Extension of a Streamwise Upwind Algorithm to a ... - NTRS - Nasa

NASA Technical

Memorandum

102800

Extension of a Streamwise Upwind Algorithm to a Moving Grid System Shigeru Obayashi MCAT Institute, San Jose, California Peter M. Goorjian and Guru P. Guruswamy Ames Research Center, Moffett Field, California

April 1990

National Aeronautics and Space Administration Ames Research Center Moffett Field, California 94()35-1000

EXTENSION

OF A STREAMWISE TO A MOVING

Shigeru

Obayashi,*

UPWIND

GRID

ALGORITHM

SYSTEM

Peter M. Goorjian, and Guru E Guruswamy Ames Research Center

SUMMARY

A new streamwise of a moving-grid

up.rind

system.

direction-implicit)

method

algorithm

has been derived

T!ae temporally

nonconservative

has been applied

for time-marching

to compute LU-ADI

unsteady

flow fields

with the use

(lower-upper-factored,

computations.

alternating-

A comparison

of the tempo-

rally nonconservative metho(_ with a time-conservative implicit upwind method indicates that the solutions are insensitive to the conservative properties of the implicit solvers when practical time-steps are used. Using this new

method,

compwations

have

been

made

for an oscillating

The computed results confinn that the present upwind central-difference scheme based on the Beam-Warming allows

larger

time-steps

time per time-step

and thus is more

efficient,

than the c_;ntral-difference

wing

at a transonic

Mach

number.

scheme captures the shock motion better than the algorithm. The new upwind option of the code

even

though

it requires

slightly

more

computational

option.

INTRODUCTION

A code,

ENSAERO,

puting

the unsteady

strated

by computing

were calculated

vortical

The

upwind

models

is sensitive

to the amount

other

upwind

Recently, lems of transonic main

feature

lead

a streamwi.,e flows

the algorithm

in comparison

swept based

flows

equations

wings

(refs.

1 and 2). The flow fields

differencing.

of the present

code.

central-difference

to stabilize

In addition,

coefficient

Institute,

scheme

is

artificial-

the CD scheme case.

On the

be specified. and applied

over a delta wing

is the use of the local stream

San Jose, California.

Such

for each

direction,

to treat steady-state (ref. 4) on fixed grids. flow velocity,

gradient. The switching of flux evaluations always takes place at sonic values, where shock ist. Therefore, this method fcllows the flow physics more closely than conventional upwind * MCAT

In this re-

(CD)

computations.

schemes.

dissipation

for com-

of the code has been demon-

on central

capability

has been developed

(ref. 3) and vortical

method

The capability

than upwind

a specific

that any coefficient

upwind algorithm

the Euler/Navier-Stokes

to the current

dissipation

solutions

and needs

do not require

over wings

of the streamwise

scheme

an artificial

dissipative

of dissipation

using

of aircraft.

flows over flexible

is to enhance

requires

to more

schemes

at Ames

finite-difference

scheme

CD scheme

dissipation hand,

and transonic

of this ,:tudy

the use of a new

investigated.

developed

and aeroelasticity

by a time-ac,zurate,

The purpose spect,

is being

aerodynamics

probThe

and pressure waves may exmethods based

on dimensionalsplitting. Thecomputedresultsconfirmthehigherresolutionof thepresentalgorithmover theCD scheme, as well as over other upwind schemes. In this paper, coordinates steady-state in order

upwind volume, applied

upwind

algorithm

has been extended

for computing flows over moving components. problems by using the lower-upper-factored,

to accelerate

nonconservative LU-ADI

the streamwise

method scheme

(ref. 4).

for computations

over

is also considered.

upwind

option,

for computing

are compared

convergence

The same

in time, is used in the present

nondimensionalized

conservation-law follows:

form

which

scheme was applied to (LU-ADI) method

is first-order

accurate

but

In order

to check

the validity

a conservative

implicit

version

of the streamwise

has been implemented

CD option.

The updated

flows over an oscillating

wing.

code

of the

in the code as a finitehas been

The computed

successfully

unsteady

pressures

data.

