Denver, CO - NTRS - NASA

AIAA

OO-2252

Multigrid Conservative

Relaxation

Discretization Flow Equations Thomas

W. Langley

David

Sidilkover

and

for

of the

Compressible

Roberts

NASA

Institute

of a Factorizable,

Research

Computer

Center

Applications

in Science

and Engineering

J. L. Thomas

NASA Hampton,

Langley

Research

Center

VA

Fluids

%

19-22

June

2000

2000/Denver,

CO

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

/

i l

MULTIGRID

RELAXATION

DISCRETIZATION

OF

OF

THE

A

Thomas

for

W.

Computer

J.

L.

Langley Hampton,

Abstract

the

factorizable

compressible

Euler

discretization

of

developed

by

equations

Sidilkover is extended to conservation era curvilinear body-fitted grids. equations

are solved

Seidel relaxation flow in a channel 0.0001

by symmetric

to a supercritical

Mach

uniform in the

lution

flow

the

semi-infinite

collective

and FAS multigrid. with Mach numbers

demonstrating loss of accuracy for

form on genThe discrete

around

parabolic

scheme

maintains

tiiining

a stagnation

number

convergence incompressible the

body

rapid

Gauss-

Solutions for ranging from are

leading

edge

demonstrates

convergence

shown,

that

for a flow

Steady

inviscid

which

subsystem tropy and bolic

can

The

operator,

hyperbolic

for supersonic

flow, space

marching

corresponding

culty well.

is true

As has

by

the

Euler

been

corresponds

to a full

for subsonic

flow and

flow.

For a purely

most

efficient

supersonic

way of solving

operator which

terms

liptic

correction Existing flow rely As such, tiveness

for certain multigrid

smooth

methods

only

gives

components for subsonic

part

of the

of the

error.

and

transonic

on the coarse grid to smooth the entire system. they are fundamentally limited by the ineffecof the

coarse

grid

in correcting

the

part

of the

Copyright Q2000 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for government purposes. All other rights are reserved by the copyright owner. * Research Scientist t Senior Staff Scientist t AIAA Fellow

1, to obtain

as

ideal

correspond

efficiently

are rapidly that

to

solved the

to be equal

The

advec-

by marching

and

the el-

with

by spfitting

parts,

a multigrid

the system

convergence

to the

and

behavior.

iteration. into

rate

slowest

its ellipof the

of the

full

two

sub-

systems. Using this approach, Brandt and Yavneh have demonstrated textbook multigrid for the incompressible equations and

of the

Ta'asan

compressible

Euler

of "canonical

Euler

equations 4.

equations

In a closely

equations.

This

variables"

which

method and that

ap-

for the

is based

express

of an elliptic

for two-dimensional

related solver

3 it. is shown

can be achieved

the

on

stead),

a hyperbolic ideal

multi-

for the compressible

Euler

subsonic

body-

flow

using

grids. work 5' 6, the

the

of the

advection

system

and unstructured the discretization.

dimensions

strated

ideal

gation collective

a multi-

operator

of equations.

from

Both

equawhich

the

elliptic

structured

grid

can achieve ideal multigrid convergence flows. The scheme has been extended

to three

This

presented

grid flow solvers were written using The results in Ref. 5 demonstrated

that the scheme rates for internal

grids--a equations

authors

for the steady, incompressible Euler on a pressure Poisson discretization

distinguishes

flows.

for a simple

a staggered-grid

a fast multigrid

in terms

In an earlier

part

are

using

equations.

In Reference

grid efficiency

results

grid,

3 presented

a set

partition

2. Their

a Cartesian

discretization proach,

grid scheme tions based

grid

Brandt

to elliptic

are treated

tion

coarse

which

correspond

terms

fitted

as the

by

diffi-

flows

advection,

those

the Euler equations. For subsonic flow, the ellipticity of the full potential factor should be effectively handled by multigrid. However multigrid is not effective for advecoperators,

out

same

ferential

ought

is elliptic

This viscous

inviscid flows, parts of the dif-

system

One

factor.

