ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
A Two-Phase Flow Model for Predicting Cavitation Dynamics Philip J. Zwart1 , Andrew G. Gerber2 , Thabet Belamri3 1: ANSYS Canada; Waterloo, Ontario, Canada;
[email protected] 2: Dept. of Mechanical Engineering; University of New Brunswick; Fredericton, NB, Canada;
[email protected] 3: ANSYS Canada; Waterloo, Ontario, Canada;
[email protected]
robust CFD methodology methodology for predicting predicting three-dimensio three-dimensional nal flows with extensive extensive cavitation cavitation is presented. presented. Abstract A robust The model is based on the multiphase flow equations, with mass transfer due to cavitation appearing as source and sink terms in the liquid and vapour continuity equations. The mass transfer rate is derived from a simplified Rayleigh-Plesset model. It is implemented into the CFX-5 software, which also features a control volume finite element discretization and a solution methodology which implicitly couples the continuity and momentum equations together. The model is validated on a range of applications including flow over a hydrofoil, an inducer, and transient cavitation in a venturi.
Introduction Cavitation is an important phenomenon which can have a profound effect on the performance of a number of devices. devices. Examples Examples include pumps, inducers, propellors, propellors, and injectors. injectors. Computational Computational Fluid Dynamics (CFD) has been extensively used to predict the flow through these devices under non-cavitating conditions. However, because of the physical and numerical challenges associated with cavitation, CFD has only recently started to be used to predict cavitating flows. Many of the CFD models developed for cavitation involve the use of a barotropic equation of state to express the mixture density as a function of local pressure. This modelling approach is attractive because it can be integrated into a basic-functionality CFD code without much effort. However, it must also be recognized that these approaches approaches are overly overly simplistic in their assumption assumption of equilibrium equilibrium thermodynamics. thermodynamics. By this we mean that as the flow conditions change, the two-phase fluid is assumed to instantaneously reach its equilibrium thermodynamic state. In reality, the finite rate effects involved in cavitation are important, and are based on complex physical processes. processes. Eisenburg Eisenburg [5] provides a summary of the transient transient dynamics of cavitation cavitation.. Cavitation Cavitation inception inception in − − 5 3 practical fluids is associated with the growth of nuclei having diameters ranging from 10 to 10 cm. These nuclei contain contain a mixture mixture of vapour vapour and noncondensible noncondensible gases. gases. As the nuclei pass through regions regions where the pressure drops below the vapour pressure, they typically grow explosively. (For the case of gaseous cavitation, growth growth is limited limited by diffusion diffusion timescales timescales and is therefore much slower.) slower.) The cavitation cavitation region is made up of a large large number of these bubbles. bubbles. Further Further downstream, downstream, as the bubbles are swept into regions of higher pressure, pressure, they collapse. collapse. The dynamics of collapse are complex complex and depend on a variety variety of factors including including surface tension, tension, viscous viscous effects, effects, and noncondensible noncondensible content. The large accelerations accelerations and pressures generated generated by the final stage of collapse are responsible for much of the noise and damage generated generated by cavitating cavitating flows. flows. Not surprisingly, there are no cavitation models which attempt to account for all of these complexities. There are models, however, which do account for nonequilibrium effects. Most of these models are based upon the Rayleigh Plesset equation [3], which describes the growth and collapse of a single bubble subjected to a far-field far-field pressure pressure disturbance. disturbance. Examples Examples of such models are given given by Schnerr Schnerr and Sauer [11], Gerber Gerber [6], and Senocak and Shyy [12]. The details of the models developed by these and other researchers vary, but they share the common feature of modelling the vapour composition by using a continuity equation having a source term which can be traced traced back to the Rayleigh Rayleigh Plesset equation. In this paper, a new Rayleigh-Plesset based cavitation model is described. The model has been implemented in CFX-5, a general-purpose three-dimensional commercial CFD package. Recognizing that cavitation fundamentally involves phase change, the model has been implemented as an interphase mass transfer process within the code’s code’s multiphase multiphase framework. framework. This has some important numerical numerical benefits, benefits, the most significant significant being that the global continuity constraint is cast in volumetric form while remaining fully conservative.
