Pressure-correction algorithm to solve Poisson system with constant coefficients for fast two-phase simulations

Transcription

1 Center or Turbulence Research Annual Research Bries Pressure-correction algorithm to solve Poisson system with constant coeicients or ast two-phase simulations By D. Kim AND P. Moin 1. Motivation and objective It is a challenging problem to numerically simulate incompressible two-phase low owing to its complexity and high computational cost. The complexity o two-phase low are caused by the moving interace between the phases, where density and viscosity abruptly change. Various numerical methods such as the marker method, the volume-o-luid (VoF) method and the level set method, have been developed to handle arbitrarily shaped interaces in a ixed grid. Each o these phase tracking methods has its merits and demerits. In the present work, the level set method is used to track the phase interace. The advantages o the level set method is that the topology and movement o the interace are automatically captured. Thus, it is easy to implement numerically. The main drawback o the level set method is that the conservation o liquid volume is not guaranteed. Hybrid methods combining the level set method with VoF Sussman (2000) have been proposed to enhance the mass conservation property. However, the delocalization method has to be used to avoid local curvature error and it makes numerical scheme more complex. High computational cost is another big issue in turbulent simulations or two-phase low because it is diicult to apply ast solvers to the Poisson equations. For incompressible two-phase low, the continuity equation o divergence-ree velocity is used to orm the Poisson equation or the pressure-correction in the ractional step method. Since the phase interace is moving in time, the coeicients o the Poisson equation is also unction o time, while the coeicients is constant in case o single phase low where density is constant. Because the coeicients o the Poisson equation change in time, the gradientbased method such as the Biconjugate Gradient Stabilized Method is oten used instead o the much aster Multigrid method. This results in large extra computational cost since the Poisson solver is the main bottleneck in numerical simulations. In the present work, a new pressure-correction algorithm is developed to improve the computational speed o the Poisson solver in two-phase low simulations. In the pressurecorrection step, the mass conservation equation or VoF equation is used to obtain the Poisson equation with constant coeicients. Using this method, the mass conservation equation is ulilled at every time step instead o the divergence-ree continuity equation. Since the Poisson equation with constant coeicients is much easier to solve than variable coeicients, the computational speed is much improved. The canonical test problems are solved to demonstrate the accuracy and perormance o the method. 2. Governing equations The governing equations or incompressible, immiscible, two-phase low are as ollows:

2 368 D. Kim and P. Moin ψ t + (ψu) = 0, (2.1) u p + (uu) = t ρ + 1 ρ (µ( u + T u)) + g + T ρ, (2.2) where ψ, u, p, µ, and g are the volume o raction o luid 1, velocity, density, pressure, viscosity, and gravitational acceleration, respectively. T is the surace tension orce, which acts only on the phase interace (x ) and vanishes at other locations; it is described as ollows: T = σκδ(x x )(n), (2.3) where σ, κ, δ, and n are the surace tension coeicient, surace curvature, delta unction, and surace normal vector, respectively. The temporal evolution o the phase interace can be predicted by solving the level-set equation: G + u G = 0, (2.4) t where the iso-surace o G = 0 deines the location o the interace whereas G > 0 and G < 0 represent two dierent phase regions, respectively. In the computational domain, G is set to be a signed distance unction to the interace such that G = 1. (2.5) The interace normal vector n and the interace curvature κ can be calculated rom G as n = G G, (2.6) κ = n. (2.7) Assuming ρ and µ are constant within each luid, the density and viscosity are deined as a unction o G: ρ = ψρ 1 + (1 ψ)ρ 2, (2.8) µ = ψµ 1 + (1 ψ)µ 2, (2.9) ψ = H(G), (2.10) where the subscripts 1 and 2 denote properties in luid 1 and 2, respectively, and H is the Heaviside unction. 3. Numerical methodology In the present study, the Reined Level Set Grid (RLSG) method (Herrmann 2008) coupled with a Lagrangian particle-tracking method (Apte et al. 2003) is used to capture the phase interace. For the low solver, an unstructured-grid inite-volume DNS/LES solver named CDP is employed (Ham & Iaccarino 2004; Mahesh et al. 2004). The RLSG uses an auxiliary Cartesian grid (G-grid) whereas the CDP uses a base structured or unstructured