Reynolds-number

pointed

the

equa-

to advection

for high

multigrid convergence rates for subsonic, the discretization nmst distinguish those

geometry

of as two snbsystems.

is the

error

advection

subsystem

which

Center

1 argams

con-

Engineering

VA

Navier-Stokes

is described

other

Research

tic and

corresponds to the equations governing envorticity advection. This subsystem is hyper-

in space.

potential

flow

and

t

Brandt

point.

be thought,

in Science

of a

Introduction

tions,

Center

Thomas

tion

rates and no limit. A so-

EQUATIONS

Roberts*

Applications

NASA

second-order

FLOW

Langleg Research D. Sidilkover t

and

The

CONSERVATIVE

COMPRESSIBLE

NASA

Institute

FACTORIZABLE,

by Sanchez

multigrid

approach

rates

has the advantage

collocated, vertex-based is used, simplifying the operations

7, who has

convergence

and

Gauss-Seidel

allowing relaxation.

also

demon-

for

internal

of non-staggered

discretization restriction and the

use

of the prolon-

of simple

Although

the

point pres-

surePoisson approach maybeextended tocompressibleseries about each grid vertex and considering the leadflows,it.is not conservative andis unsuited forsuper- ing terms of the truncation error. These terms are the criticalflowswithshocks.Furthermore, theextension artificial dissipation of the scheme. toviscous flowsislimitedto tileincompressible NavierThe starting point fox" the scheme is the twoStokes equations. dimensional Euler equations in non-conservation form. Recently, Sidilkover 8 hasdevised a discretization of Let p be the density, ff = _u + jr, be the velocity, and p thecompressible flowequations that.overcomes these be the pressure. The entropy s is defined as limitations.Thisdiscretization maybeappliedto the Eulerequations in conservative form,usingthemultidimension;d upwindscheme ofSidilkover 9' 10.In Ref.10 it is shownthatthisapproach should leadtoa scheme thatis factorizable, i.e.,thescheme distinguishes be- where p0 and po are a reference pressure and density, and "_ is the ratio of the specific heats. tweenthosepartsoftheoperator thatrepresent, advec- respectively, Then the Euler equations may be written in the varition,andthatpartof the operator that. represents poables (s, u, v,p): tential

flow.

In

Ref.

8, such

a factorizable

constructed for Cartesi,'m grids. Because tion is stable for Gauss-Seidel relaxation,

scheme

is

the diseretizait the conver-

gence rate does not depend upon a Courasat-FriedrichLewy number restriction, unlike standard discretizations of the Euler equations which must use time marching. For this reason, the same convergence rate is obtained for subsonic pressible

Mach

limit.

The

numbers

all

scheme

may

be written

plus

an

of a central-difference As

such,

it

upwind,

part

is very

similar

finite-vohune

discrete terms

must

he shows the the

dissipation.

scheme

In the

ix, the present

factorlzable nates

viscosity

scheme

presented,

artificial

the

terms

must

be

ability.

A discussion

of the next.

nel_

infetl _Iach

numbers

additional

of the

computation

how

collective

the

dissi-

factorlz-

Gauss-Seidel

for flow in a chan-

ranging

from

0.0001

flow around

the

leading

ficulties

for

the authors'

previous

point

scheme

Mathematical The can

artificial

tion,

or first

hy

first

differential

discrete

scheme.

This

is found

by expmlding

were

observed

11

of the

it

way

without

The

advection

encing in coordinate

the

differential

the difference

weakly

coupled

equations

to

the form of equa-

the

through

momen-

the

equation

may be discretized independently of the and pressure equations in any appropriate affecting

the

operator

factorizability

ff.V uses

of the

simple

Eq. (2a). Let (,_, r/) be a general system and define the contravariant

of the

velocity,

(U, F') by the

scheme.

upwind

differ-

curvilinear compo-

transformation

(;: ;,9 In this

coordinate

tions

are

with

a grid

system

discretized

first-order

spacing upwind

g.Vt7 = (?O_ + f'O,.

on

a uniform

A¢

=

difference

ATI =

grid 1.

in The

equaspace,

FDA

of the

to tT.V is

and

the

entropy

1

equation

.=

is discretized

(4) as

qs = 0.