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
The resulting model has been validated on a range of cavitating devices featuring a number of different fluids. Three test cases will be discussed in this paper: a cavitating hydrofoil, cavitation in an inducer, and transient cavitation in a venturi.
Mathematical Model Conservation Equations The equations governing multiphase flow include conservation of mass for each phase α :
∂ (rα ρα ) ∂ (rα ρα ui ) ˙α + = S ∂t ∂x i
(1)
and conservation of momentum for the mixture (assuming no interphase slip):
∂ (ρm ui ) ∂ (ρm u j ui ) ∂P ∂ (τ ji ) i + = − i + ρm rα g + , ∂t ∂x j ∂x ∂x j
(2)
˙ α respectively represent the volume fraction, Cartesian velocity components, density, and where r α , u i , ρ α , S mass generation rate of phase α; g i represents acceleration due to gravity; P is the pressure; and τ ji is the stress tensor, related to the deformation rates using Stokes’ law: ji
τ = µm
∂u i ∂ u j + ∂x j ∂x i
.
(3)
ρm and µ m are the volume-weighted mixture density and viscosity. We have assumed that the mass sources arise from interphase mass transfer, and therefore satisfy the constraint N
˙ α = 0. S
(4)
α=1
We also have the constraint that the phases must fill up the available volume: N
rα = 1.
(5)
α=1
Eqs. (1), (2), and (5) together form a closed system involving (N + 4) equations and (N + 4) unknowns, N being the number of phases. For convenience, we choose to replace one of the phasic continuity equations with the sum of all continuity equations divided by their respective densities: N
α=1
1 ρα
∂ρ α ∂ (rα ρα ui ) ˙α + − S ∂t ∂x i
= 0.
(6)
When all phases are incompressible, this equation may be interpreted as requiring the velocity divergence to balance the volume generation due to phase change. For example, cavitation involves a vapour and liquid phase, with sources related by: ˙ v = − S ˙ l = S ˙ lv (7) S Then Eq. (6) reduces to
∂u i ˙ lv = S i ∂x
1 1 − ρv ρl
.
(8)
When the flow is turbulent, the velocities in the above equations represent statistically averaged velocities, and additional Reynolds Stress terms appear in the momentum equation. These stresses are modelled using an eddy viscosity approach such as the k − ε or Shear Stress Transport (SST) models [7].
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
Cavitation Model The Rayleigh-Plesset equation describes the growth of a vapour bubble in a liquid:
d2 RB 3 RB + dt2 2
2
dRB dt
+
2σ P v − P = RB ρl
(9)
where RB represents the bubble radius, σ represents the surface tension coefficient, and P v represents the vapour pressure. Neglecting the second order terms and the surface tension yields the simplified expression
dRB = dt
2 P v − P . 3 ρl
(10)
The rate of change of mass of a single bubble follows as
dmB = 4πR 2B ρv dt
2 P v − P . 3 ρl
(11)
If there are N B bubbles per unit volume, we may express the vapour volume fraction as
4 rv = V B N B = πR 3B N B 3
(12)
(13)
and the total interphase mass transfer rate due to cavitation per unit volume is
˙ lv = 3rv ρv S RB
2 P v − P . 3 ρl
This model has been derived assuming bubble growth (vaporization). It can be generalized to include condensation as follows:
˙ lv = F 3rv ρv S RB
2 |P v − P | sign(P v − P ) 3 ρl
(14)
where F is an empirical calibration coefficient. This model works well for condensation. However, it is physically incorrect (and numerically unstable) if applied to vaporization. One of the key assumptions in its derivation is that the cavitation bubbles do not interact with each other. This is plausible only during the earliest stages of cavitation, when the cavitation bubble grows from the nucleation site. As the vapour volume fraction increases, the nucleation site density must decrease accordingly. With this in mind, we replace rv by r nuc (1 − rv ) during vaporization, where r nuc is the nucleation site volume fraction. R B is interpreted as the radius of a nucleation site. The final form of the cavitation model is: rv )ρv 2 P v −P if P < P v F vap 3rnuc (1− R 3 ρl B ˙ (15) S lv = if P > P v F cond 3rRv ρv 23 P −ρ P v
B
l
