COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. O˜ nate and E. Dvorkin (Eds.) c
CIMNE, Barcelona, Spain 1998
ALGORITHMS FOR PRESSURE CORRECTION IN THE DRIVEN CAVITY PROBLEM Rodrigo B. Platte ∗ , Haroldo F. de Campos Velho ∗∗ , Julio C. Ruiz Claeyssen ∗ and Elba O. Bravo ∗ ∗
Instituto de Matem´atica/PROMEC Universidade Federal do Rio Grande do Sul (UFRGS) Porto Alegre (RS) - BRASIL e-mail: [rbplatte, julio]@mat.ufrgs.bf and
[email protected] ∗∗
Laborat´orio Associado de Computa¸ca˜o e Matem´atica Aplicada (LAC) Instituto Nacional de Pesquisas Espaciais (INPE), Caixa Postal 515 12201-970 – S˜ao Jos´e dos Campos (SP) – BRASIL e-mail:
[email protected]
Key words: Pressure Poisson equation, incompressible fluid, driven cavity problem. Abstract. The driven cavity problem has been extensively used as a benchmark problem to evaluate numerical techniques that solve the Navier-Stokes equations. In this work, the incompressible fluid flow is integrated by calculating the velocity fields and computing the pressure field in each time step. In order to avoid approximation errors, that imply a lack of incompressibility, a dilatation term is incorporated to the Poisson equation of the pressure. The goal of this paper is twofold. The first purpose is to examinate the performance of the three different methods for the pressure correction: the Claeyssen-Bravo algorithm, the iterative scheme given in C. Hirsch (1990), and the steady state solution of the nonhomogeneous diffusion equation. The second goal is to perform numerical experimentation with deep and shallow cavities.
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Rodrigo B. Platte, Haroldo F. de Campos Velho, Julio C. Ruiz Claeyssen and Elba O. Bravo
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Introduction
The laminar incompressible flow in a cavity, whose upper wall continuosly moves with uniform velocity parallel to its plane, has been extensively used as a test problem to evaluate the numerical techniques. In this work the Navier-Stokes equations are solved for a viscous, isothermic and incompressible fluid in a rectangular cavity. The numerical solution consists on getting the velocity fields by using the Adams-Bashforth method for the time integration and by means of the central finite differences for the spatial discretization in a staggered grid. The Poisson equation for the pressure with Neumann boundary conditions is obtained following Gresho and Sani [7]. In an incompressible flow special attention must be given to the solution of this equation. The correction terms are inserted in the discretized Poisson equation for avoiding the lack of incompressibility. In order to solve this equation three schemes are used: the iterative Gauss-Seidel method, the Claeyssen-Bravo algorithm, which updates the pressure in only one step, and the asymptotic solution for the nonhomogeneous diffusion equation. The numerical results are shown for different Reynolds numbers. It is also analysed the influence of the diffusivity parameter in the final result.
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The Driven Cavity Problem
The Navier-Stokes equations are considered in its primitive variables in the bi-dimensional cavity ∂u + u.∇u ∂t ∇.u u(x, 0) u
= −∇p + ν∇2 u , t > 0 and x ∈ Ω
(1)
= 0 , (x, t) ∈ Ω × [0, ∞) = u0 (x) , x at Ω = Ω ⊕ Γ = uΓ (x, t) at Γ = ∂Ω ,
(2) (3) (4)
with the following Poisson equation for the pressure [3] ∇2 p = ∇.(ν∇2 u − u.∇u) = ∇.(u.∇u)
(5)
associated to the Neumann boundary condition [7] ∂p ∂n = ν∇2 un − ( + u.∇un ) at Γ for t ≥ 0 ∂n ∂t
(6)
where un = u.n is the normal component of the velocity field u = ux i + vy j = u i + v j = (u, v). 2
Rodrigo B. Platte, Haroldo F. de Campos Velho, Julio C. Ruiz Claeyssen and Elba O. Bravo
The goal is to determine the induced flow by shear movement of the upper wall, which is moving with uniform horizontal velocity uT = 1. The other walls remain fixed. Therefore, the boundary conditions for the tangential velocity are given by u(x, 0, t) = v(0, y, t) = v(Lx , y, t) = 0 ,
u(x, Ly , t) = uT = 1 ;
and the boundary conditions for the normal velocity u(0, y, t) = u(Lx , y, t) = v(x, 0, t) = v(x, Ly , t) = 0 where Lx and Ly are the length and the height of the cavity, respectively. In addition, it is assumed that, at the initial time, the fluid is in rest u0 (x, y) = v0 (x, y) = 0 .