has

upwind also

particular

been

cases

(5)

discretization used

shown

of

in Eq. below

the

the

(5). use

order advection operator has an insignificant the computed results. This point, is discussed

(FDA),

of the

sults

equation

which

in a Taylor

The tions,

for

of a second-

equa-

schemc

adveetion

However,

modified

equation

The (¢,t/)

approximation

1

the

factorizable

approximation is the

Therefore, momentum

A second-order

presenting

scheme depends on added to this system

of state (1). In fact, the entropy equation corresponds to one of the advection factors of the Euler equations.

operator

Formulation

dissipation

be described

that

(2c)

+ ff.Vp

to

the dif-

a stagnation

= 0.

pressure

are

edge of a semi-infinite parabola demonstrates that current scheme does not suffer from the convergence near

(2b)

is only

the

are shown, demonstrating rates of the scheme. An

for the

= 0,

multidimensional

Solutions

Mach number and convergence

entropy and

coordi-

equations

to maintain

point

is presented

with

with

tum

nents

It. is shown

relaxation a supercritical the accuracy

In addition,

body-fitted

discretized

ff-Vz7 + 1Vp P

of Sidilkover's

governing

form

dissipation.

pation

for the

limit.

curviIinear,

the

way.

generalization

First.,

including

upwind

that

the factorizability, and thus h-ellipticity of the operator

the

to

can be a modi-

may be rescaled

incompressible

is presented.

and with

(2a)

The factorizability of the of the artificial dissipation tions. The

and

the dissipation

in a specific

work,

Euler

equations, 8 shows

the artificial

viscosity.

for the

factorizability

incom-

to standard

Sidilkover

Mach number such that accuracy, as well as the

is preserved

artificial

scheme

be discretized

the

= 0,

pc2V.a

in the form

upwind

to preserve

that

to

formulation

Navier-Stokes

as a conventional

fied artificial

way

discretizations

Reynolds-Averaged written

in

the

a.Vs

attect on in the Re-

section. dissipation Eqs.

(2b)

for the momentum

and

pressure

and (2c), is the umltidimensional

equaupwind

dissipation of Sidilkover pation

is written

9.

In vector

notation,

this dissi-

adding

the following

corrections:

as a-_"/O = qL/-4- 7l [(--?[05/at -'t- l [_='l 0_0._',

a.va+ 1-vp- "'ev (p 2v.a+a.vp) =0, p

2

(6)

2pc

pc2_7.ff+ff.Vp

--pc-_.(_._ff+

_Vp)

=0,

(7)

1

The

cross-derivative

second-order where

c is the

coefficients, grid tum

speed

and

spacing. equation

residual, the

the

divergence

of the

artificial

scheme.

ap

are

pressure

momentum

of the

and

the

dissipation. second

equation

that

ttle

that

curl

The

ilarly,

because

of the

momentum

part.

irrotational

of the

factor,

If the

and

advection terms

the

h-elliptic. This central difference

advection

8.

equation, operator

Replacing

first-order

tion term second-order

(2e)

is

to the

by Eq.

accurate.

upon

covariant

operator

Note

(7)

components

by

the

the

to be

accuracy

The

form

accuis not of the term

first-order

pressure

while

of these

of the

of the velocity to the

a discretization

by

maintaining nmst be that fac-

terms

is de-

physical

(8) to find (9a)

into

and

the

/at and

(9b)

are

physical

of the

necessary

factorizability

for at the

_', the

antidis-

evaluated,

and

components.

This

form

/7.Vff

= qa

+ VD

(10)

where D= The

above

of the

gain

more

terms,

where is the

(iCrlo, a+

expression

coordinates tion

GI

the

.

is the generalization one given

insight FDA

in Ref.

about,

of Eq.

the

(10)

(11) to curvilinear

8. nature

may

of the

be rewritten

J

+ .7(c' I '1

correcas

e"lC'l

_o _ 0_t, - Ouu is the vorticity, J = av_y,7 - y{x, Jacobian of the coordinate transformation, mM

(4e, gn) are the contravariant coordinate directions.* Note t.ional,

then

identically accurate.

the first-order and

truncation

the approximation

It was pointed out upon the dissipation written

basis vectors in the ((, _/) that. if the flow is irrotaerror

terms

becomes

vanish

second-order

above thai factorizalfility depends of the momentum equation being

as the gradient

of the pressure

dissipation

of the

equation

divergence second-order

of

pressure

the momentum correction to the

equation, being

equation. adveetion

and

written

the

as the

Likewise, the terms must be

the curl operator first-order

of Eq. (6) yields q.0 = 0. Thus the acting on the vorticity is unchanged scheme.

