the immersed finite element method and closely examines and resolves several detailed numerical issues that are present in the current non-conforming techniques. It provides a more accurate and a more reliable approach to be used in the simulations of fluid–structure interactions. [8] The fluid-structure interaction in fully nonlinear setting, where different space discretization can be used. The model problem considers finite elements for structure and finite volume for fluid. The computations for by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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such interaction problem are performed by implicit schemes, and the partitioned algorithm separating fluid from structural iterations. The formal proof is given to find the condition for convergence of this iterative procedure in the fully nonlinear setting. Several validation examples are shown to confirm the proposed convergence criteria of partitioned algorithm. The model problem for testing the novel paradigm of solution procedure based upon the direct coupling of different codes developed for a particular sub problem (i.e. either solid or fluid mechanics) into a single code. In particular, we seek to provide the guarantees for the robustness of such a computation approach in fully nonlinear setting, where implicit schemes are used for each sub problem, and we derive (by a formal proof) the convergence criterion for partitioned scheme iterations. For solving fluid–structure interaction problems are mostly oriented towards the monolithic schemes, where both sub-problems are discretized in space and time in exactly the same manner resulting with a large set of (monolithic) algebraic equations to be solved simultaneously with no need to distinguish between the ―fluid‖ and the ―structure‖ part. Provided the unified discretization basis for monolithic approach, the most frequent choice is to use the stabilized finite elements for fluids (first proposed by Hughes and co-authors followed by Tezduyar and many other works The main advantage of code-coupling approach for fluid– structure interaction concerns the fact that the coupling is limited only to the fluid–structure interface. Therefore, the main difficulty is reduced to enforcing the interface matching with respect to two different discretization schemes, finite element versus finite volume, as well as two different time integration schemes and different time steps. We thus split the presentation of our work in two parts, pertaining, respectively, to time and to space discretization for fluid and for structure and their matching at the interface. We will deal with the interface matching for different space discretization, along with other related issues pertaining to the computational efficiency enhancements by nested parallelization. In present paper (Part I), we discuss how to accommodate any particular (implicit) scheme that ensures the unconditional stability for either fluid or structure motion computation, and how to ensure that the unconditional stability extends to partitioned solution of the fluid–structure interaction problem. By considering equal time step size for fluid and structure, this direct force-motion transfer algorithm is named conventional serial staggered (DFMTCSS).Also consider the so-called Sub-cycled conventional staggered scheme (DFMT-SCSS) where time steps selected for integration of fluid flow and structure motion are not the same size. Conclusions: [8] Examined partitioned solution approach for nonlinear fluid–structure interaction problems. The partitioned approach is preferred for its modularity and the possibility of re-using existing software developed for each sub-problem (see Part II). The partitioned approach used here is based on the DFMT. Both explicit and implicit coupling algorithms for multi-physics problems are detailed. An explicit strategy leads to the so-called ―added mass effect‖, and for that justifies the use of more costly implicit solvers for the case of incompressible fluid flows. In this work, the problem of enforcing the fluid– structure interface matching is handled by the fixed-point strategy (DFMT-BGS) with an adaptive relaxation parameter. This strategy shows a sufficiently robust performance, especially for the example where the flow is not highly constrained by incompressibility. In fact, we showed by direct proof the stability of the implicit DFMT-BGS algorithm which is valid for the fully nonlinear fluid–structure interaction problem. [9] New adaptive Lattice Boltzmann method (LBM) implementation within the Peano framework, with special focus on nano-scale particle transport problems. With the continuum hypothesis not holding anymore on these small scales, new physical effects—such as Brownian fluctuations—need to be incorporated. We explain the overall layout of the application, including memory layout and access, and shortly review the adaptive algorithm. The scheme is validated by different benchmark computations in two and three dimensions. An extension to dynamically changing grids and a spatially adaptive approach to fluctuating hydrodynamics, allowing for the thermalisation of the fluid in particular regions of interest, is proposed. Both dynamic adaptivity and adaptive fluctuating hydrodynamics are validated separately in simulations of particle transport problems. The application of this scheme to an oscillating particle in a nano-pore illustrates the importance of Brownian fluctuations in such setups. Presented an approach to nano-flow simulations in complex and/or moving geometries. by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Conclusions: [9] Implementated a block-structured adaptive Lattice Boltzmann solver, including its memory access, adaptivity concept and intermolecular collision models. In order to profit from both the simple-to-use adaptivity concept of the Peano framework and the simple and computationally cheap Lattice Boltzmann update rule, we proposed the usage of an application-specific grid management system handling the memory-intensive storage of the particle distribution functions. This scheme avoids costly copy operations between the Peano-internal stacks on the one hand, but leaves the handling of the adaptive grids as well as the parallelization to the Peano kernel on the other hand. We verified and validated our adaptive implementation by different benchmark computations using adaptive and non-adaptive grids in two and three dimensions. Furthermore, the extension of the adaptive scheme to dynamically changing grids has been presented, allowing for the simulation of moving structures within the flow. The new scheme was validated for particle transport problems which are of major concern in our work. An additional focus of research was on nano-flow simulations where Brownian motion effects play a crucial role. The modeling of the respective Brownian fluctuations, however, comes at high computational costs as huge numbers of Gaussian random numbers are required in this case. We proposed a multiscale approach, allowing for fluctuating effects within the fluid on fine grid levels only. On coarser grid levels, the fluctuations are cut off and a simple BGK collision kernel is applied. We used this cut-off approach to simulate the diffusion of an isolated spherical particle. The short-time diffusion of the particle is slightly underestimated by the method, the long-term behavior is captured correctly. Finally, combined our dynamic refinement approach and the cut-off mechanism for thermal fluctuations to simulate a particle which is exposed to oscillating pressure fields within a nano-pore. Similar to previous results, diffusive effects due to thermal fluctuations dictate the magnitude and the direction of the particle drift. Both our new cut-off approach and a completely thermalised fluid model show a behavior of the particle drift which is different to non-fluctuating simulations. This illustrates the importance of Brownian motion on the nano-scale for our flow scenarios. As part of future work, further studies in two and three dimensions will be carried out to completely understand the short-time behavior of the particle motion within the nano-pores. Within this contribution, we restricted our numerical experiments to the simulation of several periods in the particle oscillations. Simulations overmuch longer time intervals might be required to completely evaluate the motion of the particle inside the pore structures. Therefore, new methods need to establish. We currently work on a hybrid approach to include both thermal fluctuations on the short and long time scale in our simulations. [10]A newly developed LES flow solver to compute a true three-dimensional flow applied. The research also investigates the behavior of turbulence statistics by comparing transient simulation results to available data based on experiments and simulations. An extensive discussion on the results such as energy spectrum, velocity profiles and time trace of velocities is carried out in the research as well. Based on the results obtained, the application of the flow solver for a turbulent three-dimensional driven cavity flow by using three grids with varying densities is proven. In addition, the research successfully verifies that in many instances computational results agreed reasonably well with the reference data, and the changes in the statistical properties of turbulence with respect to time are closely related to the changes in the flow structure and strength of vortices. The focus of this study is on the prediction of a sub-grid scale Reynolds shear stress profiles and the results show that the standard model is able to reproduce general trends measured from experiments. Furthermore, in certain areas inside the cavity the computed shear stress values are in close agreement with experimental data. The dynamics of the statistical properties of turbulence as these vortices and secondary flow develop. A further novel aspect of this work is to obtain some insight into accuracy of shear stress computations using the baseline Smagorinsky model, for flows with spatially and temporally varying turbulence structures . Published data from Prasad and Kosef and Migeon et al. are used to validate the code

Conclusions: [10] Investigated in detail the dynamics of the statistical properties of turbulence as Taylor-Gortler vortices and secondary flows develop. In particular, the w v stress profiles, which are particularly difficult to predict, by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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are studied. Furthermore, some insight into the accuracy of shear stress computations using the baseline Smagorinsky model, for flows with spatially and temporally varying turbulence structures has been obtained. This paper has demonstrated the application of the flow solver for a turbulent three-dimensional driven cavity flow by using three uniform grids with varying densities. In many instances, computational results agree reasonably well with the reference data. A number of important conclusions can be drawn from this case: In general, the profiles from computation follow the trend exhibited by the reference data. The numerical setup was, to some extent, quantitatively successful in predicting the w v stress profiles. In addition, to predict the w v stress profiles more accurately, the value of the Smagorinsky constant, Cs, must be varied with location inside the domain. The turbulence kinetic energy spectrum plots show the presence of inertial sub-range eddies though the level of energy may vary with respect to the location of the monitoring points. The turbulence kinetic energy plots show that this parameter is produced in the region where the Taylor-Görtler-like vortices reside. [11]

An experimental study is presented for a flow field in a two dimensional wavy channels by PlY. This flow has two major applications such as a blood flow simulation and the enhancement of heat transfer in a heat exchanger. While the numerical flow visualization results have been limited to the fully developed cases, existing experimental results of this flow were simple qualitative ones by smoke or dye streak test Therefore, the main purpose of this study is to produce quantitative flow data for fully developed and developing flow regimes by the Correlation Based Correction PlY (CBC PlY) and to conjecture the analogy between flow characteristics and heat transfer enhancement with low pumping power. Another purpose of this paper is to examine the onset position of the transition and the global mixing, which results in transfer enhancement. PlY results on the fully developed and developing flow in a wavy channel at Re=500, 1000 and 2000 are obtained. For the case Reynolds Number equals 500, the PlY results are compared with the finite difference numerical solution.

The practical use of the particle image velocimetry (PIV), a whole flow field measurement technique, requires the use of fast, reliable, computer-based methods for tracking velocity vectors. The full search block matching, the most widely studied and applied technique both in the area of PIV and Image Coding & Compression is computationally costly. Many alternatives have been proposed and applied successfully in the area of image compression and coding, i. e. MPEG, H. 261 etc. Among others, the Three Step Search (TSS) (Jain, 1981), the New Three Step Search (NTSS) (Li et al., 1994), the Hierarchical Projection Method (HPM) (Sauer and Schuartz, 1996), the FFT-Direct Hybrid Method (HYB) and the Two Resolution Method (TRM) (Anandan, 1989) are introduced. A Correlation Based Correction technique (CBC) (Hart, 2000) is also appreciated and found to be most accurate and adequate for this flow. For the cases Reynolds number fRe) of 500, 1000 and 2000. Developing and fully developed flow data are obtained by CBC PIV with one window shifting. The global mixing phenomenon; which results in the increase in heat and mass transfer and drag, can be identified through the investigation of developing flow in beginning modules. At Re above 500, promotion to turbulence is prominent. While it happens at Re above 2300 in a straight channel. The threepoint Gaussian fit is used for a sub pixel estimator, and the Local Median Filter (LMF) is chosen to validate a vector field. (Kim, 1999) Conclusions: [11]Unlike a simple dye or smoke streak visualization, the PlY analysis can resolve the exact flow structure, even in turbulent flow situation. It can also deal with the unsteady behavior of global mixing. In this paper, fully developed and developing flow data in a wavy channel of Re 500, 1000, 2000 are obtained through the CBC PlY measurements. The analogy between flow characteristics and the enhancement of heat and mess transfer in a wavy channel can be visible through the RMS distribution near wall. The onset point of the global mixing is clearly identified through instantaneous velocity and RMS intensity distributions of a couple of beginning modules. It happens at 4th wavy module for Re= 500, and 2nd module in case Re equals 1000, which are in a good agreement of Rush et aI.'s prediction from dye streak visualization. The phase averaging of PIV results will give a more precise insight of flow structure, like instability and shedding vortices etc. by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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[12]

Analyze the fluid flow with moving boundary using a finite element method. The algorithm uses a fractional step approach that can be used to solve low-speed flow with large density changes due to intense temperature gradients. The explicit Lax-WendrofT scheme is applied to nonlinear convective terms in the momentum equations to prevent checkerboard pressure oscillations. The ALE (Arbitrary Lagrangian Eulerian) method is adopted for moving grids. The numerical algorithm in the present study is validated for two-dimensional unsteady flow in a driven cavity and a natural convection problem. To extend the present numerical method to engine simulations, a piston-driven intake flow with moving boundary is also simulated. The density, temperature and axial velocity profiles are calculated for the three-dimensional unsteady piston-driven intake flow with density changes due to high inlet fluid temperatures using the resent algorithm. The calculated results are in good agreement with other numerical and experimental ones. Conclusions: [12] Used a fractional step method with equal-order interpolation functions for the velocity components and pressure. The explicit Lax-Wendroff scheme was applied to the nonlinear convective terms in the momentum equations and the ALE (Arbitrary Lagrangian-Eulerian) method was adopted for treating the moving boundary. To validate the present algorithm, several problems have been calculated and compared with other results. As a result, the calculation results have shown good agreement with other results. In order to extend the present numerical method to engine simulations, we also investigated the basic behavior of the unsteady flow generated by an impulsively started piston movement in a piston-cylinder assembly, yielding flow separation and spatially moving vortices. The numerical results indicate that the present calculation procedure can be used to predict the behavior of periodic intake/exhaust flows and is applicable to a wide range of problems. Although the discussion has been restricted to laminar flows governed by the NavierStokes equations, the methodology proposed can readily be extended to accommodate the Reynoldsaveraged equations and turbulence models. [13] The bifurcation of the lines of a viscous-fluid flow in a rectangular cavity with a moving cover has been investigated for different ratios between the sides of the cavity and different Reynolds numbers on the basis of the qualitative theory of dynamic systems. The critical parameters of the problem, at which the type of singular points changes and other topological characteristics of a vortex flow in the indicated cavity have been determined and the corresponding bifurcation diagrams have been constructed. The topological characteristics of a flow can be determined on the basis of solution of the Navier–Stokes equations. a flow in a rectangular cavity was investigated for different ratios between the cavity sides and different velocities of travel of the upper and lower walls. The finite-element method with bunching of nodes of a grid in the neighborhood of local stagnation points was used for discretization of Navier–Stokes equations. The investigations were carried out for fairly small Reynolds numbers (Re < 100). Reasonably exact results were not obtained by the finite-element method because it, when used for solving fluid-mechanics problems, provides a lower accuracy than the finite-difference method. Conclusions: [13] The

dependence of the change in the type of singular points of a vortex flow (bifurcation of streamlines) in a rectangular cavity on the ratio between the cavity sides and on the Reynolds number has been investigated on the basis of numerical simulation of this flow. The data obtained can be used for determining the topological characteristics and features of separation flows in cavities of more complex geometries.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-III

Objectives of the work 1) To study the fluid dynamics in the lid-driven cavity. 2) To study the effect of governing parameters in terms of non dimensional numbers in Navier-Stokes equation for four different cases that are Isothermal, Forced convection, mixed convection and Natural convection. 3) To study the effect of grid sizes (coarse, medium and fine) to catch actual characteristics of the fluid flow in above said four different cases. 4) To study the effect of non dimensional numbers on the temperature in above said cases except first case. 5) To study the stream function contours for different non dimensional number in all above said cases. 6) To study the time steps for different parameters in said cases. 7) To study the convergence criteria in all above said cases. 8) To study the vertices movement in the cavity for different non dimensional numbers for said cases. 9) To study the computation time for said cases with different parameters. 10) To study the code validation.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

14

Chapter-IV

Physical description of the problem and models for simulations To study the effect of non dimensional parameters in fluid flow characterization; the lid driven cavity is one of the most widely used benchmark problems to test steady state incompressible fluid dynamics codes. Our interest will be to present this problem as a benchmark for the steady and unsteady state solution. In order to demonstrate the grid independence, code validation and other details like time step, grid size and steadiness criteria; we taken 2D Cartesian (x, y as horizontal and vertical components) square domain of size L x L (1 unit x 1 unit for simplicity of the problem) with bottom, left and right boundaries as solid walls stationary; whereas top wall is like a long conveyor-belt, moving horizontally with a constant velocity (Uo=1unit for the simplicity of the problem) shown in Fig.1.

Top long horizontal moving belt (lid) with velocity Uo units

Left side stationary solid wall

Cavity

Right side stationary solid wallwall

1 unit

Temperature Boundary Conditions:

Y

1 unit

X

Top wall at TH temperature and Left, Right & Bottom walls are at TC temperature

Bottom stationary solid wall

Boundaries of the domain for iterations steps (i,j)= (1,1) at left bottom corner; (i,j)= (imax,1) at right bottom corner (i,j)= (1,jmax) at left top corner; (i,j)= (imax, jmax) at right top corner

Fig.1 Lid driven cavity To perform a non dimensional CFD simulation for various values of non-dimensional governing parameters such as Reynolds number, Prandl number, Grashoff number etc.; the following models are made for four different situations (cases) as mentioned in the following tables.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Case-I: Isothermal fluid flow Note: All walls of the cavity are at constant temperature i.e TH=TC=Constant Reynolds Number Model No. of Grids in x direction X No. No. of Grids in y direction 32X32 100 I1 52X52 100 I2 32X32 400 I3 52X52 400 I4 32X32 1000 I5 52X52 1000 I6

Case-II: Forced convection fluid flow Note: Non-isothermal i.e top wall at TH and all other walls at TC temperatures. This corresponds to non-dimensional temperature , and is a buoyancy induced flow therefore in all models Grashoff number is taken as zero i.e Gr=0 Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 F1 1 100 32X32 F2 1 400 12X12 F3 1 400 32X32 F4 0.5 100 12X12 F5 0.5 100 32X32 F6 0.5 400 12X12 F7 0.5 400 32X32 F8 1.2 100 12X12 F9 1.2 100 32X32 F10 1.2 400 12X12 F11 1.2 400 32X32 F12 Case-III: Mixed convection fluid flow Note: All the conditions are same as in Case-II except the Grashoff number is finite values i.e +ve or –ve and ML=Mixed flow Low Grashoff number model Gr=1X105 Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 ML1 1 100 32X32 ML 2 1 400 12X12 ML 3 1 400 32X32 ML 4 0.5 100 12X12 ML 5 0.5 100 32X32 ML 6 0.5 400 12X12 ML 7 0.5 400 32X32 ML 8 1.2 100 12X12 ML 9 1.2 100 32X32 ML 10 1.2 400 12X12 ML 11 1.2 400 32X32 ML 12 by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Case-III: Mixed convection fluid flow Note: MM=Mixed flow Medium Grashoff number model Gr=1x106 Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 MM 1 1 100 32X32 MM 2 1 400 12X12 MM 3 1 400 32X32 MM 4 0.5 100 12X12 MM 5 0.5 100 32X32 MM 6 0.5 400 12X12 MM 7 0.5 400 32X32 MM 8 1.2 100 12X12 MM 9 1.2 100 32X32 MM 10 1.2 400 12X12 MM 11 1.2 400 32X32 MM 12

Case-III: Mixed convection fluid flow Note: MH=Mixed flow High Grashoff number model Gr=2X106 Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 MH 1 1 100 32X32 MH 2 1 400 12X12 MH 3 1 400 32X32 MH 4 0.5 100 12X12 MH 5 0.5 100 32X32 MH 6 0.5 400 12X12 MH 7 0.5 400 32X32 MH 8 1.2 100 12X12 MH 9 1.2 100 32X32 MH 10 1.2 400 12X12 MH 11 1.2 400 32X32 MH 12

Case-III: Mixed convection fluid flow Note: All the conditions are same as in Case-II except the Grashoff number is finite values i.e +ve or –ve and ML=mixed flow low Grashoff number model Gr=-2X106; MNL: Mixed convection Less Negative Grashoff number. * Due to shortage of time to submit the report the simulation has not been done, because that simulations are taking days together. Model No. MNL1 MNL 2* MNL 3 MNL 4 MNL 5 MNL 6* MNL 7

Prandl Number 1 1 1 1 0.5 0.5 0.5

Reynolds Number 100 100 400 400 100 100 400

No. of Grids in x direction X No. of Grids in y direction 12X12 32X32 12X12 32X32 12X12 32X32 12X12

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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MNL 8 MNL 9* MNL 10 MNL 11 MNL 12

0.5 1.2 1.2 1.2 1.2

400 100 100 400 400

32X32 12X12 32X32 12X12 32X32

Case-III: Mixed convection fluid flow Note: All the conditions are same as in Case-II except the Grashoff number is finite values i.e +ve or –ve and ML=mixed flow low Grashoff number model Gr=-1X105; MNH: Mixed convection High Negative Grashoff number Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 MNH1 1 100 32X32 MNH 2 1 400 12X12 MNH 3 1 400 32X32 MNH 4 0.5 100 12X12 MNH 5 0.5 100 32X32 MNH 6 0.5 400 12X12 MNH 7 0.5 400 32X32 MNH 8 1.2 100 12X12 MNH 9 1.2 100 32X32 MNH 10 1.2 400 12X12 MNH 11 1.2 400 32X32 MNH 12

Case-IV: Natural convection fluid flow Note: For natural convection, the flow is only due to buoyancy with no forced flow. Thus, the lid is also taken as stationary here. Thus, the physical situation corresponds to a buoyancy induced flow in a differentially heated closed square cavity. The left-wall is maintained at TH, right-wall at TC and the remaining walls are insulated; all walls are stationary. Model No. N1 N2 N3 N4

Prandl Number 0.71 0.71 0.71 0.71

Rayleigh Number 103 103 104 104

No. of Grids in x direction X No. of Grids in y direction 12X12 32X32 12X12 32X32

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-V

Mathematical modeling with boundary conditions We consider the incompressible viscous fluid flow with constant density, viscosity and thermal conductivity in the absence of an applied body force. A set of non-dimensional governing equations following ellipticparabolic characterization are as follows: General mass conservation equation:

In the eqn. (1) u and v are velocity components along x and y directions respectively. steam velocities along x and y directions and total length of the domain respectively. dimensional velocities along the x and y directions respectively; and similarly dimensional coordinates.

are free are nonare the non

General momentum equations:

In the eqn. (2) are non-dimensional temperature, Reynolds number and Prandl number respectively. is the bulk mean temperature in case of internal flow and free stream temperature in case of external flow situations. is the known temperature of the wall or surface. T is the temperature to be find during iteration and time steps. are viscosity, specific heat and thermal conductivity of the fluid respectively and these are taken as constant for our problem. This equation includes temperature term therefore it is used to discritize non-isothermal cases that are case-II to IV in our CFD simulations.

In the eqn. (3);

is non-dimensional pressure and is

; where

is free stream known pressure and

P is pressure to be find for each step of the CFD simulation. This equation does not include any temperature terms; therefore this can be used to solve isothermal type problems i.e case-I in our CFD simulation.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Boundary conditions:

The boundary condition equations from (4) to (7) are incorporated to solve isothermal type of problems; where the temperature remains constant i.e is in case-I in our CFD simulations. And the boundary condition equations from (8) to (11) are incorporated to solve non-isothermal type of problems; where the temperature varying with respect to time and coordinates; that are case-II to IV in our CFD simulations. In equations (4) to (6) and (8) to (10); the velocity components on left, right and bottom surfaces are taken as zero unit along both x and y directions because of viscous effect of the solid stationary walls on the fluid particles. Because the top surface is continuously moving in the horizontal direction only i.e x direction with a velocity ‗u‘; the fluid particles close to the bottom surface of the top plate are affected by the motion of the plate; therefore it is considered in the equations (7) and (11) as unit for simplification of the problem.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-VI

Validation study As a first step towards numerical investigation of the physical problem, we first justify our computer code. A validation study is often proceeding by an analytical assessment of the problem. To achieve this goal and, furthermore, estimate the spatial rate of convergence of the scheme employed, we consider the following transport eqn. (12) for a scalar Φ in a simple domain of two dimensions; .

In eqn.(12) left side first unsteady term will calculate scalar quantity for all time steps and second and third terms in the same side will calculate advection quantities on all geometric coordinates. The first term on the right side calculates diffusion coefficients.

Where are the pressures at east and north nodes respectively and is the pressure at the node where the pressure has to find. The eqn. (13) to (16) are used as pressure correctors to remove the difficulty in the pressure term for linear interpolation is solved by taking staggered grid solution. Prediction error due to oscillatory velocities and the so called false diffusion error grossly pollute the flow physics over the entire domain. Remedy for such discretization error is to apply pressure correction equations in semi explicit QUICK wind scheme SOU at the boundaries. The above shown equations are used to calculate the velocities along x and y direction by considering adjacent nodes pressure for the next time step. As is usual, we assessed the employed QUICK-type upwind discretization scheme by examining the prediction nodal errors. Tests on various grids were conducted to assure that the solution converged. With grid spacing being continuously refined, we could compute the rate of convergence from the computed. The test case considered and the results obtained thus far confirm the applicability of the QUICK scheme to multidimensional analyses. We now turn to examining whether or not linearization procedures and the zerodivergence constraint condition will cause the rate of convergence to deteriorate. To answer this question, we solved a Navier-Stokes problem in the same domain as that considered in the previous benchmark test by GHIA et al. (1982) JOURNAL OF COMPUTATIONAL PHYSICS VOL. 48, pp.387-411 @ Re = 100 we are assured that the proposed scheme is also applicable to analysis of incompressible Navier-Stokes equations. The good agreement from two analytical tests, as demonstrated provides us with strong confidence to proceed with investigation of the time-history of the flow evolution, which is driven by a constant upper lid, in the rectangular cavity.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

21

Chapter-VII

Results and Discussions Case-I: Isothermal fluid flow Stream function contours 32X32 Grid structure

52X52 Grid structure

Fig.I 1.1

Fig.I 2.1

Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.1

Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.1

Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 5.1

Fig.I 6.1

Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

It is observed from the above shown stream function contours that as the magnitude Reynolds number increases then the inertia forces will increase in the cavity therefore more turbulences will formed in the fluid flow; hence for more values Reynolds number, the solution will take less time steps, less computational time and less number of iterations in the mass conservation loop for the convergence even though change in by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

22

the grids in x and y directions for all figures of Case-I. Higher magnitude of Reynolds number means; top surface of the cavity moving with a higher velocity. As we observed from the stream contour figures‘ I 1.1, I 3.1 and I 5.1; the lower value stream contours decreases with increasing the magnitude of the Reynolds number means top lid moving with higher velocity and the veracity will shift towards top right corner. Same effect can be observed in the higher grid points also. It is observed from the figures‘ I1.1 & I1.2, I1.3 & I1.4 and I1.5 & I1.6 of steam function contours that as the grid size increases from 32X32 to 52X52 structure in the x and y directions; the simulation catches lower values of stream functions towards bottom surface of the top lid and also we can observed the sharp changes in the physical shape of the verticity near at top right corner of the cavity for 52X52 grid size.