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Numerical Integration
In order to solve Eq. (1), some numerical schemes were tested and compared in [11]. For the time integration it was used the explicit Euler method, the second order AdamsBashforth method, the upwind method with centered differences of first and second orders, and an Euler-Lagrange (or semi-Lagrangian) method. For convection dominant flow the Euler-Lagrange scheme presented the lower numerical diffusion, being however slower than other methods. The Adams-Bashforth method for time integration associated to the central differences for the space derivative showed good results and smaller restrictions as to the numerical stability than the explicit Euler method. Therefore the Adams-Bashforth scheme was chosen for the integration of Eq. (1), that is: ∆t uk+1 = uk + [3F(uk ) − F(uk−1 ) + 3∇pk − ∇pk−1] (7) 2 where F(uk ) is the discretization of the convective and diffusive terms from (1) by central differences of second order. This scheme is implemented with staggered grid in [8] and ∇pk is also approximated by central differences.
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Solving the Pressure Poisson Equation
The simple discretization of the pressure in Eq. (5) produces a cumulative error due to the approximation errors, which implies in a leakage of the incompressibility leading to instabilities in the simulation. These errors can be avoided by adding corrective terms in Eq. (5). This is done in [9] where it is obtained the corrective equations for the pressure on
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Rodrigo B. Platte, Haroldo F. de Campos Velho, Julio C. Ruiz Claeyssen and Elba O. Bravo
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Figure 1: Normalized velocity fields: (a) Re = 100; (b) Re = 1000; (c) Re = 4000.
the Adams-Bashforth approach. For Eq. (7) the following approximation for the pressure equation can be assumed ∇2 p k = −
2∇.uk 1 − ∇.[3F(uk ) − F(uk−1 ) − ∇pk−1 ]. 3∆t 3
(8)
Applying the central finite difference operator of second order in Eq. (8) together with boundary conditions for the pressure given by Eq. (6), which is discretized in such a way to satisfy the compatibility condition [1], the following linear algebraic equation system is obtained Ap = b (9) where b represents the right hand side of Eq. (8) and the matrix A is a singular matrix [2]. Despite the singularity of matrix A, it is possible to obtain a solution for the system by using the Gauss-Seidel method [5]. Another technique to solve Eq. (9) was proposed by Claeyssen and Bravo [3] and [4], and is refered as the Claeyssen-Bravo algorithm (C-B), where the pressure is initialized by using the singular value decomposition, and in each time step the pressure is updated in an explicit and direct way in only one step, as follows: 1 k+1 h2 k+1 k k pk+1 = + p + p + p ] − [p H(u, p) i,j i,j−1 i+1,j i,j+1 4 i−1,j 4
(10)
where h = ∆x = ∆y and H(u, p) = − 23 (∇.uk /∆t) − 13 ∇.[3F(uk ) − F(uk−1 ) − ∇pk−1]. This work presents an alternative way in order to update the pressure, differently from the above mentioned schemes. The pressure field is initialized as in the C-B scheme, and 4
Rodrigo B. Platte, Haroldo F. de Campos Velho, Julio C. Ruiz Claeyssen and Elba O. Bravo
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Figure 2: Normalized velocity fields: (a) C-B algorithm; (b) diffusion equation method; (c) iterative method.
for each time step the pressure is obtained by using the following diffusion equation 1 ∂p = ∇2 p − H(u, p), γ ∂t
(11)
where γ represents a diffusivity. It is important to note that when γ → ∞ the Poisson equation for the pressure (8) is recovered. However, the parameter γ can not have an arbitrary magnitude, since it leads to wrong results. This parameter was numerically estimated and its influence in the numerical integration will be discussed in the next section. The numerical scheme used to solve Eq. (11) is the Alternating Direction Implicit (ADI) method.