To get. full use second-order operator elliptic,

second-order accurate

in the momentum fully

second-order

= C0{

+ _'0 n

accuracy discretization equation. upwind

advection from the

it. is necessary to of the advection The

advection

FDA

of an h-

operator

(_, Ii), and

contravariant vector.

(9) are

in the form of a gradient so that the vorticity equation is is unaffected. With the definition in Eq. (10), taking

advec-

approximated

coordinates

in terms

are related

dius-

restores h-ellipticity but it. now becomes

that

the computational

be written

ant. components

and

discretized

antidissipative term_ equation in such a way

is preserved.

must

are

is second-order such a scheme

in Eq. (4) factorizable,

second-order

torizability

gradient,

(7)

in Eq. (7) continues central differences.

obtain

the

Eq.

Eq. (6). Sidilkover shows corresponds to vorticity

this

h-ellipticity, appropriate added to the momentum pendent

pressure

back

+vD=

as the gradi-

described

the scheme However,

upwind approximation and the scheme remains

by

the solenoidal

lack of h-ellipticity is a result approximation to the advection

in the momentum that this advection

they

not

it corresponds

(6) and

gives

To

by tile Jr-

equation

FDA

operator, in Eqs.

ing central differences, rate and factorizable.

To

pressure

equation,

to Sim-

this property.

vergence

only

corresponds

may be written

As the

continuity

full potentiM preserves

part

to

(7) is the divergence

field but

rotated

equation of by the arti-

only

Eq.

in Eq.

then

in Eq.

and to retain

(6) is

equations.

it is affected

of the velocity

of a potential.

a form

of Eq.

equation,

part

The

equation

of the Euler

the dissipation

rotational ent

vortieity factor

Using

terms

It

leads

of Eq.

time.

sipative

terms

accuracy

is

residual.

dissipation

Note

adveetion

same

to the

equation

identical to the curl of Eq. (2b), i.e., vorticity the governing Euler equations is unaffected ficial

scaling

proportional

dissipation of the momenof the pressure equation

dissipation

of the

property

a factorizable

am scale

Note that the is tile gradient

and

is this

of sound,

( is a length

(9a)

1

The

and

quo

covari-

components '2 1

transformation

IglO_+2sgn_'f'OeO,+_O,,]

(12a)

when 1 '1> lv'l and *, W¥iting the

the

advection

covariant

containing scheme

velocity the

may

y,_

=

operator

" in

upgraded

(6)

and

ignoring

geometric to

(8)

Eq.

components

higher-order be

v

nearly

in terms

qHO ----_0_

"k-VO,7

of - 21 k(C20_ _

terms

derivatives,

the

second-order

by

*dg_

+ 2 sg,,t 7- c-o{o,,

= _Yn - dxn, ggn = __y¢ + jx_

+ lvl o,g )

(a2b)

The

when l_?[ < I_::[. With t.his advection operator in the momentum equation, the quantity D in Eq. (11) must be modified in order to preserve the factorizability of tile scheme.

This

modified

D is given

-= D + lsgn_"

Dm_

following

difference

_" (O,L? + c3¢9)

g

=

0

o.=_

-4

sponding

operator,

dissipation

Eqs. (6), (7) and (9) is related to the standard upwind difference scheme, consider a uniform scaling the

equal

spacing

coefficients

grid

dissipation standard first-order replaces

in the

a,,

spacing,

and

and

x and

#v are

the

taken

terms

upwind

scheme

on

a nonmfiform

where

the

addition given

expressions

a six-point vertices,

may then

taking

-4

,

the

operator

of Eq.

cross-derivative

upwind

(9) on terms

a Cartesian

yields

grid.

