CFD [AE2402] Question Bank

A QUESTION BANK ON COMPUTATIONAL FLUID DYNAMICS (AE-2402)

Prepared for Anna University syllabus (2012) By  Surendra Bogadi Asst. Professor Department of Aeronautical Engineering Rajalakshmi Engineering College, Thandalam, Chennai - 602105

PART-A QUESTIONS (2 MARKS) 1. What are the important applications of CFD in engineering? 2. Distinguish between conservation and non-conservation forms of fluid flow. 3. Write down the conservative form of the continuity equation and explain the terms t erms involved. 4. What is the physical significance/meaning of the various terms in conservation form of momentum equation? 5. Write down an expression for substantial derivative in Cartesian coordinates. 6. List out advantaged of panel method. 7. Explain the difficulties of evaluating the influences of a panel at its own control point. 8. What are limitations of panel methods? 9. Define (a) Convergence and (b) Lax equivalence theorem 10. Elaborate the basic aspects of the finite f inite difference equations. 11. Write down the significance of  Taylor series expansion. 

12. Write down the second order central mixed finite difference expression for

2

u

 x y

13. Explain cell-centered method. 14. Define stability in numerical solution of fluid flow governing equations. 15. Define convergence in numerical solution of fluid flow governing equations. 16. Write down the second order central mixed finite difference expression for 1D heat conduction equation. 17. Discuss the need of upwind type discretization. 18. Name the important errors that commonly occur in numerical solution. 19. Transform the steady, incompressible continuity equation from x, y physical plane to the ξ , η computational plane. 20. What is the importance of CFL condition? 21. Compare implicit and explicit methods. 22. What are the different categories of boundary conditions? Give example for each category. 23. Differentiate between structured and unstructured grids. 24. Differentiate between surface fitted and body fitted coordinate systems. 25. What are the methods available for grid generation?

26. What is the necessity for staggered grid in control volume method? 27. List out differences between  finite volume and finite difference methods. 28. Define peclet number and state its importance? 29. What is the necessity of strong and weak formulations in boundary value problems? 30. What types of grids are generally used in Finite Volume Method?

PART-B QUESTIONS (16 MARKS) 1. (i) What is the need for classification of PDE and how do you classify second order PDE?

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(ii) What are the discretization techniques and how do you discretize the equations for subsonic and supersonic flows?

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2. Write down the elliptic, parabolic and hyperbolic partial differential equations as applicable to computational fluid dynamics. 3. (a) Derive the continuity continuity equation in differential form form for in compressible flow. [10] (b) Define & develop expression for substantial derivative.

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4. Derive the continuity equation for a inviscid flow in partial differential nonconservation form. 5. Derive the energy equation for a viscous flow in partial differential non-conservation form. 6. Discuss the source panel method for the flow past an oscillating cylinder. 7. Write down the procedure for the calculation of pressure coefficient distribution around a circular cylinder using the source panel technique. 8. Discuss the vortex panel method applied to lifting flows over a flat plate. 9. Obtain the 2D compressible continuity equation in transformed coordinates for transformation     x,   ln ln( ( y  1) . 10. (i) How is conformal mapping of a polygon carried out by Schwarz-Christoffel transformation?

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(ii) Illustrate the basic ideas of algebraic transformations of two dimensional, steady, boundary layer flow over flat plate with suitable transformation relations.

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11. Explain the grid generation techniques based on PDE and summarize the advantages of the elliptic grid generation method.

12. (i)

What is the need for grid generation? Mention the different grid generation generation technique and list down their relative merits and demerits.

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(ii) Explain how grid generation is achieved by numerical solution of elliptic Poison‟s equations.

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13. (i) What is meant by by “wiggles” in the numerical solution? Describe with an example. [6] (ii) Consider steady 1-D 1-D convection diffusion equation of a property φ d/dx (ρu φ) = d/dx {Γ d φ/dx} using control volume approach discretize the above equation and obtain the neighboring coefficients by (1) Central difference scheme (2) Upwind difference scheme

[10]

14. What is meant by hierarchy of boundary layer equations? Derive Zeroth, first and second order boundary layer equations? 15. Explain the description of Prandtl boundary layer equation and its solution methodology. 16. Write short notes on the following : (i)

Strong formulation

(ii) Weighted Residual formulation (iii) Galerkin Formulation (iv) Weak formulation

[4 X 4 = 16]

17. Consider a cylindrical fin with uniform cross-sectional area A. the base is at a temperature of 100 0C ( T  B) and the end is insulated. The fin is exposed to an ambient 0

temperature of 20 C. One-dimensional heat transfer in this situation is governed by d/dx {kA (dT/dx)}-hP(T-T  ) = 0 ∞

where „h‟ is the convective heat transfer coefficient, „P‟ the perimeter, k the thermal conductivity of the material and T ∞ the ambient temperature. Calculate the temperature distribution along the fin using five equally placed control volumes. Take 2

hp / (kA)=25m (note: kA is constant)

18. State and explain the difference between explicit and implicit methods with suitable examples. 19. How do you determine the accuracy of the discretization process? What are the uses and difficulties of approximating the derivatives with higher order finite difference schemes? How do you overcome these difficulties?

20. Study the stability behaviour of second order wave equation by Von-Neuman stability method. 21. (i) Explain explicit Lax-Wendroff scheme of time dependent methods.

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(ii) Discuss cell centered formulation in Finite Volume Techniques.

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22. What are quadrilateral Lagrange elements and isoparametric elements in FEM? 23. Draw a flow chart and describe SIMPLE algorithm for two-dimensional laminar steady flow equations in Cartesian co-ordinates. 24. Solve the simplified Sturn-Lioville equation:  2 y   y  F   x 2

With boundary conditions y(0) = 0 and

 y 1  0  x ;

using Galerkin finite

element method. 25. What is strong formulation? Explain with the help of one dimensional boundary value problem. 26. Explain Runge-Kutta and multi-stage time stepping. 27. Discuss the properties of discretization schemes and explain upwind discretization applied to FVM. 28. What is cell centered formulation? Explain with the help of using control volume, semi discretization equation, ij U  / t    F .nds nds  0. 29. State and explain the spurious modes for Runge-Kutta cell vertex formulation in FVM. 30. What is cell centered formulation? Explain with the use of control volume and semi discretization equation. 31. Derive an expression for energy equation for infinitesimally small, moving fluid element. 32. List the full procedure for the solution of Blasius equation using shooting method. 33. How characteristics lines are related to Courant number from stability view? Explain. 34. (a) Derive

 ui3,  j  4ui2,  j  5ui1,  j  2ui , j   2u       O( x 2 ) 2   x 2   x     i ,  j

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(b) Write short notes on (i) Consistency

(ii) Convergence Convergence

(iii) Stability

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35. (a) Derive the forward and central difference, approximations to the first derivative, along with the leading error term.

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st

(b) Derive the stability criterion (CFL conditions) for the I order wave equation. [8]

35. For the following equation: T   2 T     t   x 2

(a) Obtain discretized form of finite difference quotient. (b) Using explicit method, write algebraic equations for 4 ´ 4 grid. (c) Explain any numerical method to obtain solution for temperatures.

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36. Explain the following with merits and demerits. a) Explicit method b) Implicit method c) Semi-implicit method

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37. Describe the MacCormak ‟s technique for evaluating the density at time step (t + Δt).