Preconditioning for Numerical Simulation of Low Mach Number Three ...

/F-¢ / NASA

Technical

Memorandum

ICOMP-97-11

113120

_o_ co,%

'_' ICOMP i-'

i"_

% o.--.,.. _o

Preconditioning for Numerical Simulation Low Mach Number Three-Dimensional Viscous

Daniel Lewis

Turbomachinery

L. Tweedt Research

and

Rodrick

Center,

Flows

V. Chima

Cleveland,

Ohio

Eli Turkel Institute for Computational and Tel-Aviv University,

Mechanics in Propulsion, Tel-Aviv, Israel

Prepared for the 28th Fluid Dynamics Conference sponsored by the American Institute Snowmass, Colorado, June 29--July

National

Aeronautics

Space

Administration

Lewis

Research

October

1997

Center

and

of Aeronautics 2, 1997

Cleveland,

Ohio

and Astronautics

of

Available NASA Center for Aerospace 800 Elkridge Landing Road Lynthicum, MD 21090-2934 Price Code: A03

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Preconditioning for Numerical Simulation of Low Mach Number Three-Dimensional Viscous

Turbomachinery

Daniel

L. Tweedt NASA

t and

Lewis

Cleveland,

Rodrick

Research Ohio

Flows V. Chima t

Center

44135

Eli Turkel § School

of Mathematical

Tel-Aviv

University,

and ICOMP,

NASA

Sciences

Tel-Aviv,

Lewis

Israel

Research

Center

Abstract A preconditioning

scheme

a three-dimensional code

viscous

for turbomachine

allows

the code,

Introduction has been

implemented

computational

blade

originally

rows.

The

developed

The general

into

fluid dynamics

fluid

preconditioning

for simulating

devices ible

com-

devices

of the compressible

speed

coordinate

system,

along

Details

about

employed. artificial

dissipation

dissipation

with

schemes

the conservative

are was

case

the

NASA

able

pressor,

large

detailed

for comparison.

compared tion

for which

applied

with

between

generally

experiment.

accuracy,

revealing

propellers,

facet

in general,

for fluid

based

on numerical

low-speed

only

for compressible

most

state-of-the-art

data

and flow

are avail-

field data

point

of the com-

between

significant levels

and where

cost

of

code development.

codes

solution

will

expensive

known

not

the flow field Mach

cally

this

occurs the

velocity.

methods

for

algorithm

to allow

increasingly

so-called

§Professor,

Department

at low Fellow AIAA

of Mathematics,

Senior Member

acoustic ble

A1AA

to avoid

numerical

NASA

TM-113120

1

the past

convergence

this

algorithm

several

the wave

problem to

solution

too low. Mach with

that changes

years,

however, numerical numbers

[ 1-10].

These

alter the eigenvalues equations

large speeds. altogether

solve

Typi-

number

at very low Mach

in the literature methods

numbers,

CFD

flow

incompressible; constant

the

of

so as to reduce,

disparity

between

Although by

is

is worth

development,

compressible-flow

appeared

and convective

about

nearly

of compressible-flow

Mach

and

become

virtually

the

preconditioning

the system Engineer

Within altering

performed

compressible-flow

below

becomes

for this is that

the fluid dynamics

acceptable

numbers

somewhere

gas becomes

in flow

have

most

have

intended

new technology

to an

when

historical that a great

is being

research

that

converge

of

incom-

simulation

reason

where

CFD

It is well

numerical of

high

other

algorithms

One

mostly including

and (CFD)

engineering

industry

compressible,

abundance

dynamic

in the aeropropulsion the

computadifferences

for the different different

are

flows. fluids

is, the fluid density

Associate

dynamics

codes

centrifugal

and low-

this

it is an interesting

fluid

computer

The

are discussed.

CAerospace Engineer,

Despite

rigs.

developed

compared.

hydraulic

test

turbomachinery,

flows

A list of such

pumps,

fans and blowers,

of computational

0.1 where

tAerospace

others,

practical

incompress-

low-speed

experimental

pressible

of

artificial

of

involve

fluid flows.

among

been

agreement

results

include,

many

and

operating

Further,

computational

implementations,

good

incompressible

would

of turbomachinery

spectrum

to a well-documented

experimental

Performance

for the near-design

and

formulation

a wide

many of which

incompressible-flow

method

and different

discussed

code

compressor

for a rotating

the preconditioning

are provided,

preconditioned involving

equations

and machines,

turbines,

discipline

involves

or nearly

pressible flow fields, to be applied to nearly-incompressible, low Mach number flows. A brief description is given Navier-Stokes

engineering

dynamics

the

it is possi-

developing

incompressible

the flow

equations, several

the

is that many ious

preconditioned

advantages

and/or

compressible-flow

applications.

tioning advantages

to

include

flow

The

code,

been

implementing Turke117],

the

work

and

layer

pwU'

+_zp

eU'

CFD

pW'

numerical

results

for a centrifugal

sented

and compared

The

uses

the

Only

the

equations RVC3D

the final thin-layer

pressible fitted

equations

coordinate

comprehensive for RVC3D The

code

including

+

pwW'

+ _z P

Details

of the viscous

which

nents

u, v and w point

the pre-

tions,

respectively,

components

11 and 12.