V velocity along the horizontal centerline 32X32 Grid structure

52X52 Grid structure

Fig.I 1.2

Fig.I 2.2

Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.2 Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Fig.I 5.2 Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.2 Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 6.2 Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

As we observed from the velocity along the horizontal centerline figures‘ I 1.2, I 3.2 and I 5.2; the deviation in the velocity result increases with increasing magnitude of the Reynolds number with the published results for 32X32 as well as in 52X52 grid structure; therefore even more denser grid structure is required to catch by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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the small in the velocities in the center horizontal line. It is also observed from both grid structures; the more closure velocity profile with the published results for 52X52 structures; and still finer grid structure is required to match with published results. And also observed from same figures; as the magnitude of the Reynolds number increases from 100 to 1000, the inside surfaces of the left and right side walls effect decreases and velocity of the fluid particles increases; it enhances little more turbulence in the fluid particles. Hence these turbulences cannot catch in the higher magnitude Reynolds number; therefore the centerline velocity profile is almost horizontal in case of 1000 Reynolds number and small change in case of Reynolds number 400; but sharp changes in Reynolds number 100.

U velocity along the vertical centerline 32X32 Grid structure

52X52 Grid structure

Fig.I 1.3

Fig.I 2.3

Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.3 Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Fig.I 5.3 Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.3 Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 6.3 Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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It is observed from the figures I1.3, I3.3 and I5.3 of horizontal velocity component (U) along the vertical centerline that the fluid particles will changes their flow direction from right side to the left side at the 0.15 unit distance from bottom surface along the y direction of the cavity for 1000 Reynolds number in the published results and that same distance will increases for lower Reynolds numbers i.e 0.28 for Re=400 and 0.55 for Re=100. And is also observed that that sharp change in the velocity profile for higher Reynolds numbers i.e for 400 and 100 but in lower values of the same number smooth changes will happen i.e for 100. As we observed from the figures I1.3 to I6.3 that our simulation velocity profile is close with published profile for lower Reynolds number i.e 100 but this closeness is decreases with higher value of the Reynolds number; this is because our grid structure is not sufficient to catch the higher velocities of the particles of higher Reynolds number. Therefore finer grid structure may be necessary to catch the higher Reynolds number turbulences.

V velocity contours 32X32 Grid structure

Fig.I 1.4 Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.4 Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Fig.I 5.4 Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

52X52 Grid structure

Fig.I 2.4 Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.4 Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 6.4 Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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It is observed from the figures from I1.4to I6.4 that primary higher velocity verticity at top left corner and secondary lower velocity verticity at the top right corner are changing their characterization when the magnitude of the Reynolds numbers varies. The magnitude of the primary and secondary verticities will decreases with increasing Reynolds number. This is because; when the velocity of the top lid increases then the turbulence in the fluid flow will also increases; so more distribution of the velocity contours. Commonly in all figures it is also observed that the moment of the lid is highly at top left corner fluid particles of the cavity on vertical velocity contours whereas it reverses at right top corner fluid particles of the cavity. By comparing the figures for 32X32 grid structure I1.4, I3.4. I5.4 and 52X52 grid structure I2.4, I4.4, I6.4; more uniform v-velocity contours in higher grid structures. This is because more fluid particles will cover in dense grid structure. As the Reynolds number increases from 100 to 400 for both grid structure; the higher value velocity contours in Re=400 than Re=100 in the domain. It is also observed from the figures; as the Reynolds number increases from 100 to 1000 through 400, the magnitude of the primary and secondary verticities is decreasing and it may disappear for even higher values of the Reynolds number.

U velocity contours 32X32 Grid structure

Fig.I 1.5 Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.5 Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Fig.I 5.5 Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

52X52 Grid structure

Fig.I 2.5 Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.5 Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 6.5 Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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It is observed from the figures I1.5 to I6.5 that the higher value U velocity contours at the top surface of the domain because top lid is moving horizontally. As the Reynolds number is low higher value U-velocity contours is more at bottom of the lid whereas these are decreases as the Reynolds number increases in both 32X32 and 52X52 grid structure. The lower velocity verticity is observed in all figures at the right side of the domain and it is towards top right corner as the Reynolds number increases. And also the magnitude of the verticity also decreases with increasing Reynolds number. It is also clearly observed from two different grid structures; the change of the characteristic of the verticity different in 52X52 than the 32X32. The effect of the top lid velocity caught in dense grid structure than the coarse grid structure. Because less viscosity effect at the bottom side top lid; the particles immediately bottom of the plate are in the same velocity of the plate i.e the highest velocity in the problem. For denser grid structure; the verticity at the top right corner of the domain may disappear.

Case-II: Forced convection fluid flow (where Gr=0) U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.F 2.1

Fig.F 1.1 Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 3.1

Fig.F 4.1

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6.1

Fig.F 5.1 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.F 8.1

Fig.F 7.1 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Fig.F 9.1

Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

Fig.F 10.1

Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12.1 Fig.F 11.1 Re:400 Pr:1.2 Computation time: 20minTime steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

It is observed from the figures from F 1.1 to F12.1; higher velocity verticity is small and it is shifting towards top right corner of the cavity of the domain for dense grid structure than the coarse grid structure. The same verticity is shifting towards bottom of the cavity for higher Reynolds number i.e 400 as compared to the 100. The time steps required for to converge the solution are more for higher Reynolds number (400) than the lower (100) for the same Prandl number irrespective of the grid structure. As the Prandl number increases from 0.5 to 1.2 through 1; the solidity of the fluid is more, therefore the velocity of the top lid is more in case of lower Prandl number (0.5) than the higher number (1.2). The physical characteristic of the verticity is slightly different in case of higher Reynolds number towards the bottom surface of the cavity. As we know top lid is moving; therefore the higher velocity contours are appears at the top.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

28

V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.F 2.2

Fig.F 1.2 Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 4.2

Fig.F 3.2 Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6.2

Fig.F 5.2 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Fig.F 7.2 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8.2 Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

29

Fig.F 9.2

Fig.F 10.2

Re:100 Pr:1.2 Computation time: 25minTime steps:109 No.of iterations in the mass conservation loop:456

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 11.2

Fig.F 12.2

Re:400 Pr:1.2 Computation time: 20minTime steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

It is observed from the figures of higher Reynolds number (400); the higher V-velocity verticity at the top left corner is stretched towards right surface of the cavity as compared to the lower number (100) and this character is clearly observed in the dense grid structure(32X32) than the coarse grid structure (12X12). The magnitude of the verticity is slightly higher in case of lower grid structure (12X12) than the higher (32X32) irrespective of Prandl and Reynolds number. In all figures from F 1.2 to F 12.2 one higher primary verticity at the top left corner and another lower secondary verticity at the right top corner of the cavity. It is observed from higher Reynolds number contours (400) having lower velocity contours than the lower Reynolds number (100) irrespective of Prandl number and grid structure. It observed from the figures; as the Reynolds number increases from 100 to 400 the magnitude of the primary higher verticity increase with decreasing secondary lower verticity. The magnitude of the secondary verticity is decreasing with Prandl number for same Reynolds number irrespective of the grid structure. The number of time steps required are more for higher Reynolds number than the lower.

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.F 1.3 Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Fig.F 2.3 Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

30

Fig.F 4.3

Fig.F 3.3 Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Fig.F 6.3

Fig.F 5.3 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Fig.F 9.3

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8.3

Fig.F 7.3 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:100 Pr:1.2 Computation time:25min Time steps:109 No.of iterations in the mass conservation loop:456

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

Fig.F 10.3 Re:100 Pr:1.2 Computation time:22 hours Time steps:137 No.of iterations in the mass conservation loop:1801

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

31

Fig.F 12.3

Fig.F 11.3

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

Re:400 Pr:1.2 Computation time: 20minTime steps:126 No.of iterations in the mass conservation loop:353

It is observed from the stream line contours; the verticity is shifting towards top right corner of the cavity in case of dense grid structure than the coarse grid structure and its magnitude also decreasing with dense grid structure. It is observed from the figures‘ F1.1 & F1.2, F1.3 & F1.4 and F1.5 & F1.6 of steam function contours that as the grid size increases from 32X32 to 52X52 structure in the x and y directions; the simulation catches lower values of stream functions towards bottom surface of the top lid and also we can observed the sharp changes in the physical shape of the verticity near at top right corner of the cavity for 52X52 grid size.

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.F 1.4

Fig.F 2.4

Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 3.4

Fig.F 4.4

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

32

Fig.F 5.4

Fig.F 6.4

Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 7.4

Fig.F 8.4

Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

Fig.F 10.4

Fig.F 9.4 Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12.4

Fig.F 11.4 Re:400 Pr:1.2 Computation time:20min Time steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

33

It observed from the figures; the characteristic of the temperature contours at the top right corner of the cavity is changes as the Reynolds number changes from 100 to 400. As the grid structure increases from 12X12 to 32X32; the sensitive variation in the temperature are also catches and it is clearly seen in the figures. It is also observed from the figures; as the Prandl number increases from 0.5 to 1.2, the higher temperature contours are available near bottom of the top lid.

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.F 1.5

Fig.F 2. 5

Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 3. 5

Fig.F 4. 5

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Fig.F 5. 5

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6. 5

Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8. 5

Fig.F 7. 5 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:400 Pr:0.5 Computation time:12 hours Time steps:115 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

34

Fig.F 9. 5

Fig.F 10. 5

Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Fig.F 11. 5

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12. 5

Re:400 Pr:1.2 Computation time: 20min Time steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

It is observed from the figures; the horizontal velocity component (U) along the vertical centerline that the fluid particles will changes their flow direction from right side to the left side at the 0.15 unit distance from bottom surface along the y direction of the cavity for 1000 Reynolds number in the published results and that same distance will increases for lower Reynolds numbers i.e 0.28 for Re=400 and 0.55 for Re=100. And is also observed that that sharp change in the velocity profile for higher Reynolds numbers i.e for 400 and 100 but in lower values of the same number smooth changes will happen i.e for 100. As we observed from the figures F1.3 to F6.3 that our simulation velocity profile is close with published profile for lower Reynolds number i.e 100 but this closeness is decreases with higher value of the Reynolds number; this is because our grid structure is not sufficient to catch the higher velocities of the particles of higher Reynolds number. Therefore finer grid structure may be necessary to catch the higher Reynolds number turbulences.

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.F 1.6

Fig.F 2. 6

Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

35

Fig.F 3. 6

Fig.F 4. 6

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6. 6

Fig.F 5. 6 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8. 6

Fig.F 7. 6 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

Fig.F 10. 6

Fig.F 9. 6 Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Fig.F 11. 6

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12. 6

Re:400 Pr:1.2 Computation time:20min Time steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

36

I is observed from same figures; as the magnitude of the Reynolds number increases from 100 to 1000, the inside surfaces of the left and right side walls effect decreases and velocity of the fluid particles increases; it enhances little more turbulence in the fluid particles. Hence these turbulences cannot catch in the higher magnitude Reynolds number; therefore the centerline velocity profile is almost horizontal in case of 1000 Reynolds number and small change in case of Reynolds number 400; but sharp changes in Reynolds number 100.

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.F 1.7

Fig.F 2. 7

Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Fig.F 3. 7

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 4. 7

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6. 7

Fig.F 5. 7 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Fig.F 7. 7 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8. 7 Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

37

Fig.F 9. 7 Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Fig.F 10. 7 Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12. 7

Fig.F 11. 7 Re:400 Pr:1.2 Computation time: 20min Time steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

It is observed from the figures; the validation of the simulated data will be more for coarse grid structure (12X12) than the dense grid structure(32X32) compared to the published data and same effect can be observed for the lower value Reynolds number(100) than the higher value Reynolds number (400). It is also observed that the sharp changes in the temperature from higher Prandl number (1.2) than the lower(0.5).

Case-III: Mixed convection fluid flow (where Gr=105) U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.ML 1.1 Re:100

Pr:1

Fig.ML 2.1 Time steps: 49

Re:100

Pr:1

Time steps:78

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

38

Fig.ML 4.1

Fig.ML 3.1 Re:400

Pr:1

Time steps:100

Re:400

Pr:1

Re:100

Pr:0.5

Re:400

Pr:0.5

Fig.ML 6.1

Fig.ML 5.1 Re:100

Pr:0.5

Time steps:44

Pr:0.5

Pr:1.2

Time steps:98

Time steps:79

Fig.ML 9.1 Re:100

Time steps:77

Fig.ML 8.1

Fig.ML 7.1 Re:400

Time steps: 130

Fig.ML 10.1 Time steps:52

Re:100

Pr:1.2

Time steps:82

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

39

Fig.ML 11.1 Re:400

Pr:1.2

Fig.ML 12.1 steps:103

Re:400

Pr:1.2

Time steps:125

The U velocity contour verticity is moving towards bottom surface of the cavity as the Prandl number decreases from 1.2 to 0.5 with higher Reynolds number irrespective of the grid structure. This is because the top lid velocity increases with liquidity of the fluid inside the cavity.

V velocity contours 12X12 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

32X32 Grid structure

Time steps:49

Re:100

Pr:1

Re:400

Pr:1

Fig.ML 3.1 Re:400

Pr:1

Time steps:100

Time steps:78

Fig.ML 4.1 Time steps: 130

Fig.ML 6.1

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

40

Fig.ML 5.1 Re:100

Pr:0.5

Re:100

Pr:0.5

Re:400

Pr:0.5

Fig.ML 7.1 Re:400

Pr:0.5

Fig.ML 8.1

Time steps:79

Pr:1.2

Time steps:52

Re:100

Fig.ML 11.1 Re:400

Pr:1.2

Time steps:98

Fig.ML 10.1

Fig.ML 9.1 Re:100

Time steps:77

Time steps:44

Pr:1.2

Time steps:82

Fig.ML 12.1 steps:103 Re:400

Pr:1.2

Time steps:125

It is observed from the above shown figures; for lower Reynolds number i.e 100 with lower grid structure showing three verticities; two at left and right faces and one in between them. But these three verticities are disappears in case of lower Prandl number (0.5) with higher grid structures (32X32). This is because as the viscosity will decreases then the Prandl number also decreases with increasing liquidity of the fluid inside the cavity; this makes uniform in a short duration and it is clearly catches in dense grid structure.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

41

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

Time steps: 49

Re:100

Fig.ML 3.1 Re:400

Pr:1

Fig.ML 5.1 Pr:0.5

Time steps:44

Pr:0.5

Fig.ML 9.1 Pr:1.2

Time steps:52

Time steps: 103

Pr:0.5

Time steps:77

Fig.ML 8.1

Time steps:79 Re:400

Re:100

Pr:1

Fig.ML 6.1 Re:100

Fig.ML 7.1 Re:400

Time steps:78

Fig.ML 4.1

Time steps:100 Re:400

Re:100

Pr:1

Pr:0.5

Time steps:98

Fig.ML 10.1 Re:100

Pr:1.2

Time steps:82

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

42

Fig.ML 12.1

Fig.ML 11.1 Re:400

Pr:1.2

steps:103 Re:400

Pr:1.2

Time steps:125

It is observed from the above sown graphs that that the Prandl number increases ( from 0.5 to 1.2) with lower Reynolds number; the fluid particles will takes a sharp at very close to the left side wall of the cavity. This is because the top lid is moving with lower velocity and liquidity of the fluid is less for less Reynolds number (100).

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

Time steps: 49

Re:100

Fig.ML 3.1 Re:400

Pr:1

Pr:0.5

Time steps:78

Fig.ML 4.1 Time steps:100

Re:400

Pr:1

Time steps: 130

Fig.ML 6.1

Fig.ML 5.1 Re:100

Pr:1

Time steps:44

Re:100

Pr:0.5

Time steps:77

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

43

Fig.ML 7.1 Re:400

Pr:0.5

Fig.ML 8.1 Time steps:79

Re:400

Pr:0.5

Fig.ML 9.1 Re:100

Pr:1.2

Fig.ML 10.1 Time steps:52

Re:100

Pr:1.2

Time steps:82

Fig.ML 12.1

Fig.ML 11.1 Re:400

Time steps:98

Pr:1.2

steps:103 Re:400

Pr:1.2

Time steps:125

It is observed from the figure; the fluid with lower Reynolds number (100) with lower Prandl number (0.5) will take two sharp turns at the bottom side of the cavity wall and these sharp turns will shift towards top surface for higher Reynolds number (400) with higher Prandl number (1.2). But more variations in the graphs when the grid structure changes from coarse to dense.

Stream function contours 12X12 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

32X32 Grid structure

Time steps: 49

Re:100

Pr:1

Time steps:78

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

44

Fig.ML 3.1 Re:400

Pr:1

Fig.ML 4.1 Time steps:100

Re:400

Pr:1

Re:100

Pr:0.5

Fig.ML 6.1

Fig.ML 5.1 Re:100

Pr:0.5

Time steps:44

Pr:0.5

Time steps:79

Re:400

Fig.ML 9.1 Re:100

Pr:1.2

Time steps:77

Fig.ML 8.1

Fig.ML 7.1 Re:400

Time steps: 130

Pr:0.5

Time steps:98

Fig.ML 10.1 Time steps:52

Re:100

Pr:1.2

Time steps:82

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

45

Fig.ML 11.1 Re:400

Pr:1.2

Fig.ML 12.1 steps:103 Re:400

Pr:1.2

Time steps125:

It is observed from the figure; two verticities are formed for lower Reynolds number (100) fluid in less dense grid structure (12X12) and almost only one verticity for verticity for higher Reynolds number (400) for the same grid structure but in dense grid structure (32X32) only one verticity in all the Reynolds number.

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.ML 1.1 Re:100

Pr:1

Fig.ML 2.1 Time steps: 49

Re:100

Pr:1

Time steps:78

Fig.ML 4.1

Fig.ML 3.1 Re:400

Pr:1

Time steps:100

Re:400

Pr:1

Time steps: 130

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

46

Fig.ML 5.1 Re:100

Pr:0.5

Fig.ML 6.1 Time steps:44

Pr:0.5

Time steps:79

Re:400

Pr:1.2

Time steps:52

Re:100

Fig.ML 11.1 Re:400

Pr:1.2

Time steps:77

Pr:0.5

Time steps:98

Fig.ML 10.1

Fig.ML 9.1 Re:100

Pr:0.5

Fig.ML 8.1

Fig.ML 7.1 Re:400

Re:100

steps:103

Pr:1.2

Time steps:82

Fig.ML 12.1 Re:400

Pr:1.2

Time steps:125

It is observed from the figures; the temperature contours with higher Reynolds number are having different characteristic and magnitude at right top corner of the cavity. This is because the top lid is moving with higher velocity and it will affect the fluid particles at bottom side of the top surface. by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

47

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

Time steps: 49

Re:100

Time steps:100

Re:400

Pr:1

Pr:0.5

Time steps:44

Re:100

Fig.ML 7.1 Re:400

Pr:0.5

Time steps:79

Fig.ML 9.1 Re:100

Pr:1.2

Pr:1

Time steps: 130

Fig.ML 6.1

Fig.ML 5.1 Re:100

Time steps:78

Fig.ML 4.1

Fig.ML 3.1 Re:400

Pr:1

Time steps:52

Pr:0.5

Time steps:77

Fig.ML 8.1 Re:400

Pr:0.5

Time steps:98

Fig.ML 10.1

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

48

Re:100

Pr:1.2

Fig.ML 12.1

Fig.ML 11.1 Re:400

Pr:1.2

Time steps:82

steps:103 Re:400

Pr:1.2

Time steps:125

It is observed from the above shown figures; the simulation results are closer with published data for higher value of the Reynolds number (400) than the lower (100).

Case-III: Mixed convection fluid flow (where Gr=106) U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MM 2.1

Fig.MM 1.1 Re:100

Pr:1

Time steps: 46

Re:100

Time steps:81

Re:400

Fig.MM 3.1 Re:400

Pr:1

Pr:0.5

Time steps:81

Fig.MM 4.1 Pr:1

Time steps: 88

Fig.MM 6.1

Fig.MM 5.1 Re:100

Pr:1

Time steps:58

Re:100

Pr:0.5

Time steps:112

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

49

Fig.MM 8.1

Fig.MM 7.1 Re:400

Pr:0.5

Re:400

Time steps:64

Pr:0.5

Fig.MM 10.1

Fig.MM 9.1 Re:100

Pr:1.2

Time steps:72

Re:100

Time steps:50

Pr:1.2

Time steps:85

Fig.MM 12.1 Fig.MM 11.1 Re:400

Pr:1.2

steps:84

Re:400

Pr:1.2

Time steps:99

It is observed from the figure; more vertices are found lower Reynolds number. This is because as the Grashoff number increases, the temperature difference between the cavity walls also increases; this increases the kinetic energy of the fluid particles.

V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MM 1.1 Re:100

Pr:1

Time steps: 46

Fig.MM 2.1 Re:100

Pr:1

Time steps:81

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

50

Fig.MM 4.1

Fig.MM 3.1 Re:400

Pr:1

Time steps:81

Re:400

Fig.MM 5.1 Re:100

Pr:0.5

Pr:0.5

Re:100

Pr:1.2

Pr:0.5

Time steps:112

Fig.MM 8.1

Time steps:64

Re:400

Fig.MM 9.1 Re:100

Time steps: 88

Fig.MM 6.1 Time steps:58

Fig.MM 7.1 Re:400

Pr:1

Pr:0.5

Time steps:72

Fig.MM 10.1 Time steps:50

Re:100

Pr:1.2

Time steps:85

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MM 11.1 Re:400

Pr:1.2

Fig.MM 12.1 steps:84

Re:400

Pr:1.2

Time steps:99

It is observed that the Groshoff number increases with the increasing kinetic energy of the fluid particles due to its higher temperature; it creates more number of verticities in the cavity. The verticities at left and right side walls of the cavity will shift towards bottom wall for lower Prandl number. This is because the liquidity of the fluid particles decreases with increasing Prandl number.

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MM 1.1 Re:100

Pr:1

Fig.MM 2.1 Time steps: 46

Re:100

Time steps:81

Re:400

Pr:1

Pr:0.5

Pr:1

Time steps: 88

Fig.MM 6.1

Fig.MM 5.1 Re:100

Time steps:81

Fig.MM 4.1

Fig.MM 3.1 Re:400

Pr:1

Time steps:58

Re:100

Pr:0.5

Time steps:112

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

52

Fig.MM 8.1

Fig.MM 7.1 Re:400

Pr:0.5

Re:400

Time steps:64

Pr:1.2

Re:100

Time steps:50

Pr:1.2

Pr:1.2

Time steps:85

Fig.MM 12.1

Fig.MM 11.1 Re:400

Time steps:72

Fig.MM 10.1

Fig.MM 9.1 Re:100

Pr:0.5

steps:84

Re:400

Pr:1.2

Time steps:99

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.MM 1.1 Re:100

Pr:1

Fig.MM 2.1 Time steps: 46

Re:100

Fig.MM 3.1 Re:400

Pr:1

Time steps:81

Pr:1

Time steps:81

Fig.MM 4.1 Re:400

Pr:1

Time steps: 88

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

53

Fig.MM 6.1

Fig.MM 5.1 Re:100

Pr:0.5

Time steps:58

Re:100

Pr:0.5

Fig.MM 8.1

Fig.MM 7.1 Re:400

Pr:0.5

Time steps:64

Pr:1.2

Re:400

Pr:0.5

Time steps:72

Fig.MM 10.1

Fig.MM 9.1 Re:100

Time steps:112

Time steps:50

Re:100

Pr:1.2

Time steps:85

Fig.MM 12.1 Fig.MM 11.1 Re:400

Pr:1.2

steps:84

Re:400

Pr:1.2

Time steps:99

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.MM 1.1

Fig.MM 2.1

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

54

Re:100

Pr:1

Time steps: 46

Re:100

Time steps:81

Re:400

Fig.MM 3.1 Re:400

Pr:1

Pr:0.5

Pr:0.5

Pr:1.2

Time steps: 88

Re:100

Pr:0.5

Time steps:112

Fig.MM 8.1

Time steps:64

Re:400

Pr:0.5

Time steps:72

Fig.MM10.1

Fig.MM 9.1 Re:100

Pr:1

Fig.MM 6.1 Time steps:58

Fig.MM 7.1 Re:400

Time steps:81

Fig.MM 4.1

Fig.MM 5.1 Re:100

Pr:1

Time steps:50

Re:100

Pr:1.2

Time steps:85

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

55

Fig.MM 12.1 Fig.MM 11.1 Re:400

Pr:1.2

steps:84

Re:400

Pr:1.2

Time steps:99

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.MM 2.1

Fig.MM 1.1 Re:100

Pr:1

Time steps: 46

Re:100

Fig.MM 3.1 Re:400

Pr:1

Pr:0.5

Time steps:81

Fig.MM 4.1 Time steps:81

Re:400

Pr:1

Time steps: 88

Fig.MM 6.1

Fig.MM 5.1 Re:100

Pr:1

Time steps:58

Re:100

Pr:0.5

Time steps:112

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

56

Fig.MM 8.1

Fig.MM 7.1 Re:400

Pr:0.5

Re:400

Time steps:64

Pr:1.2

Re:100

Time steps:50

Pr:1.2

Pr:1.2

Time steps:85

Fig.MM 12.1

Fig.MM 11.1 Re:400

Time steps:72

Fig.MM 10.1

Fig.MM 9.1 Re:100

Pr:0.5

Re:400

steps:84

Pr:1.2

Time steps: 99

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MM 2.1

Fig.MM 1.1 Re:100

Pr:1

Time steps: 46

Re:100

Pr:1

Time steps:81

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MM 3.1 Re:400

Pr:1

Fig.MM 4.1 Time steps:81

Pr:0.5

Pr:1

Re:100

Time steps:58

Pr:0.5

Fig.MM 7.1 Re:400

Pr:0.5

Pr:1.2

Re:400

Pr:0.5

Pr:1.2

Time steps:72

Fig.MM 10.1 Time steps:50

Re:100

Pr:1.2

Time steps:85

Fig.MM 12.1

Fig.MM 11.1 Re:400

Time steps:112

Fig.MM 8.1

Time steps:64

Fig.MM 9.1 Re:100

Time steps: 88

Fig.MM 6.1

Fig.MM 5.1 Re:100

Re:400

Re:400

steps:84

Pr:1.2

Time steps:99

Case-III: Mixed convection fluid flow (where Gr=2x106) U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Re:100

