KUMAR D. Santhosh

School of Mechanical and Building Science, Vellore Institute of Technology, Vellore-632014, India

DASS Anoop K.

Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati-781039, India

DEWAN Anupam

Department of Applied Mechanics, Indian Institute of Technology Delhi Hauz Khas, New Delhi-110016, India, E-mail: adewan@am.iitd.ac.in

A MULTIGRID-ACCELERATED THREE-DIMENSIONAL TRANSIENTFLOW CODE AND ITS APPLICATION TO A NEW TEST PROBLEM*

KUMAR D. Santhosh

School of Mechanical and Building Science, Vellore Institute of Technology, Vellore-632014, India

DASS Anoop K.

Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati-781039, India

DEWAN Anupam

Department of Applied Mechanics, Indian Institute of Technology Delhi Hauz Khas, New Delhi-110016, India, E-mail: adewan@am.iitd.ac.in

(Received August 26, 2009, Revised August 10, 2010)

A multigrid-assisted solver for the three-dimensional time-dependent incompressible Navier-Stokes equations on graded Cartesian meshes is developed. The spatial accuracy is third-order for the convective terms and fourth-order for the viscous terms, and a fractional-step strategy ensures second-order time accuracy. To achieve good time-wise efficiency a multigrid technique is used to solve the highly time-consuming pressure-Poisson equation that requires to be solved at every time step. The speed-up achieved by multigrid is shown in tabular form. The performance and accuracy of the code are first ascertained by computing the flow in a single-sided lid-driven cubic cavity with good grid-economy and comparing the results available in the literature. The code, thus validated, is then applied to a new test problem we propose and various transient and asymptotically obtained steady-state results are presented. Given the care taken to establish the credibility of the code and the good spatio-temporal accuracy of the discretization, these results are accurate and may be used for ascertaining the performance of any computational algorithm applied to this test problem.

incompressible flows, three-dimensional cavity flow, fractional-step method, multigrid method, Taylor-Gotler-Like (TGL) vortices, grid transformation

1. Introduction

The flow of an incompressible fluid in a lid-driven cubic cavity has been used extensively by numerical modelers as a test case for newly developed computational schemes. Its simple geometry and easily posed boundary conditions have made this flow a popular test case for computational schemes, giving rise to a need for an accurate database against which such schemes may be validated.

The transformed governing equations contain a larger number of terms than the original equations, which may result in some increase in the computational cost. Particularly, in the computation of transient flows, the pressure-Poisson equation that requires very accurate solution at every time step and consumes more than eighty percent of the total computational time needs special attention and an efficient algorithm for its numerical solution is highly desirable. The multigrid method which is arguably the best general convergence acceleration technique[13]is used in the present article to solve the pressure-Poisson equation numerically. Here the performance of multigrid method in the numerical solution of the pressure-Poisson equation on graded Cartesian meshes is examined in some details. Expectedly multigrid accelerates the convergence of the Gauss-Seidel iterations significantly, which in turn brings about a substantial reduction in the computational cost in the overall transient flow computation.

Thus it can be summarized that the main purpose of the present article is to develop a useful and efficient numerical tool for computing incompressible transient-viscous flows using multigrid method on graded Cartesian meshes and also to record our experiences on the application of multigrid to nonuniform meshes through the two problems studied, including the one-sided and two-sided 3-D lid-driven cavity problems. Other aims of this work are to carry out a comparison of the transient and steady-state results between 2-D and 3-D geometries and apply the present code to produce accurate transient and steady results for a new test case in the shape of the two-sided 3-D lid-driven cavity problem.

The article is organized in seven sections. Section 2 describes the governing equations in the physical plane (x, y, z) and computational plane (ξ, η, ζ). Section 3 gives a brief description of the fractional-step method employed and Section 4 provides some basics of the multigrid algorithm. Section 5 deals with the numerical method and Section 6 with the new test problem. Finally in Section 7 we list our observations and conclusions.

2. Governing equations

The 3-D unsteady, incompressible N-S equations in the primitive variable formulation and non-dimensional form can be written as

Here u, v and w denote the velocities along the x, y and z directions respectively, t the time and p the pressure. The solution of governing Eqs.(1) - (4) on the graded Cartesian meshes using the finite difference method makes it necessary to transform the equations from the physical to the computational planes. The details of the transformation are presented in the next subsection.