The following model parameters have found to work well for a variety of fluids and devices: RB = 10−6 m, rnuc = 5 × 10−4 , F vap = 50, and F cond = 0.01. For unsteady cavitating flows, it has been observed in the literature that standard turbulence models fail to properly predict the oscillating behaviour of the flow. This has also been observed with the cavitation model we have proposed. We follow the example of [4] and use a modified formulation for turbulent viscosity. In the standard k − ε model, the eddy viscosity for the mixture is
µtm = ρ m C µ
k2 ε
(16)
The modified expression effectively reduces the eddy viscosity in the cavitating regions by using
k2 µtm = f (ρ)C µ ε where
f (ρ) = ρ v +
ρv − ρm ρv − ρl
(17)
n
(ρl − ρv ).
(18)
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
Figure 1: Element-based finite volume discretization of the spatial domain. Solid lines define element boundaries and dashed lines divide elements into sectors. Solution unknowns are colocated at the nodal points (•), and surface fluxes are evaluated at integration points ( ◦). Control volumes are constructed as unions of element sectors (shaded region).
Numerical Model Discretization Scheme The conservation equations described above are discretized using an element-based finite volume method [10]. The mesh may consist of tetrahedral, prismatic, pyramid, and hexahedral elements. A control volume is constructed around each nodal point of the mesh, as illustrated in Figure 1. The subface between two control volumes within a particular element is called an integration point (ip); it is at integration points that the fluxes are discretized. Integration point quantities such as pressure and velocity gradients are obtained from nodal values using finite element shape functions, with the exception of advected variables which are obtained using an upwind-biased discretization. We now consider the discretization of the conservation equations at each control volume. The discretization is fully conservative and time-implicit. The conservation equations are integrated over each control volume, volume integrals are converted to surface integrals using Gauss’ divergence theorem, and surface fluxes are evaluated in exactly the same manner for the two control volumes adjacent to an integration point. In the following discussion V represents the volume of a control volume, A iip the area vector of an integration point, δt the time step, and the superscripts n + 1 and n mean that the quantity is evaluated at the new and old time step, respectively. The discrete conservation equations for the phasic continuity may be viewed as evolution equations for the volume fractions:
V (ρα rα )n+1 − (ρα rα )n + δt
(ρα ui Ai )nip+1 (rα,ip )n+1 = 0,
(19)
ip
The advection scheme used to evaluate rα,ip in terms of neighbouring vertex values must give solutions which are both bounded and accurate. We write it in the form
rα,ip = r α,up + β ∇rα · R,
(20)
is the vector from the upwind vertex to the integration point. If where r α,up is the upwind vertex value and R β = 0, this scheme recovers the first-order upwind scheme, which is bounded but excessively diffusive. If β = 1, this scheme is a second-order upwind-biased scheme, but unbounded. A bounded high-resolution scheme
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
can be obtained by making β as close to 1 as possible, but reducing where necessary to prevent overshoots and undershoots from occurring. We use a method similar to that described by Barth and Jesperson [2]. The mass flows must be discretized in a careful manner to avoid pressure-velocity decoupling. This is performed by generalizing the interpolation scheme proposed by Rhie and Chow [9], such that the advecting velocity is evaluated as follows:
uiip = u i ip +
dip
∂P ∂P − ∂x i ∂x i
,
(21)
ip
where
dip ∝ −V /a a ∝ ρ m V/δt + b
(22)
(23)
and b represents the sum of advection and viscous coefficients in the discretized momentum equation. The overbar denotes the average of the control volume values adjacent to the integration point. The discretized phasic momentum equations may be viewed as an evolution equation for the phasic velocity field:
V (ρm ui )n+1 − (ρm ui )n + δt
(ρm u j A j )n+1 (ui )n+1 = −
ip
n+1 i P ip A + ρnm+1g i V +
ip
((τ ji )n+1 A j )ip .