description

can be found

equations

since

to a body-fitted using

by arbitrary

go"

that the _-coordinate surface

is almost

the relative

reference

normal

around

to the blade

compo-

contravariant

direcvelocity

;x

+ rl Y v" + rlzW' u +

velocity

;yV'

+ ;z

(2)

W"

components

=

are:

U

body-

v' =

v-

w' =

w+_y

then _1 ( = cp/c

veloc-

pressure

_Z

(3)

gas

with

v ) is constant,

constant

specific

and the energy

and static

by 2

sime = p

a0,

+

(4)

and

is roughly

paral-

it, while

the rl-

p = (_t-1)

and the _-

resulting

+ v

where

the sonic

density

equa-

I e-_p(u2+v 1

velocity

by the equation

a is related

2 +w 2) -

to static

(5)

pressure

The (1)

tions:

where

NASA

TM-113120

= J-l

lp,

pu,

pv,

pw,

el T

2

metric

terms

and

of state: a 2 = _' -- P P

O = j-lq

heats,

have been

Po,

surface The

an ideal

are given

and nondimen-

direction

direction runs along the blade span. tions can be summarized as follows:

relative

a more

system,

quantities

and wraps

=

U'

for a Carte-

equations

approximation,

sionalized

velocity

by

com-

with angular

(_, _, _) coordinate

the thin-layer

lel to the blade

rotating

are given

in the references

are written

The Cartesian

direction

where

Assuming

ity £_ about the x-axis.

Note

inte-

[11, 12].

Navier-Stokes

absolute

(_v

below.

to a generalized here

and

v' = _xu+ _yv"+ _zw'

W'

of the viscous

are presented

The

and the

V" = _xu

are numerically

as transformed system

J-1L'_

in the x, y, and z coordinate

are given

data.

are summarized

formulation

=

flux vectors/_v

in References

are then pre-

experimental

code

I'I

_y p

pvW'

and

Computed

impeller

+ rlz p

eV" + pV

The

et al. [10].

dissipation

system

plified

pwV'

p

eW' + pW

by

artificial

sian (x, y, z) coordinate mapped

pvV,+rly

Precondi-

the

encountered.

compressor to detailed

governing using

= j-t

+ fix

puW" + _xP

Governing Equations grated

p

+ pU

(_ = j-Â

a thin-

[3, 4], Radespiel

method,

problems

code

[13] to simu-

into

different

some

and

turbulence.

of Turkei

scheme,

schemes,

to

three-dimen-

12],

et al. [6], and Turkel

the

conditioning

+ _r p

compressible

solves

model

incorporated

Radespiel

describes

[11,

turbulence

of boundary

has

RVC3D,

equations

algebraic

the effects

pvU'

Other

of the compressblade row is

of the Reynolds-averaged

Baldwin-Lomax tioning

_=j-1

the capability both

viscous

designated

Navier-Stokes

paper

and

involving

code.

pV', puV

puU" + _,

exist for var-

existing

versatility

fields

U'

to add a precondi-

a three-dimensional

formulation

late

easy

developed for the simulation field within a turbomachine

considered. sional

already

have

of which

flow regions.

In this paper, originally ible flow

codes

already

code

simulate

and incompressible

layer

an

codes

not the least

It is relatively

scheme

directly

compressible benefits,

are defined

(6)

by

the following

rela-

stage

scheme,

the physical

are calculated

The time linear

(7) ynZ¢

-yCz n yCz¢ - y{z¢

XnY ¢ -xCy n xCy_-

xgy¢

J and its inverse

j-I

= x{YnZ¢ -x{ycz

Al'_l -t- At;l+

+ XnY;Z{

A* is the maximum

ular

multistage

time

steps

system

of equations

distributions lute

involves

of total

(8)

condition

specification

lated

upstream At

the

variables,

eigenvalue

without

The

component

from the interior boundary

magnitudes

namely

p,

to the

pu,

boundary,

specified

at the inner

spanwise

direction

(hub)

using

v 0 . The

five

conservation

and

pw,

are

extrapolated

and

the

static

pressure

boundary

simple

and integrated

radial

_

the

to the largest

inviscid

equations

=

+ae k

IU'I

a_rl

(11)

At_l = X¢ = IW'I +ao¢ where

iu'i : ÷ [_yV'/+ I zw'l iv'l = In,u!+ ]nyV'l ÷ In w'l iw'l= i;xu!+l;yv'l+lg=w'!

is

in the

2

:

Pv0 r

(12)

and

equilibrium:

-_2

dp dr

(for

AtnI -- Z.n -- IV'l ÷

(non-

the

pv,

one-dimensional

are equal

and abso-

Riemann-invariant component is extrapoof

inverse

grid direction

for the partic-

above

to the inlet boundary.

four

number

preconditioning):

of the spanwise

velocity

exit

downstream

for the

total temperature,

preconditioned) upstream-running based on the meridional velocity

bitl/'

_

pressure,

circumferential

A,_I+

Courant

scheme.

for each

Ate 1 = k{ boundary

as follows:

(10)

n

are

n - XrlY_Z _ - x;ynz

(inflow)

from a three-dimensional

can be expressed

x{y n - xny _

J-I

+ x;Y{Zn

analysis

A*

local

inlet

terms

At