3 Pressure-correction algorithm or ast two-phase simulations 369 grid or luid low. Because separate grids are employed in the RLSG and CDP computations, an eective parallel coupling strategy is required or inormation exchange. The present numerical technique utilizes a coupling inrastructure, called CHIMPS (Alonso et al. 2006), to perorm the parallel data exchange between CDP and RLSG. In the present study, the velocities u n+1 i 3.1. Level set transport equation is used to advance Eq. 3.1 as G n+3/2 G n+1/2 + u n+1 G n+1/2 i = 0. (3.1) The velocities are obtained rom CDP and an iterative procedure is needed to solve both G n+3/2 and u n+1. This procedure will be explained in section 3.5. The numerical algorithm or the level set transport equation is based on the RLSG method (Herrmann 2008). We use a ith-order WENO scheme or Hamilton-Jacobi equations (Jiang & Peng 2000) in conjunction with a local Lax-Friedrichs entropy correction (Shu and Osher, 1989). Time integration is perormed using a third-order TVD Runge-Kutta method (Shu & Osher 1989) Reinitialization Due to numerical errors, the solution o Eq. 3.1 will not preserve G as a signed distance unction, i.e., G = 1. Thus, a reinitialization procedure has to be applied to reset G such that G = 1. There are several dierent methods to reinitialize G. Here, we solve the perturbed Hamilton-Jacobi PDE to a steady state (Chang et al. 1995). G τ + (V o V (τ))( P + κ)g = 0 (3.2) where V 0 is the total liquid volume at the initial state, and V (τ) is the liquid mass at pseudo time τ. κ is the local curvature and P is a positive stabilizing constant. Unortunately, this correction moves the G = 0 iso-surace, which is inevitable in a PDE-based reinitialization. However, we can conserve the global mass up to any accuracy depending on the criterion used to achieve the steady state solution. The global mass conservation should be always guaranteed by solving the Poisson equation deduced rom the continuity-constraint RLSG-CDP coupling The volume raction ψ cv o the cell volume V cv is calculated using a step unction developed by Van Der Pill et al. (2005) as, ψ cv = 1 H(G)dv. (3.3) V cv V cv The integration is done through CHIMPS. The density and viscosity at the cell center o the cell volume V cv are deined as ρ cv = ψ cv ρ 1 + (1 ψ cv )ρ 2, (3.4) µ cv = ψ cv µ 1 + (1 ψ cv )µ 2. (3.5)