ip
(24) A standard second-order or high-resolution scheme may be used for the advected velocity in this equation, and finite-element shape functions are used to evaluate the gradients for the pressure and viscous forces. Finally it remains to derive a discrete equation for pressure. This is obtained by integrating Eq. (6) over the control volume:
N
α=1
1 ρα
V n+1 ρ − ρnα + δt α
ip
ρα rα ui Ai
n+1 ip
˙ α V − S
= 0,
(25)
which yields a diagonally-dominant equation for pressure because of the special interpolation used for uiip and
˙ α /∂P . through the derivative ∂ S
Solution Strategy The set of algebraic equations (19), (24), (25), and (5) represent equations for the volume fraction, velocity, and pressure fields. With two phases, these equations form a 6 ×6 coupled system of equations at each nodal point. Equation (5) is an algebraic equation which may be decoupled from the active set. In addition, equation (19) is currently also decoupled from the pressure-velocity system and is treated in a segregated manner. The linear system of equations is solved using the coupled algebraic multigrid technique developed by Raw [8].
Validation Hydrofoil Cavitation This test case involves a cavitating hydrofoil [13] at two angles of attack. At a one-degree angle, cavitation is induced along the midchord. At a four-degree angle, cavitation is induced at the leading edge. The mesh is solved as a three-dimensional problem with two layers of nodes in the depth direction and symmetry boundaries applied to these two planes. There are 15,288 nodes on each of the two planes, and symmetry conditions are applied to these two planes. A close-up of the mesh in the vicinity of the hydrofoil is shown in Figure 2. Along the hydrofoil surface a no-slip condition is applied. Free-slip conditions are used for the far-field boundaries above and below the hydrofoil. A velocity-specified boundary is applied to the inlet to yield a specified Reynolds number. At the outlet, the pressure is imposed to yield a specified cavitation number. The fluid properties are taken to be saturated water conditions at 25 C. For the one-degree angle of attack, cavitation numbers of 0.43, 0.38, and 0.34 are considered. The Reynolds number based on chord length is 3 × 106 . A comparison between experimental and computed pressure coefficients for these situations is plotted in Figure 3. The plot shows excellent agreement with the data.
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
Figure 2: Hydrofoil mesh For the four-degree angle of attack, cavitation numbers of 1.00, 0.91, and 0.84 are considered. The Reynolds number based on chord length is 2 × 106 . The calcalated and experimental pressure coefficients for these cases are plotted in Figure 3. In this case, the trends compare well with experiment but but the length of the cavitation zone is somewhat underpredicted at the lower cavitation numbers. Inducer Cavitation A second validation test case involves flow in an inducer tested at LEMFI [1]. The cavitating flow through a single blade passage is modelled at a range of flow rates and cavitation numbers. The mesh, consisting of 250,000 hexahedral elements for the blade passage, was generated using CFX-TurboGrid. The total pressure is specified at the inlet, and the mass flow rate at the outlet. A periodic boundary condition is used to connect the sides of the domain together. No-slip walls are used for all other boundaries. Each calculation was declared converged when the maximum normalized residual dropped below 10−4 . The head drop-off curve was predicted for a range of flow rates. The flow rate is characterized by the actual-to-nominal flow ratio, Q/Qn . For each curve, a noncavitating solution was first obtained. The cavitation model was then activated and the inlet total pressure decreased by 10,000 Pa for each point on the curve. When the head dropoff became significant, the inlet pressure was dropped in smaller increments of 1000 Pa. The experimental and predicted dropoff curves (nondimensional pressure rise as a function of cavitation number) are plotted for three different flow rates in Figure 5. At low flow rates ( Q/Qn = 0.79), the predicted dropoff curve occurs smoothly and slightly before the experimental curve. Close to the design flow rate (Q/Qn = 1.03), dropoff occurs more rapidly and simultaneously with the experimental measurements. Agreement between the two results is very satisfactory. At high flow rates ( Q/Qn = 1.15), the dropoff curve occurs rapidly and slightly