the standard

-

2

the subscript h to the corre-

of the

overbar The

in flux

a two-point

( )

secondform

on an edge

by

between

difference

of

on a vertex. The subscripts e difference operators centered

respectively. as

f is

-

above.

centered

centered to denote

on an edge or a vertex, scheme is then written

,

be expressed

difference

and

-20 -

qhs

The

fully

discrete

= 0,

(14at

the The

further

p

the grid = O,

spacings in the two coordinate directions. The multidimensional upwind scheme uses a single length scale, and retains the cross-derivative terms. Consider the advection

condition:

-

scheme, difference

differences

in the

grid

f_ and (u are

and

"wide"

those expressions and v are used

If the

terms of Eqs. (6) and (7) are ignored, first-order upwind sdmme is obtained. (_" by g_.0_ + lyon,

two

first-order Cartesian

to be one,

cross-derivative

taking

of

g directions.

the

difference

when IUI < I% Taking tit(" curl of the momentum equation now gives qno w = 0, i. e., the vorticity equation is now second-order accurate. upwind

this

0-

write the complete discrete is used to denote a standaxd denotes

with

_

To

f-7 (O,7_ -4- O I1"1and DHC) = D -4- lsgn_>

satisfy

' '[i' i] , ,[!,!] '[! !] _

by

( 13at

To see how the

operators

Dropping

p c2

_Z h . ff

___ ff

._

(14b)

h p

first-order

discretization.

.

h-

1

(V__F,_)p)

=0

'

(1@)

Discretization The The factorizability of the FDA is a necessary condition for the factorizability of the difference scheme, but. it is not torizat,le, same

sufficient.

the

the difference

way as the

difference

For

difference

operators

differential

must

opera!ors

approximations

scheme

to the partial

8.

Introducing derivative

in the

advection

the oper-

part

of DH-_ given

wide,

six-point

way,

qHo

and

must

depend

be

upon

stencils

for the general

the appendix The

0_,,=0_+...,

to this

scaling

in Eq.

difference

discretized

the

curvilinear

(ll)

stencil

terms of DHO and for the second-order flow

grids

in a very

direction

the relative magnitudes of Cr and V. The described in Ref. 12 for uniform Cartesian

0,7 =0,_+---,

4', =0_ +...,

the

operator

precise

at, ors, =0¢+..-,

in that using

for 0_/_ and 0_V. The additional the corresponding dissipation terms

to be fac-

commute

derivatives

are discretized

and

details grids.

are The

are presented

in

paper.

coefficients

are 1

0-,_ = max(._/, where the following

conditions

must

Mach

hold

_I

chosen • _h

.-,h

, h

• h

stagnation

cJ_cJ,m = OCnO_n h

h

,h

is the

number

local

to be

O(h),

points.

skewed

4

Laplacian

_rv -

Mach

to prevent

pressure equation tic factor in the

,h

Me),

max(M,

number,

division

and

essentially

The

purpose

M_)

and

/ff_ is a cutoff

by zero.

The

becomes of the

(15)

cutoff

active

rescaling

dissipation, crp, is to prevent the discrete equations from becoming operator

in the

incompressible

is

near of the ellipthe limit.

Currently, we take Mc and AM is the two-point

= v IAM[, where difference in Mach

on an edge.

flow cases

For

tile

channel

u = 5 number

shox_m below,

never becomes larger than M. For the leading-edge the cutoff does become active at the stagnation When

Me

patton

> 3/I, it is necessary

to the

dissipation

advection

where tion.

momentum

coordinate

to both

as_d is cast

maintain

conservation.

optimize

either

ator

of tile

is added

equation, the

coefficient

(16)

transforma-

the entropy

in flux form

No at tenq_t

wilt.ten, the Eq. This is because

artificial

dissipation

in order

to

made

to

ha._ been

v or the form

and the

of the oper-

(14) is valid tile pressure

terms

are

only for subsonic differences in the

not. fully

upwinded

in a

supersonic zone. A simple modification to Eq. (14) can be made by rescaling the gradients of the pressure when the flow becomes defined as

sonic.

