APPLICATIONS OF MULTI-GRID METHODS FOR

TRANSONIC FLOW CALCULATIONS

Wolfgang Schmidt Dornier GmbH D-7990 Friedrichshafen

Antony Jameson Princeton University Princeton, NY 08544

Abstract

Multiple grid methods are discussed on the basis of the Poisson equa-

tion. In the second part, routine-type application of a multi-grid

solver for the full potential equation in combination with a boundary

layer integral method is demonstrated. Finally, the importance of

multi-grid techniques for systems of partial differential equations

(Euler, Navier Stokes) is discussed and some first results are presen-

ted for a scheme presently under consideration.

Introduction

Very fast and accurate methods for transonic flow computations are

needed by design engineers to improve present day and to develop fu-

ture transport as well as military aircraft. Such methods permit

detailed transonic flow studies within a short time at low cost.

Multiple grid methods have been demonstrated to be very powerful for

non-elliptic two-dimensional, transonic flow problems by South and

Brandt I as well as Jameson 2. Since practical flow analysis implies

contour-conformal grid systems, which in general will be non-ortho-

gonal and stretched, fast flow solvers must be insensitive against

mesh spacing.

In order to understand the basic properties and the efficiency of

multi-grid methods, a study of such methods has been performed on the

basis of the Poisson equation. These numerical experiments indicated

the combination of multi-grid plus an ADI-scheme to be the most robust

procedure. Therefore the combination of such a solver for the full

potential equation as proposed by Jameson 2 is presented in the second

part of the present paper. Comparison with measurements is done after

inclusion of viscous effects by means of integral boundary layer meth-

ods.

600

While such potential flow solutions prove to be useful for flows with

moderate shock strength, the isentropic and irrotational flow assump-

tion does lead to significant errors in the upper transonic speed

range. Additional deficiencies occur in transonic lifting flow as

pointed out in Ref. 3 due to the Kutta condition for potential flow

theory. These problems can be overcome by solving either the time

dependent exact inviscid equations (Euler equations) or the time

dependent viscid equations (Navier Stokes equations). However, tech-

niques being developed for the full potential equation (quasi-linear

second order equation) cannot be simply applied for the Euler or

Navier Stokes equations, since they represent systems of first order

partial differential equations of hyperbolic nature in time. Since

standard solution methods for these equations require very large com-

puter time, efficient multi-grid solvers are highly desirable for

such equations. In the last part of the present paper two different

approaches for multi-grid schemes for the Euler equations are dis-

cussed. For the one similar for the Ni-scheme 4 first results are pre-

sented.

Multi-Grid Study for the Poisson Equation

The model case which will be considered is a test function

d I f = ci (x+y) sin(a I x) sin(b I y)

d 2 + c2(x+y) sin(a 2 x) sin(b 2 y)

(1)

which should be equal to the solution g of the two-dimensional

Poission equation

Lg = W (2)

where the operator L on the solution is the central second order dif-

ference approximation in each point (i,j)

(Lg)i,j = gi+1,j - 2gi,j + gi-l,j

+ (Ax/Ay) 2 (gi,j+1 - 2gi,j + gi,j-1 )

(3

and the right hand side is the same approximation of the test function

601

Wi,j = fi+1,j - 2fi,j + fi-l,j

+ (Ax/~y)2 (fi,j+1 - 2fi,j + fi,j-1 )

(4)

such that in the converged state

g = f

The multi-grid equation is formulated as follows. Let

Lh g - W = O (5)

be the equation with the grid width h. Then L h approximates the

linear differential operator L on the grid with a spacing proportio-

nal to the parameter h. Let U be the present estimate of g, and let V

be the required correction to U such that U+V satisfies (5). Then the

basis of the multi-grid method is to replace (5) and determine V by

h L2h V + I2h (LhU-W) = 0 (6)

where L2h is the same approximation to L on a grid in which the spac-

ing has been doubled, and I~h is an operator which transfers to each

grid point of the coarse grid the residual LhU-W of the coincident

point of the fine mesh. After the solution of Eq. (6) the approxima-

tion on the fine grid is updated by interpolating the correction cal-

culated on the coarse grid to the fine grid, so that U is replaced by

2h U new = U + I h V (7)

2h where I h is an interpolation operator.