after the experimental measurements. At this flow rate, the pressure rise is slightly underestimated, and cavitation-induced blockage through the blade passage is seen to be more extensive in the experiments than the model predictions. Overall, the agreement is very encouraging. For one of the flow rates ( Q/Qn = 1.03), the predicted shape of the cavitation pockets was compared to experimental visualizations. The comparison is illustrated in Figure 6. The general development of cavitation in the inducer is illustrated at three operating points, corresponding to cavitation numbers of 0.09, 0.06, and 0.045. The vapour first appears near the shroud at the leading edge of the suction side of the blade . As the cavitation number decreases, the cavity remains attached to the blade but grows into the blade-to-blade channel and down towards the hub. Finally, the vapour passes to the pressure side of the blade, the blockage becomes extensive, and the performance breaks down. The plot also illustrates the predicted vapour bubbles corresponding to 10% volume fraction in red. The hub is coloured in green and the blade in gray. The agreement is quite reasonable,
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
0.5
0.4
0.3 p C
0.2
0.1
Calculated, 0.43 Experiment, 0.43 Calculated, 0.38 Experiment, 0.38 Calculated, 0.34 Experiment, 0.34
0
-0.1 0
0.2
0.4
0.6
0.8
1
x/L Figure 3: Comparison of experimental and computed surface pressure coefficients for midchord cavitation on hydrofoil for three cavitation numbers
1.2 1 0.8 0.6 p C
0.4 Calculated, 1.00 Experiment, 1.00 Calculated, 0.91 Experiment, 0.91 Calculated, 0.84 Experiment, 0.84
0.2 0 -0.2 0
0.2
0.4
0.6
0.8
1
x/L Figure 4: Comparison of experimental and computed surface pressure coefficients for leading edge cavitation on hydrofoil for three cavitation numbers
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
Figure 5: Predicted and experimental head dropoff curves at Q/Qn = 0.79, 1.03, 1.15.
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
Figure 6: Inducer flow visualizations at Q/Qn = 1.03 at cavitation numbers of 0.09, 0.06, and 0.045. especially the location and evolution of the vapour region. Transient Venturi Cavitation The cavitation model was also tested for transient cavitation conditions. A venturi-type geometry was used for the test approximating that used in the experiments of Stutz and Reboud [14]. In their experiments the venturi was operated with unsteady shedding of vapour clouds just past the venturi throat. The shedding frequency is characterized by a Strouhal number, defined as
St =
fLcav V ref
(26)
where f is the frequency, Lcav is the average cavity length, and V ref is the inlet velocity. The Strouhal number reported in the experiments is 0.27, based on a shedding frequency of 50 Hz, an average cavity length of 45 mm, and a velocity of 8 m/s. The current calculations use an inlet velocity of 7.9 m/s, an outlet pressure of 30 kPa, symmetry planes at the front and back, and a no-slip condition on the venturi walls. A noncavitating solution was used to provide initial conditions. The cavitation number (based on the average inlet total pressure) is 1.9. A time step of 10−4 s is used. The predicted cavity length is 40 mm and average frequency is 55 Hz, yielding an average Strouhal number of 0.278. The cavitation model parameters were set to F vap = 0.4 and F cond = 0.001. In addition, the turbulence model modification described in Eq. (17) was used with n = 5. Figure 7 shows a typical pressure response at a point 0.26 m downstream of the throat, indicating a nearly regular perturbation of the flow field as vapour clouds are advected downstream. When a vapour cavity passes the probe, the vapour pressure is plotted. The volume fraction variation with time at the same location is also provided. Figure 8 shows a typical vapor distribution at a fixed time, along with a cross-hair indicating the location where the readings for Figure 7 are taken. It should be noted that with the turbulence model modifications required to induce the unsteadiness as well as the cavitation model constants, there is still a
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
Figure 7: Pressure and volume fraction response in venturi at a point 0.26 m downstream of the throat. significant amount of tuning involved in the prediction of transient flow behavior of this kind. However, it is encouraging to see that the general flow features and shedding frequency can be accurately predicted.