Introducing

the

parameter

n,

This

=max(l, final

form

of the

M2),

scheme q

h

artificial

dissipation

in

as

1

h .

(17)

is a conservative

terms

of the

square

bracket

form

primitive fluxes.

(20)

_

h

,

s - d_ps = O,

dissipation

The

and

These

terms

the

antidissipative

recast

terms

in conservation

are

The

length

scale

ing min((e¢,

/_h(/_,9)

form

discretized

( is evaluated

In addition (14b) terms

to the and

on the

are

discretized

(20)

o-v are

that.

using

the

a term pressure

be

by tak-

grid direction. evaluated

using

examination

the

pressure

wide

part of the conservation to account for these

may

cell face

terms,

shows

differ@ and

on cell faces.

in each

dissipation

(14e)

central-difference must be corrected

in Eq.

length

The scaling coefficients a,,, and the Mach number on the face. Eqs.

in the

on cell

or narrow operator

and evaluated

g,1), the shortest

cast inside

(21) are now interpreted

terms

faces according to the appropriate wide ences. The first-order upwind advection

This is done by adding for the momentum and

is

of the

variables.

in Eqs.

as dissipative

ent

the

The

can be rewritten

of this

d_p.

As flows.

(14).

(14c)

+V,_

dissi-

form

Eq.

Me

c( (0_ + 0_), 7

Mc-M)

Jacobian

operator

Tile

and

(14b)

pseudo-Laplacian,

1 _max(0,

J is the This

qHo.

in

Eqs.

flow, point.

additional

h

operator

is a five-point

d_p =

to add

operators

of gradi-

stencils.

The

equations (19) wide differences.

to the dissipative fluxes equations as follows:

(18a) p

qhl(,a

-- d_p_ + V,,D_,,

+ _lvhp P

O'm(_Th " (pc2_72"_-I-

pC2_Th.

([

+

and

o_+_

the

(14c),

_"u'_rep)2pc

can

+v_

difference be

)

(18e)

(14a),

(14b)

p =o.

equations

written

as

a central

equations

finite-volume

are diseretized

using

difference

by the

transformation

+

vhp

The

e = dissipative

in terms

a.ff/2

+ P/(O(_ fluxes

of (s, u,v,p)

on

each

using

the

conservation

p o

o

eval-

variables

,,/d] ,,lal

,_. _,, "

h/cV \41

0,,

where

h = e + p/p

is the

total

Solution The

solution

Procedure

of the discrete point found

ent.halpy.

equations

is performed

collective Gauss-Seidel t.o be a very effective

us-

iteration, smoother.

= 0,

(19a)

The grid point.s are ordered such that. the forward GaussSeidel sweep is in the streamwise direction. This means

=

(19b)

forward

(19e)

relaxed

0 ....

v_.[(p_ +v)a] = 0

where

to the

-P"I" -p,,Is

ing a symmetric which has been

a central-difference

(Ss, 5u, Jr, 5p) have been

converted

(24)

that l_rh.(pff_)

are

fluxes

they

part a conof the

approximation, Vh-(pa)

the dissipative

uated,

I L_o_'i = I

plus a dissipation, it is straightforward to obtain servative discretization. The conservation form Euler

(lSb)

ff._rhp

-peT,,,.._u. Because

Once

= o,

1)) cell

is the face

appropriate

total are

the

entropy

equation

sweep. with

The a factor

of the

conservation

puted

at each

is marched

reverse of 0.5 using

computed

in the previous formed to the

section. residuals

difference

inverse

transformation

energy.

of the

sweep

during

The

(r¢,, ro,,

ro,,, rp_) are

the

discretization

These residuals of the primitive in Eq.

(24).

the

is under-

for stability.

equations

vertex

in space

Gauss-Seidel

residuals com-

presented

are then transvariables by the

flow

ff

[_

p

.-ighl

=

I

1 1

Ic,_gth

Figure

At each

point,

the

is the

matrix

collecting

the contributions

entropy this

equation

is a block

decouples

is easily

The

of the

coefficients

froln

matrix

entry, and multiplying

found

relaxation

faces.

Because

At

the

the

rest

of the system,

the

upper

is accelerated Scheme

(FAS)

block

is

using

dissipation

a standard

multigrid.