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Numerical Experiments
The numerical tests were carried out for the bi-dimensional cavity. The velocity fields obtained for Re = 100, Re = 1000, and Re = 4000 flows are shown in figure 1. The velocity fields displayed in this figure express the steady state for the flow. Particularly in this case, the three methods for calculating the pressure yielded identical results. The difference occurs in the transient state of the flow. This can be seen in figure 2, where it is shown a simulation with Re = 1000, in the time t = 5, for a grid with 80 × 80 points. The behaviour of C-B algorithm was studied in [9] and [11], and these works show that ∂p this method presents an error of order O( ∆t ). From this analysis and from numerical h2 ∂t experiments, it was verified that this scheme leads to oscillations when computing the pressure. For lower Reynolds numbers the flow presents initial oscillations that decay 5
Rodrigo B. Platte, Haroldo F. de Campos Velho, Julio C. Ruiz Claeyssen and Elba O. Bravo
Table 1: Parameters used in the simulations. no 1 2 3 4
h 0.02 0.02 0.0125 0.0125
Re = 100 ∆t 0.005 0.005 0.005 0.005
γ 0.08 0.09 0.03 0.04
no 5 6 7 8
Re = 400 ∆t 0.005 0.005 0.005 0.005
h 0.02 0.02 0.0125 0.0125
γ 0.08 0.09 0.03 0.04
no 9 10 11 12
h 0.02 0.02 0.01 0.01
Re = 1000 ∆t 0.01 0.01 0.007 0.007
γ 0.04 0.05 0.014 0.02
fastly. For dominant convective flow (high Re) the oscillations present smaller amplitudes, but they decay slowly. A similar behaviour is found when the pressure is computed by the diffusion equation method, which depends on γ. For a better understanding of the differences between these methods, it is presented a plot of the pressure versus time at the central point in the square cavity for the flow with Re = 1000 in figure 3. It can observed that the diffusion equation method has closer results with the iterative method than the C-B algorithm. In this simulation, it was used a grid of 80 × 80 points, time-discretization ∆t = 0.005, and γ = 0.03 for the diffusion equation method.
Figure 3: Pessure at the center of the square cavity (Re = 1000).
The choice of γ has fundamental importance to obtain the pressure field by the diffusion equation method. Figure 4 shows the problems that may arise when using values of γ that are too much low or too much high. It can be seen in Eq. (11) that γ shall be as high as possible. However, special precaution must be taken against instability when calculating the pressure field. It has been observed in numerical simulations that: γ ∝ h2 /∆t, what led to the following rule in order to guarantee stable results: γ≤ 6
h2 ∆t
(12)
Rodrigo B. Platte, Haroldo F. de Campos Velho, Julio C. Ruiz Claeyssen and Elba O. Bravo
The Reynolds number does not have influence in the estimation of the parameter γ. In order to illustrate this non dependence, the results for Re = 400 are shown in figure 5.
Figure 4: Pressure in the cavity center: (a) γ = 0.001; (b) γ = 0.037.
Besides the presented simulations, the expression (12) was verified through simulations with the parameters shown in Table 1. In tests 1, 3, 5, 7 and 9 the results presented small oscillations, however in tests 2, 4, 6, 8 and 10 the results were numerically unstable.
5.1
Deep and Shallow Cavities
An example where the C-B algorithm and the diffusion equation method show good performance, that is, where they can be applied with advantages over the iterative method, is the study of the stationary flow in deep and shallow cavities. The aspect ratio of the deep cavity is defined as A = width . Therefore, deeper cavities have A > 1 and shallow cavities A < 1. Numerical results were obtained for flows with different aspect ratios and different Reynolds numbers. The deep and shallow cavities require a higher computational effort than a square cavity, because a higher number of points in the grid is necessary. In shallow cavities it was noted lower scale phenomena than with square cavities for the same Reynolds number. For deeper cavities, it can be seen low energy regions appearing in the bottom of the cavity, making the convergence to the steady state condition very slow. Figure 6 presents results for cavities with A = 1.5 and A = 0.666 . . .. These results were obtained with the C-B algorithm in a grid with 100 × 150 and 150 × 100 points, respectively, and Re = 4000. It is observed that the diameter of the bigger vortice (principal vortice) is characterized by the smaller dimension of the cavity. The number 7
Rodrigo B. Platte, Haroldo F. de Campos Velho, Julio C. Ruiz Claeyssen and Elba O. Bravo
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Figure 5: Pressure in the cavity center: grid 50×50, ∆t = 0.01, Re = 400 and (a) γ = 0.04; (b) γ = 0.05.
of the vortices inside the domain tend to decrease for more viscous flow [10]. It was also observed for higher Re that the asymptotic state can become oscillatory [6].