GOVERNING

The

method,

to moving

computations.

algorithm

to the previous

transonic

with the experimental

grids,

The resulting

in addition

unsteady

moving

The streamwise upwind alternating-direction-implicit

LU-ADI

unsteady

from fixed coordinates

thin-layer

EQUATIONS

Navier-Stokes

in a generalized

body-conforming

equations

used in this study

curvilinear

coordinate

can be written

system

in

( _, _7, _ ) as

1

aTQ + o':3,_#+ a,7/_ + a_;G = _..-_-oq(G -----3t' /-._e

The Euler equations are obtained by setting the viscous flux vector served quantities Q and the inviscid flux vectors/_, F, and G are

r

G_ equal

20

(l)

to zero.

The vector

of con-

l

I p,,u+_,pI

' I

+ 1

# = 7 1p,,,u+_,pI LpHU

- (tpJ A

A

._w

2v

[puy+n_pl

1/,vv

?= 7/p_

LpHV where H is the total enthalpy, are defined as

[ puW

/

+,7,r'/

I p,,,w+ 0,

Cl Cm

(6)

sr = (1 - era)(1

-

er)

where 1 [l+sign(M el,m,_ = _-

2 l,_,_-

1)]

and M = _/c. A simple however, Thus,

_r/_

expansion

way

to evaluate

it is important

to detect

is replaced

by M • V/_

shocks,

the rotation

the rotation whether

angle

is to use cos0

the velocity

= V/c.

If V/c

angle is determined

projected

becomes by a mixture

= V'/_.

In supersonic

to the grid line is beyond

larger

than one, cos0

of averaged

flow

the Mach

is set to one.

(m) and pointwise

fields, cone.

To avoid

(l, r) values:

-2 COS2Ol,r

=

mini

( 1 - ¢) c-_'_+ w _2 , 1 ] _l,r

(7)

The following

relation

is used here for evaluating

[ _ 27

= max

where

pl and p2 denote

arithmetic

average

sin20

= 1-

V and A are backward _ = _-, Koren's

a small

difference

limiter

proposed

R represents

Applied

operators,

family,

_, of interpolations

of the

movements

is added

splitting

(ref.

to prevent

(ref. 7).

The limiter

For the third-order

_b is calculated

as (10)

+ e

the division

by zero.

The same formulas

Method

for the present

authors

to equation

using

10). For example,

scheme

(ref. 6). The LU-ADI

(3). The change form

side,

is the LU-ADI

method

is written

1) x (T(LcDcUcT_

On the left-hand

the diagonal

upwind

(3), this method

side of equation

of the grids.

is rewritten

(9)

respectively

3VpjAp] + e Vpj) 2 + 3_TpyApj

-

x (T_LBDBUsTff

the right-hand

method

flux-vector

methods

by one of the present

(T_LADAUAT(1)

Warming

by an

variables.

of the time-marching

ing translational

sine is determined

(1 + _)Al}pj

(ref. 8) is used here.

e, e = 10 -6 typically, primitive

ADI and LU factorization.

where

The

4

LU-ADI

method

respectively.

CJ+'[( 1 + _)_7 + (1 - _)AI}pj+I

and forward

differentiable

constant

are used for the other

One

(8)

0]

+ c0s20_).

a one-parameter

- _)V+

_bj = 2(Apj where

pressures,

from

of the smoothness:

p, u, v, w, and p. For example,

v_ = {1

scheme,

because

1 + (q + 1)P2]}, pl

½(cos20t

Pt = {1 + --_[(1

where

2-_[q-

are constructed

schemes

variables,

-

and downstream

of the cosines:

Higher-order primitive

upstream

(1

_b 6 [0,1]

factorization

is a compromise

between

as, 1) x A_) '_= AtR"

of volume

(11)

in time is neglected,

the original

ADI

(ref. 9) and the first-order

operator

accurate

assum-

of the Beam-

Steger-Warming

in the rl-direction,

A

I + zXtSnB

= T.( r_+ zxtV,3_ + zxt_9_)T_ = T,( I-

At/5_lj + AtV,3_)(I

= TnLnDnUBT_ This factorization simple

+ Atl3BIj)-_(I

+ At3+BIi + Atz_9[_)T;

1 (12)

I

is the approximate

LU factorization

_

LDU (lower-diagonal-upper)

since the diagonal

element

always

6

factorization.

has the absolute

This is more

stable

value of the eigenvalues.

than

Approximate

The

LU-ADI

diagonalization.

method

To investigate

proach

is considered

similar

to the Beam-Warming

algorithm

described

here.

are expensive

To construct

method.

=2

7