Full-

A sequence

ator, u be the vector be the fine-to-coarse

of the conservation variables, grid restriction operator, and

be the coarse-to-fine

grid

tion

equations,

to the

of

grid

solving

This

procedure

=

(2G)

equation

for Uk-!,

the fine-

Multigrid tion.

performed pletely.

the

recursively

coarsest

to

to solve

insure

grid, that

A conventional

to the

many the

same

relaxation

fine-grid

equation.

coarse-grid

relaxation

equation

Iz cycle

Solutions

are

shown

channel,

is solved

are com-

is used.

Pig.

shape

1. The

of the

lower

wall

is shown

between

in

0 < z
O.

.h-

l 1

Sidilkover,

D., optimally

the inviseid vol.

Roberts,

approaches

efficient

4-5,

pp.

551-557,

Multigrid

Paper

99-3337,

for

and Fluids,

1999.

R. C.,

to

1

( _/'i+

take

Ideally

Airfoil

(Pi+2,j + "9,+2,j-1 - P,+l,_ - 13',+l,_-i ),

o2v = 97 Fourth,

"Extending

Methods

,

con-

solvers

Computers

T. W., Swanson,

Converging

towards

multigrid

flow equations,"

28, nos.

AL4A

"Some

- 1Ai+to-_)

O,U = 97(lgi+,o

1994.

structing

11.

Equations,"

Third,

2,3q_

l

--

"171__2,3__1)

.

(r < 0, 17"< 0.

1

Flows,"

(90h/_

=

_

(/_/+l,j+2

--

a/+l,3

),

1999. ;, c, c, 0_'P = 1_ (L+2,_+_ + L+2,_ - I;,+_,_+_ -

12.

Sidilkover,

D.,

Steady-State in Two

"Factorizable

Equations

and Three

Sdaemes

of Compressible

Dimensions,",

for Fluid

ICASE

the Flow

Report

The

Discretization

diseretization

for DHO, given

of the

in Eqs.

(13)

was so as to preserve scheme. In Ref. 12, sions

for a uniform

in

(13a),

derivatives must

Cartesian

in the expression

repeated

here

Ohol_ = 21 (lli,j

- ll,,j__

(i,j + 1/2}. the following

+ _,__.a

First we difference

-/_,-_o-1)

,

be done in a very. precise

grid.

here.

Now consider the o-cell edge, take (Tr > 0, 1-"> 0, in which case fornmlas are used.

DH¢_

the factorizability Sidilkover presents

of I/.771> ]_7"1is presented by Eq.

of

1

a._ = _ ff"+_,,+' - _,+_,,__).

preparation. Appendix:

9i+1,3)

of the these

The

discrete expres-

particular

In this case,

1 (_i,j

case

- "13',__,j) ,

-,_ah_'-= 971('_i-l,j+l--_i--l,3+_)i-";+l--_)i--2,3)

DHo is given

for convenience.

Second,

take

(r > 0, {-: < 0.

1 O_a

Duo

=-- O + lsgn(;

_" (OJ2

_-

2

(l_i'Jq-2

--

U"3"_I

-{- l_i--l'jq'2

=

1 (V_,,+_ - P,-_j+_),

--

/_t--l'3-}-I

)'

+ _¢_')

1

_,_,_ = -_(v,_,,.+_ - _',_,,. + 9___,,+, - P___,.) The

term

tion

section,

D has already and

been

what

discussed

remains

above.

Each

of these

a _-cell

edge,

(i + 1/2, j),

terms

is discretized

and

where i and j are the indices directions, respectively.

in the Discretiza-

is 0,1/_,

an ,t-cell

0_'

and

differently edge,

of the vertices

Third,

0_' on

in the _ and

*1

(o,,, -

1

a29 = ,hr,

_ f._,,.

l

- V,_l,,_, ).