4 370 D. Kim and P. Moin The properties at the ace o the cell volume are computed by averaging two neighboring properties at the cell center. The velocities, u i, in the level set equation (Eq. 3.1) are interpolated rom CDP grid using tri-linear interpolation Computation o low ields CDP is based on the ractional-step method or collocated variables on unstructured grids, which is especially suitable or large-scale parallel simulations o turbulent lows in complex geometries (Ham & Iaccarino 2004; Mahesh et al. 2004). The spatial discretization is perormed using a low dissipation second-order central-dierence scheme while time integration is perormed using a ully implicit Crank-Nicholson method. An upwind-biased scheme is employed to discretize nonlinear terms only at interace cells (Kim et al. 2007). The pressure gradient and surace tension orces are predicted by a balanced orce algorithm, which guarantees a discrete balance between them (Francois et al. 2006; Herrmann 2008). In the ollowing, the new pressure-correction algorithm is presented, which satisies the mass conservation equation (Eq. 2.1). The numerical algorithm is ψ n+3/2 cv ψcv n+1/2 + ψn+1/2 cv u n+1 = 0, (3.6) V cv u un + u n+1/2 u n+1/2 + u n+1/2 i,nbr A = V cv g + F viscous, (3.7) 2 ρ n+1/2 cv u n+1 ρn+1/2 cv u = pn+1/2 + T n+1/2. (3.8) where A is the ace area, u the ace normal velocity, g the gravity, T n+1/2 the surace tension orce. The viscous term F viscous is ully implemented and solved in the lux orm with the viscosity at the cell ace calculated by the harmonic mean o two neighboring viscosities at the cell volumes. Ater solving Eq. 3.7 to obtain u, the cell ace normal velocities u are calculated, u = 1 2 (u + u i,nbr)n i,. (3.9) Taking the divergence o Eq. 3.8 results in the ollowing Poisson equation, 2 p n+1/2 x 2 = 1 i ( ρn+1/2 cv u n+1 ρn+1/2 cv u ) + T n+1/2. (3.10) By using Eq. 3.4 and Eq. 3.6, the mass lux terms in the right hand side become ρ n+1/2 cv u n+1 = (ρ 1 ρ 2 ) ψ n+1/2 cv u n+1 = (ρ 1 ρ 2 ) ψn+3/2 cv ψcv n+1/2, (3.11) ρ n+1/2 cv u = (ρ 1 ρ 2 )u ψ n+1/2 cv + (ρ 2 + (ρ 1 ρ 2 )ψ n+1/2 cv ) u

5 Pressure-correction algorithm or ast two-phase simulations 371 = (ρ 1 ρ 2 ) ψ cv ψn+1/2 cv + ρ n+1/2 u cv, (3.12) where ψcv is ψcv = 1 H(G )dv. (3.13) V cv V cv G is computed rom the level set equation as G G n+1/2 + u i Then, Eq is written as and 2 p n+1/2 cv x 2 i p n+1/2 n = 1 A = 1 {(ρ 1 ρ 2 ) ψn+3/2 cv ψcv {(ρ 1 ρ 2 ) ψn+3/2 cv ψ cv G n+1/2 = 0, (3.14) + ρ n+1/2 cv V cv + ρ n+1/2 cv in terms o the inite volume ormulation, where T n+1/2 The surace tension T n+1/2 at the ace is deined, T n+1/2 = σκ n+1/2 ( ) ψ n+1/2 u } n+1/2 T +, (3.15) u A } + T n+1/2 A (3.16) is the ace normal surace tension.. (3.17) Pressure can be solved rom Eq by using Multigrid method because it has constant coeicients. Then, the ace normal velocity components to satisy the mass conservation equation (Eq. 3.6) is obtained, where P is deined as u n+1 = u P, (3.18) P = 1 ρ n+1/2 ( p n+1/2 ) 1 ρ n+1/2 T n+1/2. (3.19) The pressure gradient at the cell center, P is calculated rom the pressure gradient at the cell ace, P, using the ace-area weighted least-squares method (Mahesh et al. 2004) by minimizing ǫ = (P n i, P ) 2 A. (3.20) The velocities at the cell center are inally corrected, u n+1 = u P. (3.21) In the present method, the velocities at the ace do not necessarily satisy the divergence ree equation ( u = 0). Instead, the mass conservation equation (3.6) is guaranteed by solving Poisson solver.