Pr:1

Time steps:73

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MH 3.1 Re:400

Fig.MH 4.1

Pr:1

Time steps:82

Re:400

Fig.MH 5.1 Re:100

Pr:0.5

Time steps:55

Pr:0.5

Pr:0.5

Time steps:111

Fig.MH 8.1 Time steps:62

Pr:1.2

Re:400

Pr:0.5

Time steps:66

Fig.MH 10.1

Fig.MH 9.1 Re:100

Time steps: 114

Fig.MH 6.1 Re:100

Fig.MH 7.1 Re:400

Pr:1

Time steps:53

Re:100

Pr:1.2

Time steps:77

Fig.MH12.1 Fig.MH 11.1 Re:400

Pr:1.2

steps:87

Re:400

Pr:1.2

Time steps:120

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

59

V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MH 1.1 Re:100

Pr:1

Fig.MH 2.1 Time steps: 53

Re:100

Time steps:82

Re:400

Pr:1

Pr:0.5

Time steps:55

Re:100

Fig.MH 7.1 Re:400

Pr:0.5

Pr:1

Time steps: 114

Fig.MH 6.1

Fig.MH 5.1 Re:100

Time steps:73

Fig.MH 4.1

Fig.MH 3.1 Re:400

Pr:1

Time steps:62

Pr:0.5

Time steps:111

Fig.MH 8.1 Re:400

Pr:0.5

Time steps:66

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

60

Fig.MH 10.1

Fig.MH 9.1 Re:100

Pr:1.2

Re:100

Fig.MH 11.1 Re:400

Pr:1.2

Pr:1.2

Time steps:77

Time steps:53

Fig.MH 12.1 steps:87 Re:400

Pr:1.2

Time steps:120

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Fig.MH 2.1 Re:100

Fig.MH 3.1 Re:400

Pr:1

Pr:0.5

Time steps:73

Fig.MH 4.1 Time steps:82

Re:400

Pr:1

Time steps: 114

Fig.MH 6.1

Fig.MH 5.1 Re:100

Pr:1

Time steps:55

Re:100

Pr:0.5

Time steps:111

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MH 8.1

Fig.MH 7.1 Re:400

Pr:0.5

Re:400

Time steps:62

Fig.MH 9.1 Re:100

Pr:1.2

Pr:0.5

Time steps:66

Fig.MH 10.1 Time steps:53

Re:100

Pr:1.2

Time steps:77

Fig.MH 12.1 Fig.MH 11.1 Re:400

Pr:1.2

Re:400

steps:87

Pr:1.2

Time steps:120

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Re:100

Time steps:82

Re:400

Pr:1

Time steps:73

Fig.MH 4.1

Fig.MH 3.1 Re:400

Pr:1

Pr:1

Time steps: 114

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

62

Fig.MH 6.1

Fig.MH 5.1 Re:100

Pr:0.5

Re:100

Time steps:55

Pr:0.5

Fig.MH 7.1 Re:400

Pr:0.5

Fig.MH 8.1 Time steps:62

Re:400

Pr:0.5

Fig.MH 9.1 Re:100

Pr:1.2

Pr:1.2

Time steps:66

Fig.MH 10.1 Time steps:53

Re:100

Pr:1.2

Time steps:77

Fig.MH 12.1

Fig.MH 11.1 Re:400

Time steps:111

steps:87

Re:400

Pr:1.2

Time steps:120

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Re:100

Pr:1

Time steps:73

Time steps: 53

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MH 3.1 Re:400

Pr:1

Time steps:82

Pr:0.5

Time steps:55

Re:100

Pr:0.5

Time steps:62

Re:400

Pr:0.5

Time steps:111

Pr:1.2

Pr:0.5

Time steps:66

Fig.MH10.1

Fig.MH 9.1 Re:100

Time steps: 114

Fig.MH 8.1

Fig.MH 7.1 Re:400

Pr:1

Fig.MH 6.1

Fig.MH 5.1 Re:100

Fig.MH 4.1 Re:400

Time steps:53

Re:100

Pr:1.2

Time steps:77

Fig.MH 12.1 Fig.MH 11.1 Re:400

Pr:1.2

steps:87

Re:400

Pr:1.2

Time steps:120

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

64

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Re:100

Pr:1

Time steps:82

Pr:0.5

Re:400

Time steps:55

Re:100

Pr:0.5

Time steps: 114

Pr:0.5

Time steps:111

Fig.MH 8.1

Fig.MH7.1 Re:400

Pr:1

Fig.MH 6.1

Fig.MH 5.1 Re:100

Time steps:73

Fig.MH 4.1

Fig.MH 3.1 Re:400

Pr:1

Time steps:62

Re:400

Pr:0.5

Time steps:66

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

65

Fig.MH 10.1

Fig.MH 9.1 Re:100

Pr:1.2

Re:100

Time steps:53

Pr:1.2

Time steps:77

Fig.MH 12.1 Fig.MH 11.1 Re:400

Pr:1.2

steps:87

Re:400

Pr:1.2

Time steps:120

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Re:100

Fig.MH 3.1 Re:400

Pr:1

Pr:0.5

Time steps:73

Fig.MH 4.1 Time steps:82

Re:400

Pr:1

Time steps: 114

Fig.MH 6.1

Fig.MH 5.1 Re:100

Pr:1

Time steps:55

Re:100

Pr:0.5

Time steps:111

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MH 7.1 Re:400

Pr:0.5

Time steps:62

Pr:1.2

Pr:0.5

Time steps:53

Re:100

Pr:1.2

Pr:1.2

Time steps:77

Fig.MH 12.1

Fig.MH 11.1 Re:400

Time steps:66

Fig.MH 10.1

Fig.MH 9.1 Re:100

Fig.MH 8.1 Re:400

steps:87 Re:400

Pr:1.2

Time steps:120

Case-III: Mixed convection fluid flow (where Gr=-105) U velocity contours 12X12 Grid structure

Fig.MNH 2.1

Fig.MNH 1.1 Re:100

Pr:0.5

32X32 Grid structure

Time steps:243

Re:100

Pr:0.5

Time steps:68

Fig.MNH 4.1

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Re:400

Fig.MNH 3.1 Re:400

Pr:0.5

Fig.MNH 5.1 Re:100

Pr:1

Pr:1

Re:100

Pr:1.2

Time steps:57

Pr:1.2

Time steps:68

Pr:1

Time steps: 53

Fig.MNH 10.1 Time steps:249

Re:100

Fig.MNH 11.1 Re:400

Pr:1

Fig.MNH 8.1 Re:400

Fig.MH 9.1 Re:100

Time steps:60

Fig.MNH 6.1 Time steps: 54

Fig.MNH 7.1 Re:400

Pr:0.5

Time steps:78

Pr:1.2

Time steps:66

Fig.MNH 12.1 Time steps:48

Re:400

Pr:1.2

Time steps:52

The U velocity contour verticity is moving towards bottom surface of the cavity as the Prandl number decreases from 1.2 to 0.5 with higher Reynolds number irrespective of the grid structure. This is because the top lid velocity increases with liquidity of the fluid inside the cavity.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

68

V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MNH 2.2

Fig.MNH 1.2 Re:100

Pr:0.5

Re:100

Time steps:243

Fig.MNH 3.2 Re:400

Pr:0.5

Pr:1

Re:400

Time steps:60

Pr:1

Time steps:68

Fig.MNH 8.2

Fig.MNH 7.2 Pr:1

Pr:0.5

Fig.MNH 6.2

Time steps:54 Re:100

Re:400

Time steps:68

Fig.MNH 4.2

Time steps:78

Fig.MNH 5.2 Re:100

Pr:0.5

Time steps:57

Re:400

Pr:1

Time steps: 53

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

69

Fig.MNH 10.2

Fig.MH 9.2 Re:100

Pr:1.2

Time steps:249

Re:100

Pr:1.2

Time steps:66

Fig.MNH 12.2

Fig.MNH 11.2 Re:400

Pr:1.2

steps:48

Re:400

Pr:1.2

Time steps:52

It is observed from the above shown figures; for lower Reynolds number i.e 100 with lower grid structure showing three verticities; two at left and right faces and one in between them. But these three verticities are disappears in case of lower Prandl number (0.5) with higher grid structures (32X32). This is because as the viscosity will decreases then the Prandl number also decreases with increasing liquidity of the fluid inside the cavity; this makes uniform in a short duration and it is clearly catches in dense grid structure.

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.3 Re:100

Pr:0.5

Fig.MNH 2.3 Time steps:243

Re:100

Fig.MNH 3.3 Re:400

Pr:0.5

Time steps:78

Pr:0.5

Time steps:68

Fig.MNH 4.3 Re:400

Pr:0.5

Time steps:60

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MNH 5.3 Re:100

Pr:1

Fig.MNH 6.3 Time steps: 54

Re:100

Time steps:57

Re:400

Fig.MNH 7.3 Re:400

Pr:1

Pr:1.2

Pr:1.2

Pr:1

Time steps: 53

Fig.MNH 10.3 Time steps:249

Re:100

Pr:1.2

Time steps:66

Fig.MNH12.3

Fig.MNH 11.3 Re:400

Time steps:68

Fig.MNH 8.3

Fig.MNH 9.3 Re:100

Pr:1

steps:48

Re:400

Pr:1.2

Time steps:52

It is observed from the above sown graphs that that the Prandl number increases ( from 0.5 to 1.2) with lower Reynolds number; the fluid particles will takes a sharp at very close to the left side wall of the cavity. This is because the top lid is moving with lower velocity and liquidity of the fluid is less for less Reynolds number (100).

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.4 Re:100

Pr:0.5

Fig.MNH 2.4 Time steps:248

Re:100

Fig.MNH 3.4 Re:400

Pr:0.5

Pr:1

Re:400

Pr:1

Re:100

Time steps:57

Re:400

Pr:1.2

Time steps:60

Pr:1

Time steps:68

Fig.MNH 8.4

Fig.MNH 9.4 Re:100

Pr:0.5

Fig.MNH 6.4

Time steps: 54

Fig.MNH 7.4 Re:400

Time steps:68

Fig.MNH 4.4

Time steps:78

Fig.MNH 5.4 Re:100

Pr:0.5

Time steps:249

Pr:1

Time steps: 53

Fig.MNH 10.4 Re:100

Pr:1.2

Time steps:66

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MNH 11.4 Re:400

Pr:1.2

steps:48

Fig.MNH 12.4 Re:400

Pr:1.2

Time steps:52

It is observed from the figure; the fluid with lower Reynolds number (100) with lower Prandl number (0.5) will take two sharp turns at the bottom side of the cavity wall and these sharp turns will shift towards top surface for higher Reynolds number (400) with higher Prandl number (1.2). But more variations in the graphs when the grid structure changes from coarse to dense.

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.5 Re:100

Pr:0.5

Time steps:243

Pr:0.5

Time steps:78

Pr:1

Time steps:68

Re:400

Pr:0.5

Time steps:60

Fig.MNH 6.5

Fig.MNH 5.5 Re:100

Pr:0.5

Fig.MNH 4.5

Fig.MNH 3.5 Re:400

Fig.MNH 2.5 Re:100

Time steps: 54

Re:100

Pr:1

Time steps:68

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MNH 8.5

Fig.MNH 7.5 Re:400

Pr:1

Time steps:57

Pr:1.2

Pr:1

Time steps:249

Re:100

Pr:1.2

Fig.MNH 11.5 Re:400

Pr:1.2

Time steps:53

Fig.MNH 10.5

Fig.MNH 9.5 Re:100

Re:400

Time steps:66

Fig.MNH 12.5 steps:48

Re:400

Pr:1.2

Time steps:52

It is observed from the figure; two verticities are formed for lower Reynolds number (100) fluid in less dense grid structure (12X12) and almost only one verticity for verticity for higher Reynolds number (400) for the same grid structure but in dense grid structure (32X32) only one verticity in all the Reynolds number.

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.6 Re:100

Pr:0.5

Fig.MNH 2.6 Time steps:243

Re:100

Pr:0.5

Time steps:68

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

74

Fig.MNH 3.6 Re:400

Pr:0.5

Fig.MNH 4.6

Time steps:78

Re:400

Pr:1

Time steps: 54

Re:100

Time steps:57

Re:400

Pr:1

Pr:1.2

Time steps:249

Re:100

Fig.MNH 11.6 Re:400

Pr:1.2

Time steps:68

Pr:1

Time steps: 53

Fig.MNH 10.6

Fig.MNH 9.6 Re:100

Pr:1

Fig.MNH 8.6

Fig.MNH 7.6 Re:400

Time steps:60

Fig.MNH 6.6

Fig.MNH 5.6 Re:100

Pr:0.5

Pr:1.2

Time steps:66

Fig.MNH 12.6 steps:48

Re:400

Pr:1.2

Time steps:52

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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It is observed from the figures; the temperature contours with higher Reynolds number are having different characteristic and magnitude at right top corner of the cavity. This is because the top lid is moving with higher velocity and it will affect the fluid particles at bottom side of the top surface.

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.7 Re:100

Pr:0.5

Fig.MNH 2.7

Time steps:243

Re:100

Fig.MNH 3.7 Re:400

Pr:0.5

Pr:1

Re:400

Pr:1

Re:100

Time steps:57

Re:400

Pr:1.2

Time steps:60

Pr:1

Time steps:68

Fig.MNH 8.7 Pr:1

Time steps: 53

Fig.MNH 10.7

Fig.MNH 9.7 Re:100

Pr:0.5

Fig.MNH 6.7 Time steps: 54

Fig.MNH 7.7 Re:400

Time steps:68

Fig.MNH 4.7

Time steps:78

Fig.MNH 5.7 Re:100

Pr:0.5

Time steps:249

Re:100

Pr:1.2

Time steps:66

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MNH 12.7

Fig.MNH 11.7 Re:400

Pr:1.2

Re:400

Pr:1.2

Time steps:52

steps:48

It is observed from the above shown figures; the simulation results are closer with published data for higher value of the Reynolds number (400) than the lower (100).

Case-III: Mixed convection fluid flow (where Gr=-2X106)

Note: Due to time restriction to submit the report online; it not possible do some simulation in mixed fluid flow with 100 Reynolds number with 32X32 grid structure in the VII chapter ‗Results and discussion‘ ; because these simulations will take days together. Therefore the some contours and graphs are predicted based on the trends of the fluid flow and characteristics of the fluid flow already simulated in the previous cases.

U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.1 Re:100

Fig.MNL 1.1 Re:100

Pr:0.5

Fig.MNL 3.1 Re:400

Pr:0.5

Pr:0.5

Time steps:

Time steps:217

Time steps:317

Fig.MNL 4.1 Re:400

Pr:0.5

Time steps:61

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MNL 6.1

Fig.MNL 5.1 Re:100

Pr:1

Time steps: 320

Re:100

Fig.MNL 7.1 Re:400

Pr:1

Time steps:340

Pr:1.2

Pr:1.2

Pr:1

Time steps:54

Fig.MNL 10.1 Time steps:341

Re:100

Fig.MNL 11.1 Re:400

Time steps:

Fig.MNL 8.1 Re:400

Fig.ML 9.1 Re:100

Pr:1

Time steps:344

Pr:1.2

Time steps:

Fig.MNL 12.1 Re:400

Pr:1.2

Time steps:52

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.2

Fig.MNL 1.2 Re:100

Pr:0.5

Re:100

Time steps:217

Fig.MNL 3.2 Re:400

Pr:0.5

Pr:1

Re:400

Pr:1

Pr:0.5

Time steps:61

Fig.MNL 6.2 Re:100

Pr:1

Time steps:

Time steps:320

Fig.MNL 7.2 Re:400

Time steps:

Fig.MNL 4.2

Time steps:317

Fig.MNL 5.2 Re:100

Pr:0.5

Time steps:340

Fig.MNL 8.2 Re:400

Pr:1

Time steps: 54

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.ML 9.2 Re:100

Pr:1.2

Time steps:

Pr:1.2

Pr:1.2

Time steps:52

Fig.MNL 12.2

Fig.MNL 11.2 Re:400

Fig.MNL 10.2 Re:100

steps:344

Re:400

Pr:1.2

Time steps:

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.3

Fig.MNL 1.3 Re:100

Pr:0.5

Time steps:217

Re:100

Fig.MNL 3.3 Re:400

Pr:0.5

Time steps:317

Pr:0.5

Time steps:

Fig.MNL 4.3 Re:400

Pr:0.5

Time steps:61

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Fig.MNL 5.3 Re:100

Pr:1

Fig.MNL 6.3 Time steps: 320

Re:100

Fig.MNL 7.3 Re:400

Pr:1

Time steps:340

Pr:1.2

Time steps:

Fig.MNL 8.3 Re:400

Pr:1

Time steps: 54

Fig.MNL 10.3

Fig.MNL 9.3 Re:100

Pr:1

Time steps:341

Re:100

Pr:1.2

Time steps:

Fig.MNL 12.3 Fig.MNL 11.3 Re:400

Pr:1.2

Re:400

Pr:1.2

Time steps:52

steps:344

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

81

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNL 1.4 Re:100

Pr:0.5

Fig.MNL 2.4 Time steps:217

Re:100

Fig.MNL 3.4 Re:400

Pr:0.5

Time steps:320

Re:100

Time steps:340

Re:400

Fig.MNL 7.4 Re:400

Pr:1

Pr:1.2

Time steps:61

Pr:1

Time steps:

Fig.MNL 8.4 Pr:1

Time steps: 54

Fig.MNL 10.4

Fig.MNL 9.4 Re:100

Pr:0.5

Fig.MNL 6.4

Fig.MNL 5.4 Pr:1

Time steps:

Fig.MNL 4.4

Time steps:317 Re:400

Re:100

Pr:0.5

Re:100

Pr:1.2

Time steps:

Time steps:341

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Fig.MNL 12.4

Fig.MNL 11.4 Re:400

Pr:1.2

Re:400

Pr:1.2

Time steps:52

steps:344

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.MNL 1.5 Re:100

Pr:0.5

Time steps:217

Fig.MNL 2.5 Re:100

Pr:0.5

Time steps:317

Re:400

Fig.MNL 5.5 Re:100

Pr:1

Time steps:

Fig.MNL 4.5

Fig.MNL 3.5 Re:400

Pr:0.5

Time steps:

Pr:0.5

Time steps:61

Fig.MNL 6.5 Re:100

Pr:1

Time steps:

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Fig.MNL 7.5 Re:400

Pr:1

Fig.MNL 8.5 Time steps:340

Re:400

Pr:1

Fig.MNL 9.5 Re:100

Pr:1.2

Time steps:341

Time steps:54

Fig.MNL 10.5 Re:100

Pr:1.2

Time steps:

Fig.MNL 12.5 Re:400

Fig.MNL 11.5 Re:400

Pr:1.2

Pr:1.2

Time steps:52

steps:344

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.6

Fig.MNL 1.6 Re:100

Pr:0.5

Time steps:217

Re:100

Pr:0.5

Time steps:

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Fig.MNL 3.6 Re:400

Pr:0.5

Fig.MNL 4.6

Time steps:317

Re:400

Fig.MNL 5.6 Re:100

Pr:1

Pr:1

Re:100

Time steps:340

Pr:1.2

Pr:1

Time steps:

Fig.MNL 8.6 Re:400

Pr:1

Time steps: 54

Fig.MNL 10.6

Fig.MNL 9.6 Re:100

Time steps:61

Fig.MNL 6.6 Time steps:

Fig.MNL 7.6 Re:400

Pr:0.5

Time steps:341

Re:100

Pr:1.2

Time steps:

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Fig.MNL 12.6

Fig.MNL 11.6 Re:400

Pr:1.2

Re:400

steps:344

Pr:1.2

Time steps:52

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.7

Fig.MNL 1.7 Re:100

Pr:0.5

Re:100

Time steps:217

Pr:0.5

Re:400

Pr:1

Pr:1

Time steps:61

Fig.MNL 6.7

Time steps:

Re:100

Time steps:340

Re:400

Fig.MNL 7.7 Re:400

Pr:0.5

Time steps:317

Fig.MNL 5.7 Re:100

Time steps:

Fig.MNL 4.7

Fig.MNL 3.7 Re:400

Pr:0.5

Pr:1

Time steps:

Fig.MNL 8.7 Pr:1

Time steps: 54

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Fig.MNL 9.7 Re:100

Pr:1.2

Fig.MNL 10.7

Time steps:341

Re:100

Pr:1.2

Fig.MNL 11.7 Re:400

Pr:1.2

Time steps:

Fig.MNL 12.7

steps:344

Re:400

Pr:1.2

Time steps:52

Case-IV: Natural convection fluid flow U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig. N 1.1 Gr:103

Pr:0.71

Fig. N 2.1 Time steps: 204

Fig. N 3.1 Gr:104

Pr:0.71

Time steps:230

Gr:103 Pr:0.71 Computation time: 21hours

Time steps: 665

Fig. N 4.1 Gr:104 Pr:0.71 Computation time: 42hours

Time steps: 1198

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V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig. N 2.2

Fig. N 1.2 Gr:103

Pr:0.71

Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Fig. N 4.2

Fig. N 3.2 Gr:10

4

Pr:0.71

Time steps: 665

Time steps: 1198

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig. N 2.3

Fig. N 1.3 Gr:10

3

Pr:0.71

Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Fig. N 4.3

Fig. N 3.3 Gr:104

Pr:0.71

Time steps: 665

Time steps: 1198

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V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig. N 2.4

Fig. N 1.4 Gr:10

3

Pr:0.71

Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Fig. N 3.4 Gr:104

Pr:0.71

Time steps:230

Time steps: 665

Fig. N 4.4 Gr:104 Pr:0.71 Computation time: 42hours

Time steps:1198

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig. N 1.5 Gr:103

Pr:0.71

Fig. N 2.5 Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Fig. N 3.5 Gr:104

Pr:0.71

Time steps: 665

Fig. N 4.5 Time steps: 1198

Temperature contours by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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12X12 Grid structure

32X32 Grid structure

Fig. N 1.6 Gr:103

Pr:0.71

Fig. N 2.6 Time steps: 204

Pr:0.71

Time steps: 665

Fig. N 4.6

Fig. N 3.6 Gr:104

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Time steps: 1198

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig. N 2.7

Fig. N 1.7 Gr:103

Pr:0.71

Time steps: 204

Gr:10

Pr:0.71

Time steps: 665

Fig. N 4.7

Fig. N 3.7 4

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Time steps: 1198

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Temperature along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig. N 1.8 Gr:103

Pr:0.71

Fig. N 2.8 Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Fig. N 4.8

Fig. N 3.8 Gr:10

4

Pr:0.71

Time steps: 665

Time steps: 1198

It is observed from the figure; the Grashoff number increases from 103 to 104, the computation time and time steps required for the convergence of the solution increases as compared to the forced and mixed convection flow for dense grid structure. This is because the temperature difference increases with increasing Grashoff number. For the less value of Grashoff number in dense grid structure (32X32); the temperature values along the horizontal center line not close to the published data. This difference may be decreases with the high dense grid structure.

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Chapter-VIII

Scilab CFD codes used in the problem and Algorithm The code is written in non-dimensional form, with Reynolds number (Re=ρU0L/μ) as the governing parameter for isothermal flow. For convective heat transfer problems, Prandtl number (Pr=ν/α), Grashoff number (Gr=gβ(TH-TC)L3/ν2) and Rayleigh Number (Ra=gβ(TH-TC)L3/να) comes as an additional governing parameters. However, Gr=0 for forced and RePr=1 for natural convection heat transfer. Note that the characteristic velocity considered here for natural convection is equal to α/L; thus, the diffusion-coefficient is Pr for momentum and 1 for energy equation.

Solution Algorithm: 1) Enter the inputs: material properties, geometric parameters (L1 & L2) and maximum number of CVs in the X and Y directions, B.Cs input and εs. 2) Grid generation: calculate all the geometric parameters of all the CVs. 3) Set ∆ from the stability criteria. 4) Set the initial condition for . 5) Set the boundary condition for 6) Set = for all CVs 7) For =u, calculate fluxes (mass, advection, diffusion) in the X and Y direction at the u-CV faces using velocity of previous time step and S(j,i)=(Pold(j,i+1))* Y 8) Calculate total advection at all centers Ai,j. 9) Calculate total diffusion at all centers, Di,j. 10) For each ―interior‖ CVs, Predict velocity as = (-Aj,i+Dj,i+Sj,i) 11) For =v, calculate the fluxes at the v-CV faces using velocity of previous time step with S(j,i)=(Pold(j,i))-Pold(j+1,i)* X and repeat steps 7-10 12) If max(Divi,j)< ε, then go to step 16 else continue 13) Compute P‘i,j at interior nodes using the mass imbalance Divi,j 14) Pn+1= Pn+ P‘ for all interior grid points 15) Compute velocity correction using pressure correction, update the predicted star velocity and go to step 12. 16) The star velocity becomes the velocity for next time step. Solve the energy equation. 17) Go to step 5 continue all steady state.

