11-0492

7/27/2019 11-0492

1/10

Appl. Math. Mech. -Engl. Ed., 33(8), 9911000 (2012)DOI 10.1007/s10483-012-1600-6cShanghai University and Springer-Verlag

Berlin Heidelberg 2012

Applied Mathematicsand Mechanics(English Edition)

Calculation of cell face velocity of non-staggered grid system

Wang LI ( )1, Bo YU ( )1, Xin-ran WANG ()1,

Shu-yu SUN ()2

( 1. Beijing Key Laboratory of Urban Oil and Gas Distribution Technology,

China University of Petroleum, Beijing 102249, P. R. China;

2. Computational Transport Phenomena Laboratory, Division of Physical Science

and Engineering, King Abdullah University of Science and Technology,

Thuwal 23955-6900, Kingdom of Saudi Arabia)

Abstract In this paper, the cell face velocities in the discretization of the continu-ity equation, the momentum equation, and the scalar equation of a non-staggered gridsystem are calculated and discussed. Both the momentum interpolation and the linearinterpolation are adopted to evaluate the coefficients in the discretized momentum andscalar equations. Their performances are compared. When the linear interpolation is usedto calculate the coefficients, the mass residual term in the coefficients must be droppedto maintain the accuracy and convergence rate of the solution.

Key words collocated grid, staggered grid, momentum interpolation

Chinese Library Classification O302, O357.12010 Mathematics Subject Classification 65M12, 76D05

1 Introduction

The momentum interpolation method and its various modified versions for non-staggeredgrids are now being widely employed in the computational heat transfer [110], for which theunphysical checkerboard pressure can be prevented and the calculation coding can be mademore easily for a non-staggered grid than for a staggered grid, especially for an unstructuredgrid. The usually momentum interpolation is used to calculate the cell face velocity at anyoccasion. Date[1112] stated that the problem of the checkerboard prediction of pressure couldbe eliminated by interpolating the pressure-gradient in the nodal momentum equations whilethe cell face velocity was still evaluated by linear interpolation. Wang et al.[13] and Nie etal.[14] calculated the cell face velocity on a collocated grid system by utilizing the momentuminterpolation and the linear interpolation in the continuity equation and the momentum equa-tion, respectively, and they obtained satisfactory results. The use of the linear interpolation inthe momentum equation and the scalar equation can simplify the coding. But the difference

between the linear interpolation and the momentum interpolation employed in these equationshas not been clarified. In this study, we adopted both the momentum interpolation and the lin-

Received Nov. 28, 2011 / Revised Mar. 31, 2012Project supported by the National Natural Science Foundation of China (Nos. 51176204 and51134006)Corresponding author Bo YU, Professor, Ph. D., E-mail: [email protected]

7/27/2019 11-0492

2/10

992 Wang LI, Bo YU, Xin-ran WANG, and Shu-yu SUN

ear interpolation to evaluate the coefficients in the discretized momentum and scalar equations,and compare their performances on the numerical accuracy and convergence rate.

2 Discretization of governing equation

For simplicity, a two-dimensional general incompressible governing equation listed below is

considered to illustrate the numerical discretization of the transport equation,

(u)

x +

(v)

y =

x

x

+

y

y

+S, (1)

where is any dependent variable, u and v are the velocity components in the x- and y-directions, and, , andSrepresent the density, the diffusion coefficient, and the source term,respectively. A finite volume method is used to discretize the governing equation. IntegratingEq. (1) over a control volume as shown in Fig. 1, we have

ns

ew

(u)

x dxdy+

ew

ns

(v)

y dydx

= n

s e

w

x

xdxdy+

e

w n

s

y

ydydx+

n

s e

w

Sdxdy. (2)

Fig. 1 Non-staggered grid arrangement

The diffusion term is discretized by the central difference scheme, and the source term istreated by linearization. The discretized equation can be obtained as follows:

((u)e (u)w)y+ ((v)n (v)s)x

= e

(x)e(E P)

w(x)w

(P W)

y+ n

(y)n(N P)

s(y)s

(P S)

x+ (SC+ SPP)xy. (3)

7/27/2019 11-0492

3/10

Calculation of cell face velocity of non-staggered grid system 993

By rearranging the above equation, we obtain the following discretized algebraic equation:

APP =AEE+ AWW+ANN+ ASS+ bP, (4)

where

AE=ey

(x)e+ max((u)ey, 0),

AW =wy

(x)w+ max((u)wy, 0),

AN=nx

(y)n+ max((v)nx, 0),

AS=sx

(y)s+ max((v)sx, 0),

AP =AE+ AW+AN+ AS+ Ab SPxy,

Ab = (u)ey (u)wy+ (v)nx (v)sx,

bP =SCxy+b1,

b1 = max((u)ey, 0)(e P) + max((u)ey, 0)(e E)

max((u)wy, 0)(w P) + max((u)wy, 0)(w W)

max((v)nx, 0)(n P) + max((v)nx, 0)(n N)

max((v)sx, 0)(s P) + max((v)sx, 0)(s S).

Here, ue, uw, vn, and vs are the interface values which can be evaluated by the differenceschemes, e.g., the central difference scheme and the QUICK scheme [15]. b1 results from theadoption of the deferred-correction procedure[16]. For the momentum equation, the pressuregradient term is included in SC.

The continuity equation can be discretized as follows:

(u)ey (u)wy+ (v)nx (v)sx= 0. (5)

From Eqs. (4) and (5), we can clearly see that the cell face velocities (ue, uw, vn, and vs)are used in the continuity equation, in the cell face flow rate computation for the determinationof the coefficients in the discretization equation, and in the calculation for the mass residualAb in the coefficientAP. We can use both linear interpolation and momentum interpolation tocalculate the cell face velocity. The expressions ofue are

ue= f+e uE+ (1 f

+e )uP. (6)

The momentum interpolation is[1]

ue=

u

i

Aiui+bPe

(AP)e

uy(pEpP)

(AP)e, (7)

where

f+e = x

2(x)e,

1

(AP)e=f+e

1

(AP)E+ (1 f+e )

1

(AP)P,

i

Aiui+bP

AP

e

=f+e

i

Aiui+bP

AP

E

+ (1 f+e )

i

Aiui+bP

AP

P

.

7/27/2019 11-0492

4/10


It should be emphasized that if momentum interpolation is used in the continuity equationwhile linear interpolation is used to determine the coefficients of the discretization equation,due to the inconsistence, Ab is not zero.

All cell face velocities of the control volumes are often calculated by the momentum interpo-lation method when the pressure-velocity coupling computation is conducted under collocatedgrids[110]. Recently, Wang et al.[13] and Nie et al.[14] used linear interpolation to determine the

coefficients in the discretization equation. Apparently, the utilization of linear interpolation cansimplify coding. However, the effects of this treatment on the accuracy and convergence ratehave not been illustrated in these papers. To clarify the effects, both momentum interpolationand linear interpolation are utilized to evaluate the coefficients in the discretized momentumand scalar equations, and their performances are compared. For the comprehensive compar-ison, we design five practices named A, B, C, D, and E as shown in Table 1. In the designof practices, Ab is treated in two manners, i.e., remain and dropped. From Table 1, it is seenthat in all the practices, momentum interpolation is used in the continuity equation to preventthe unphysical pressure field, while either momentum interpolation or linear interpolation isemployed for the momentum equation or the scalar equation. Apparently, the mass residualterm (Ab) has great effects on the solution, especially for the computation with coarse meshes.

Table 1 Interface velocity interpolation method and disposal ofAb

Practice Continuity equation Momentum equation Scalar/energy equation Ab

A MI MI MI Remain

B MI MI LI Dropped

C MI LI LI Dropped

D MI MI LI Remain

E MI LI LI Remain

*MI represents momentum interpolation; LI represents linear interpolation

3 Results and discussion

We compare the performances of five practices (A, B, C, D, and E) in two computationalexamples. The first example is a lid-driven cavity flow[17], which is shown in Fig. 2. The wallsare held at a constant scalar = 100 and the dimensionless diffusion coefficient is set as 1 /Rewhere the Reynolds number Re is defined by Re = UlidL/. The other example is a mixedconvection example, whose schematic diagram is shown in Fig. 3. The side length of the cavityisL= 1.0 m, while the left and right temperatures are 1 and 0, respectively. Both the bottom

Fig. 2 Lid-driven cavity flow Fig. 3 Schematic diagram of mixed convection

7/27/2019 11-0492

5/10


and the top walls are thernmally insulated. The non-slip boundary condition is employed for allthe walls. The lid velocity is 1.0 m/s. The Reynolds number Re, the Prandtl number P r, andthe Grashof number Gr are set as 103, 0.71, and 106, respectively. The semi-implicit methodfor pressure-linked equations (SIMPLE) is used to couple the velocity and pressure, and theQUICK scheme is employed for the convective term in those computational examples.