2.1 Transformation of governing equations

It may be noted that when the governing Eqs.(1)- (4) are transformed from the physical plane to a computational plane in which the mesh is uniform, the mathematical characteristics of the equations do not change. The transformed version of the governing equations is then solved in the computational plane by a higher order approximation and mapping of the results back to the physical plane is done. The transformation from the physical (x, y, z) plane to the computational (ξ, η, ζ) plane is of the form

Using the transformations the governing equations in the computational plane can now be written as

3. Fractional-step procedure

The four-step fractional-step method[10]employed in the present article is now described in some detail. In the first step the momentum Eq.(10) is solved for an auxiliary velocity fieldiu using the pressure from the previous time step, the convective terms and viscous terms are solved explicitly using the second-order Adams-Bashforth method.

where H is an operator representing the discretized convective and diffusive terms. In the second step Eq.(11) is solved to advance the auxillary velocity field ?iu toiu?which is, of course, not divergence-free.

In the third step, the incompressibility condition is enforced by solving the pressure Poisson Eq.(12), which is obtained by taking the divergence of Eq.(13) satisfying continuity equation for the next time step.

Finally in the fourth step the velocity which satisfies the incompressibility condition is obtained by using the following correction step:

The Poisson equation in the physical plane is given by Eq.(12) in the tensor form. This equation in thetransformed plane, solved using the multigrid method at each time step to satisfy the divergence-free condition, may be written in the expanded form as

where σdenotes the source term with the form

The pressure Poisson equation (Eq.(12)) is derived by taking the divergence of the intermediate velocity field Eq.(13) denoted by “?” subjected to the continuity constraint for the next time step. For computing the transient flow accurately the pressure Poisson equation has to be solved accurately at every time step. As is opposed to the solution of the transient momentum Eqs.(7) - (9), which is a relatively simple matter with an explicit technique, the solution of the pressure Poisson Eq.(14) is a nontrivial task that consumes almost 90% of the total CPU time required to update the primitive variables over one time step. For an efficient computation of transient flow using the pressure-velocity formulation it is therefore imperative that this equation is solved with an algorithm involving low computational cost. Here we use a four-level V-cycle multigrid method to solve the pressure Poisson equation.

4. Description of multigrid algorithm

The multigrid method has been chosen to accelerate the convergence rate, as it is highly suitable for elliptic equations such as the Poisson equation. There are two broad categories of multigrid methods: (1) Correction Schemes (CS) and (2) Full Approximation Schemes (FAS). The correction scheme is applicable to linear systems only. It involves corrections to the solution on coarse grids, which are eventually added to the fine-grid solution. As the Poisson equation is a linear equation, the correction scheme is used in the present work.

4.1 Linear two-grid algorithm

A two-grid algorithm for linear problems consists of smoothing on the fine grid, approximation of the required correction on the coarse grid, prolongation of the coarse grid correction to the fine grid and again smoothing on the fine grid. Let the equation to be solved iswhere S denotes the source term. This equation after discretization by the finite difference method can be written as L(p)i,j=Si,j, where L denotes the Laplacian operator. A two-grid correction scheme is described below:

(1) Perform n iterations on a fine grid, L(p)i,j=Si,j.

(2) Compute the residual Rfi,j=L(p)i,j?Si,j

and store at each point. This residual is restricted to the next coarser grid so that their lower frequency error components can be smoothened and the restricted residual is denoted by IcRf.

fi,j

(3) The correction equation, given by

is iterated few times on the coarser grid using zero as the initial guess while keeping the residual fixed at each grid point. The solution Δpi,jrepresents the correction to the fine-grid solution.

(4) The corrections obtained on the coarsest grid are prolongated on to the next finer grid and denoted by Ifc(Δp)ci,j

(5) Correct the fine grid approximations using pnewi,j=poldi,j+Ifc(Δp)ci,j. It is necessary to repeat the above two-level cycle until desired convergence is achieved.

5. Numerical method

The governing equations are discretized in space, using a finite difference formulation on a staggered, Cartesian grid, where velocities and pressure are calculated at different locations. The main advantage of the staggered grid arrangement is the strong coupling between the pressure and velocities without requiring special interpolation techniques. This helps avoid convergence problems and oscillations in pressure fields. Another advantage of using the staggered grid for incompressible flows is that the pressure boundary conditions are not required when the momentum equations are evaluated. For the pressure Poisson equation, the pressure gradient normal to the walls is assumed zero, i.e.,

The convective terms of the momentum equations are approximated with Kuwahara’s third-order upwind scheme[14](Eq.(17)) and viscous terms are discretized with a fourth-order central-difference scheme (Eq.(18))as given below:

6. Computations, results and discussions

The code is based on the fractional-step method for the time integration of 3-D N-S equations. The performance of the code is already evaluated by the same authors in regard to time-wise efficiency and accuracy[15]. The results show that the 2-D code is capable of capturing transient results accurately and the methods used to develop the code seems to be sound. Initially the transient 3-D incompressible laminar lid-driven flow in a cubic cavity is computed through the developed code for code validation and to show the performance of multigrid.