Scilab codes for Case-I: // ***************************************************************************** // Codes developed by Vishesh Aggarwal // Under the supervision of Dr.Atul Sharma, IIT Bombay // ***************************************************************************** clc; printf("\n"); printf("*******************************************************************\n"); printf(" LID DRIVEN CAVITY PROBLEM USING 2D STAGGERED GRID NS SOLVER\n"); printf("*******************************************************************\n"); printf("\nGOVERNING PARAMETERS:"); printf("\n\tREYNOLDS NUMBER (Re) BASED ON TOP PLATE VELOCITY\n"); printf("\nBENCHMARK DATA AVAILABLE AT Re = 100, 400, 1000\n"); Re = input("ENTER Re (Must be 100 or 400 or 1000 for benchmarking): ")

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// **************************** PROBLEM PARAMETERS ***************************** // NOTE: The parameters are based on non-dimensional governing equation U = 1; //Top-plate velocity (characteristic velocity scale) Lx = 1; //Length of domain in x-direction (characteristic length scale) Ly = 1; //Length of domain in y-direction dens = 1; //Fluid density vis = 1/Re; //Fluid viscosity // ***************************** DEFINE GRID SIZE ****************************** printf("\nENTER THE NO. OF GRID POINTS\n"); // NOTE: The entered value includes the boundary grid points // This number is based on the pressure cell centre locations imax = input("IN THE X-DIRECTION : "); jmax = input("IN THE Y-DIRECTION : "); dx = Lx/(imax-2); // Grid spacing in x-direction dy = Ly/(jmax-2); // Grid spacing in y-direction dV = dx*dy; // ************ TIME STEP EVALUATION (BASED ON STABILITY CRITERION) ************ // NOTE: Courant–Friedrichs–Lewy (CFL) and Grid Fourier Criterion are used below // These are only neccessary but not sufficient condition for stability // since they are obtained from pure convection and pure diffusion, but not for // the NS equation which is a convection-diffusion equation with a source term // Furthermore, the maximum velocity needed here to obtained minimum time-step // is equal to lid velocity. // If the maximum velocity occurs inside the domain and changes with time, // then this expression needs to be used after each transient computation. dt = min(0.5*dx/U, 0.25*((dx*dy)*(dx*dy)/((vis/dens)*(dx*dx + dy*dy)))); // ************************* OTHER CONTROL PARAMETERS ************************** steady_state_criteria = 1e-3; // Used to stop outer time loop mass_div_criteria = 1e-8; // Used to stop inner mass divergence loop time_step = 0; total_time = 0; // ***************** DEFINING ARRAYS TO HOLD PROBLEM VARIABLES ***************** x = zeros(jmax-1,imax-1); y = zeros(jmax-1,imax-1); x_p = zeros(jmax,imax); y_p = zeros(jmax,imax); x_u = zeros(jmax,imax-1); y_u = zeros(jmax,imax-1); x_v = zeros(jmax-1,imax); y_v = zeros(jmax-1,imax); u = zeros(jmax,imax-1); v = zeros(jmax-1,imax); p = zeros(jmax,imax); pc = zeros(jmax,imax); uold = zeros(jmax,imax-1); vold = zeros(jmax-1,imax); ustar = zeros(jmax,imax-1); vstar = zeros(jmax-1,imax); Div = zeros(jmax-1,imax-1); mx1 = zeros(jmax,imax-2); ax1 = zeros(jmax,imax-2); dx1 = zeros(jmax,imax-2); my1 = zeros(jmax,imax-2); ay1 = zeros(jmax,imax-2); dy1 = zeros(jmax,imax-2);

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mx2 = zeros(jmax-2,imax); ax2 = zeros(jmax-2,imax); dx2 = zeros(jmax-2,imax); my2 = zeros(jmax-2,imax); ay2 = zeros(jmax-2,imax); dy2 = zeros(jmax-2,imax); // ******************** ASSIGNING STAGGERED GRID INFORMATION ******************* // Corner vertices of each p-cell for i=1:1:imax-1 for j=1:1:jmax-1 x(j,i) = (i-1)*dx; y(j,i) = (j-1)*dy; end end // Cell center of interior p-cell for i=2:1:imax-1 for j=2:1:jmax-1 x_p(j,i) = 0.5*(x(j,i) + x(j,i-1)); y_p(j,i) = 0.5*(y(j,i) + y(j-1,i)); end end // Cell center of boundary p-cell for i=2:1:imax-1 x_p(1,i) = 0.5*(x(1,i)+x(1,i-1)); y_p(1,i) = 0; x_p(jmax,i) = 0.5*(x(jmax-1,i)+x(jmax-1,i-1)); y_p(jmax,i) = Ly; end for j=2:1:jmax-1 x_p(j,1) = 0; y_p(j,1) = 0.5*(y(j,1)+y(j-1,1)); x_p(j,imax) = Lx; y_p(j,imax) = 0.5*(y(j,imax-1)+y(j-1,imax-1)); end // Corner p-cells of domain x_p(1,1) = 0; y_p(1,1) = 0; x_p(1,imax) = Lx; y_p(1,imax) = 0; x_p(jmax,1) = 0; y_p(jmax,1) = Ly; x_p(jmax,imax) = Lx; y_p(jmax,imax) = Ly; // Cell center of interior u-cell for i=2:1:imax-2 for j=2:1:jmax-1 x_u(j,i) = x(j,i); y_u(j,i) = 0.5*(y(j,i)+y(j-1,i)); end end // Cell center of boundary u-cell for i=2:1:imax-2 x_u(1,i) = x(1,i); y_u(1,i) = 0;

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x_u(jmax,i) = x(jmax-1,i); y_u(jmax,i) = Ly; end for j=2:1:jmax-1 x_u(j,1) = 0; y_u(j,1) = 0.5*(y(j,1)+y(j-1,1)); x_u(j,imax-1) = Lx; y_u(j,imax-1) = 0.5*(y(j,imax-1)+y(j-1,imax-1)); end // Corner u-cells of domain x_u(1,1) = 0; y_u(1,1) = 0; x_u(1,imax-1) = Lx; y_u(1,imax-1) = 0; x_u(jmax,1) = 0; y_u(jmax,1) = Ly; x_u(jmax,imax-1) = Lx; y_u(jmax,imax-1) = Ly; // Cell center of interior v-cell for i=2:1:imax-1 for j=2:1:jmax-2 x_v(j,i) = 0.5*(x(j,i)+x(j,i-1)); y_v(j,i) = y(j,i); end end // Cell center of boundary v-cell for i=2:1:imax-1 x_v(1,i) = 0.5*(x(1,i)+x(1,i-1)); y_v(1,i) = 0; x_v(jmax-1,i) = 0.5*(x(jmax-1,i)+x(jmax-1,i-1)); y_v(jmax-1,i) = Ly; end for j=2:1:jmax-2 x_v(j,1) = 0; y_v(j,1) = y(j,1); x_v(j,imax) = Lx; y_v(j,imax) = y(j,imax-1); end // Corner v-cells of domain x_v(1,1) = 0; y_v(1,1) = 0; x_v(1,imax) = Lx; y_v(1,imax) = 0; x_v(jmax-1,1) = 0; y_v(jmax-1,1) = Ly; x_v(jmax-1,imax) = Lx; y_v(jmax-1,imax) = Ly; // ************************ APPLYING INITIAL CONDITIONS ************************ for i=1:1:imax

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for j=1:1:jmax u(j,i) = 0; v(j,i) = 0; ustar(j,i) = 0; vstar(j,i) = 0; p(j,i) = 0; end end // ******************** FUNCTION: APPLY BOUNDARY CONDITION ********************* // NOTE: Boundary condition application is encapsulated in a function // It allows ease in modification of boundary conditions based on problem setup // It can be called within the main loop repeatedly if the problem demands function [u, v, ustar, vstar, p]=APPLY_BC(u, v, ustar, vstar, p) funcprot(0); // Bottom Boundary for i=1:1:imax u(1,i) = 0; v(1,i) = 0; ustar(1,i) = 0; vstar(1,i) = 0; p(1,i) = p(2,i); end // Top Boundary for i=1:1:imax u(jmax,i) = U; v(jmax-1,i) = 0; ustar(jmax,i) = U; vstar(jmax-1,i) = 0; p(jmax,i) = p(jmax-1,i); end // Left Boundary for j=1:1:jmax u(j,1) = 0; v(j,1) = 0; ustar(j,1) = 0; vstar(j,1) = 0; p(j,1) = p(j,2); end // Right Boundary for j=1:1:jmax u(j,imax-1) = 0; v(j,imax) = 0; ustar(j,imax-1) = 0; vstar(j,imax) = 0; p(j,imax) = p(j,imax-1); end endfunction // ************* FUNCTION: PRESSURE CORRECTION INITIAL CONDITION *************** function [pc]=APPLYIC_PCORR(pc) funcprot(0); for j=1:1:jmax for i=1:1:imax pc(j,i) = 0; end end endfunction // ************* FUNCTION: PRESSURE CORRECTION BOUNDARY CONDITION ************** function [pc]=APPLYBC_PCORR(pc)

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for j=1:1:jmax pc(j,1) = pc(j,2); pc(j,imax) = pc(j,imax-1); end for i=1:1:imax pc(1,i) = pc(2,i); pc(jmax,i) = pc(jmax-1,i); end endfunction // ********************* MAIN TIME LOOPING BEGINS HERE ************************* unsteadiness = 1e6; while unsteadiness > steady_state_criteria // Apply boundary conditions [u,v,ustar,vstar,p] = APPLY_BC(u,v,ustar,vstar,p); // Store old time level data uold = u; vold = v; //****************************************************************** // Predict new time level velocities // Fluxes across u-velocity cell faces for j=2:1:jmax-1 for i=1:1:imax-2 mx1(j,i) = dens*0.5*(u(j,i)+u(j,i+1)); ax1(j,i) = max(mx1(j,i),0)*u(j,i) - max(-mx1(j,i),0)*u(j,i+1); dx1(j,i) = vis*(u(j,i+1)-u(j,i))/(x_u(j,i+1)-x_u(j,i)); end end for j=1:1:jmax-1 for i=2:1:imax-2 my1(j,i) = dens*0.5*(v(j,i)+v(j,i+1)); ay1(j,i) = max(my1(j,i),0)*u(j,i) - max(-my1(j,i),0)*u(j+1,i); dy1(j,i) = vis*(u(j+1,i)-u(j,i))/(y_u(j+1,i)-y_u(j,i)); end end // Fluxes across v-velocity cell faces for j=2:1:jmax-2 for i=1:1:imax-1 mx2(j,i) = dens*0.5*(u(j,i)+u(j+1,i)); ax2(j,i) = max(mx2(j,i),0)*v(j,i) - max(-mx2(j,i),0)*v(j,i+1); dx2(j,i) = vis*(v(j,i+1)-v(j,i))/(x_v(j,i+1)-x_v(j,i)); end end for j=1:1:jmax-2 for i=2:1:imax-1 my2(j,i) = dens*0.5*(v(j,i)+v(j+1,i)); ay2(j,i) = max(my2(j,i),0)*v(j,i) - max(-my2(j,i),0)*v(j+1,i); dy2(j,i) = vis*(v(j+1,i)-v(j,i))/(y_v(j+1,i)-y_v(j,i)); end end // Predict cell center velocities for j=2:1:jmax-1 for i=2:1:imax-2 Au = (ax1(j,i)-ax1(j,i-1))*dy + (ay1(j,i)-ay1(j-1,i))*dx; Du = (dx1(j,i)-dx1(j,i-1))*dy + (dy1(j,i)-dy1(j-1,i))*dx; Su = (p(j,i)-p(j,i+1))*dy; ustar(j,i) = u(j,i) + (dt/(dens*dV))*(Du-Au+Su);

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end end for j=2:1:jmax-2 for i=2:1:imax-1 Av = (ax2(j,i)-ax2(j,i-1))*dy + (ay2(j,i)-ay2(j-1,i))*dx; Dv = (dx2(j,i)-dx2(j,i-1))*dy + (dy2(j,i)-dy2(j-1,i))*dx; Sv = (p(j,i)-p(j+1,i))*dx; vstar(j,i) = v(j,i) + (dt/(dens*dV))*(Dv-Av+Sv); end end //****************************************************************** // Divergence term (mass error) evaluation per cell RMS_Div = 1e6; [pc] = APPLYIC_PCORR(pc); count = 0; while (RMS_Div > mass_div_criteria) // NOTE: It may be needed to restrict the maximum no. of iterations // besides checking convergence for some flow problems // Further, applying boundary conditions for USTAR and VSTAR // within this loop is also useful for channel flow problems RMS_Div = 0; for j=2:1:jmax-1 for i=2:1:imax-1 Div(j,i) = (ustar(j,i)-ustar(j,i-1))*dens*dy + (vstar(j,i)-vstar(j-1,i))*dens*dx; if (RMS_Div mass_div_criteria) // NOTE: It may be needed to restrict the maximum no. of iterations // besides checking convergence for some flow problems // Further, applying boundary conditions for USTAR and VSTAR // within this loop is also useful for channel flow problems RMS_Div = 0; for j=2:1:jmax-1 for i=2:1:imax-1 Div(j,i) = (ustar(j,i)-ustar(j,i-1))*dens*dy + (vstar(j,i)-vstar(j-1,i))*dens*dx; if (RMS_Div steady_state_criteria // Apply boundary conditions [u,v,ustar,vstar,p,T] = APPLY_BC(u,v,ustar,vstar,p,T); // Store old time level data uold = u; vold = v; Told = T; //****************************************************************** // Solving temperature equation // Fluxes in x-direction for T for j=2:1:jmax-1 for i=1:1:imax-1 mxT(j,i) = dens*Cp*u(j,i); axT(j,i) = max(mxT(j,i),0)*T(j,i) - max(-mxT(j,i),0)*T(j,i+1); dxT(j,i) = dif*(T(j,i+1)-T(j,i))/(x_p(j,i+1)-x_p(j,i)); end end // Fluxes in y-direction for T for j=1:1:jmax-1 for i=2:1:imax-1 myT(j,i) = dens*Cp*v(j,i);

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ayT(j,i) = max(myT(j,i),0)*T(j,i) - max(-myT(j,i),0)*T(j+1,i); dyT(j,i) = dif*(T(j+1,i)-T(j,i))/(y_p(j+1,i)-y_p(j,i)); end end // Get new time level temperatures for j=2:1:jmax-1 for i=2:1:imax-1 DTp = (dxT(j,i)-dxT(j,i-1))*dy + (dyT(j,i)-dyT(j-1,i))*dx; ATp = (axT(j,i)-axT(j,i-1))*dy + (ayT(j,i)-ayT(j-1,i))*dx; T(j,i) = T(j,i) + dt*(DTp - ATp)/dV; end end //****************************************************************** // Predict new time level velocities // Fluxes across u-velocity cell faces for j=2:1:jmax-1 for i=1:1:imax-2 mx1(j,i) = dens*0.5*(u(j,i)+u(j,i+1)); ax1(j,i) = max(mx1(j,i),0)*u(j,i) - max(-mx1(j,i),0)*u(j,i+1); dx1(j,i) = vis*(u(j,i+1)-u(j,i))/(x_u(j,i+1)-x_u(j,i)); end end for j=1:1:jmax-1 for i=2:1:imax-2 my1(j,i) = dens*0.5*(v(j,i)+v(j,i+1)); ay1(j,i) = max(my1(j,i),0)*u(j,i) - max(-my1(j,i),0)*u(j+1,i); dy1(j,i) = vis*(u(j+1,i)-u(j,i))/(y_u(j+1,i)-y_u(j,i)); end end // Fluxes across v-velocity cell faces for j=2:1:jmax-2 for i=1:1:imax-1 mx2(j,i) = dens*0.5*(u(j,i)+u(j+1,i)); ax2(j,i) = max(mx2(j,i),0)*v(j,i) - max(-mx2(j,i),0)*v(j,i+1); dx2(j,i) = vis*(v(j,i+1)-v(j,i))/(x_v(j,i+1)-x_v(j,i)); end end for j=1:1:jmax-2 for i=2:1:imax-1 my2(j,i) = dens*0.5*(v(j,i)+v(j+1,i)); ay2(j,i) = max(my2(j,i),0)*v(j,i) - max(-my2(j,i),0)*v(j+1,i); dy2(j,i) = vis*(v(j+1,i)-v(j,i))/(y_v(j+1,i)-y_v(j,i)); end end // Predict cell center velocities for j=2:1:jmax-1 for i=2:1:imax-2 Au = (ax1(j,i)-ax1(j,i-1))*dy + (ay1(j,i)-ay1(j-1,i))*dx; Du = (dx1(j,i)-dx1(j,i-1))*dy + (dy1(j,i)-dy1(j-1,i))*dx; Su = (p(j,i)-p(j,i+1))*dy; ustar(j,i) = u(j,i) + (dt/(dens*dV))*(Du-Au+Su); end end for j=2:1:jmax-2 for i=2:1:imax-1 Av = (ax2(j,i)-ax2(j,i-1))*dy + (ay2(j,i)-ay2(j-1,i))*dx; Dv = (dx2(j,i)-dx2(j,i-1))*dy + (dy2(j,i)-dy2(j-1,i))*dx; Sv = (p(j,i)-p(j+1,i))*dx + Ra*Pr*0.5*(T(j,i)+T(j+1,i))*dV; vstar(j,i) = v(j,i) + (dt/(dens*dV))*(Dv-Av+Sv);

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end end //****************************************************************** // Divergence term (mass error) evaluation per cell RMS_Div = 1e6; [pc] = APPLYIC_PCORR(pc); count = 0; while (RMS_Div > mass_div_criteria) // NOTE: It may be needed to restrict the maximum no. of iterations // besides checking convergence for some flow problems // Further, applying boundary conditions for USTAR and VSTAR // within this loop is also useful for channel flow problems RMS_Div = 0; for j=2:1:jmax-1 for i=2:1:imax-1 Div(j,i) = (ustar(j,i)-ustar(j,i-1))*dens*dy + (vstar(j,i)-vstar(j-1,i))*dens*dx; if (RMS_Div

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“Solution of Non-dimensional Navier-Stokes equations for Flow and Heat transfer by using Scilab CFD codes in LID DRIVEN CAVITY and study of variation of non-dimensional, convergence criteria parameters with different grid structure” Submitted by

S.N.Topannavar [email protected] Cell: +91 9480397798 Sub center: KIT, Kolhapur

Ten Day ISTE Main Workshop on Computational Fluid Dynamics (CFD) Conducted by

Indian Institute of Technology Bombay

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

1

Index Chapter

Content

Page No.

I

Introduction

3-4

II

Literature survey with conclusions

5-13

III

Objectives of the work

14

IV

Physical description of the problem and models for simulations

15-18

V

Mathematical modeling with boundary conditions

19-20

VI

Validation study

21

VII

Results and Discussions

22-91

VIII

Scilab CFD codes used in the problem and algorithm

92-125

IX

Conclusion

126

X

References

127

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-I

Introduction Over the last three decades, the so-called lid-driven cavity flow problem has received considerable attention mainly because of its geometric simplicity, physical abundance, and its close relevance to some fundamental engineering flows. While some fundamental flow phenomena have become clear to us through twodimensional solutions. The recent progress in numerical analyses and computer hardware have made it possible to numerically analyze unsteady flow problems by solving their corresponding Navier-Stokes equations with a large number of grid points within a three-dimensional domain. In a parallel development, a considerable number of experimental studies on this problem have been done since the early 1980s. Due to the relatively inexpensive high speed computers, numerical simulation approach, such as computational fluid dynamics (CFD), is widely adopted for investigating realistic and research problems. Numerical simulation has full control on computing the parameters of problems of different complexities. Therefore, it is able to provide a compromising solution among cost, efficiency and complexity to engineering problems. Although high speed computers and robust numerical techniques have been developed rapidly, the computation of turbulence at high Reynolds number using direct numerical simulation (DNS) is too expensive for practical problems. The large-eddy simulation (LES) is an alternative that demands relatively less computational load. However, it still requires huge amount of computation resources for simulations conducted on sequential computers. The recent advance of supercomputers provides a possibility for conducting these large scale computations. Sequential computer codes could be parallelized directly by compilers but it is unable to fully utilize supercomputers. Therefore, innovative parallel solution techniques are necessary for exploring the power of parallel computing. To facilitate parallel computation the domain is usually divided into several sub-domains according to the structure of the mesh Flow in a lid driven cavity is one of the most widely used benchmark problems to test steady state incompressible fluid dynamics codes. Our interest will be to present this problem as a benchmark for the steady and unsteady state solution. In order to demonstrate the grid independence, code validation and other details like time step, grid size and steadiness criteria; we taken 2D Cartesian (x, y as horizontal and vertical components) square domain of size L x L (1 unit x 1 unit for simplicity of the problem) with bottom, left and right boundaries as solid walls stationary; whereas top wall is like a long conveyor-belt, moving horizontally with a constant velocity (Uo=1unit for the simplicity of the problem). To study the influence of grid sizes, computational time steps on the convergence of the governing equation codes and to catch the oscillations in the contours of different governing parameters; we categorized problem into three models; firstly, a coarse grids i.e. 12 x 12 model secondly, a medium grids i.e. 32 x 32 model and finally, a fine grids i.e. 52 x 52 model. To study the x and y velocity component contours and steam functions in the conservation of mass and momentum; we employed Non-dimensional Navier-Stokes solver with Reynolds number, Prandl number, Grashoff number and Richardson number for the simulation of the problem. And whole study is mainly concentrated on Four different cases like Isothermal fluid flow (all walls of the are at same temperature in all time steps),Non-isothermal forced convection fluid flow ( where Grashoff number is almost negligible because buoyancy induced flow exists), Mixed Convection fluid flow ( where buoyancy as well as inertia induced; the Grashoff number is non zero value) and Natural convection fluid flow ( where inertia is negligible therefore Reynolds not taken into consideration) The study of the stability of two dimensional vortex flows of a viscous fluid is one of the fundamental problems of hydrodynamics, which concerns the problems of control of separation flows. The topological characteristics of a flow can be determined on the basis of solution of the Navier–Stokes equations. However, the number of problems that can be exactly solved in this way is limited; this being so, numerical methods are used in the majority of cases. The problem on the flow of an incompressible viscous fluid in a rectangular cavity with a moving wall is a classical fluid-mechanics problem with closed boundaries. The by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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main structural peculiarities of this flow are characteristic of other separation flows having a more complex geometry; therefore, solution of the problem on the indicated flow is used for testing and comparison of different numerical methods of integrating the Navier–Stokes equations. Comprehensive data on the vortex structure and characteristics of a flow in a rectangular cavity. On the basis of systematization and analysis of the data, steady state convergence criteria 10-4 have been adopted for estimating the quality of the discrete model used. The computation of turbulence at high Reynolds number using direct numerical simulation (DNS) is too expensive for practical problems. The large-eddy simulation (LES) is an alternative that demands relatively less computational load. However, it still requires huge amount of computation resources for simulations conducted on sequential computers. But we facilitated Microsoft Windows XP professional version 2002 operating system, Inter(R) Core(TM) Duo CPU T6670 @ 2.20 GHz processor with small memory of 1.96G, 2.19GHz speed RAM Dell Vastro lap top for large scale computations. Due to time restriction of submission of the report we skipped some fine grid computation of simulations because, it is observed that some simulations will take one and half day computation time. .

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-II

Literature survey with conclusions [1] A numerical study of the vertical flow structure in a confined lid-driven cavity which is defined by a depth-to-width aspect ratio of 1:1 and a span-to-width aspect ratio of 3:1 SAR (Spanwise Aspect Ratio) ( L : B . 1; 2; 3). A simple discretization technique ; third-order QUICK (Leonard 1979) upwind scheme, formulated on the non-uniform basis, to the nonlinear advective fluxes was applied to study carefully examined the computed data that the useful to gain an in-depth knowledge of the complex interactions among secondary eddies, primary eddies, and spiraling span wise motions. Chief of conclusions drawn from this study is to explain how the secondary eddies are intimately coupled with the primary re-circulating flow. Also enlighten in this paper why spiraling vortices inside the upstream secondary eddy tend to destabilize the incompressible flow system and aid development of laminar instabilities. Prior to describing the appearance of TGL (Taylor-GoÈrtler) vortices are studied in detail how eddies of different sizes and attributes are intimately coupled. And same is permitted a systematic approach to understanding the complex interaction among spiraling eddies. The separation surface plotted in this paper furthermore helps to show that fluid flows present in the narrow wavy trough of the separation surface have a higher propensity to develop into TGL vortices. Conclusions: [1] The geometry of the cavity examined is extraordinarily simple; the flow physics in the cavity are nevertheless rich. The physical complexity is attributable to the eddies which are characterized as possessing different sizes and characteristics. Also, how interaction proceeds among the eddies is crucial to the development into laminar instabilities. In the entire flow evolution, the transport mechanism is rooted largely in the spiraling nature of the flow motion established inside the secondary eddies and, of course, in the primary core. According to the finite volume solutions concluded with some important findings from the numerical simulation. The three-dimensional lid-driven cavity flow is manifested by the presence of a spanwise velocity component which arises due to the presence of two vertical end walls. Accompanying the span-wise motion, the flow exhibiting the dominant recirculation flow pattern is prone to spiral. It is interpreted that the presence of USE particles, which are engulfed from regions fairly near the two end walls into the primary core and then spiral monotonically towards the symmetry plane, as being the main cause leading to the flow instability because the two flow streams moving in opposite directions tend to collide with each other at the symmetry plane. This instability causes the surface separating the primary core and the upstream secondary eddy to detach from the upstream side wall. It is this distorted detachment which disrupts the well-balanced force between the centrifugal and pressure-gradient forces established inside the primary re-circulating cell. This paves the way for the onset of Taylor-GoÈrtler vortices. As the end wall is approached, particles in the downstream secondary eddy begin to be engulfed into the primary core and this is followed by suction of particles in the upstream secondary eddy, which is closer to the end wall, into the primary core through the spiral-saddle point. There exists a higher possibility that instabilities will result at spatial locations where the width of the upstream secondary eddy becomes appreciably larger than the width of the downstream secondary eddy. Computational experience from this study reveals that the size of the upstream secondary eddy and the contour lines of zero span-wise velocity at the surface, and the separation surface are closely related.. In the vicinity of the distorted v = 0 contour surface, the sign-switching span-wise velocity induces a free-shear vortex. The pressure field established to support the existence of this vortex further affects the boundary layer of the outward-running spiraling flow in the sense that a wall-shear vortex is formed near the floor of the cavity. This pair of well-established vortices, referred to as Taylor-GoÈrtler vortices, bursts from the spatial location which has the local maximum kinetic energy.

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[3] The work on stabilized finite element formulation proposed by Tezduyar applied to solve steady viscoplastic incompressible flows on unstructured grids. The formulation, originally proposed for Newtonian fluids, allows that equal-order-interpolation velocity-pressure elements are employed, circumventing the Babuska-Brezzi stability condition by introducing two stabilization terms. The first term used is the streamline upwind/Petrov-Galerkin (SUPG) introduced by Brooks and Hughes and the other one is the ressurestabilizing/ Petrov-Galerkin (PSPG) stabilization proposed initially by Hughes for Stokes flows and generalized by Tezduyar to the Navier–Stokes equations. The inexact-Newton methods associated with iterative Krylov solvers have been used to reduce computational efforts related to non-linearities in many problems of computational fluid dynamics, offering a trade-off between accuracy and the amount of computational effort spent per iteration.

A parallel edge-based solution of three dimensional viscoplastic flows governed by the steady Navier– Stokes equations is presented. The governing partial differential equations are discritized using the SUPG (streamline upwind/Petrov-Galerkin)/PSPG (pressure stabilizing/Petrov-Galerkin) stabilized finite element method on unstructured grids. The highly nonlinear algebraic system arising from the convective and material effects is solved by an inexact Newton-Krylov method. The locally linear Newton equations are solved by GMRES with nodal block diagonal pre-conditioner. Matrix-vector products within GMRES are computed edge-by-edge (EDE), diminishing flop counts and memory requirements. A comparison between EDE and element-by-element data structures is presented. The parallel computations were based in a message passing interface standard. Performance tests were carried out in representative three dimensional problems, the sudden expansion for power-law fluids and the flow of Bingham fluids in a lid-driven cavity. Results have shown that edge based schemes requires less CPU time and memory than element based solutions. The SUPG/PSPG finite element formulation with the inexact nonlinear method.

Conclusions: [3] The nonlinear character due to the non-Newtonian viscous and convective terms of the Navier–Stokes equations was treated by an inexact-nonlinear method allowing a good tradeoff between convergence and computational effort. At the beginning of the solution procedure the large linear tolerances produced fast nonlinear steps, and as the solution progresses, the inexact nonlinear method adapts the tolerances to reach the desired accuracy. The linear systems of equations within the nonlinear solution procedure were solved with the nodal block diagonal preconditioned GMRES. An edge-based data structure was introduced and successfully employed to improve the performance of the matrix-vector products within the iterative solver. The results showed that the computing time when using EDE data structure was on the average 2.5 times faster than for those problems using standard EBE. The computations were performed in a message passing parallelism environment presenting good speedup and scalability. [4] A general, efficient, accurate and reliable algorithm developed with emphasis on high Reynolds number flows that still maintains a simple algorithmic structure and which is not hampered by the diffusive time step limit. A new semi-implicit finite element algorithm for time-dependent viscous incompressible flows. The algorithm is of a general type and can handle both low and high Reynolds number flows, although the emphasis is on convection dominated flows. An explicit three-step method is used for the convection term and an implicit trapezoid method for the diffusion term. The consistent mass matrix is only used in the momentum phase of the fractional step algorithm while the lumped mass matrix is used in the pressure phase and in the pressure Poisson equation. An accuracy and stability analysis of the algorithm is provided for the pure convection equation. Two different types of boundary conditions for the end-of-step velocity of the fractional step algorithm have been investigated. Numerical tests for the lid-driven cavity at Re = 1 and Re= 7500 and flow past a circular cylinder at Re =100 are presented to demonstrate the usefulness of the method. Finite element method for predicting time-dependent viscous incompressible flows over a wide range of inertial conditions has been presented. The method is mainly aimed at solving convection dominated flows and employs an explicit three-step algorithm for the convection terms, which gives not only high accuracy by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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but also high efficiency since it allows large Courant numbers. To further improve accuracy for this kind of flows, the consistent mass matrix has also been included. Two variants of the method have been used; one fully explicit scheme with lumped mass matrix and one semi-implicit scheme with the consistent mass matrix in the momentum phase of the fractional step algorithm but with the lumped mass matrix in the pressure phase and in the pressure Poisson equation. The latter of these variants requires an extra system of linear equations to be solved at every time step. This was done in a simple and efficient way by using just a few Jacobi iterations and it was shown that this worked well even for very low Reynolds number flows. Two different kinds of velocity boundary conditions for the end-of-step velocity of the fractional step algorithm have been investigated, one which excludes checker boarding (type 1 B.C.) and one simpler version which does not exclude the checkerboard mode (type 2 b.c.). The type 1 B.C. was found to be slightly more accurate and it was also found to initiate the vortex shedding behind the circular cylinder much earlier than the type 2 b.c.