6 372 D. Kim and P. Moin Figure 1. Inviscid column at equilibrium Iteration procedure at each time step The iteration process at each time step is ollowed: 1. Intermediate velocities u i are obtained rom the momentum equation. 2. Estimate G n+3/2 by solving the level set transport equation using u i. 3. Integrate ψcv n+3/2, κ n+3/2 cv and transer to CDP using CHIMPS. 4. Calculate ρ n+3/2 cv, µ n+3/2 cv rom ψcv n+3/2. 5. The corrected velocities u n+1 i and pressure p n+1/2 are obtained by solving the Poisson equation. 6. Interpolate u n+1 i on the CDP-grid to G-grid. 7. Repeat the process 2 using updated u n+1 i until convergence. 4. Results 4.1. Equilibrium inviscid column with exact curvature The equilibrium inviscid column is considered in order to demonstrate the balanced orce algorithm (Francois et al. 2006). As explained in Francois et al. (2006), no spurious currents should be produced around a column at equilibrium when the balanced orce algorithm is used. Here, we applied the parameters used in Francois et al. (2006). A column o radius R = 2 is placed at the center o 8 8 domain (igure 1). The surace tension coeicient is σ = 73 and the theoretical pressure jump across the interace is p ex = The densities inside and outside the column are ρ 1 and ρ 2, respectively. The same equidistance Cartesian grid is used or the CDP-grid and G-grid, and the grid size is set to 0.2. Table 1. shows the errors in velocities and pressure ater a single time step o = 10 6 when the exact curvature is used. The errors are machine precision zero even or large density ratios. In the practical simulations, however, the non-zero spurious current is produced because o errors in the curvature calculation Long-time evolution o the viscous sphere In the case o section 4.1, the exact curvature produces machine-zero spurious currents. However, the numerical errors in curvature evaluation result in spurious currents which are small, but non-zero. The long-time behavior o the equilibrium sphere, thus, has been simulated to check errors in velocities ater long time steps. A sphere o diameter D = 0.25 is placed at the center o a unit sized Cartesian box with Neumann boundary

7 Pressure-correction algorithm or ast two-phase simulations 373 ρ 1/ρ 2 L (u) L (p max p min p ex) Table 1. Errors in velocity and pressure ater single time step = 10 6 or dierent density ratios using the exact curvature. h, h G 1/128 1/160 1/192 Ca Ca (Renardy & Renardy) Table 2. Spurious current capillary numbers o the viscous sphere with those o Renardy & Renardy (2002). conditions at the sides. The surace tension coeicient is set to σ = 0.357, density and viscosity in both luids are ρ = 4 and µ = 1. The time step size is = Table 2 shows the maximum spurious current capillary number at t = compared with Renardy & Renardy (2002). Both methods exhibit good results Oscillation o column The oscillation period or the column without gravity has been perormed to veriy our method. The theoretical period in the linear regime is given by (Lamb 1945) as w 2 = n(n2 1)σ (ρ 1 + ρ 2 )R0 3. (4.1) A column o radius R 0 = 2 is placed at the center o domain and the parameters are ρ 1 = 1, ρ 2 = 1000, µ 1 = 0.01, µ 2 = , and σ = 1. The column is initially perturbed by a mode n = 2 perturbation with initial amplitude o A 0 = 0.01R 0. The same equidistance Cartesian grid is used or CDP-grid and G-grid. Figure 2 shows the error o the oscillation period with respect to the number o grids. The second order grid convergence is achieved as shown Broken dam problem The well-known broken problem is computed to compare with the experimental result o Martin & Moyce (1952). A water column with a = 0.06m wide and 2a = 0.12m high is initially located at rest. The column suddenly breaks down under the gravity g = 9.81m/s 2. No slip conditions are applied or all side boundaries. The Cartesian grid o = m is used here. Figure 3 shows the time history o the phase interace. In

8 374 D. Kim and P. Moin Figure 2. Error in oscillation period with respect to the grid size. Figure 3. Time history o the phase interace in the broken dam problem at the dierent nondimensional times t 2g/a=0, 2.0, 3.0, 5.0. order to compare with the experimental data (Martin & Moyce 1952), the ront position and the height o the column are presented in igure 4. It shows good agreement with the experimental data Computational perormance Since the algebraic multigrid method (AMG) can be used or the Poisson solver having constant coeicients, the present method shows much aster perormance than the previous method which uses the biconjugate gradient stabilized method (BCGSTAB) or the Poisson solver. Table 3 shows the computational costs measured in CPU seconds per time step on a tetrahedral mesh having 1 million control volumes. In the previous method using BCGSTAB, the level set advection takes only 0.06% o the total time. About 98.8%