Equation (6) can in turn be solved by introducing an approximation on

a yet coarser grid, so that a multiple sequence of grids may be used,

leading to a rapid solution procedure for two reasons. First, the

number of operations required for a relaxation sweep on one of the

coarse grids is much smaller than the number required on the fine

grid. Second, the rate of convergence is faster on a coarse grid, re-

flecting the fact that corrections can be propagated from one end of

the grid to the other in a smaller number of steps.

To extend this idea to nonlinear equations and to avoid a perturbation

form of the equation, Eq. (6)is reorganised by adding and subtracting

602

the current solution U to give

h L2h (U+V) - L2h U + 12h (LhU-W) = 0

or L2h U - W = 0 (8)

where U is the improved estimate of the solution to be determined

on the coarse grid, and W is an appropriately modified right-hand

side

h = L2h U - I2h (LhU-W) (9)

The updating formula (7) now becomes

2h (U-U) (10) U new = U + I h

2h As updating operator I h fourth order interpolation formulas are used

(third order at boundaries).

The success of the multiple grid method generally depends on the use

of a relaxation algorithm which rapidly reduces the high frequency

components of error on any given grid. To analyse the influence of

this smoothing algorithm, three different relaxation schemes were

tested:

- horizontal line relaxation (SLOR)

- alternating direction scheme AFI 5

(~ - 6 ) (~ - ~ ) 6g = ~ ? Lg (11)

where

r = (~x/Ay) 2, e is a parameter to be chosen, ~ is an over-

relaxation factor, and the residual Lg is calculated using the

result of the previous iteration

alternating direction scheme AF2 6

(~r6y- 6 2 ) (~ - 6y) 6g = ~ Lg (12)

using one-sided difference operators 6- and 6 + . Y Y

803

On an orthogonal mesh with equidistant spacing in Ax and Ay the com-

putational results indicate that horizontal line relaxation and AF2

work well in the multi-grid mode when

Ax < Ay (128 x 32 grid)

but not so well when

Ax > Ay (32 x 128 grid)

On an equally spaced grid (Ax = Ay) all three schemes worked, but AFt

was best. The average rate of reduction of residual per work unit

(measured as fine grid iteration) that could be achieved with AF1 in

multi-grid mode was

- 0.230 on a 128 x 128 grid

- 0.259 on a 64 x 64 grid

- 0.272 on a 32 x 32 grid

It is interesting that the observed convergence rate gets faster as

the grid gets finer. The result on the 128x128 grid represents a re-

duction from an initial average residual of 0.108.10 -I to a final

average residual of O.576-10 -13 in eleven multi-grid cycles (17.66

work units).

The average error If-gl was reduced from 0.57~3.10 -I to 0.554.10 -13

for a mean rate of reduction of O.162 per work unit. For comparison

the rate of convergence with line relaxation on a 64x64 grid (using

I grid) was 0.982 for the residual and 0.993 for the error. In multi-

grid mode line relaxation gave rates of 0.576 for the residual and

0.511 for the error.

An extensive study was made for the rate of convergence of the AFI

scheme in multigrid mode with different choices of the parameters

and ~. Fig. I shows curves of the rate of convergence for the 128x

128 grid plotted against ~ for different values of ~. Optimal con-

vergence rates can be obtained for certain combinations of ~, ~ only,

however these combinations do depend on grid spacing and the optimum

is very narrow. In production-type codes automatic adjustment for

optimal e-~ combination would be highly desirable, however the logics

for that are not understood yet.

604

MAD-Full Potential Solver

In the case of transonic flow we have to allow for a change from

elliptic to hyperbolic type as the flow becomes locally supersonic.

In the model problem

a ~ + b ¢ = O (13) xx yy

this corresponds to one of the coefficients, say a, to become negative.

The classical alternating direction scheme AFI

(~ - ~) (~ B~) ~ : ~L~ (14)

then has the disadvantage 7 that if one regards the iterations to re-

present time steps At in an artificial time direction t, it simulates

the time dependent equation

At ~ = ~ % + b ¢ (15) z xx yy

When ~ < O and Cauchy data is given at x = O - corresponding to super-

sonic inflow - this leads to an ill posed problem which admits oscil-

latory solutions which are undamped in time and grow in the x-direc-

tion.