_ 9_-_,_-,),

take

U > 0, V < 0.

h-

Fourth,

take

r,

(r < 0, 17"< 0.

|

1

r.'__,,._,)

0,+_,_+_ +0,o+z-

&o+,)

-,

4"v = 97(v,+2,,+1 - _',,,+,), 1 0._ ' = 97(_,+_,,+, - _',+_,. + 9,+_,.+, - _+,,,) above

definition

= 7 (&,j+_ - &,,),

1 (>,_,

- >,,,),

1

Analogous

a.u

+ a,,, - a,,,_,),

O,V = _ (_'_÷_,_÷_- _,'_+_,_+ V_÷_,,÷_- _;_+_,_)

The Second,

1

0,,_ _ - = 971 (&+_o+_-

1 (_.,. + _. • h -,

U" < 0, 17_> 0.

= (a,+,,, - u,+

(i, j + 1/2),

Consider the _-cell edge, (i+1/2, j). First. we take U > 0, 17_> 0, in which case the following difference formulas are used.

o,, v = _ (_',_,,,+,

takc

corresponding

differences

are

of the second-order expressions to lf:l

may >

[/?[.

also

the

advection be developed

ones

used

operator for Eq.

in the qno. (lab),

1,1E-11

1.0E-

11

doe_

-0.0002

1

9.0E-12

8.0E-12

_5

--o

Entropy

t-0.0004

Drag

-I

-0.0006

7,0E-12 -4

-_ -0.0008 o _

6.0E-12

v

5,0E-12

-0.0m 0

j_

-0.0012 -6

4.0E-12

-....

L,(rho) L,(r_o-u)

.......

L2(rho-v

)

L2(rbo-e) I @oe_ 3.0E-12

2.0E-12

iI

-0.0014

o _° t

I

' ii

-0,0016

O.OE+O00

-8o

-O.O01B

1.0E-12

.... ,'o.... 4 .... 4 .... 4'0.... ;0 V(1,1)cycles

10

20

30

40

5O

V(1,1) cycles Figure 7. Convergence of t.he L2 entropy error drag on the lower wall for a M = 10 -4 inlet using a FMG cycle with a 385 × 129 fine grid.

Figure 9. flow using

and flow

Convergence rates a FMG cycle with

for a M = 0.73 inlet a 385 × 129 fine grid.

0.0024

5.0E-04

0.0022 4.5E-04 0,002 4.0E-04

0.0018 _

--o

Entropy Drag

] 0.0016

3.5E-04 0.0014

\

3.0E-04

00012 I

_

2.5E-04

_r'

2.0E-04

Do.o

0.001 0 0.0008

-, %g/

0.0006

1,5E-04

0.0004

1,0E-04

o

I p _I

5.0E-05

I i O.OE+O0

0.0002

,Z' g _ r

0

i l 10

i/

$ i

-0.0002 i

,

i 20

....

V(1,1

Figure 8. Mach number contours for a ,_I = 0.73 inlet, contour increment _,"ll = 0.025, for a 385 × 129

l 30

, , ,

, I .... 40

Figure 10. Convergence of the L.. entropy drag on the lower wall for a M = 0.7a

grid.

using

]0

a FMG

cycle

with

0,0004 50"

) cycles

error and inlet flow

a 385 × 129 fine grid.

m

p v

-4

o

-k,(rho) .... k,(rho-u) ....... k,(rho-v) ...................... k,(rho-e)

10

20

30

V(1,1

Figure edge

Figure

11.

Semi-infinite

parabola,

13. using

Convergence a FMG

rates

cycle

40

50

) cycles

for a M

with

= 0.1

leading-

a 129 x 129 fine

grid.

33 × 33 grid.

7.0E-04 6,5E-04 6.0E-04 5,5E-04

i

--o

Entropy

]

5.0E-04 4.5E-04 A 4.0E-04

E

3.5E-04

Q 3.0E-04 2,5E-04 2.0E-04

C;7

1,5E-04 1.0E-04 50E-05

O.OE+O0

o

10

20 V(1,1

Figure Figure M

=

a 129

12. 0.1

Pressure leading-edge,

coefiicient increment

contours AC' v =

around 0.05,

a

M

for

=

14. 0.1

Convergence leading-edge

a 129 x 129 fine

× 129 grid.

ll

grid.

30

40

50

) cycles

of the flow

L2 entropy

using

a FMG

error

for

cycle

with

a