Conclusions: [4] The simple algorithmic structure and that no extra terms or new higher-order derivatives are needed. In spite of the simplicity, the method is of a general nature and can easily handle complex geometries. Numerical tests show good agreement with other numerical solutions and experimental data and suggest that the proposed method is competitive in terms of both accuracy and efficiency. [5] Fixed point iteration idea employed to linearize the coarse and fine scale sub-problems that arise in the variational multi scale frame work and it lead to a stabilized method for the incompressible Navier–Stokes equations. In the current work we present a consistent linearization of the nonlinear coarse and fine scale sub-problems, and substitution of the fine scales extracted from the fine-scale problem into the coarse-scale variational form leads to the new stabilized method. The solution of the fine-scale or the sub-grid scale problem which is an integral component of the proposed procedure for developing stabilized methods automatically yields an explicit definition of the stabilization operator τ. Another significant contribution of the paper is a numerical technique for evaluating the advection part of the stabilization operator τ that brings in the notion of up-winding in the resulting method. Presented a variational multi-scale-based stabilized formulation for the incompressible Navier–Stokes equations. A novel feature of our method is that fine scales are solved in a direct nonlinear fashion. Consistent linearization of the nonlinear equations in the context of the variational multi scale framework leads to the design of the stabilization terms in the new method A variational multi-scale residual-based stabilized finite element method for the incompressible Navier– Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation furnished by the variational multi-scale framework. A significant feature of the new method is that the fine scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the advection part of the stabilization tensor. The new method circumvents the Babuska–Brezzi condition and yields a stable formulation for high Reynolds number flows. A family of equal-order pressure-velocity elements comprising 4- and 10-node tetrahedral elements and 8- and 27-node hexahedral elements is developed. Convergence rates are reported and accuracy properties of the method are presented via the liddriven cavity flow problem. Presented the strong form and the classical weak form of the incompressible Navier–Stokes equations. Consistent linearization of the nonlinear equations performed in the vartiational multi-scale setting leads to the new multi-scale /stabilized formulation that is developed. The structure of the stabilization tensor and a numerical scheme to evaluate its advection part are presented; a convergence study for a family of 3D tetrahedral and hexahedral elements. An extensive set of numerical simulations of liddriven cavity flows for various Reynolds number are also presented.

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Conclusions: [5] The VMS based stabilized form possesses additional stabilization terms than are present in the classical stabilization methods alone. An important feature of the new method is that a definition of the stabilization operator τ appears naturally via the solution of the fine-scale problem. This stabilization operator is a second order tensor and leads to a full matrix that brings in cross coupling effects in the stabilization terms. A computationally economic scheme is proposed that incorporates up-winding effects in the calculation of the advection part of the stabilization operator τ. Good stability and accuracy properties of the new method are shown for a family of linear and quadratic tetrahedral and hexahedral elements. [6]A scalable numerical model to solve the unsteady incompressible Navier–Stokes equations is developed using the Galerkin finite element method. The coupled equations are decoupled by the fractional-step method and the systems of equations are inverted by the Krylov subspace iterations. The data structure makes use of a domain decomposition of which each processor stores the parameters in its sub-domain, while the linear equations solvers and matrices constructions are parallelized by a data parallel approach. The accuracy of the model is tested by modeling laminar flow inside a two-dimensional square lid-driven cavity for Reynolds numbers at 1,000 as well as three-dimensional turbulent plane and wavy Couette flow and heat transfer at high Reynolds numbers. The parallel performance of the code is assessed by measuring the CPU time taken on an IBM SP2 supercomputer. The speed up factor and parallel efficiency show a satisfactory computational performance. The innovative parallel solution techniques are adopted for exploring the power of parallel computing. Domain decomposition or the Schwarz method that is commonly adopted by CFD analysts. The discretized information is distributed to each processor which is responsible for the calculation in the corresponding sub-domain. The boundary conditions are obtained from the neighboring sub-domains during computations. To facilitate parallel computation the domain is usually divided into several sub-domains according to the structure of the mesh. A semi-implicit second-order accurate fractional-step method is used to decouple unsteady incompressible Navier–Stokes equation. The quasi-minimal residual (QMR) and the conjugate gradient (CG) methods are used to solve the non-symmetric and symmetric systems of equations, respectively. These are non-stationary iterations that involve some constants to be calculated at each iteration. Typically these constants are calculated by either taking products of matrices and vectors, or inner products of the vectors. Hence, the iterations are parallelized once the above two products are able to do so. A data parallel approach is adopted to perform these two parallelizations in the study. Conclusions: [6]A computation model based on equal-order FEM interpolating polynomials is developed for solving both velocity and pressure of the Navier–Stokes equations. The governing equations are decoupled by a four-step fractional method. The spatial domain is solved by the Galerkin FEM while the temporal domain is integrated by the Crank–Nicolson scheme, both of second-order accuracy. The main advantage of the current model is its simplicity in prescribing the boundary conditions for the velocity and pressure formulation. The proposed solution procedure is parallelized for porting on distributed-memory machines. Those expensive computational loads such as data storage, matrices/vectors constructions and linear equations solvers are parallelized by employing either domain decomposition or data parallel approaches. The developed parallel model is implemented on a distributed memory IBM SP2 supercomputer which is a SPMD type model. Improvement on solution accuracy of an equal-order FEM is shown by comparing laminar flow solution inside a two-dimensional square cavity at a Reynolds number of 1000. In addition, the current model is validated by a three-dimensional DNS of fluid turbulence in plane Couette flow at a Reynolds number of 5000. The capability of the current numerical scheme in large-scale scientific computation is further demonstrated through DNS of turbulent Couette flow over wavy surface. The parallel performance of the proposed parallel strategy is tested by analyzing the CPU time taken on an IBM SP2 supercomputer. Two scales, namely the small and large, of computations consisting of millions of elements are performed on different numbers of processors and improved computational performance is obtained by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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upon parallelism. By measuring the speed up factor and the parallel efficiency, the large scale calculation shows better parallel performance and scalability compared with its small scale counterpart. [7]The solution accuracy is compared with the existing interpolation functions such as the discretized Dirac delta function and the reproducing kernel interpolation function. The finite element shape function is easy to implement and it naturally satisfies the reproducing condition. They are interpolated through only one element layer instead of smearing to several elements. A pressure jump is clearly captured at the fluid–solid interface. Two example problems are studied and results are compared with other numerical methods. A convergence test is thoroughly conducted for the independent fluid and solid meshes in a fluid–structure interaction system. The required mesh size ratio between the fluid and solid domains is obtained. The discretized Dirac delta function used in the immersed boundary method and the reproducing particle method used in the immersed finite element method satisfy this condition. Propose and implemented a finite element interpolation function for non-uniform background fluid grid to capture a sharper fluid–structure interface than the reproducing kernel interpolation function used in the immersed finite element method and the Dirac delta function used in the immersed boundary method. The solutions are examined thoroughly and compared with other published results. The convergence test will be performed and a range of allowable mesh size ratios between the fluid and solid domains will be identified. A comprehensive convergence test is performed using this example. We pay special attention to the allowable fluid– solid mesh size ratios that can be used to yield convergent solutions. For a coupled fluid– structure problem, the convergence rate is computed independently with Lagrangian mesh element size and Eulerian grid spacing. Since there is no analytical solution for this problem, the errors of fluid velocity and solid displacement are calculated based on the solution obtained from a finely discretized system. The convergence of the solid displacement is calculated by refining the Lagrangian mesh while keeping the Eulerian mesh fixed at a refined state. Similarly, the convergence of the N-S solver is studied by refining the fluid mesh while keeping the solid mesh at a very fine resolution. Both components are performed with uniform mesh spacings for consistencies. Errors in the fluid velocity and solid displacement are calculated in L2 norms for steady state solutions. Conclusions: [7] The interpolation functions used in the immersed boundary method and the immersed finite element method, i.e. the discretized Dirac delta function and the reproducing kernel function. Proposed a straightforward finite element interpolation function that is capable of producing sharper interface that preserves the accuracy in interface solutions and to be used on unstructured background fluid meshes. The finite element interpolation function naturally satisfies the reproducing condition and it is easy to implement. Comparing to the previously mentioned techniques, the thickness of the interface can be narrowed by approximately 65% when using uniform grids, and can be improved even further when non uniform or unstructured grids are used. Through the example problems, we performed a thorough convergence test and examined the mesh size compatibility requirement for the fluid and solid domains. We found a mesh size ratio of 0.5 is required for the fluid and solid discretization to avoid numerical issues. If the fluid mesh size is less than half of the solid mesh size, then a leaking phenomenon would occur and lead the solutions to diverge. This value is consistent for several mesh resolutions. We also observed a relatively large volume change when the solid comes near a moving fluid boundary that generates large velocity gradient. A volume correction algorithm is imposed to enforce this incompressibility constraint. This correction algorithm can dramatically improve the durability of the incompressibility assumption and enhance the performance of the simulation. In summary, this paper introduces a finite element interpolation function to be used in the immersed finite element method and closely examines and resolves several detailed numerical issues that are present in the current non-conforming techniques. It provides a more accurate and a more reliable approach to be used in the simulations of fluid–structure interactions. [8] The fluid-structure interaction in fully nonlinear setting, where different space discretization can be used. The model problem considers finite elements for structure and finite volume for fluid. The computations for by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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such interaction problem are performed by implicit schemes, and the partitioned algorithm separating fluid from structural iterations. The formal proof is given to find the condition for convergence of this iterative procedure in the fully nonlinear setting. Several validation examples are shown to confirm the proposed convergence criteria of partitioned algorithm. The model problem for testing the novel paradigm of solution procedure based upon the direct coupling of different codes developed for a particular sub problem (i.e. either solid or fluid mechanics) into a single code. In particular, we seek to provide the guarantees for the robustness of such a computation approach in fully nonlinear setting, where implicit schemes are used for each sub problem, and we derive (by a formal proof) the convergence criterion for partitioned scheme iterations. For solving fluid–structure interaction problems are mostly oriented towards the monolithic schemes, where both sub-problems are discretized in space and time in exactly the same manner resulting with a large set of (monolithic) algebraic equations to be solved simultaneously with no need to distinguish between the ―fluid‖ and the ―structure‖ part. Provided the unified discretization basis for monolithic approach, the most frequent choice is to use the stabilized finite elements for fluids (first proposed by Hughes and co-authors followed by Tezduyar and many other works The main advantage of code-coupling approach for fluid– structure interaction concerns the fact that the coupling is limited only to the fluid–structure interface. Therefore, the main difficulty is reduced to enforcing the interface matching with respect to two different discretization schemes, finite element versus finite volume, as well as two different time integration schemes and different time steps. We thus split the presentation of our work in two parts, pertaining, respectively, to time and to space discretization for fluid and for structure and their matching at the interface. We will deal with the interface matching for different space discretization, along with other related issues pertaining to the computational efficiency enhancements by nested parallelization. In present paper (Part I), we discuss how to accommodate any particular (implicit) scheme that ensures the unconditional stability for either fluid or structure motion computation, and how to ensure that the unconditional stability extends to partitioned solution of the fluid–structure interaction problem. By considering equal time step size for fluid and structure, this direct force-motion transfer algorithm is named conventional serial staggered (DFMTCSS).Also consider the so-called Sub-cycled conventional staggered scheme (DFMT-SCSS) where time steps selected for integration of fluid flow and structure motion are not the same size. Conclusions: [8] Examined partitioned solution approach for nonlinear fluid–structure interaction problems. The partitioned approach is preferred for its modularity and the possibility of re-using existing software developed for each sub-problem (see Part II). The partitioned approach used here is based on the DFMT. Both explicit and implicit coupling algorithms for multi-physics problems are detailed. An explicit strategy leads to the so-called ―added mass effect‖, and for that justifies the use of more costly implicit solvers for the case of incompressible fluid flows. In this work, the problem of enforcing the fluid– structure interface matching is handled by the fixed-point strategy (DFMT-BGS) with an adaptive relaxation parameter. This strategy shows a sufficiently robust performance, especially for the example where the flow is not highly constrained by incompressibility. In fact, we showed by direct proof the stability of the implicit DFMT-BGS algorithm which is valid for the fully nonlinear fluid–structure interaction problem. [9] New adaptive Lattice Boltzmann method (LBM) implementation within the Peano framework, with special focus on nano-scale particle transport problems. With the continuum hypothesis not holding anymore on these small scales, new physical effects—such as Brownian fluctuations—need to be incorporated. We explain the overall layout of the application, including memory layout and access, and shortly review the adaptive algorithm. The scheme is validated by different benchmark computations in two and three dimensions. An extension to dynamically changing grids and a spatially adaptive approach to fluctuating hydrodynamics, allowing for the thermalisation of the fluid in particular regions of interest, is proposed. Both dynamic adaptivity and adaptive fluctuating hydrodynamics are validated separately in simulations of particle transport problems. The application of this scheme to an oscillating particle in a nano-pore illustrates the importance of Brownian fluctuations in such setups. Presented an approach to nano-flow simulations in complex and/or moving geometries. by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Conclusions: [9] Implementated a block-structured adaptive Lattice Boltzmann solver, including its memory access, adaptivity concept and intermolecular collision models. In order to profit from both the simple-to-use adaptivity concept of the Peano framework and the simple and computationally cheap Lattice Boltzmann update rule, we proposed the usage of an application-specific grid management system handling the memory-intensive storage of the particle distribution functions. This scheme avoids costly copy operations between the Peano-internal stacks on the one hand, but leaves the handling of the adaptive grids as well as the parallelization to the Peano kernel on the other hand. We verified and validated our adaptive implementation by different benchmark computations using adaptive and non-adaptive grids in two and three dimensions. Furthermore, the extension of the adaptive scheme to dynamically changing grids has been presented, allowing for the simulation of moving structures within the flow. The new scheme was validated for particle transport problems which are of major concern in our work. An additional focus of research was on nano-flow simulations where Brownian motion effects play a crucial role. The modeling of the respective Brownian fluctuations, however, comes at high computational costs as huge numbers of Gaussian random numbers are required in this case. We proposed a multiscale approach, allowing for fluctuating effects within the fluid on fine grid levels only. On coarser grid levels, the fluctuations are cut off and a simple BGK collision kernel is applied. We used this cut-off approach to simulate the diffusion of an isolated spherical particle. The short-time diffusion of the particle is slightly underestimated by the method, the long-term behavior is captured correctly. Finally, combined our dynamic refinement approach and the cut-off mechanism for thermal fluctuations to simulate a particle which is exposed to oscillating pressure fields within a nano-pore. Similar to previous results, diffusive effects due to thermal fluctuations dictate the magnitude and the direction of the particle drift. Both our new cut-off approach and a completely thermalised fluid model show a behavior of the particle drift which is different to non-fluctuating simulations. This illustrates the importance of Brownian motion on the nano-scale for our flow scenarios. As part of future work, further studies in two and three dimensions will be carried out to completely understand the short-time behavior of the particle motion within the nano-pores. Within this contribution, we restricted our numerical experiments to the simulation of several periods in the particle oscillations. Simulations overmuch longer time intervals might be required to completely evaluate the motion of the particle inside the pore structures. Therefore, new methods need to establish. We currently work on a hybrid approach to include both thermal fluctuations on the short and long time scale in our simulations. [10]A newly developed LES flow solver to compute a true three-dimensional flow applied. The research also investigates the behavior of turbulence statistics by comparing transient simulation results to available data based on experiments and simulations. An extensive discussion on the results such as energy spectrum, velocity profiles and time trace of velocities is carried out in the research as well. Based on the results obtained, the application of the flow solver for a turbulent three-dimensional driven cavity flow by using three grids with varying densities is proven. In addition, the research successfully verifies that in many instances computational results agreed reasonably well with the reference data, and the changes in the statistical properties of turbulence with respect to time are closely related to the changes in the flow structure and strength of vortices. The focus of this study is on the prediction of a sub-grid scale Reynolds shear stress profiles and the results show that the standard model is able to reproduce general trends measured from experiments. Furthermore, in certain areas inside the cavity the computed shear stress values are in close agreement with experimental data. The dynamics of the statistical properties of turbulence as these vortices and secondary flow develop. A further novel aspect of this work is to obtain some insight into accuracy of shear stress computations using the baseline Smagorinsky model, for flows with spatially and temporally varying turbulence structures . Published data from Prasad and Kosef and Migeon et al. are used to validate the code

Conclusions: [10] Investigated in detail the dynamics of the statistical properties of turbulence as Taylor-Gortler vortices and secondary flows develop. In particular, the w v stress profiles, which are particularly difficult to predict, by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

11

are studied. Furthermore, some insight into the accuracy of shear stress computations using the baseline Smagorinsky model, for flows with spatially and temporally varying turbulence structures has been obtained. This paper has demonstrated the application of the flow solver for a turbulent three-dimensional driven cavity flow by using three uniform grids with varying densities. In many instances, computational results agree reasonably well with the reference data. A number of important conclusions can be drawn from this case: In general, the profiles from computation follow the trend exhibited by the reference data. The numerical setup was, to some extent, quantitatively successful in predicting the w v stress profiles. In addition, to predict the w v stress profiles more accurately, the value of the Smagorinsky constant, Cs, must be varied with location inside the domain. The turbulence kinetic energy spectrum plots show the presence of inertial sub-range eddies though the level of energy may vary with respect to the location of the monitoring points. The turbulence kinetic energy plots show that this parameter is produced in the region where the Taylor-Görtler-like vortices reside. [11]

An experimental study is presented for a flow field in a two dimensional wavy channels by PlY. This flow has two major applications such as a blood flow simulation and the enhancement of heat transfer in a heat exchanger. While the numerical flow visualization results have been limited to the fully developed cases, existing experimental results of this flow were simple qualitative ones by smoke or dye streak test Therefore, the main purpose of this study is to produce quantitative flow data for fully developed and developing flow regimes by the Correlation Based Correction PlY (CBC PlY) and to conjecture the analogy between flow characteristics and heat transfer enhancement with low pumping power. Another purpose of this paper is to examine the onset position of the transition and the global mixing, which results in transfer enhancement. PlY results on the fully developed and developing flow in a wavy channel at Re=500, 1000 and 2000 are obtained. For the case Reynolds Number equals 500, the PlY results are compared with the finite difference numerical solution.

The practical use of the particle image velocimetry (PIV), a whole flow field measurement technique, requires the use of fast, reliable, computer-based methods for tracking velocity vectors. The full search block matching, the most widely studied and applied technique both in the area of PIV and Image Coding & Compression is computationally costly. Many alternatives have been proposed and applied successfully in the area of image compression and coding, i. e. MPEG, H. 261 etc. Among others, the Three Step Search (TSS) (Jain, 1981), the New Three Step Search (NTSS) (Li et al., 1994), the Hierarchical Projection Method (HPM) (Sauer and Schuartz, 1996), the FFT-Direct Hybrid Method (HYB) and the Two Resolution Method (TRM) (Anandan, 1989) are introduced. A Correlation Based Correction technique (CBC) (Hart, 2000) is also appreciated and found to be most accurate and adequate for this flow. For the cases Reynolds number fRe) of 500, 1000 and 2000. Developing and fully developed flow data are obtained by CBC PIV with one window shifting. The global mixing phenomenon; which results in the increase in heat and mass transfer and drag, can be identified through the investigation of developing flow in beginning modules. At Re above 500, promotion to turbulence is prominent. While it happens at Re above 2300 in a straight channel. The threepoint Gaussian fit is used for a sub pixel estimator, and the Local Median Filter (LMF) is chosen to validate a vector field. (Kim, 1999) Conclusions: [11]Unlike a simple dye or smoke streak visualization, the PlY analysis can resolve the exact flow structure, even in turbulent flow situation. It can also deal with the unsteady behavior of global mixing. In this paper, fully developed and developing flow data in a wavy channel of Re 500, 1000, 2000 are obtained through the CBC PlY measurements. The analogy between flow characteristics and the enhancement of heat and mess transfer in a wavy channel can be visible through the RMS distribution near wall. The onset point of the global mixing is clearly identified through instantaneous velocity and RMS intensity distributions of a couple of beginning modules. It happens at 4th wavy module for Re= 500, and 2nd module in case Re equals 1000, which are in a good agreement of Rush et aI.'s prediction from dye streak visualization. The phase averaging of PIV results will give a more precise insight of flow structure, like instability and shedding vortices etc. by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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[12]

Analyze the fluid flow with moving boundary using a finite element method. The algorithm uses a fractional step approach that can be used to solve low-speed flow with large density changes due to intense temperature gradients. The explicit Lax-WendrofT scheme is applied to nonlinear convective terms in the momentum equations to prevent checkerboard pressure oscillations. The ALE (Arbitrary Lagrangian Eulerian) method is adopted for moving grids. The numerical algorithm in the present study is validated for two-dimensional unsteady flow in a driven cavity and a natural convection problem. To extend the present numerical method to engine simulations, a piston-driven intake flow with moving boundary is also simulated. The density, temperature and axial velocity profiles are calculated for the three-dimensional unsteady piston-driven intake flow with density changes due to high inlet fluid temperatures using the resent algorithm. The calculated results are in good agreement with other numerical and experimental ones. Conclusions: [12] Used a fractional step method with equal-order interpolation functions for the velocity components and pressure. The explicit Lax-Wendroff scheme was applied to the nonlinear convective terms in the momentum equations and the ALE (Arbitrary Lagrangian-Eulerian) method was adopted for treating the moving boundary. To validate the present algorithm, several problems have been calculated and compared with other results. As a result, the calculation results have shown good agreement with other results. In order to extend the present numerical method to engine simulations, we also investigated the basic behavior of the unsteady flow generated by an impulsively started piston movement in a piston-cylinder assembly, yielding flow separation and spatially moving vortices. The numerical results indicate that the present calculation procedure can be used to predict the behavior of periodic intake/exhaust flows and is applicable to a wide range of problems. Although the discussion has been restricted to laminar flows governed by the NavierStokes equations, the methodology proposed can readily be extended to accommodate the Reynoldsaveraged equations and turbulence models. [13] The bifurcation of the lines of a viscous-fluid flow in a rectangular cavity with a moving cover has been investigated for different ratios between the sides of the cavity and different Reynolds numbers on the basis of the qualitative theory of dynamic systems. The critical parameters of the problem, at which the type of singular points changes and other topological characteristics of a vortex flow in the indicated cavity have been determined and the corresponding bifurcation diagrams have been constructed. The topological characteristics of a flow can be determined on the basis of solution of the Navier–Stokes equations. a flow in a rectangular cavity was investigated for different ratios between the cavity sides and different velocities of travel of the upper and lower walls. The finite-element method with bunching of nodes of a grid in the neighborhood of local stagnation points was used for discretization of Navier–Stokes equations. The investigations were carried out for fairly small Reynolds numbers (Re < 100). Reasonably exact results were not obtained by the finite-element method because it, when used for solving fluid-mechanics problems, provides a lower accuracy than the finite-difference method. Conclusions: [13] The

dependence of the change in the type of singular points of a vortex flow (bifurcation of streamlines) in a rectangular cavity on the ratio between the cavity sides and on the Reynolds number has been investigated on the basis of numerical simulation of this flow. The data obtained can be used for determining the topological characteristics and features of separation flows in cavities of more complex geometries.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-III

Objectives of the work 1) To study the fluid dynamics in the lid-driven cavity. 2) To study the effect of governing parameters in terms of non dimensional numbers in Navier-Stokes equation for four different cases that are Isothermal, Forced convection, mixed convection and Natural convection. 3) To study the effect of grid sizes (coarse, medium and fine) to catch actual characteristics of the fluid flow in above said four different cases. 4) To study the effect of non dimensional numbers on the temperature in above said cases except first case. 5) To study the stream function contours for different non dimensional number in all above said cases. 6) To study the time steps for different parameters in said cases. 7) To study the convergence criteria in all above said cases. 8) To study the vertices movement in the cavity for different non dimensional numbers for said cases. 9) To study the computation time for said cases with different parameters. 10) To study the code validation.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-IV

Physical description of the problem and models for simulations To study the effect of non dimensional parameters in fluid flow characterization; the lid driven cavity is one of the most widely used benchmark problems to test steady state incompressible fluid dynamics codes. Our interest will be to present this problem as a benchmark for the steady and unsteady state solution. In order to demonstrate the grid independence, code validation and other details like time step, grid size and steadiness criteria; we taken 2D Cartesian (x, y as horizontal and vertical components) square domain of size L x L (1 unit x 1 unit for simplicity of the problem) with bottom, left and right boundaries as solid walls stationary; whereas top wall is like a long conveyor-belt, moving horizontally with a constant velocity (Uo=1unit for the simplicity of the problem) shown in Fig.1.