Therefore Jameson 2 proposed the following generalised alternating

direction scheme. Let the scalar parameter ~ in Eq. (14) be replaced

by a difference operator

S £ e + ~i 6- + ~2 6- (16) O x y

where 6- and 6- denote one sided difference operators in the x and y x y

directions. This yields the scheme

(s - A6~) (s - B6~) 6~ = ~sL~ (17)

in which the residual L# is differenced by the operator S. The cor-

responding time dependent equation is now a hyperbolic one of the form

80 ~t + 81 ~xt + 82 ~yt = a #xx + b #yy (18)

where the coefficients 8o, 81, 82 depend on the paramters So, ~i, e2.

605

Two remarks should be made to this scheme:

- There are additional error terms because the operator S does

not commute with a~, bS~, and the order of the factors may

matter.

- Applied on a single grid it is necessary to use a sequence of

the parameters So, ~i, ~2 to reduce all frequency bands of the

error.

From the different possible strategies a very simple one has been

chosen. Each cycle begins on the fine grid. The alternating direction

iteration is performed once on each grid until the coarsest grid is

reached. Then it is performed once on each grid going back up to the

second finest grid, and the cycle terminates with the interpolation of

the correction from the second finest to the fine grid. For viscous

flow computations, this result is being used for a boundary layer

computation, the displacement thickness of which is entering the next

multi-grid cycle to provide finally a converged viscid solution. The

additional effort for the viscid analysis is less than the time of

one MAD-cycle per cycle, without increasing the total amount of MAD-

cycles needed for convergence.

Some typical results are presented here. Further results are given in

Ref. 8. All the examples were calculated on a circular domain generat-

ed by conformal mapping of the airfoil to a unit circle, with 16Ox32

in the 0 and r direction on the fine grid. Five grids were used in the

multi-grid scheme, giving a coarse grid of IOx2 cells; Fig. 2 shows a

typical result for the RAE 2822 airfoil including viscous effects. The

result was obtained after 12 MAD-cycles reducing the average residual

from O.1194.10 -3 to O.7355.10 -5 . Total CPU-time on an IBM 3031 computer

for this case is 120 sec.

Fig. 3 shows similar results for the DO-AI airfoil (CAST 7). Again

there is very good agreement with the experimental data. Finally,

Fig. 4 portrays inviscid results for the cylinder flow at M = 0.50

after 29 cycles and a reduction in average residual from O.7876.10 -2

to O.71.10 -6 .

606

Multiple Grid Schemes for Systems of First Order Partial

Differential Equations

The example which will be considered is two-dimensional unsteady in-

viscid flow which is described by the unsteady two-dimensional Euler

equations

~ ~ + ~ = 0 (19)

where ~ = pu ~ = pv

pE i vl pU2+p , ~ = pvU (20)

pUv I pv2+p

puH J P vH

and p, p, u, v, E and H denote the static pressure, density, Cartesian

velocity components, total energy and total enthalpy. For a perfect

gas is

E = P +I (T--1)p ~ (U2+V 2 ) , H = E + P/P (21)

If only homoenergetic steady flow (H = H = const) is of interest, the o

transient phase does not have to be time-accurate, thus allowing for

accelerating techniques. Two possible ones are described in Ref. 9,

the local time stepping and a forcing term proportional to H-H . Both o

acceleration techniques basically permit the additional use of multi-

ple grid techniques.

All three schemes presently under multi-grid evaluation are based on

the integral conservation formulation of Eq. (19)

I I ~-~ ~ d v o l + ~ ~ fi ds = 0 (22)

describing the change of the flow vector ~ in time in one volume to

be equal to fluxes of the flow in mass, momentum, energy through

the surfaces of this volume.

The three schemes differ in the approximation of the flow in each cell

- central point Runge Kutta Stepping (Ref. 9)

where the flow is discretised by values in the volume center

607

nodal point Runge Kutta Stepping

where the flow is discretised by values in the volume corners

one step distribution formula scheme (Ref.4)

The nodal point and the distribution formula schemes are basically

better suited to multi-grid techniques since after mesh halving points

will remain mesh points.