9 Pressure-correction algorithm or ast two-phase simulations 375 Figure 4. (a) time history o the height o the column (b) time history o the ront position o the column. : present result; : experimental data (Martin & Moyce 1952) Previous method (BCGSTAB) Present method (AMG) G advection Reinitialization Momentum solver Poisson solver Total CPU time Table 3. Comparison o computational cost measured in CPU seconds per time step on a tetrahedral mesh o 1 millions. o the total time step is spent on solving the Poisson equation. However, in case o the present method using AMG, it is more than 10 times aster than the previous method even with several iterations. 5. Conclusions and uture work A new pressure correction algorithm is presented or ast and accurate two-phase simulations. A ractional step method is used to solve the Navier-Stokes equations, and the pressure is used to correct the velocity ield to satisy the mass conservation equation instead o divergence-ree velocity. This procedure yields the Poisson system having constant coeicients, which saves computational cost greatly. In the present method, the mass conservation equation is ulilled at every time step, but the incompressibility is not ully guaranteed due to the numerical errors. These errors can be eiciently reduced by reining the grid. The canonical test problems are solved to demonstrate the accuracy and much improved perormance o the method. The main disadvantage o the present method is that the incompressibility is not ulilled owing to the numerical errors. In the uture, we will ocus on reducing or eliminating the errors in incompressibility. Finally, the present method will be used to study air layer drag phenomenon in turbulent low using more than 500 millions grid points.

10 376 D. Kim and P. Moin Acknowledgments The work presented in this paper was supported by the Oice o Naval Research. REFERENCES Alonso, J. J., Hahn, S., Ham, F., Herrmann, M., Iaccarino, G., Kalitzin, G., LeGresley, P., Mattsson, K., Medic, G., Moin, P., Pitsch, H., Schlüter, J., Svärd, M., van der Weide, E., You, D. & Wu, X Chimps: A highperormance scalable module or multi-physics simulations. AIAA Paper Apte, S. V., Gorokhovski, M. & Moin, P Les o atomizing spray with stochastic modeling o secondary breakup. Int. J. Mult. Flow 29, Chang, Y. C., Hou, T. Y., Merriman, B. & Osher, S A level set ormulation o eulerian interace capturing methods or incompressible luid lows. Journal o Computational Physics 124, Francois, M. M., Cummins, S. J., Dendy, E. D., Kothe, D. B., Sicilian, J. M. & Williams, M. W A balanced-orce algorithm or continuous and sharp interacial surace tension models within a volume tracking ramework. Journal o Computational Physics 213, Ham, F. & Iaccarino, G Energy conservation in collocated discretization schemes on unstructured meshes. Annual Research Brie 2004, Herrmann, M A balanced orce reined level set grid method or two-phase lows on unstructured low solver grids. Journal o Computational Physics 227, Jiang, G.-S. & Peng, D Weighted eno schems or hamilton-jacobi equations. SIAM J. Sci. Comp. 21, Kim, D., Herrmann, M. & Moin, P Numerical simulation o high shear rate two-phase low with high density ratio. Bulletin o the American Physical Society. Lamb, H Hydrodynamics. Dover, New York. Mahesh, K., Constantinescu, G. & Moin, P A numerical method or large eddy simulation in complex geometries. Journal o Computational Physics 197, Martin, J. & Moyce, W An experimental study o the collapse o liquid columns on a rigid horizontal plane. Philos. Trans. R. Soc. A224, Renardy, Y. & Renardy, M Prost: a parabolic reconstruction o surace tension or the volume-o-luid method. Journal o Computational Physics 183, Shu, C.-W. & Osher, S Eicient implementation o essentially non-oscillatory shock-capturing schemes. Journal o Computational Physics 77, Sussman, M. & Puckett, E A coupled level set and volume-o-luid method or computing 3d and axisymmetric incompressible two-phase lows. Journal o Computational Physics 162, Van Der Pill, S. P., Segal, A. & Vuik, C A mass-conserving level-set method or modelling o multi-phase lows. Int. J. Numer. Meth. Fluids 47,