Top long horizontal moving belt (lid) with velocity Uo units

Left side stationary solid wall

Cavity

Right side stationary solid wallwall

1 unit

Temperature Boundary Conditions:

Y

1 unit

X

Top wall at TH temperature and Left, Right & Bottom walls are at TC temperature

Bottom stationary solid wall

Boundaries of the domain for iterations steps (i,j)= (1,1) at left bottom corner; (i,j)= (imax,1) at right bottom corner (i,j)= (1,jmax) at left top corner; (i,j)= (imax, jmax) at right top corner

Fig.1 Lid driven cavity To perform a non dimensional CFD simulation for various values of non-dimensional governing parameters such as Reynolds number, Prandl number, Grashoff number etc.; the following models are made for four different situations (cases) as mentioned in the following tables.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Case-I: Isothermal fluid flow Note: All walls of the cavity are at constant temperature i.e TH=TC=Constant Reynolds Number Model No. of Grids in x direction X No. No. of Grids in y direction 32X32 100 I1 52X52 100 I2 32X32 400 I3 52X52 400 I4 32X32 1000 I5 52X52 1000 I6

Case-II: Forced convection fluid flow Note: Non-isothermal i.e top wall at TH and all other walls at TC temperatures. This corresponds to non-dimensional temperature , and is a buoyancy induced flow therefore in all models Grashoff number is taken as zero i.e Gr=0 Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 F1 1 100 32X32 F2 1 400 12X12 F3 1 400 32X32 F4 0.5 100 12X12 F5 0.5 100 32X32 F6 0.5 400 12X12 F7 0.5 400 32X32 F8 1.2 100 12X12 F9 1.2 100 32X32 F10 1.2 400 12X12 F11 1.2 400 32X32 F12 Case-III: Mixed convection fluid flow Note: All the conditions are same as in Case-II except the Grashoff number is finite values i.e +ve or –ve and ML=Mixed flow Low Grashoff number model Gr=1X105 Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 ML1 1 100 32X32 ML 2 1 400 12X12 ML 3 1 400 32X32 ML 4 0.5 100 12X12 ML 5 0.5 100 32X32 ML 6 0.5 400 12X12 ML 7 0.5 400 32X32 ML 8 1.2 100 12X12 ML 9 1.2 100 32X32 ML 10 1.2 400 12X12 ML 11 1.2 400 32X32 ML 12 by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Case-III: Mixed convection fluid flow Note: MM=Mixed flow Medium Grashoff number model Gr=1x106 Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 MM 1 1 100 32X32 MM 2 1 400 12X12 MM 3 1 400 32X32 MM 4 0.5 100 12X12 MM 5 0.5 100 32X32 MM 6 0.5 400 12X12 MM 7 0.5 400 32X32 MM 8 1.2 100 12X12 MM 9 1.2 100 32X32 MM 10 1.2 400 12X12 MM 11 1.2 400 32X32 MM 12

Case-III: Mixed convection fluid flow Note: MH=Mixed flow High Grashoff number model Gr=2X106 Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 MH 1 1 100 32X32 MH 2 1 400 12X12 MH 3 1 400 32X32 MH 4 0.5 100 12X12 MH 5 0.5 100 32X32 MH 6 0.5 400 12X12 MH 7 0.5 400 32X32 MH 8 1.2 100 12X12 MH 9 1.2 100 32X32 MH 10 1.2 400 12X12 MH 11 1.2 400 32X32 MH 12

Case-III: Mixed convection fluid flow Note: All the conditions are same as in Case-II except the Grashoff number is finite values i.e +ve or –ve and ML=mixed flow low Grashoff number model Gr=-2X106; MNL: Mixed convection Less Negative Grashoff number. * Due to shortage of time to submit the report the simulation has not been done, because that simulations are taking days together. Model No. MNL1 MNL 2* MNL 3 MNL 4 MNL 5 MNL 6* MNL 7

Prandl Number 1 1 1 1 0.5 0.5 0.5

Reynolds Number 100 100 400 400 100 100 400

No. of Grids in x direction X No. of Grids in y direction 12X12 32X32 12X12 32X32 12X12 32X32 12X12

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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MNL 8 MNL 9* MNL 10 MNL 11 MNL 12

0.5 1.2 1.2 1.2 1.2

400 100 100 400 400

32X32 12X12 32X32 12X12 32X32

Case-III: Mixed convection fluid flow Note: All the conditions are same as in Case-II except the Grashoff number is finite values i.e +ve or –ve and ML=mixed flow low Grashoff number model Gr=-1X105; MNH: Mixed convection High Negative Grashoff number Prandl Reynolds No. of Grids in x direction X Model Number Number No. of Grids in y direction No. 1 100 12X12 MNH1 1 100 32X32 MNH 2 1 400 12X12 MNH 3 1 400 32X32 MNH 4 0.5 100 12X12 MNH 5 0.5 100 32X32 MNH 6 0.5 400 12X12 MNH 7 0.5 400 32X32 MNH 8 1.2 100 12X12 MNH 9 1.2 100 32X32 MNH 10 1.2 400 12X12 MNH 11 1.2 400 32X32 MNH 12

Case-IV: Natural convection fluid flow Note: For natural convection, the flow is only due to buoyancy with no forced flow. Thus, the lid is also taken as stationary here. Thus, the physical situation corresponds to a buoyancy induced flow in a differentially heated closed square cavity. The left-wall is maintained at TH, right-wall at TC and the remaining walls are insulated; all walls are stationary. Model No. N1 N2 N3 N4

Prandl Number 0.71 0.71 0.71 0.71

Rayleigh Number 103 103 104 104

No. of Grids in x direction X No. of Grids in y direction 12X12 32X32 12X12 32X32

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-V

Mathematical modeling with boundary conditions We consider the incompressible viscous fluid flow with constant density, viscosity and thermal conductivity in the absence of an applied body force. A set of non-dimensional governing equations following ellipticparabolic characterization are as follows: General mass conservation equation:

In the eqn. (1) u and v are velocity components along x and y directions respectively. steam velocities along x and y directions and total length of the domain respectively. dimensional velocities along the x and y directions respectively; and similarly dimensional coordinates.

are free are nonare the non

General momentum equations:

In the eqn. (2) are non-dimensional temperature, Reynolds number and Prandl number respectively. is the bulk mean temperature in case of internal flow and free stream temperature in case of external flow situations. is the known temperature of the wall or surface. T is the temperature to be find during iteration and time steps. are viscosity, specific heat and thermal conductivity of the fluid respectively and these are taken as constant for our problem. This equation includes temperature term therefore it is used to discritize non-isothermal cases that are case-II to IV in our CFD simulations.

In the eqn. (3);

is non-dimensional pressure and is

; where

is free stream known pressure and

P is pressure to be find for each step of the CFD simulation. This equation does not include any temperature terms; therefore this can be used to solve isothermal type problems i.e case-I in our CFD simulation.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Boundary conditions:

The boundary condition equations from (4) to (7) are incorporated to solve isothermal type of problems; where the temperature remains constant i.e is in case-I in our CFD simulations. And the boundary condition equations from (8) to (11) are incorporated to solve non-isothermal type of problems; where the temperature varying with respect to time and coordinates; that are case-II to IV in our CFD simulations. In equations (4) to (6) and (8) to (10); the velocity components on left, right and bottom surfaces are taken as zero unit along both x and y directions because of viscous effect of the solid stationary walls on the fluid particles. Because the top surface is continuously moving in the horizontal direction only i.e x direction with a velocity ‗u‘; the fluid particles close to the bottom surface of the top plate are affected by the motion of the plate; therefore it is considered in the equations (7) and (11) as unit for simplification of the problem.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-VI

Validation study As a first step towards numerical investigation of the physical problem, we first justify our computer code. A validation study is often proceeding by an analytical assessment of the problem. To achieve this goal and, furthermore, estimate the spatial rate of convergence of the scheme employed, we consider the following transport eqn. (12) for a scalar Φ in a simple domain of two dimensions; .

In eqn.(12) left side first unsteady term will calculate scalar quantity for all time steps and second and third terms in the same side will calculate advection quantities on all geometric coordinates. The first term on the right side calculates diffusion coefficients.

Where are the pressures at east and north nodes respectively and is the pressure at the node where the pressure has to find. The eqn. (13) to (16) are used as pressure correctors to remove the difficulty in the pressure term for linear interpolation is solved by taking staggered grid solution. Prediction error due to oscillatory velocities and the so called false diffusion error grossly pollute the flow physics over the entire domain. Remedy for such discretization error is to apply pressure correction equations in semi explicit QUICK wind scheme SOU at the boundaries. The above shown equations are used to calculate the velocities along x and y direction by considering adjacent nodes pressure for the next time step. As is usual, we assessed the employed QUICK-type upwind discretization scheme by examining the prediction nodal errors. Tests on various grids were conducted to assure that the solution converged. With grid spacing being continuously refined, we could compute the rate of convergence from the computed. The test case considered and the results obtained thus far confirm the applicability of the QUICK scheme to multidimensional analyses. We now turn to examining whether or not linearization procedures and the zerodivergence constraint condition will cause the rate of convergence to deteriorate. To answer this question, we solved a Navier-Stokes problem in the same domain as that considered in the previous benchmark test by GHIA et al. (1982) JOURNAL OF COMPUTATIONAL PHYSICS VOL. 48, pp.387-411 @ Re = 100 we are assured that the proposed scheme is also applicable to analysis of incompressible Navier-Stokes equations. The good agreement from two analytical tests, as demonstrated provides us with strong confidence to proceed with investigation of the time-history of the flow evolution, which is driven by a constant upper lid, in the rectangular cavity.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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Chapter-VII

Results and Discussions Case-I: Isothermal fluid flow Stream function contours 32X32 Grid structure

52X52 Grid structure

Fig.I 1.1

Fig.I 2.1

Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.1

Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.1

Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 5.1

Fig.I 6.1

Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

It is observed from the above shown stream function contours that as the magnitude Reynolds number increases then the inertia forces will increase in the cavity therefore more turbulences will formed in the fluid flow; hence for more values Reynolds number, the solution will take less time steps, less computational time and less number of iterations in the mass conservation loop for the convergence even though change in by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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the grids in x and y directions for all figures of Case-I. Higher magnitude of Reynolds number means; top surface of the cavity moving with a higher velocity. As we observed from the stream contour figures‘ I 1.1, I 3.1 and I 5.1; the lower value stream contours decreases with increasing the magnitude of the Reynolds number means top lid moving with higher velocity and the veracity will shift towards top right corner. Same effect can be observed in the higher grid points also. It is observed from the figures‘ I1.1 & I1.2, I1.3 & I1.4 and I1.5 & I1.6 of steam function contours that as the grid size increases from 32X32 to 52X52 structure in the x and y directions; the simulation catches lower values of stream functions towards bottom surface of the top lid and also we can observed the sharp changes in the physical shape of the verticity near at top right corner of the cavity for 52X52 grid size.

V velocity along the horizontal centerline 32X32 Grid structure

52X52 Grid structure

Fig.I 1.2

Fig.I 2.2

Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.2 Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Fig.I 5.2 Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.2 Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 6.2 Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

As we observed from the velocity along the horizontal centerline figures‘ I 1.2, I 3.2 and I 5.2; the deviation in the velocity result increases with increasing magnitude of the Reynolds number with the published results for 32X32 as well as in 52X52 grid structure; therefore even more denser grid structure is required to catch by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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the small in the velocities in the center horizontal line. It is also observed from both grid structures; the more closure velocity profile with the published results for 52X52 structures; and still finer grid structure is required to match with published results. And also observed from same figures; as the magnitude of the Reynolds number increases from 100 to 1000, the inside surfaces of the left and right side walls effect decreases and velocity of the fluid particles increases; it enhances little more turbulence in the fluid particles. Hence these turbulences cannot catch in the higher magnitude Reynolds number; therefore the centerline velocity profile is almost horizontal in case of 1000 Reynolds number and small change in case of Reynolds number 400; but sharp changes in Reynolds number 100.

U velocity along the vertical centerline 32X32 Grid structure

52X52 Grid structure

Fig.I 1.3

Fig.I 2.3

Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.3 Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Fig.I 5.3 Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.3 Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 6.3 Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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It is observed from the figures I1.3, I3.3 and I5.3 of horizontal velocity component (U) along the vertical centerline that the fluid particles will changes their flow direction from right side to the left side at the 0.15 unit distance from bottom surface along the y direction of the cavity for 1000 Reynolds number in the published results and that same distance will increases for lower Reynolds numbers i.e 0.28 for Re=400 and 0.55 for Re=100. And is also observed that that sharp change in the velocity profile for higher Reynolds numbers i.e for 400 and 100 but in lower values of the same number smooth changes will happen i.e for 100. As we observed from the figures I1.3 to I6.3 that our simulation velocity profile is close with published profile for lower Reynolds number i.e 100 but this closeness is decreases with higher value of the Reynolds number; this is because our grid structure is not sufficient to catch the higher velocities of the particles of higher Reynolds number. Therefore finer grid structure may be necessary to catch the higher Reynolds number turbulences.

V velocity contours 32X32 Grid structure

Fig.I 1.4 Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.4 Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Fig.I 5.4 Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

52X52 Grid structure

Fig.I 2.4 Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.4 Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 6.4 Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

25

It is observed from the figures from I1.4to I6.4 that primary higher velocity verticity at top left corner and secondary lower velocity verticity at the top right corner are changing their characterization when the magnitude of the Reynolds numbers varies. The magnitude of the primary and secondary verticities will decreases with increasing Reynolds number. This is because; when the velocity of the top lid increases then the turbulence in the fluid flow will also increases; so more distribution of the velocity contours. Commonly in all figures it is also observed that the moment of the lid is highly at top left corner fluid particles of the cavity on vertical velocity contours whereas it reverses at right top corner fluid particles of the cavity. By comparing the figures for 32X32 grid structure I1.4, I3.4. I5.4 and 52X52 grid structure I2.4, I4.4, I6.4; more uniform v-velocity contours in higher grid structures. This is because more fluid particles will cover in dense grid structure. As the Reynolds number increases from 100 to 400 for both grid structure; the higher value velocity contours in Re=400 than Re=100 in the domain. It is also observed from the figures; as the Reynolds number increases from 100 to 1000 through 400, the magnitude of the primary and secondary verticities is decreasing and it may disappear for even higher values of the Reynolds number.

U velocity contours 32X32 Grid structure

Fig.I 1.5 Re:100 Computation time:5 hours Time steps:49 No.of iterations in the mass conservation loop:1938

Fig.I 3.5 Re:400 Computation time:4 hours Time steps:46 No.of iterations in the mass conservation loop:1072

Fig.I 5.5 Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

52X52 Grid structure

Fig.I 2.5 Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 4.5 Re:400 Computation time:18 hours Time steps:46 No.of iterations in the mass conservation loop:2623

Fig.I 6.5 Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

26

It is observed from the figures I1.5 to I6.5 that the higher value U velocity contours at the top surface of the domain because top lid is moving horizontally. As the Reynolds number is low higher value U-velocity contours is more at bottom of the lid whereas these are decreases as the Reynolds number increases in both 32X32 and 52X52 grid structure. The lower velocity verticity is observed in all figures at the right side of the domain and it is towards top right corner as the Reynolds number increases. And also the magnitude of the verticity also decreases with increasing Reynolds number. It is also clearly observed from two different grid structures; the change of the characteristic of the verticity different in 52X52 than the 32X32. The effect of the top lid velocity caught in dense grid structure than the coarse grid structure. Because less viscosity effect at the bottom side top lid; the particles immediately bottom of the plate are in the same velocity of the plate i.e the highest velocity in the problem. For denser grid structure; the verticity at the top right corner of the domain may disappear.

Case-II: Forced convection fluid flow (where Gr=0) U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.F 2.1

Fig.F 1.1 Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 3.1

Fig.F 4.1

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6.1

Fig.F 5.1 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

27

Fig.F 8.1

Fig.F 7.1 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Fig.F 9.1

Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

Fig.F 10.1

Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12.1 Fig.F 11.1 Re:400 Pr:1.2 Computation time: 20minTime steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

It is observed from the figures from F 1.1 to F12.1; higher velocity verticity is small and it is shifting towards top right corner of the cavity of the domain for dense grid structure than the coarse grid structure. The same verticity is shifting towards bottom of the cavity for higher Reynolds number i.e 400 as compared to the 100. The time steps required for to converge the solution are more for higher Reynolds number (400) than the lower (100) for the same Prandl number irrespective of the grid structure. As the Prandl number increases from 0.5 to 1.2 through 1; the solidity of the fluid is more, therefore the velocity of the top lid is more in case of lower Prandl number (0.5) than the higher number (1.2). The physical characteristic of the verticity is slightly different in case of higher Reynolds number towards the bottom surface of the cavity. As we know top lid is moving; therefore the higher velocity contours are appears at the top.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

28

V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.F 2.2

Fig.F 1.2 Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 4.2

Fig.F 3.2 Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6.2

Fig.F 5.2 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Fig.F 7.2 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8.2 Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

29

Fig.F 9.2

Fig.F 10.2

Re:100 Pr:1.2 Computation time: 25minTime steps:109 No.of iterations in the mass conservation loop:456

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 11.2

Fig.F 12.2

Re:400 Pr:1.2 Computation time: 20minTime steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

It is observed from the figures of higher Reynolds number (400); the higher V-velocity verticity at the top left corner is stretched towards right surface of the cavity as compared to the lower number (100) and this character is clearly observed in the dense grid structure(32X32) than the coarse grid structure (12X12). The magnitude of the verticity is slightly higher in case of lower grid structure (12X12) than the higher (32X32) irrespective of Prandl and Reynolds number. In all figures from F 1.2 to F 12.2 one higher primary verticity at the top left corner and another lower secondary verticity at the right top corner of the cavity. It is observed from higher Reynolds number contours (400) having lower velocity contours than the lower Reynolds number (100) irrespective of Prandl number and grid structure. It observed from the figures; as the Reynolds number increases from 100 to 400 the magnitude of the primary higher verticity increase with decreasing secondary lower verticity. The magnitude of the secondary verticity is decreasing with Prandl number for same Reynolds number irrespective of the grid structure. The number of time steps required are more for higher Reynolds number than the lower.

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.F 1.3 Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Fig.F 2.3 Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

30

Fig.F 4.3

Fig.F 3.3 Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Fig.F 6.3

Fig.F 5.3 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Fig.F 9.3

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8.3

Fig.F 7.3 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:100 Pr:1.2 Computation time:25min Time steps:109 No.of iterations in the mass conservation loop:456

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

Fig.F 10.3 Re:100 Pr:1.2 Computation time:22 hours Time steps:137 No.of iterations in the mass conservation loop:1801

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

31

Fig.F 12.3

Fig.F 11.3

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

Re:400 Pr:1.2 Computation time: 20minTime steps:126 No.of iterations in the mass conservation loop:353

It is observed from the stream line contours; the verticity is shifting towards top right corner of the cavity in case of dense grid structure than the coarse grid structure and its magnitude also decreasing with dense grid structure. It is observed from the figures‘ F1.1 & F1.2, F1.3 & F1.4 and F1.5 & F1.6 of steam function contours that as the grid size increases from 32X32 to 52X52 structure in the x and y directions; the simulation catches lower values of stream functions towards bottom surface of the top lid and also we can observed the sharp changes in the physical shape of the verticity near at top right corner of the cavity for 52X52 grid size.

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.F 1.4

Fig.F 2.4

Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 3.4

Fig.F 4.4

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

32

Fig.F 5.4

Fig.F 6.4

Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 7.4

Fig.F 8.4

Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

Fig.F 10.4

Fig.F 9.4 Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12.4

Fig.F 11.4 Re:400 Pr:1.2 Computation time:20min Time steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

33

It observed from the figures; the characteristic of the temperature contours at the top right corner of the cavity is changes as the Reynolds number changes from 100 to 400. As the grid structure increases from 12X12 to 32X32; the sensitive variation in the temperature are also catches and it is clearly seen in the figures. It is also observed from the figures; as the Prandl number increases from 0.5 to 1.2, the higher temperature contours are available near bottom of the top lid.

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.F 1.5

Fig.F 2. 5

Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 3. 5

Fig.F 4. 5

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Fig.F 5. 5

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6. 5

Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8. 5

Fig.F 7. 5 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:400 Pr:0.5 Computation time:12 hours Time steps:115 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

34

Fig.F 9. 5

Fig.F 10. 5

Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Fig.F 11. 5

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12. 5

Re:400 Pr:1.2 Computation time: 20min Time steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

It is observed from the figures; the horizontal velocity component (U) along the vertical centerline that the fluid particles will changes their flow direction from right side to the left side at the 0.15 unit distance from bottom surface along the y direction of the cavity for 1000 Reynolds number in the published results and that same distance will increases for lower Reynolds numbers i.e 0.28 for Re=400 and 0.55 for Re=100. And is also observed that that sharp change in the velocity profile for higher Reynolds numbers i.e for 400 and 100 but in lower values of the same number smooth changes will happen i.e for 100. As we observed from the figures F1.3 to F6.3 that our simulation velocity profile is close with published profile for lower Reynolds number i.e 100 but this closeness is decreases with higher value of the Reynolds number; this is because our grid structure is not sufficient to catch the higher velocities of the particles of higher Reynolds number. Therefore finer grid structure may be necessary to catch the higher Reynolds number turbulences.

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.F 1.6

Fig.F 2. 6

Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

35

Fig.F 3. 6

Fig.F 4. 6

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6. 6

Fig.F 5. 6 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8. 6

Fig.F 7. 6 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

Fig.F 10. 6

Fig.F 9. 6 Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Fig.F 11. 6

Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12. 6

Re:400 Pr:1.2 Computation time:20min Time steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

36

I is observed from same figures; as the magnitude of the Reynolds number increases from 100 to 1000, the inside surfaces of the left and right side walls effect decreases and velocity of the fluid particles increases; it enhances little more turbulence in the fluid particles. Hence these turbulences cannot catch in the higher magnitude Reynolds number; therefore the centerline velocity profile is almost horizontal in case of 1000 Reynolds number and small change in case of Reynolds number 400; but sharp changes in Reynolds number 100.

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.F 1.7

Fig.F 2. 7

Re:100 Pr:1 Computation time:15 mins Time steps:92 No.of iterations in the mass conservation loop:456

Fig.F 3. 7

Re:100 Pr:1 Computation time:11 hours Time steps:126 No.of iterations in the mass conservation loop:2029

Fig.F 4. 7

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 6. 7

Fig.F 5. 7 Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Fig.F 7. 7 Re:400 Pr:0.5 Computation time:15 mins Time steps:104 No.of iterations in the mass conservation loop:353

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 8. 7 Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

37

Fig.F 9. 7 Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Fig.F 10. 7 Re:100 Pr:1.2 Computation time: 22hours Time steps:137 No.of iterations in the mass conservation loop:1801

Fig.F 12. 7

Fig.F 11. 7 Re:400 Pr:1.2 Computation time: 20min Time steps:126 No.of iterations in the mass conservation loop:353

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

It is observed from the figures; the validation of the simulated data will be more for coarse grid structure (12X12) than the dense grid structure(32X32) compared to the published data and same effect can be observed for the lower value Reynolds number(100) than the higher value Reynolds number (400). It is also observed that the sharp changes in the temperature from higher Prandl number (1.2) than the lower(0.5).

Case-III: Mixed convection fluid flow (where Gr=105) U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.ML 1.1 Re:100

Pr:1

Fig.ML 2.1 Time steps: 49

Re:100

Pr:1

Time steps:78

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

38

Fig.ML 4.1

Fig.ML 3.1 Re:400

Pr:1

Time steps:100

Re:400

Pr:1

Re:100

Pr:0.5

Re:400

Pr:0.5

Fig.ML 6.1

Fig.ML 5.1 Re:100

Pr:0.5

Time steps:44

Pr:0.5

Pr:1.2

Time steps:98

Time steps:79

Fig.ML 9.1 Re:100

Time steps:77

Fig.ML 8.1

Fig.ML 7.1 Re:400

Time steps: 130

Fig.ML 10.1 Time steps:52

Re:100

Pr:1.2

Time steps:82

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

39

Fig.ML 11.1 Re:400

Pr:1.2

Fig.ML 12.1 steps:103

Re:400

Pr:1.2

Time steps:125

The U velocity contour verticity is moving towards bottom surface of the cavity as the Prandl number decreases from 1.2 to 0.5 with higher Reynolds number irrespective of the grid structure. This is because the top lid velocity increases with liquidity of the fluid inside the cavity.

V velocity contours 12X12 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

32X32 Grid structure

Time steps:49

Re:100

Pr:1

Re:400

Pr:1

Fig.ML 3.1 Re:400

Pr:1

Time steps:100

Time steps:78

Fig.ML 4.1 Time steps: 130

Fig.ML 6.1

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

40

Fig.ML 5.1 Re:100

Pr:0.5

Re:100

Pr:0.5

Re:400

Pr:0.5

Fig.ML 7.1 Re:400

Pr:0.5

Fig.ML 8.1

Time steps:79

Pr:1.2

Time steps:52

Re:100

Fig.ML 11.1 Re:400

Pr:1.2

Time steps:98

Fig.ML 10.1

Fig.ML 9.1 Re:100

Time steps:77

Time steps:44

Pr:1.2

Time steps:82

Fig.ML 12.1 steps:103 Re:400

Pr:1.2

Time steps:125

It is observed from the above shown figures; for lower Reynolds number i.e 100 with lower grid structure showing three verticities; two at left and right faces and one in between them. But these three verticities are disappears in case of lower Prandl number (0.5) with higher grid structures (32X32). This is because as the viscosity will decreases then the Prandl number also decreases with increasing liquidity of the fluid inside the cavity; this makes uniform in a short duration and it is clearly catches in dense grid structure.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

41

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

Time steps: 49

Re:100

Fig.ML 3.1 Re:400

Pr:1

Fig.ML 5.1 Pr:0.5

Time steps:44

Pr:0.5

Fig.ML 9.1 Pr:1.2

Time steps:52

Time steps: 103

Pr:0.5

Time steps:77

Fig.ML 8.1

Time steps:79 Re:400

Re:100

Pr:1

Fig.ML 6.1 Re:100

Fig.ML 7.1 Re:400

Time steps:78

Fig.ML 4.1

Time steps:100 Re:400

Re:100

Pr:1

Pr:0.5

Time steps:98

Fig.ML 10.1 Re:100

Pr:1.2

Time steps:82

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

42

Fig.ML 12.1

Fig.ML 11.1 Re:400

Pr:1.2

steps:103 Re:400

Pr:1.2

Time steps:125

It is observed from the above sown graphs that that the Prandl number increases ( from 0.5 to 1.2) with lower Reynolds number; the fluid particles will takes a sharp at very close to the left side wall of the cavity. This is because the top lid is moving with lower velocity and liquidity of the fluid is less for less Reynolds number (100).

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

Time steps: 49

Re:100

Fig.ML 3.1 Re:400

Pr:1

Pr:0.5

Time steps:78

Fig.ML 4.1 Time steps:100

Re:400

Pr:1

Time steps: 130

Fig.ML 6.1

Fig.ML 5.1 Re:100

Pr:1

Time steps:44

Re:100

Pr:0.5

Time steps:77

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

43

Fig.ML 7.1 Re:400

Pr:0.5

Fig.ML 8.1 Time steps:79

Re:400

Pr:0.5

Fig.ML 9.1 Re:100

Pr:1.2

Fig.ML 10.1 Time steps:52

Re:100

Pr:1.2

Time steps:82

Fig.ML 12.1

Fig.ML 11.1 Re:400

Time steps:98

Pr:1.2

steps:103 Re:400

Pr:1.2

Time steps:125

It is observed from the figure; the fluid with lower Reynolds number (100) with lower Prandl number (0.5) will take two sharp turns at the bottom side of the cavity wall and these sharp turns will shift towards top surface for higher Reynolds number (400) with higher Prandl number (1.2). But more variations in the graphs when the grid structure changes from coarse to dense.