Best results in multi-grid so far have been obtained using the one

step distribution formula scheme. Therefore this procedure is descri-

bed briefly.

The basic idea again is to use the coarser grids to propagate the fine

grid connections properly and rapidly throughout the field, thus im-

proving convergence rate to steady state while maintaining low trunca-

tion errors by using the fine grid discretisation.

The changes AU2h in the coarse grid, obtained by removing every other

line from the fine grid, are determined by

h ~U h (23) AU2h = I2h

h Where I2h is an operator which transfers to each control volume of

the coarse grid the correction ~U h of the fine grid using specific

distribution formulas.

After computing the corrections 6U2h on all coarse grid points, the

flow properties at the finest grid are updated by

2h U new = U + I h ~U2h (24)

2h where I h is a linear interpolation operator which interpolates the

coarse grid corrections to give the corrections at each fine grid

point of the finest mesh.

The scheme is quite sensitive to the choice of distribution formulas

and the boundary condition treatment. Also, compared with the Runge-

Kutta stepping schemes in Ref. 9, a fairly large dissipation has to be

incorporated to make multi-grid converging efficiently. Fig. 5 por-

trays the converged pressure distribution C for the NACA 0012 airfoil P

at M = 0,80 and a = 0 O. The added convergence history with an average

608

reduction rate per cycle for 500 cycles of 0.9686 is quite good.

Fig. 6 shows similar results for the cylinder at M = 0.35. Here, the

reduction in ~p/~t is equal to 0.9785. However, both results show ef-

fects of numerical viscosity. The cylinder-case gives no fully symme-

tric solution, the NACA airfoil case shows errors in the shock region

compared with fully converged solutions of the very accurate Runge

Kutta stepping scheme in Ref. 9.

Conclusions

Engineering application of computational methods requires fast and

reliable numerical schemes. Multi-grid techniques are very well suited

to meet those requirements, and have proved to be very useful for

problems governed by quasi-linear equations. However, more has to be

done to develop fast and accurate schemes for systems of first order

partial differential equations since these equations are the most re-

levant ones for computational fluid mechanics.

609

References

South, J.C., and Brandt, A.: Application of a multi-level grid

method to transonic flow calculations. Transonic Flow Problems

in Turbomachinery, edited by T.C. Adamson and M.F. Platzer,

Hemisphere, Washington, 1977

Jameson, A.: Acceleration of transonic potential flow calcula-

tions on arbitrary meshes by the multiple grid method. AIAA-

Paper 79-1458, AIAA 4th CFD Conference, Williamsburg, July 1979

Schmidt, W., Jameson, A., Whitfield, D.: Finite volume solution

for the Euler equations for transonic flow over airfoils and

wings including viscous effects. AIAA-Paper 81-1265, AIAA 14th

FPD Conference, Palo Alto, June 1981

Ni, R.H.: A multiple grid scheme for solving the Euler equations.

5th AIAA CFD Conference, AIAA-Paper 81-1025, June 1981,

Palo Alto

Jameson, A.: An alternating direction method for the solution of

the transonic small disturbance equation. New York Univ.,

ERDA Report COO-3077-96, 1975.

Ballhaus, W.F., Jameson, A., Albert, J.: Implicit approximate

factorization schemes for the efficient solution of steady

transonic flow problems. 3rd AIAA CFD Conference, Albuquerque,

June 1977.

Jameson, A.: Iterative solution of transonic flows over airfoils

and wings, including flows at Mach I. Comm. Pure Appl. Math.,

Vol. 27, 1974, pp. 283-309.

Longo, J., Schmidt, W., Jameson, A. : Viscous transonic airfoil

flow simulation. DGLR Symposium "Str~mung mit Abl~sung",

Stuttgart, 1981.

Jameson, A., Schmidt, W., Turkel, E.: Numerical Solutions of

the Euler Equations by Finite Volume Methods Using Runge-

Kutta Time-Stepping Schemes. AIAA-Paper 81-1259, AIAA 14th

FPD Conference, Palo Alto, June 1981.

610

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