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Final Comments
This work compares three numerical techniques to solve the Poisson equation for the pressure. The scheme that requires the lowest computational effort is the C-B algorithm. The C-B algorithm and the diffusion equation method presented oscillations, but the latter method presented a lower level. However, the diffusion equation method requires a special attention in the choice of parameter γ. The expression (12) presents a constraint for this parameter based on the space and time discretization, which was obtained by empirical experiments. The last technique used, the Gauss-Seidel iterative technique, did not show oscillations, but requires the highest computational effort. The influence of the geometry over the flow was shown in this study with deep and shallow cavities. For deep cavities it is always observed low energy regions, and the number of the primary vortices is determined by the smaller geometrical dimension. This characteristic scale determines the diameter of the primary vortices. For shallow cavities the number of vortices depends on the characteristic geometrical scale and the Reynolds number. For instance, for the aspect ratio A = 0.25 and Re = 100 the flow presented only one vortice, and for A = 0.25 and Re = 1000 the flow presents two vortices [10]. The simulations were performed with Re < 4000 to assure that eventual oscillations in the pressure or velocity fields are due to numerical effects and not to physical phenomena, 8
Rodrigo B. Platte, Haroldo F. de Campos Velho, Julio C. Ruiz Claeyssen and Elba O. Bravo
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Figure 6: Normalized velocity fields: (a) A = 1.5; (b) A = 0.666 . . .
which can happen for higher Reynolds [12].
Acknowledgments The first author acknowledges the CNPq (Brazilian Council for Scientific and Technological Development) for the financial support and the Laboratory for Computing and Applied Mathematics of the Brazilian Institute for Space Research (LAC-INPE) for the 4-months technical visit, during which this work was partially done.
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Rodrigo B. Platte, Haroldo F. de Campos Velho, Julio C. Ruiz Claeyssen and Elba O. Bravo
can Conference on Computational Methods for Engineering, Curitiba (PR), Brazil, pp. 592-600. [3] E.O. BRAVO (1997):. “Incompressible Flow with Neumann Pressure Condition: Simulation and Matrix Formulation in Primitive Variables”, PhD Thesis, PROMEC-UFRGS, Porto Alegre, Brasil. [4] E.O. BRAVO, J.C.R. CLAEYSSEN, A. CASTRO (1997): “A Modified VelocityPressure Algorithm with Neumann Pressure Conditions for Incompressible Flow on a Staggered Grid”, Proceedings II Italian - Latinamerican Conference on Applied and Industrial Mathematics (ITLA’97), Roma, Italy, pp. 31. [5] V. CASULLI (1988): “Eulerian–Lagrangian Methods for the Navier-Stokes Equations at High Reynolds Number”. International Journal for Numerical Methods in Fluids, New York, vol.8, pp. 1349-1360. [6] J.W. GOODRICH, K. GUSTAFSON, K. HALASI (1990): “Hopf Bifurcation in the Driven Cavity”, Journal of Computational Physics, vol. 90, pp. 219–261. [7] P.M. GRESHO, R.L. SANI (1987): “On Pressure Boundary Conditions for the Incompressible Navier-Stokes Equations”, International Journal for Numerical Methods in Fluids, vol. 7, pp. 1111-1145. [8] C. HIRSCH (1990): “Numerical Computation of Internal and External Flows”, John Wiley & Sons Ltd.. [9] R.B. PLATTE, E.O. BRAVO, J.C.R. CLAEYSSEN, H.F. CAMPOS VELHO, (1997): “The Behaviour of the Velocity-pressure Algoritms in Incompressible Flow with Pressure Neumman Condition”, XVI Iberian Latin American Conference on Computational Methods for Engineering, Bras´ılia (DF), Brazil, vol. II, pp. 10051012, in portuguese. [10] R.B. PLATTE, H.F. CAMPOS VELHO, J.C.R. CLAEYSSEN, E.O. BRAVO (1997): “Numerical Experiments in Bidimensional Cavity Flow”, Technical Report, INPE, in portuguese (to be published). [11] R.B. PLATTE (1998): “The Numerical Study of the Bidimensional Cavity Flow”, Master Degree Thesis, Institute of Mathematics, UFRGS, Porto Alegre, Brasil (in portuguese). [12] M. POLIASHENKO, C.K. AIDUN (1995): “A Direct Method for Computation of Simple Bifurcation”, Journal of Computational Physics, vol. 121, pp. 246-260.
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