Stream function contours 12X12 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

32X32 Grid structure

Time steps: 49

Re:100

Pr:1

Time steps:78

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

44

Fig.ML 3.1 Re:400

Pr:1

Fig.ML 4.1 Time steps:100

Re:400

Pr:1

Re:100

Pr:0.5

Fig.ML 6.1

Fig.ML 5.1 Re:100

Pr:0.5

Time steps:44

Pr:0.5

Time steps:79

Re:400

Fig.ML 9.1 Re:100

Pr:1.2

Time steps:77

Fig.ML 8.1

Fig.ML 7.1 Re:400

Time steps: 130

Pr:0.5

Time steps:98

Fig.ML 10.1 Time steps:52

Re:100

Pr:1.2

Time steps:82

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

45

Fig.ML 11.1 Re:400

Pr:1.2

Fig.ML 12.1 steps:103 Re:400

Pr:1.2

Time steps125:

It is observed from the figure; two verticities are formed for lower Reynolds number (100) fluid in less dense grid structure (12X12) and almost only one verticity for verticity for higher Reynolds number (400) for the same grid structure but in dense grid structure (32X32) only one verticity in all the Reynolds number.

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.ML 1.1 Re:100

Pr:1

Fig.ML 2.1 Time steps: 49

Re:100

Pr:1

Time steps:78

Fig.ML 4.1

Fig.ML 3.1 Re:400

Pr:1

Time steps:100

Re:400

Pr:1

Time steps: 130

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

46

Fig.ML 5.1 Re:100

Pr:0.5

Fig.ML 6.1 Time steps:44

Pr:0.5

Time steps:79

Re:400

Pr:1.2

Time steps:52

Re:100

Fig.ML 11.1 Re:400

Pr:1.2

Time steps:77

Pr:0.5

Time steps:98

Fig.ML 10.1

Fig.ML 9.1 Re:100

Pr:0.5

Fig.ML 8.1

Fig.ML 7.1 Re:400

Re:100

steps:103

Pr:1.2

Time steps:82

Fig.ML 12.1 Re:400

Pr:1.2

Time steps:125

It is observed from the figures; the temperature contours with higher Reynolds number are having different characteristic and magnitude at right top corner of the cavity. This is because the top lid is moving with higher velocity and it will affect the fluid particles at bottom side of the top surface. by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

47

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.ML 2.1

Fig.ML 1.1 Re:100

Pr:1

Time steps: 49

Re:100

Time steps:100

Re:400

Pr:1

Pr:0.5

Time steps:44

Re:100

Fig.ML 7.1 Re:400

Pr:0.5

Time steps:79

Fig.ML 9.1 Re:100

Pr:1.2

Pr:1

Time steps: 130

Fig.ML 6.1

Fig.ML 5.1 Re:100

Time steps:78

Fig.ML 4.1

Fig.ML 3.1 Re:400

Pr:1

Time steps:52

Pr:0.5

Time steps:77

Fig.ML 8.1 Re:400

Pr:0.5

Time steps:98

Fig.ML 10.1

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

48

Re:100

Pr:1.2

Fig.ML 12.1

Fig.ML 11.1 Re:400

Pr:1.2

Time steps:82

steps:103 Re:400

Pr:1.2

Time steps:125

It is observed from the above shown figures; the simulation results are closer with published data for higher value of the Reynolds number (400) than the lower (100).

Case-III: Mixed convection fluid flow (where Gr=106) U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MM 2.1

Fig.MM 1.1 Re:100

Pr:1

Time steps: 46

Re:100

Time steps:81

Re:400

Fig.MM 3.1 Re:400

Pr:1

Pr:0.5

Time steps:81

Fig.MM 4.1 Pr:1

Time steps: 88

Fig.MM 6.1

Fig.MM 5.1 Re:100

Pr:1

Time steps:58

Re:100

Pr:0.5

Time steps:112

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

49

Fig.MM 8.1

Fig.MM 7.1 Re:400

Pr:0.5

Re:400

Time steps:64

Pr:0.5

Fig.MM 10.1

Fig.MM 9.1 Re:100

Pr:1.2

Time steps:72

Re:100

Time steps:50

Pr:1.2

Time steps:85

Fig.MM 12.1 Fig.MM 11.1 Re:400

Pr:1.2

steps:84

Re:400

Pr:1.2

Time steps:99

It is observed from the figure; more vertices are found lower Reynolds number. This is because as the Grashoff number increases, the temperature difference between the cavity walls also increases; this increases the kinetic energy of the fluid particles.

V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MM 1.1 Re:100

Pr:1

Time steps: 46

Fig.MM 2.1 Re:100

Pr:1

Time steps:81

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

50

Fig.MM 4.1

Fig.MM 3.1 Re:400

Pr:1

Time steps:81

Re:400

Fig.MM 5.1 Re:100

Pr:0.5

Pr:0.5

Re:100

Pr:1.2

Pr:0.5

Time steps:112

Fig.MM 8.1

Time steps:64

Re:400

Fig.MM 9.1 Re:100

Time steps: 88

Fig.MM 6.1 Time steps:58

Fig.MM 7.1 Re:400

Pr:1

Pr:0.5

Time steps:72

Fig.MM 10.1 Time steps:50

Re:100

Pr:1.2

Time steps:85

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

51

Fig.MM 11.1 Re:400

Pr:1.2

Fig.MM 12.1 steps:84

Re:400

Pr:1.2

Time steps:99

It is observed that the Groshoff number increases with the increasing kinetic energy of the fluid particles due to its higher temperature; it creates more number of verticities in the cavity. The verticities at left and right side walls of the cavity will shift towards bottom wall for lower Prandl number. This is because the liquidity of the fluid particles decreases with increasing Prandl number.

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MM 1.1 Re:100

Pr:1

Fig.MM 2.1 Time steps: 46

Re:100

Time steps:81

Re:400

Pr:1

Pr:0.5

Pr:1

Time steps: 88

Fig.MM 6.1

Fig.MM 5.1 Re:100

Time steps:81

Fig.MM 4.1

Fig.MM 3.1 Re:400

Pr:1

Time steps:58

Re:100

Pr:0.5

Time steps:112

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

52

Fig.MM 8.1

Fig.MM 7.1 Re:400

Pr:0.5

Re:400

Time steps:64

Pr:1.2

Re:100

Time steps:50

Pr:1.2

Pr:1.2

Time steps:85

Fig.MM 12.1

Fig.MM 11.1 Re:400

Time steps:72

Fig.MM 10.1

Fig.MM 9.1 Re:100

Pr:0.5

steps:84

Re:400

Pr:1.2

Time steps:99

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.MM 1.1 Re:100

Pr:1

Fig.MM 2.1 Time steps: 46

Re:100

Fig.MM 3.1 Re:400

Pr:1

Time steps:81

Pr:1

Time steps:81

Fig.MM 4.1 Re:400

Pr:1

Time steps: 88

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

53

Fig.MM 6.1

Fig.MM 5.1 Re:100

Pr:0.5

Time steps:58

Re:100

Pr:0.5

Fig.MM 8.1

Fig.MM 7.1 Re:400

Pr:0.5

Time steps:64

Pr:1.2

Re:400

Pr:0.5

Time steps:72

Fig.MM 10.1

Fig.MM 9.1 Re:100

Time steps:112

Time steps:50

Re:100

Pr:1.2

Time steps:85

Fig.MM 12.1 Fig.MM 11.1 Re:400

Pr:1.2

steps:84

Re:400

Pr:1.2

Time steps:99

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.MM 1.1

Fig.MM 2.1

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

54

Re:100

Pr:1

Time steps: 46

Re:100

Time steps:81

Re:400

Fig.MM 3.1 Re:400

Pr:1

Pr:0.5

Pr:0.5

Pr:1.2

Time steps: 88

Re:100

Pr:0.5

Time steps:112

Fig.MM 8.1

Time steps:64

Re:400

Pr:0.5

Time steps:72

Fig.MM10.1

Fig.MM 9.1 Re:100

Pr:1

Fig.MM 6.1 Time steps:58

Fig.MM 7.1 Re:400

Time steps:81

Fig.MM 4.1

Fig.MM 5.1 Re:100

Pr:1

Time steps:50

Re:100

Pr:1.2

Time steps:85

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

55

Fig.MM 12.1 Fig.MM 11.1 Re:400

Pr:1.2

steps:84

Re:400

Pr:1.2

Time steps:99

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.MM 2.1

Fig.MM 1.1 Re:100

Pr:1

Time steps: 46

Re:100

Fig.MM 3.1 Re:400

Pr:1

Pr:0.5

Time steps:81

Fig.MM 4.1 Time steps:81

Re:400

Pr:1

Time steps: 88

Fig.MM 6.1

Fig.MM 5.1 Re:100

Pr:1

Time steps:58

Re:100

Pr:0.5

Time steps:112

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

56

Fig.MM 8.1

Fig.MM 7.1 Re:400

Pr:0.5

Re:400

Time steps:64

Pr:1.2

Re:100

Time steps:50

Pr:1.2

Pr:1.2

Time steps:85

Fig.MM 12.1

Fig.MM 11.1 Re:400

Time steps:72

Fig.MM 10.1

Fig.MM 9.1 Re:100

Pr:0.5

Re:400

steps:84

Pr:1.2

Time steps: 99

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MM 2.1

Fig.MM 1.1 Re:100

Pr:1

Time steps: 46

Re:100

Pr:1

Time steps:81

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

57

Fig.MM 3.1 Re:400

Pr:1

Fig.MM 4.1 Time steps:81

Pr:0.5

Pr:1

Re:100

Time steps:58

Pr:0.5

Fig.MM 7.1 Re:400

Pr:0.5

Pr:1.2

Re:400

Pr:0.5

Pr:1.2

Time steps:72

Fig.MM 10.1 Time steps:50

Re:100

Pr:1.2

Time steps:85

Fig.MM 12.1

Fig.MM 11.1 Re:400

Time steps:112

Fig.MM 8.1

Time steps:64

Fig.MM 9.1 Re:100

Time steps: 88

Fig.MM 6.1

Fig.MM 5.1 Re:100

Re:400

Re:400

steps:84

Pr:1.2

Time steps:99

Case-III: Mixed convection fluid flow (where Gr=2x106) U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Re:100

Pr:1

Time steps:73

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

58

Fig.MH 3.1 Re:400

Fig.MH 4.1

Pr:1

Time steps:82

Re:400

Fig.MH 5.1 Re:100

Pr:0.5

Time steps:55

Pr:0.5

Pr:0.5

Time steps:111

Fig.MH 8.1 Time steps:62

Pr:1.2

Re:400

Pr:0.5

Time steps:66

Fig.MH 10.1

Fig.MH 9.1 Re:100

Time steps: 114

Fig.MH 6.1 Re:100

Fig.MH 7.1 Re:400

Pr:1

Time steps:53

Re:100

Pr:1.2

Time steps:77

Fig.MH12.1 Fig.MH 11.1 Re:400

Pr:1.2

steps:87

Re:400

Pr:1.2

Time steps:120

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

59

V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MH 1.1 Re:100

Pr:1

Fig.MH 2.1 Time steps: 53

Re:100

Time steps:82

Re:400

Pr:1

Pr:0.5

Time steps:55

Re:100

Fig.MH 7.1 Re:400

Pr:0.5

Pr:1

Time steps: 114

Fig.MH 6.1

Fig.MH 5.1 Re:100

Time steps:73

Fig.MH 4.1

Fig.MH 3.1 Re:400

Pr:1

Time steps:62

Pr:0.5

Time steps:111

Fig.MH 8.1 Re:400

Pr:0.5

Time steps:66

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

60

Fig.MH 10.1

Fig.MH 9.1 Re:100

Pr:1.2

Re:100

Fig.MH 11.1 Re:400

Pr:1.2

Pr:1.2

Time steps:77

Time steps:53

Fig.MH 12.1 steps:87 Re:400

Pr:1.2

Time steps:120

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Fig.MH 2.1 Re:100

Fig.MH 3.1 Re:400

Pr:1

Pr:0.5

Time steps:73

Fig.MH 4.1 Time steps:82

Re:400

Pr:1

Time steps: 114

Fig.MH 6.1

Fig.MH 5.1 Re:100

Pr:1

Time steps:55

Re:100

Pr:0.5

Time steps:111

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

61

Fig.MH 8.1

Fig.MH 7.1 Re:400

Pr:0.5

Re:400

Time steps:62

Fig.MH 9.1 Re:100

Pr:1.2

Pr:0.5

Time steps:66

Fig.MH 10.1 Time steps:53

Re:100

Pr:1.2

Time steps:77

Fig.MH 12.1 Fig.MH 11.1 Re:400

Pr:1.2

Re:400

steps:87

Pr:1.2

Time steps:120

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Re:100

Time steps:82

Re:400

Pr:1

Time steps:73

Fig.MH 4.1

Fig.MH 3.1 Re:400

Pr:1

Pr:1

Time steps: 114

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

62

Fig.MH 6.1

Fig.MH 5.1 Re:100

Pr:0.5

Re:100

Time steps:55

Pr:0.5

Fig.MH 7.1 Re:400

Pr:0.5

Fig.MH 8.1 Time steps:62

Re:400

Pr:0.5

Fig.MH 9.1 Re:100

Pr:1.2

Pr:1.2

Time steps:66

Fig.MH 10.1 Time steps:53

Re:100

Pr:1.2

Time steps:77

Fig.MH 12.1

Fig.MH 11.1 Re:400

Time steps:111

steps:87

Re:400

Pr:1.2

Time steps:120

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Re:100

Pr:1

Time steps:73

Time steps: 53

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

63

Fig.MH 3.1 Re:400

Pr:1

Time steps:82

Pr:0.5

Time steps:55

Re:100

Pr:0.5

Time steps:62

Re:400

Pr:0.5

Time steps:111

Pr:1.2

Pr:0.5

Time steps:66

Fig.MH10.1

Fig.MH 9.1 Re:100

Time steps: 114

Fig.MH 8.1

Fig.MH 7.1 Re:400

Pr:1

Fig.MH 6.1

Fig.MH 5.1 Re:100

Fig.MH 4.1 Re:400

Time steps:53

Re:100

Pr:1.2

Time steps:77

Fig.MH 12.1 Fig.MH 11.1 Re:400

Pr:1.2

steps:87

Re:400

Pr:1.2

Time steps:120

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

64

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Re:100

Pr:1

Time steps:82

Pr:0.5

Re:400

Time steps:55

Re:100

Pr:0.5

Time steps: 114

Pr:0.5

Time steps:111

Fig.MH 8.1

Fig.MH7.1 Re:400

Pr:1

Fig.MH 6.1

Fig.MH 5.1 Re:100

Time steps:73

Fig.MH 4.1

Fig.MH 3.1 Re:400

Pr:1

Time steps:62

Re:400

Pr:0.5

Time steps:66

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

65

Fig.MH 10.1

Fig.MH 9.1 Re:100

Pr:1.2

Re:100

Time steps:53

Pr:1.2

Time steps:77

Fig.MH 12.1 Fig.MH 11.1 Re:400

Pr:1.2

steps:87

Re:400

Pr:1.2

Time steps:120

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MH 2.1

Fig.MH 1.1 Re:100

Pr:1

Time steps: 53

Re:100

Fig.MH 3.1 Re:400

Pr:1

Pr:0.5

Time steps:73

Fig.MH 4.1 Time steps:82

Re:400

Pr:1

Time steps: 114

Fig.MH 6.1

Fig.MH 5.1 Re:100

Pr:1

Time steps:55

Re:100

Pr:0.5

Time steps:111

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

66

Fig.MH 7.1 Re:400

Pr:0.5

Time steps:62

Pr:1.2

Pr:0.5

Time steps:53

Re:100

Pr:1.2

Pr:1.2

Time steps:77

Fig.MH 12.1

Fig.MH 11.1 Re:400

Time steps:66

Fig.MH 10.1

Fig.MH 9.1 Re:100

Fig.MH 8.1 Re:400

steps:87 Re:400

Pr:1.2

Time steps:120

Case-III: Mixed convection fluid flow (where Gr=-105) U velocity contours 12X12 Grid structure

Fig.MNH 2.1

Fig.MNH 1.1 Re:100

Pr:0.5

32X32 Grid structure

Time steps:243

Re:100

Pr:0.5

Time steps:68

Fig.MNH 4.1

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

67

Re:400

Fig.MNH 3.1 Re:400

Pr:0.5

Fig.MNH 5.1 Re:100

Pr:1

Pr:1

Re:100

Pr:1.2

Time steps:57

Pr:1.2

Time steps:68

Pr:1

Time steps: 53

Fig.MNH 10.1 Time steps:249

Re:100

Fig.MNH 11.1 Re:400

Pr:1

Fig.MNH 8.1 Re:400

Fig.MH 9.1 Re:100

Time steps:60

Fig.MNH 6.1 Time steps: 54

Fig.MNH 7.1 Re:400

Pr:0.5

Time steps:78

Pr:1.2

Time steps:66

Fig.MNH 12.1 Time steps:48

Re:400

Pr:1.2

Time steps:52

The U velocity contour verticity is moving towards bottom surface of the cavity as the Prandl number decreases from 1.2 to 0.5 with higher Reynolds number irrespective of the grid structure. This is because the top lid velocity increases with liquidity of the fluid inside the cavity.

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

68

V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MNH 2.2

Fig.MNH 1.2 Re:100

Pr:0.5

Re:100

Time steps:243

Fig.MNH 3.2 Re:400

Pr:0.5

Pr:1

Re:400

Time steps:60

Pr:1

Time steps:68

Fig.MNH 8.2

Fig.MNH 7.2 Pr:1

Pr:0.5

Fig.MNH 6.2

Time steps:54 Re:100

Re:400

Time steps:68

Fig.MNH 4.2

Time steps:78

Fig.MNH 5.2 Re:100

Pr:0.5

Time steps:57

Re:400

Pr:1

Time steps: 53

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

69

Fig.MNH 10.2

Fig.MH 9.2 Re:100

Pr:1.2

Time steps:249

Re:100

Pr:1.2

Time steps:66

Fig.MNH 12.2

Fig.MNH 11.2 Re:400

Pr:1.2

steps:48

Re:400

Pr:1.2

Time steps:52

It is observed from the above shown figures; for lower Reynolds number i.e 100 with lower grid structure showing three verticities; two at left and right faces and one in between them. But these three verticities are disappears in case of lower Prandl number (0.5) with higher grid structures (32X32). This is because as the viscosity will decreases then the Prandl number also decreases with increasing liquidity of the fluid inside the cavity; this makes uniform in a short duration and it is clearly catches in dense grid structure.

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.3 Re:100

Pr:0.5

Fig.MNH 2.3 Time steps:243

Re:100

Fig.MNH 3.3 Re:400

Pr:0.5

Time steps:78

Pr:0.5

Time steps:68

Fig.MNH 4.3 Re:400

Pr:0.5

Time steps:60

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

70

Fig.MNH 5.3 Re:100

Pr:1

Fig.MNH 6.3 Time steps: 54

Re:100

Time steps:57

Re:400

Fig.MNH 7.3 Re:400

Pr:1

Pr:1.2

Pr:1.2

Pr:1

Time steps: 53

Fig.MNH 10.3 Time steps:249

Re:100

Pr:1.2

Time steps:66

Fig.MNH12.3

Fig.MNH 11.3 Re:400

Time steps:68

Fig.MNH 8.3

Fig.MNH 9.3 Re:100

Pr:1

steps:48

Re:400

Pr:1.2

Time steps:52

It is observed from the above sown graphs that that the Prandl number increases ( from 0.5 to 1.2) with lower Reynolds number; the fluid particles will takes a sharp at very close to the left side wall of the cavity. This is because the top lid is moving with lower velocity and liquidity of the fluid is less for less Reynolds number (100).

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

71

V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.4 Re:100

Pr:0.5

Fig.MNH 2.4 Time steps:248

Re:100

Fig.MNH 3.4 Re:400

Pr:0.5

Pr:1

Re:400

Pr:1

Re:100

Time steps:57

Re:400

Pr:1.2

Time steps:60

Pr:1

Time steps:68

Fig.MNH 8.4

Fig.MNH 9.4 Re:100

Pr:0.5

Fig.MNH 6.4

Time steps: 54

Fig.MNH 7.4 Re:400

Time steps:68

Fig.MNH 4.4

Time steps:78

Fig.MNH 5.4 Re:100

Pr:0.5

Time steps:249

Pr:1

Time steps: 53

Fig.MNH 10.4 Re:100

Pr:1.2

Time steps:66

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

72

Fig.MNH 11.4 Re:400

Pr:1.2

steps:48

Fig.MNH 12.4 Re:400

Pr:1.2

Time steps:52

It is observed from the figure; the fluid with lower Reynolds number (100) with lower Prandl number (0.5) will take two sharp turns at the bottom side of the cavity wall and these sharp turns will shift towards top surface for higher Reynolds number (400) with higher Prandl number (1.2). But more variations in the graphs when the grid structure changes from coarse to dense.

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.5 Re:100

Pr:0.5

Time steps:243

Pr:0.5

Time steps:78

Pr:1

Time steps:68

Re:400

Pr:0.5

Time steps:60

Fig.MNH 6.5

Fig.MNH 5.5 Re:100

Pr:0.5

Fig.MNH 4.5

Fig.MNH 3.5 Re:400

Fig.MNH 2.5 Re:100

Time steps: 54

Re:100

Pr:1

Time steps:68

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

73

Fig.MNH 8.5

Fig.MNH 7.5 Re:400

Pr:1

Time steps:57

Pr:1.2

Pr:1

Time steps:249

Re:100

Pr:1.2

Fig.MNH 11.5 Re:400

Pr:1.2

Time steps:53

Fig.MNH 10.5

Fig.MNH 9.5 Re:100

Re:400

Time steps:66

Fig.MNH 12.5 steps:48

Re:400

Pr:1.2

Time steps:52

It is observed from the figure; two verticities are formed for lower Reynolds number (100) fluid in less dense grid structure (12X12) and almost only one verticity for verticity for higher Reynolds number (400) for the same grid structure but in dense grid structure (32X32) only one verticity in all the Reynolds number.

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.6 Re:100

Pr:0.5

Fig.MNH 2.6 Time steps:243

Re:100

Pr:0.5

Time steps:68

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

74

Fig.MNH 3.6 Re:400

Pr:0.5

Fig.MNH 4.6

Time steps:78

Re:400

Pr:1

Time steps: 54

Re:100

Time steps:57

Re:400

Pr:1

Pr:1.2

Time steps:249

Re:100

Fig.MNH 11.6 Re:400

Pr:1.2

Time steps:68

Pr:1

Time steps: 53

Fig.MNH 10.6

Fig.MNH 9.6 Re:100

Pr:1

Fig.MNH 8.6

Fig.MNH 7.6 Re:400

Time steps:60

Fig.MNH 6.6

Fig.MNH 5.6 Re:100

Pr:0.5

Pr:1.2

Time steps:66

Fig.MNH 12.6 steps:48

Re:400

Pr:1.2

Time steps:52

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

75

It is observed from the figures; the temperature contours with higher Reynolds number are having different characteristic and magnitude at right top corner of the cavity. This is because the top lid is moving with higher velocity and it will affect the fluid particles at bottom side of the top surface.

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNH 1.7 Re:100

Pr:0.5

Fig.MNH 2.7

Time steps:243

Re:100

Fig.MNH 3.7 Re:400

Pr:0.5

Pr:1

Re:400

Pr:1

Re:100

Time steps:57

Re:400

Pr:1.2

Time steps:60

Pr:1

Time steps:68

Fig.MNH 8.7 Pr:1

Time steps: 53

Fig.MNH 10.7

Fig.MNH 9.7 Re:100

Pr:0.5

Fig.MNH 6.7 Time steps: 54

Fig.MNH 7.7 Re:400

Time steps:68

Fig.MNH 4.7

Time steps:78

Fig.MNH 5.7 Re:100

Pr:0.5

Time steps:249

Re:100

Pr:1.2

Time steps:66

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

76

Fig.MNH 12.7

Fig.MNH 11.7 Re:400

Pr:1.2

Re:400

Pr:1.2

Time steps:52

steps:48

It is observed from the above shown figures; the simulation results are closer with published data for higher value of the Reynolds number (400) than the lower (100).

Case-III: Mixed convection fluid flow (where Gr=-2X106)

Note: Due to time restriction to submit the report online; it not possible do some simulation in mixed fluid flow with 100 Reynolds number with 32X32 grid structure in the VII chapter ‗Results and discussion‘ ; because these simulations will take days together. Therefore the some contours and graphs are predicted based on the trends of the fluid flow and characteristics of the fluid flow already simulated in the previous cases.