In the first example, the calculations are carried out for Re = 100 and Re = 1 000, and two

sets of uniform grids, i.e., 13 13 and 41 41, are employed. Apparently, for the lid-drivencavity flow with the uniform scalar boundary condition, it is expected that the uniform scalarfield should be 100 by solving the scalar equation. Uniform fields can be obtained by practicesA, B, and C. Therefore, it is not necessary to show their scalar contours. Figures 47 show thescalar contours of practices D and E. It can be clearly seen that unphysical non-uniform fieldsare predicted by the two practices, and both the Reynolds number and the grid size have animportant effect on the scalar field. For the grid of 1313, the scalar field is further non-uniformwith larger Reynolds numbers and wider scalar distribution in a wider range. When the mesh isdenser, the scalar field becomes much uniform, but it still greatly deviates the real field. Theseunphysical predictions are due to the non-zero mass residual term Ab. For practices A, B, andC, Ab is always equal to zero, either automatically for practice A or compulsively for practicesB and C. Therefore, reasonable results are obtained by the three practices.

Fig. 4 Contours of for practice D withRe= 100

Fig. 5 Contours of for practice D withRe= 1000

7/27/2019 11-0492

6/10


Fig. 6 Contours of for practice E withRe= 100

Fig. 7 Contours of for practice E withRe= 1 000

Moreover, Figs. 8 and 9 show the scalar profiles along the vertical centerline of the squarecavity. For different Reynolds numbers and different grid numbers, the scalar values for practicesA, B, and C are all exactly 100, but the scalar values for practices D and E have large oscillations.The oscillation produced by practice E is more serious because the mass residual term in thiscase is not equal to zero in the discrtized coefficients for both the momentum equation and thescalar equation.

The convergence process of the continuity equation, the U momentum equation, the Vmomentum equation, and the scalar equation are shown in Figs. 1013, in whichRm, RU, RV,and R present the residuals of the continuity equation, the Umomentum equation, the V

momentum equation, and the scalar equation, respectively. As it can be seen, the convergencerates of practices A, B, and C are almost the same and are much faster than those of practicesD and E, especially for larger Reynolds numbers. This indicates that Ab affects not only thesolution but also its convergence rate. WhenAb = 0, either automatically or compulsively, theconvergence rate is similar regardless of the interpolation used in the momentum and scalarequations.

In the second example, four sets of uniform grids, i.e., 12 12, 2222, 4242, and 6262,

7/27/2019 11-0492

7/10


are employed for the calculation. When we calculate the example by using the 12 12 mesh,practices A, B, and C are convergent with the relaxation factor of 0.5, practice D is convergentwhen the relaxation factor is below 0.1, while practice E is divergent in any relaxation factor.

Fig. 8 profiles along vertical cavity centerline for different practices with grid of 13 13

Fig. 9 profiles along vertical cavity centerline for different practices with grid of 41 41

Fig. 10 Convergence curves of continuity equation for different practices with grid of 13 13

7/27/2019 11-0492

8/10


Fig. 11 Convergence curves ofUmomentum equation for different practices with grid of 13 13

Fig. 12 Convergence curves ofVmomentum equation for different practices with grid of 13 13

Fig. 13 Convergence curves of scalar equation for different practices with grid of 13 13

When we calculate the example by using the grids of 2222, 4242, and 6262, practices A,B, C, D, and E are all convergent with the relaxation factor of 0.5. Figure 14 shows the Nusseltnumber on the left wall of practices A, B, C, D, and E, from which it can be observed thatthe Nusselt number is reasonable for practices A, B, and C while is unreasonable for practicesD and E. The only difference between practices D, E and practices B, C is that Ab does notcompulsively set to be zero. Therefore, the unreasonable solution must be caused by Ab.