Conclusions A new multiphase flow algorithm for predicting cavitation has been presented. The mass transfer rate between liquid and vapour phases is computed using a model based upon the Rayleigh Plesset equation. It has been implemented in a conservative manner in an element-based finite volume method, and features careful linearization and coupling behaviour in order to obtain good convergence behaviour. Three validation examples have been provided. First, flow around a hydrofoil, with cavitation induced at both the leading edge and midchord, has been successfully predicted. Second, the flow through an inducer has been modelled, including the head-dropoff curves. Finally, the transient shedding of vapour bubbles through a venturi has been modelled. This final case required some retuning of various model coefficients, including a modification to the turbulent viscosity, but nonetheless indicates that transient cavitating flows can also be simulated by this model.
References [1] F. Bakir, R. Rey, A. G. Gerber, T. Belamri, and B. Hutchinson. Numerical and experimental investigations of the cavitating behaviour of an inducer. International Journal for Rotating Machinery, 10:15–25, 2004. [2] T. J. Barth and D. C. Jesperson. The design and application of upwind schemes on unstructured meshes. AIAA Paper 89-0366 , 1989. [3] C. E. Brennen. Cavitation and Bubble Dynamics. Oxford University Press, 1995. [4] O. Coutier-Delgosha, R. Fortes-Patella, and J. L. Reboud. Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation. Journal of Fluids Engineering, 125:38–45, 2003. [5] P. Eisenberg. Cavitation. In Handbook of Fluid Mechanics. McGraw Hill, 1961. [6] A. G. Gerber. A CFD model for devices operating under extensive cavitation conditions. In International Mechanical Engineering Congress and Exhibit , 2002.
ICMF 2004 International Conference on Multiphase Flow Yokohama, Japan, May 30-June 3, 2004 Paper No.152
Figure 8: Snapshot of volume fraction in venturi. The crosshair indicates the probe location used for Figure 7. [7] F. R. Menter and H. Grotjans. Application of advanced turbulence models to complex industrial flows. In G. Tzabiras, editor, Advances in Fluid Mechanics: Calculation of Complex Turbulent Flows. WIT Press, 2000. [8] M. Raw. Robustness of coupled algebraic multigrid for the Navier-Stokes equations. AIAA Paper 96-0297 , 1996. [9] C. M. Rhie and W. L. Chow. Numerical study of the turbulent flow past an airfoid with trailing edge separation. AIAA Journal, 21:1525–1532, 1983. [10] G. E. Schneider and M. J. Raw. Control volume finite-element method for heat transfer and fluid flow using colocated variables — 1. Computational procedure. Numerical Heat Transfer , 11:363–390, 1987. [11] G. H. Schnerr and J. Sauer. Physical and numerical modelling of unsteady cavitation dynamics. In 4th International Conference on Multiphase Flow, 2001. [12] I. Senocak and W. Shyy. Evaluation of cavitation models for Navier-Stokes computations. In ASME 2002 Fluids Engineering Division Summer Meeting, 2002. [13] Y. T. Shen and P. E. Dimitakis. The influence of surface cavitation on hydrodynamic forces. In Proc. 22nd ATTC , 1989. [14] B. Stutz and J. L. Reboud. Experiments on unsteady cavitation. Experiments in Fluids, 22:191–198, 1997.