U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.1 Re:100

Fig.MNL 1.1 Re:100

Pr:0.5

Fig.MNL 3.1 Re:400

Pr:0.5

Pr:0.5

Time steps:

Time steps:217

Time steps:317

Fig.MNL 4.1 Re:400

Pr:0.5

Time steps:61

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Fig.MNL 6.1

Fig.MNL 5.1 Re:100

Pr:1

Time steps: 320

Re:100

Fig.MNL 7.1 Re:400

Pr:1

Time steps:340

Pr:1.2

Pr:1.2

Pr:1

Time steps:54

Fig.MNL 10.1 Time steps:341

Re:100

Fig.MNL 11.1 Re:400

Time steps:

Fig.MNL 8.1 Re:400

Fig.ML 9.1 Re:100

Pr:1

Time steps:344

Pr:1.2

Time steps:

Fig.MNL 12.1 Re:400

Pr:1.2

Time steps:52

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V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.2

Fig.MNL 1.2 Re:100

Pr:0.5

Re:100

Time steps:217

Fig.MNL 3.2 Re:400

Pr:0.5

Pr:1

Re:400

Pr:1

Pr:0.5

Time steps:61

Fig.MNL 6.2 Re:100

Pr:1

Time steps:

Time steps:320

Fig.MNL 7.2 Re:400

Time steps:

Fig.MNL 4.2

Time steps:317

Fig.MNL 5.2 Re:100

Pr:0.5

Time steps:340

Fig.MNL 8.2 Re:400

Pr:1

Time steps: 54

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Fig.ML 9.2 Re:100

Pr:1.2

Time steps:

Pr:1.2

Pr:1.2

Time steps:52

Fig.MNL 12.2

Fig.MNL 11.2 Re:400

Fig.MNL 10.2 Re:100

steps:344

Re:400

Pr:1.2

Time steps:

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.3

Fig.MNL 1.3 Re:100

Pr:0.5

Time steps:217

Re:100

Fig.MNL 3.3 Re:400

Pr:0.5

Time steps:317

Pr:0.5

Time steps:

Fig.MNL 4.3 Re:400

Pr:0.5

Time steps:61

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Fig.MNL 5.3 Re:100

Pr:1

Fig.MNL 6.3 Time steps: 320

Re:100

Fig.MNL 7.3 Re:400

Pr:1

Time steps:340

Pr:1.2

Time steps:

Fig.MNL 8.3 Re:400

Pr:1

Time steps: 54

Fig.MNL 10.3

Fig.MNL 9.3 Re:100

Pr:1

Time steps:341

Re:100

Pr:1.2

Time steps:

Fig.MNL 12.3 Fig.MNL 11.3 Re:400

Pr:1.2

Re:400

Pr:1.2

Time steps:52

steps:344

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V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNL 1.4 Re:100

Pr:0.5

Fig.MNL 2.4 Time steps:217

Re:100

Fig.MNL 3.4 Re:400

Pr:0.5

Time steps:320

Re:100

Time steps:340

Re:400

Fig.MNL 7.4 Re:400

Pr:1

Pr:1.2

Time steps:61

Pr:1

Time steps:

Fig.MNL 8.4 Pr:1

Time steps: 54

Fig.MNL 10.4

Fig.MNL 9.4 Re:100

Pr:0.5

Fig.MNL 6.4

Fig.MNL 5.4 Pr:1

Time steps:

Fig.MNL 4.4

Time steps:317 Re:400

Re:100

Pr:0.5

Re:100

Pr:1.2

Time steps:

Time steps:341

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Fig.MNL 12.4

Fig.MNL 11.4 Re:400

Pr:1.2

Re:400

Pr:1.2

Time steps:52

steps:344

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig.MNL 1.5 Re:100

Pr:0.5

Time steps:217

Fig.MNL 2.5 Re:100

Pr:0.5

Time steps:317

Re:400

Fig.MNL 5.5 Re:100

Pr:1

Time steps:

Fig.MNL 4.5

Fig.MNL 3.5 Re:400

Pr:0.5

Time steps:

Pr:0.5

Time steps:61

Fig.MNL 6.5 Re:100

Pr:1

Time steps:

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Fig.MNL 7.5 Re:400

Pr:1

Fig.MNL 8.5 Time steps:340

Re:400

Pr:1

Fig.MNL 9.5 Re:100

Pr:1.2

Time steps:341

Time steps:54

Fig.MNL 10.5 Re:100

Pr:1.2

Time steps:

Fig.MNL 12.5 Re:400

Fig.MNL 11.5 Re:400

Pr:1.2

Pr:1.2

Time steps:52

steps:344

Temperature contours 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.6

Fig.MNL 1.6 Re:100

Pr:0.5

Time steps:217

Re:100

Pr:0.5

Time steps:

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Fig.MNL 3.6 Re:400

Pr:0.5

Fig.MNL 4.6

Time steps:317

Re:400

Fig.MNL 5.6 Re:100

Pr:1

Pr:1

Re:100

Time steps:340

Pr:1.2

Pr:1

Time steps:

Fig.MNL 8.6 Re:400

Pr:1

Time steps: 54

Fig.MNL 10.6

Fig.MNL 9.6 Re:100

Time steps:61

Fig.MNL 6.6 Time steps:

Fig.MNL 7.6 Re:400

Pr:0.5

Time steps:341

Re:100

Pr:1.2

Time steps:

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Fig.MNL 12.6

Fig.MNL 11.6 Re:400

Pr:1.2

Re:400

steps:344

Pr:1.2

Time steps:52

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig.MNL 2.7

Fig.MNL 1.7 Re:100

Pr:0.5

Re:100

Time steps:217

Pr:0.5

Re:400

Pr:1

Pr:1

Time steps:61

Fig.MNL 6.7

Time steps:

Re:100

Time steps:340

Re:400

Fig.MNL 7.7 Re:400

Pr:0.5

Time steps:317

Fig.MNL 5.7 Re:100

Time steps:

Fig.MNL 4.7

Fig.MNL 3.7 Re:400

Pr:0.5

Pr:1

Time steps:

Fig.MNL 8.7 Pr:1

Time steps: 54

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Fig.MNL 9.7 Re:100

Pr:1.2

Fig.MNL 10.7

Time steps:341

Re:100

Pr:1.2

Fig.MNL 11.7 Re:400

Pr:1.2

Time steps:

Fig.MNL 12.7

steps:344

Re:400

Pr:1.2

Time steps:52

Case-IV: Natural convection fluid flow U velocity contours 12X12 Grid structure

32X32 Grid structure

Fig. N 1.1 Gr:103

Pr:0.71

Fig. N 2.1 Time steps: 204

Fig. N 3.1 Gr:104

Pr:0.71

Time steps:230

Gr:103 Pr:0.71 Computation time: 21hours

Time steps: 665

Fig. N 4.1 Gr:104 Pr:0.71 Computation time: 42hours

Time steps: 1198

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V velocity contours 12X12 Grid structure

32X32 Grid structure

Fig. N 2.2

Fig. N 1.2 Gr:103

Pr:0.71

Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Fig. N 4.2

Fig. N 3.2 Gr:10

4

Pr:0.71

Time steps: 665

Time steps: 1198

U velocity along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig. N 2.3

Fig. N 1.3 Gr:10

3

Pr:0.71

Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Fig. N 4.3

Fig. N 3.3 Gr:104

Pr:0.71

Time steps: 665

Time steps: 1198

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V velocity along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig. N 2.4

Fig. N 1.4 Gr:10

3

Pr:0.71

Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Fig. N 3.4 Gr:104

Pr:0.71

Time steps:230

Time steps: 665

Fig. N 4.4 Gr:104 Pr:0.71 Computation time: 42hours

Time steps:1198

Stream function contours 12X12 Grid structure

32X32 Grid structure

Fig. N 1.5 Gr:103

Pr:0.71

Fig. N 2.5 Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Fig. N 3.5 Gr:104

Pr:0.71

Time steps: 665

Fig. N 4.5 Time steps: 1198

Temperature contours by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

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12X12 Grid structure

32X32 Grid structure

Fig. N 1.6 Gr:103

Pr:0.71

Fig. N 2.6 Time steps: 204

Pr:0.71

Time steps: 665

Fig. N 4.6

Fig. N 3.6 Gr:104

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Time steps: 1198

Temperature along vertical centerline 12X12 Grid structure

32X32 Grid structure

Fig. N 2.7

Fig. N 1.7 Gr:103

Pr:0.71

Time steps: 204

Gr:10

Pr:0.71

Time steps: 665

Fig. N 4.7

Fig. N 3.7 4

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Time steps: 1198

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Temperature along horizontal centerline 12X12 Grid structure

32X32 Grid structure

Fig. N 1.8 Gr:103

Pr:0.71

Fig. N 2.8 Time steps: 204

Gr:103 Pr:0.71 Computation time: 21hours

Time steps:230

Gr:104 Pr:0.71 Computation time: 42hours

Fig. N 4.8

Fig. N 3.8 Gr:10

4

Pr:0.71

Time steps: 665

Time steps: 1198

It is observed from the figure; the Grashoff number increases from 103 to 104, the computation time and time steps required for the convergence of the solution increases as compared to the forced and mixed convection flow for dense grid structure. This is because the temperature difference increases with increasing Grashoff number. For the less value of Grashoff number in dense grid structure (32X32); the temperature values along the horizontal center line not close to the published data. This difference may be decreases with the high dense grid structure.

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Chapter-VIII

Scilab CFD codes used in the problem and Algorithm The code is written in non-dimensional form, with Reynolds number (Re=ρU0L/μ) as the governing parameter for isothermal flow. For convective heat transfer problems, Prandtl number (Pr=ν/α), Grashoff number (Gr=gβ(TH-TC)L3/ν2) and Rayleigh Number (Ra=gβ(TH-TC)L3/να) comes as an additional governing parameters. However, Gr=0 for forced and RePr=1 for natural convection heat transfer. Note that the characteristic velocity considered here for natural convection is equal to α/L; thus, the diffusion-coefficient is Pr for momentum and 1 for energy equation.

Solution Algorithm: 1) Enter the inputs: material properties, geometric parameters (L1 & L2) and maximum number of CVs in the X and Y directions, B.Cs input and εs. 2) Grid generation: calculate all the geometric parameters of all the CVs. 3) Set ∆ from the stability criteria. 4) Set the initial condition for . 5) Set the boundary condition for 6) Set = for all CVs 7) For =u, calculate fluxes (mass, advection, diffusion) in the X and Y direction at the u-CV faces using velocity of previous time step and S(j,i)=(Pold(j,i+1))* Y 8) Calculate total advection at all centers Ai,j. 9) Calculate total diffusion at all centers, Di,j. 10) For each ―interior‖ CVs, Predict velocity as = (-Aj,i+Dj,i+Sj,i) 11) For =v, calculate the fluxes at the v-CV faces using velocity of previous time step with S(j,i)=(Pold(j,i))-Pold(j+1,i)* X and repeat steps 7-10 12) If max(Divi,j)< ε, then go to step 16 else continue 13) Compute P‘i,j at interior nodes using the mass imbalance Divi,j 14) Pn+1= Pn+ P‘ for all interior grid points 15) Compute velocity correction using pressure correction, update the predicted star velocity and go to step 12. 16) The star velocity becomes the velocity for next time step. Solve the energy equation. 17) Go to step 5 continue all steady state.

Scilab codes for Case-I: // ***************************************************************************** // Codes developed by Vishesh Aggarwal // Under the supervision of Dr.Atul Sharma, IIT Bombay // ***************************************************************************** clc; printf("\n"); printf("*******************************************************************\n"); printf(" LID DRIVEN CAVITY PROBLEM USING 2D STAGGERED GRID NS SOLVER\n"); printf("*******************************************************************\n"); printf("\nGOVERNING PARAMETERS:"); printf("\n\tREYNOLDS NUMBER (Re) BASED ON TOP PLATE VELOCITY\n"); printf("\nBENCHMARK DATA AVAILABLE AT Re = 100, 400, 1000\n"); Re = input("ENTER Re (Must be 100 or 400 or 1000 for benchmarking): ")

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// **************************** PROBLEM PARAMETERS ***************************** // NOTE: The parameters are based on non-dimensional governing equation U = 1; //Top-plate velocity (characteristic velocity scale) Lx = 1; //Length of domain in x-direction (characteristic length scale) Ly = 1; //Length of domain in y-direction dens = 1; //Fluid density vis = 1/Re; //Fluid viscosity // ***************************** DEFINE GRID SIZE ****************************** printf("\nENTER THE NO. OF GRID POINTS\n"); // NOTE: The entered value includes the boundary grid points // This number is based on the pressure cell centre locations imax = input("IN THE X-DIRECTION : "); jmax = input("IN THE Y-DIRECTION : "); dx = Lx/(imax-2); // Grid spacing in x-direction dy = Ly/(jmax-2); // Grid spacing in y-direction dV = dx*dy; // ************ TIME STEP EVALUATION (BASED ON STABILITY CRITERION) ************ // NOTE: Courant–Friedrichs–Lewy (CFL) and Grid Fourier Criterion are used below // These are only neccessary but not sufficient condition for stability // since they are obtained from pure convection and pure diffusion, but not for // the NS equation which is a convection-diffusion equation with a source term // Furthermore, the maximum velocity needed here to obtained minimum time-step // is equal to lid velocity. // If the maximum velocity occurs inside the domain and changes with time, // then this expression needs to be used after each transient computation. dt = min(0.5*dx/U, 0.25*((dx*dy)*(dx*dy)/((vis/dens)*(dx*dx + dy*dy)))); // ************************* OTHER CONTROL PARAMETERS ************************** steady_state_criteria = 1e-3; // Used to stop outer time loop mass_div_criteria = 1e-8; // Used to stop inner mass divergence loop time_step = 0; total_time = 0; // ***************** DEFINING ARRAYS TO HOLD PROBLEM VARIABLES ***************** x = zeros(jmax-1,imax-1); y = zeros(jmax-1,imax-1); x_p = zeros(jmax,imax); y_p = zeros(jmax,imax); x_u = zeros(jmax,imax-1); y_u = zeros(jmax,imax-1); x_v = zeros(jmax-1,imax); y_v = zeros(jmax-1,imax); u = zeros(jmax,imax-1); v = zeros(jmax-1,imax); p = zeros(jmax,imax); pc = zeros(jmax,imax); uold = zeros(jmax,imax-1); vold = zeros(jmax-1,imax); ustar = zeros(jmax,imax-1); vstar = zeros(jmax-1,imax); Div = zeros(jmax-1,imax-1); mx1 = zeros(jmax,imax-2); ax1 = zeros(jmax,imax-2); dx1 = zeros(jmax,imax-2); my1 = zeros(jmax,imax-2); ay1 = zeros(jmax,imax-2); dy1 = zeros(jmax,imax-2);

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mx2 = zeros(jmax-2,imax); ax2 = zeros(jmax-2,imax); dx2 = zeros(jmax-2,imax); my2 = zeros(jmax-2,imax); ay2 = zeros(jmax-2,imax); dy2 = zeros(jmax-2,imax); // ******************** ASSIGNING STAGGERED GRID INFORMATION ******************* // Corner vertices of each p-cell for i=1:1:imax-1 for j=1:1:jmax-1 x(j,i) = (i-1)*dx; y(j,i) = (j-1)*dy; end end // Cell center of interior p-cell for i=2:1:imax-1 for j=2:1:jmax-1 x_p(j,i) = 0.5*(x(j,i) + x(j,i-1)); y_p(j,i) = 0.5*(y(j,i) + y(j-1,i)); end end // Cell center of boundary p-cell for i=2:1:imax-1 x_p(1,i) = 0.5*(x(1,i)+x(1,i-1)); y_p(1,i) = 0; x_p(jmax,i) = 0.5*(x(jmax-1,i)+x(jmax-1,i-1)); y_p(jmax,i) = Ly; end for j=2:1:jmax-1 x_p(j,1) = 0; y_p(j,1) = 0.5*(y(j,1)+y(j-1,1)); x_p(j,imax) = Lx; y_p(j,imax) = 0.5*(y(j,imax-1)+y(j-1,imax-1)); end // Corner p-cells of domain x_p(1,1) = 0; y_p(1,1) = 0; x_p(1,imax) = Lx; y_p(1,imax) = 0; x_p(jmax,1) = 0; y_p(jmax,1) = Ly; x_p(jmax,imax) = Lx; y_p(jmax,imax) = Ly; // Cell center of interior u-cell for i=2:1:imax-2 for j=2:1:jmax-1 x_u(j,i) = x(j,i); y_u(j,i) = 0.5*(y(j,i)+y(j-1,i)); end end // Cell center of boundary u-cell for i=2:1:imax-2 x_u(1,i) = x(1,i); y_u(1,i) = 0;

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x_u(jmax,i) = x(jmax-1,i); y_u(jmax,i) = Ly; end for j=2:1:jmax-1 x_u(j,1) = 0; y_u(j,1) = 0.5*(y(j,1)+y(j-1,1)); x_u(j,imax-1) = Lx; y_u(j,imax-1) = 0.5*(y(j,imax-1)+y(j-1,imax-1)); end // Corner u-cells of domain x_u(1,1) = 0; y_u(1,1) = 0; x_u(1,imax-1) = Lx; y_u(1,imax-1) = 0; x_u(jmax,1) = 0; y_u(jmax,1) = Ly; x_u(jmax,imax-1) = Lx; y_u(jmax,imax-1) = Ly; // Cell center of interior v-cell for i=2:1:imax-1 for j=2:1:jmax-2 x_v(j,i) = 0.5*(x(j,i)+x(j,i-1)); y_v(j,i) = y(j,i); end end // Cell center of boundary v-cell for i=2:1:imax-1 x_v(1,i) = 0.5*(x(1,i)+x(1,i-1)); y_v(1,i) = 0; x_v(jmax-1,i) = 0.5*(x(jmax-1,i)+x(jmax-1,i-1)); y_v(jmax-1,i) = Ly; end for j=2:1:jmax-2 x_v(j,1) = 0; y_v(j,1) = y(j,1); x_v(j,imax) = Lx; y_v(j,imax) = y(j,imax-1); end // Corner v-cells of domain x_v(1,1) = 0; y_v(1,1) = 0; x_v(1,imax) = Lx; y_v(1,imax) = 0; x_v(jmax-1,1) = 0; y_v(jmax-1,1) = Ly; x_v(jmax-1,imax) = Lx; y_v(jmax-1,imax) = Ly; // ************************ APPLYING INITIAL CONDITIONS ************************ for i=1:1:imax

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for j=1:1:jmax u(j,i) = 0; v(j,i) = 0; ustar(j,i) = 0; vstar(j,i) = 0; p(j,i) = 0; end end // ******************** FUNCTION: APPLY BOUNDARY CONDITION ********************* // NOTE: Boundary condition application is encapsulated in a function // It allows ease in modification of boundary conditions based on problem setup // It can be called within the main loop repeatedly if the problem demands function [u, v, ustar, vstar, p]=APPLY_BC(u, v, ustar, vstar, p) funcprot(0); // Bottom Boundary for i=1:1:imax u(1,i) = 0; v(1,i) = 0; ustar(1,i) = 0; vstar(1,i) = 0; p(1,i) = p(2,i); end // Top Boundary for i=1:1:imax u(jmax,i) = U; v(jmax-1,i) = 0; ustar(jmax,i) = U; vstar(jmax-1,i) = 0; p(jmax,i) = p(jmax-1,i); end // Left Boundary for j=1:1:jmax u(j,1) = 0; v(j,1) = 0; ustar(j,1) = 0; vstar(j,1) = 0; p(j,1) = p(j,2); end // Right Boundary for j=1:1:jmax u(j,imax-1) = 0; v(j,imax) = 0; ustar(j,imax-1) = 0; vstar(j,imax) = 0; p(j,imax) = p(j,imax-1); end endfunction // ************* FUNCTION: PRESSURE CORRECTION INITIAL CONDITION *************** function [pc]=APPLYIC_PCORR(pc) funcprot(0); for j=1:1:jmax for i=1:1:imax pc(j,i) = 0; end end endfunction // ************* FUNCTION: PRESSURE CORRECTION BOUNDARY CONDITION ************** function [pc]=APPLYBC_PCORR(pc)

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for j=1:1:jmax pc(j,1) = pc(j,2); pc(j,imax) = pc(j,imax-1); end for i=1:1:imax pc(1,i) = pc(2,i); pc(jmax,i) = pc(jmax-1,i); end endfunction // ********************* MAIN TIME LOOPING BEGINS HERE ************************* unsteadiness = 1e6; while unsteadiness > steady_state_criteria // Apply boundary conditions [u,v,ustar,vstar,p] = APPLY_BC(u,v,ustar,vstar,p); // Store old time level data uold = u; vold = v; //****************************************************************** // Predict new time level velocities // Fluxes across u-velocity cell faces for j=2:1:jmax-1 for i=1:1:imax-2 mx1(j,i) = dens*0.5*(u(j,i)+u(j,i+1)); ax1(j,i) = max(mx1(j,i),0)*u(j,i) - max(-mx1(j,i),0)*u(j,i+1); dx1(j,i) = vis*(u(j,i+1)-u(j,i))/(x_u(j,i+1)-x_u(j,i)); end end for j=1:1:jmax-1 for i=2:1:imax-2 my1(j,i) = dens*0.5*(v(j,i)+v(j,i+1)); ay1(j,i) = max(my1(j,i),0)*u(j,i) - max(-my1(j,i),0)*u(j+1,i); dy1(j,i) = vis*(u(j+1,i)-u(j,i))/(y_u(j+1,i)-y_u(j,i)); end end // Fluxes across v-velocity cell faces for j=2:1:jmax-2 for i=1:1:imax-1 mx2(j,i) = dens*0.5*(u(j,i)+u(j+1,i)); ax2(j,i) = max(mx2(j,i),0)*v(j,i) - max(-mx2(j,i),0)*v(j,i+1); dx2(j,i) = vis*(v(j,i+1)-v(j,i))/(x_v(j,i+1)-x_v(j,i)); end end for j=1:1:jmax-2 for i=2:1:imax-1 my2(j,i) = dens*0.5*(v(j,i)+v(j+1,i)); ay2(j,i) = max(my2(j,i),0)*v(j,i) - max(-my2(j,i),0)*v(j+1,i); dy2(j,i) = vis*(v(j+1,i)-v(j,i))/(y_v(j+1,i)-y_v(j,i)); end end // Predict cell center velocities for j=2:1:jmax-1 for i=2:1:imax-2 Au = (ax1(j,i)-ax1(j,i-1))*dy + (ay1(j,i)-ay1(j-1,i))*dx; Du = (dx1(j,i)-dx1(j,i-1))*dy + (dy1(j,i)-dy1(j-1,i))*dx; Su = (p(j,i)-p(j,i+1))*dy; ustar(j,i) = u(j,i) + (dt/(dens*dV))*(Du-Au+Su);

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

97

end end for j=2:1:jmax-2 for i=2:1:imax-1 Av = (ax2(j,i)-ax2(j,i-1))*dy + (ay2(j,i)-ay2(j-1,i))*dx; Dv = (dx2(j,i)-dx2(j,i-1))*dy + (dy2(j,i)-dy2(j-1,i))*dx; Sv = (p(j,i)-p(j+1,i))*dx; vstar(j,i) = v(j,i) + (dt/(dens*dV))*(Dv-Av+Sv); end end //****************************************************************** // Divergence term (mass error) evaluation per cell RMS_Div = 1e6; [pc] = APPLYIC_PCORR(pc); count = 0; while (RMS_Div > mass_div_criteria) // NOTE: It may be needed to restrict the maximum no. of iterations // besides checking convergence for some flow problems // Further, applying boundary conditions for USTAR and VSTAR // within this loop is also useful for channel flow problems RMS_Div = 0; for j=2:1:jmax-1 for i=2:1:imax-1 Div(j,i) = (ustar(j,i)-ustar(j,i-1))*dens*dy + (vstar(j,i)-vstar(j-1,i))*dens*dx; if (RMS_Div mass_div_criteria) // NOTE: It may be needed to restrict the maximum no. of iterations // besides checking convergence for some flow problems // Further, applying boundary conditions for USTAR and VSTAR // within this loop is also useful for channel flow problems RMS_Div = 0; for j=2:1:jmax-1 for i=2:1:imax-1 Div(j,i) = (ustar(j,i)-ustar(j,i-1))*dens*dy + (vstar(j,i)-vstar(j-1,i))*dens*dx; if (RMS_Div steady_state_criteria // Apply boundary conditions [u,v,ustar,vstar,p,T] = APPLY_BC(u,v,ustar,vstar,p,T); // Store old time level data uold = u; vold = v; Told = T; //****************************************************************** // Solving temperature equation // Fluxes in x-direction for T for j=2:1:jmax-1 for i=1:1:imax-1 mxT(j,i) = dens*Cp*u(j,i); axT(j,i) = max(mxT(j,i),0)*T(j,i) - max(-mxT(j,i),0)*T(j,i+1); dxT(j,i) = dif*(T(j,i+1)-T(j,i))/(x_p(j,i+1)-x_p(j,i)); end end // Fluxes in y-direction for T for j=1:1:jmax-1 for i=2:1:imax-1 myT(j,i) = dens*Cp*v(j,i);

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

118

ayT(j,i) = max(myT(j,i),0)*T(j,i) - max(-myT(j,i),0)*T(j+1,i); dyT(j,i) = dif*(T(j+1,i)-T(j,i))/(y_p(j+1,i)-y_p(j,i)); end end // Get new time level temperatures for j=2:1:jmax-1 for i=2:1:imax-1 DTp = (dxT(j,i)-dxT(j,i-1))*dy + (dyT(j,i)-dyT(j-1,i))*dx; ATp = (axT(j,i)-axT(j,i-1))*dy + (ayT(j,i)-ayT(j-1,i))*dx; T(j,i) = T(j,i) + dt*(DTp - ATp)/dV; end end //****************************************************************** // Predict new time level velocities // Fluxes across u-velocity cell faces for j=2:1:jmax-1 for i=1:1:imax-2 mx1(j,i) = dens*0.5*(u(j,i)+u(j,i+1)); ax1(j,i) = max(mx1(j,i),0)*u(j,i) - max(-mx1(j,i),0)*u(j,i+1); dx1(j,i) = vis*(u(j,i+1)-u(j,i))/(x_u(j,i+1)-x_u(j,i)); end end for j=1:1:jmax-1 for i=2:1:imax-2 my1(j,i) = dens*0.5*(v(j,i)+v(j,i+1)); ay1(j,i) = max(my1(j,i),0)*u(j,i) - max(-my1(j,i),0)*u(j+1,i); dy1(j,i) = vis*(u(j+1,i)-u(j,i))/(y_u(j+1,i)-y_u(j,i)); end end // Fluxes across v-velocity cell faces for j=2:1:jmax-2 for i=1:1:imax-1 mx2(j,i) = dens*0.5*(u(j,i)+u(j+1,i)); ax2(j,i) = max(mx2(j,i),0)*v(j,i) - max(-mx2(j,i),0)*v(j,i+1); dx2(j,i) = vis*(v(j,i+1)-v(j,i))/(x_v(j,i+1)-x_v(j,i)); end end for j=1:1:jmax-2 for i=2:1:imax-1 my2(j,i) = dens*0.5*(v(j,i)+v(j+1,i)); ay2(j,i) = max(my2(j,i),0)*v(j,i) - max(-my2(j,i),0)*v(j+1,i); dy2(j,i) = vis*(v(j+1,i)-v(j,i))/(y_v(j+1,i)-y_v(j,i)); end end // Predict cell center velocities for j=2:1:jmax-1 for i=2:1:imax-2 Au = (ax1(j,i)-ax1(j,i-1))*dy + (ay1(j,i)-ay1(j-1,i))*dx; Du = (dx1(j,i)-dx1(j,i-1))*dy + (dy1(j,i)-dy1(j-1,i))*dx; Su = (p(j,i)-p(j,i+1))*dy; ustar(j,i) = u(j,i) + (dt/(dens*dV))*(Du-Au+Su); end end for j=2:1:jmax-2 for i=2:1:imax-1 Av = (ax2(j,i)-ax2(j,i-1))*dy + (ay2(j,i)-ay2(j-1,i))*dx; Dv = (dx2(j,i)-dx2(j,i-1))*dy + (dy2(j,i)-dy2(j-1,i))*dx; Sv = (p(j,i)-p(j+1,i))*dx + Ra*Pr*0.5*(T(j,i)+T(j+1,i))*dV; vstar(j,i) = v(j,i) + (dt/(dens*dV))*(Dv-Av+Sv);

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur

119

end end //****************************************************************** // Divergence term (mass error) evaluation per cell RMS_Div = 1e6; [pc] = APPLYIC_PCORR(pc); count = 0; while (RMS_Div > mass_div_criteria) // NOTE: It may be needed to restrict the maximum no. of iterations // besides checking convergence for some flow problems // Further, applying boundary conditions for USTAR and VSTAR // within this loop is also useful for channel flow problems RMS_Div = 0; for j=2:1:jmax-1 for i=2:1:imax-1 Div(j,i) = (ustar(j,i)-ustar(j,i-1))*dens*dy + (vstar(j,i)-vstar(j-1,i))*dens*dx; if (RMS_Div