7/27/2019 11-0492

9/10


Fig. 14 Distribution of Nusselt number on left wall

4 Conclusions

In the discretization of governing equations, the calculation of the cell face velocity is oftenencountered. In this paper, its treatment in various situations is discussed on a collocated

grid system. It is found that momentum interpolation should be always used to calculate thecell face velocity in the discretized continuity equation to prevent the checkerboard pressurefield, while both momentum interpolation and linear interpolation can be used in the otherdiscretization equations. When linear interpolation is utilized to obtain the cell face velocity,employed to calculate the coefficients of other equations, mass conservation cannot be satisfiedwhich results in an additional mass residual term in the coefficients. Due to the effect of themass residual term, the convergence rate decreases and the solutions accuracy deteriorates. Ifthe mass residual term in the coefficients is enforced to be zero, both the convergence rate andthe solution accuracy are the same as those obtained by the use of momentum interpolation.In a word, on a collocated grid system, linear interpolation can be used to calculate the cellface velocity in the momentum equation and other scalar equations on the condition that themass residual term is forced to be zero. This treatment simplifies coding, and maintains thesolution accuracy and convergence rate.

References

[1] Rhie, C. M. and Chow, W. L. A numerical study of the turbulent flow past an isolated airfoil withtrailing edge separation. AIAA Journal, 21(11), 15251552 (1983)

[2] Peric, M.A Finite Volume Method for the Prediction of Three-Dimensional Fluid Flow in ComplexDucts, Ph. D. dissertation, University of London, UK (1985)

[3] Majumdar, S. Development of a Finite-Volume Procedure for Prediction of Fluid Flow Problemswith Complex Irregular Boundaries, SFB-210/T-29, University of Karlsruhe, Germany (1986)

[4] Peric, M., Kessler, R., and Scheuerer, G. Comparison of finite-volume numerical methods withstaggered and collocated grids. Computers and Fluids, 16(4), 389403 (1988)

[5] Majumdar, S. Role of underrelaxation in momentum interpolation for calculation of flow withnon-staggered grids. Numerical Heat Transfer, Part B, 13(1), 125132 (1988)

[6] Rahman, M. M., Miettinen, A., and Siikonen, T. Modified simple formulation on a collocated gridwith an assessment of the simplified QUICK scheme. Numerical Heat Transfer, Part B, 30(3),291314 (1996)

[7] Choi, S. K. Note on the use of momentum interpolation method for unsteady flows. NumericalHeat Transfer, Part A, 36(5), 545550 (1999)

[8] Barton, I. E. and Kirby, R. Finite difference scheme for the solution of fluid flow problems on non-staggered grids. International Journal of Numerical Methods in Fluids, 33(7), 939959 (2000)

7/27/2019 11-0492

10/10


[9] Yu, B., Kawaguchi, Y., Tao, W. Q., and Ozoe, H. Checkerboard pressure predictions due to theunder-relaxation factor and time step size for a nonstaggered grid with momentum interpolationmethod. Numerical Heat Transfer, Part B, 41(1), 8594 (2002)

[10] Yu, B., Tao, W. Q., Wei, J. J., Kawaguchi, Y., Tagawa, T., and Ozoe, H. Discussion on momentuminterpolation method for collocated grids of incompressible flow. Numerical Heat Transfer, PartB, 42(2), 141166 (2002)

[11] Date, A. W. Solution of Navier-Stokes equations on non-staggered at all speeds.InternationalJournal of Heat and Mass Transfer, 36(4), 19131922 (1993)

[12] Date, A. W. Complete pressure correction algorithm for solution of incompressible Navier-Stokesequations on a non-staggered grid. Numerical Heat Transfer, Part B, 29(4), 441458 (1996)

[13] Wang, Q. W., Wei, J. G., and Tao, W. Q. An improved numerical algorithm for solution ofconvective heat transfer problems on non-staggered grid system. Heat and Mass Transfer, 33(4),273288 (1998)

[14] Nie, J. H., Li, Z. Y., Wang, Q. W., and Tao, W. Q. A method for viscous incompressible flowswith simplified collocated grid system. Proceedings of Symposium on Energy and Engineering inthe21st Century, 1, 177183 (2000)

[15] Leonard, B. P. A stable and accurate convective modeling procedure based on quadratic upstreaminterpolation. Computer Methods in Applied Mechanics and Engineering, 19(1), 5998 (1979)

[16] Khosla, P. K. and Rubin, S. G. A diagonally dominant second-order accurate implicit scheme.

Computers and Fluids, 2(2), 207218 (1974)[17] Botella, O. and Peyret, R. Benchmark spectral results on the lid-driven cavity flow. Computers

and Fluids, 27(4), 421433 (1998)

Documents

11-0492