In the present study a 4-level V-cycle multigrid shown in Fig.1(a) has been used in all the computations to solve the pressure Poisson equation and the number of sweeps used in various grid-levels is given inside the corresponding circles. At Re=1000 on a 65 × 65 × 65 grid, it is observed that using four levels is good enough and a further increase of level produces no time-wise gain. To gain an insight into the convergence behavior after 100 time steps, at t=0.1 a plot between the pressure error and number of iterations required to solve the pressure Poisson equation is shown in Fig.1(b). Before examining overall quantitative performance data, it is instructive to consider the convergence histories in Fig.1(b) obtained with the single grid and with the sequences of 2- and 4-levels. Expectedly the convergence curves relating to multigrid are considerably steeper than those relating to the single grid. The ratio of the computational effort in work units for the single grid to that for the multigrid may be termed speed-up. As the convergence limit of error is lowered speed-up is seen to increase. This is because on a single grid, error-reduction rate becomes smaller and smaller as the iteration progresses whereas multigrid maintains more or less the same error-reduction rate throughout. The speed-up achieved by the 4-level multigrid is represented inTable 1. The CPU times of multigrid and single-grid computations to reach t=0.1 and t=0.5 for Re=1000 are shown in Table 1. For the same fall of residual, time gain by multigrid is impressive. Figure 2 shows the steady-state velocity profile of the u-component on the vertical centreline at the plane z=0.5 at Re=1000. The results show an excellent match with those of Ku etal.[4]. The 2-D results are also shown for comparison.

The results presented in the preceding section show the accuracy and the capability of the present code to accurately compute transient viscous flows. After having thus gained confidence in the code, it is then applied to compute a hitherto unexplored flow configuration. The problem consists of a cubic cavity filled with a stationary fluid that is set into motion by the sudden simultaneous movement of the top and the bottom walls from the left to right. The physical boundary conditions of the geometry are specified for t≥0 as

(1) At x=0 and 1, u=v=w=0.

(2) At z=0 and 1, u=v=w=0.

(3) At y=0, u=1, v=0, w=0.

(4) At y=1, u=1, v=0, w=0.

The reason for selecting this flow configuration for numerical study is that this geometry involves interesting flow features including gradual development of a free-shear layer and associated off-corner secondary vortices. The use of a grid with appropriate clustering and the use of multigrid suggest that the present computations combine accuracy, space-grid economy and time-wise efficiency. As has already been mentioned the flow in a cubic cavity with two-sided parallel wall movement has not been examined so far.

7. Conclusion

The present work deals with the development and application of a multigrid-accelerated 3-D code that accurately and efficiently computes incompressible transient viscous flows especially in geometries that allow the graded Cartesian meshes to be used conveniently to resolve the sharp gradients in shear layers by locally clustering the grid there. A fractional-step method is used to obtain second-order temporal accuracy and spatial discretizations of the convective and viscous terms are of third-and fourth-order spatial accuracy, respectively. The time-consuming process of computation of the Poisson equation at every time step is accelerated by using a V-cycle multigrid approach. For accurate implementation of the multigrid method, the finite difference discretization and grid transformation of the graded Cartesian meshes are employed. First, computations of transient and steady-state flows are carried out for the single-sided lid-driven cubical cavity. After having used the code to compute the flow in the single-sided cubical cavity to establish its credibility, it is then used to compute transient and steady-state flows in a hitherto unexplored configuration, namely, the two-sided cubical cavity flow induced by the motion of two facing walls in the same direction. The grid here is clustered not only near the walls but also in the free-shear layer midway between the moving walls to obtain a good grid-economy. It is observed that for the nonfacing two-sided cubic cavity, flow becomes periodic at Re=540. For the facing wall two-sided cubic cavity, however, steady solutions are obtained much beyond this Reynolds number and steady and transient results are presented here also for such values of Re. Various observations are made about this untested flow configuration. As all the steady-state results are grid independent, the transient results are time-step independent produced on the same grids and the accuracy of the spatio-temporal discretization is sufficiently high, all the steady and transient results reported for this new flow configuration are likely to be highly reliable and accurate. As has already been mentioned the results are also produced with good grid economy and time-wise efficiency. These results can thus serve as a means of verifying the performance of various algorithms that can be applied to the new test problem we propose. The code developed for the present transient computations can also be effectively used for various other 3-D rectangular geometries. It will, therefore, not be out of place to mention here that the code may have the potential of being gainfully used for LES and DNS in those geometries that allow effective use of the graded Cartesian meshes.

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10.1016/S1001-6058(09)60124-4

* Biography: KUMAR D. Santhosh (1979-), Male, Ph. D., Associate Professor