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Dec 23, 1993 - Figure. 3.1. Domain partitioning into a discretization, a) hexahedra on R3, b) composite hexahedra ...... Figure. 7.20b) presents in perspective.
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dE- I;;-9/Y AN ARBITRARY GRID CFD ALGORITHM FOR CONFIG URA TION AERODYNAMICS ANALYSIS Vol. Prepared by: A. J. Baker, G. S. Iannelli, COMPUTATIONAL

1.

P.D. Manhardt

MECHANICS

Theory

and

Validations

and J. A. Orzechowski

CORPORATION

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Field,

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Prepared for: NASA Ames Research Moffett

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0 0

Center

California

r7

FINAL REPORT FOR SBIR PHASE CONTRACT NO.: NAS2-12568

II

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>-0

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¢1[ tJ.. Z_Z

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Report: CMC TR2.1 December 1993

COMPUTATIONAL 601 Concord Knoxville,

Street, TN

Suite

37919-3382

Phone:

(615)

FAX:

(615)

emaih

ajbaker

- 94

MECHANICS

546-3664 546.7463 @comco3.akcess.com

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u'_ "r tr_

.--

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0

CORPORATION

116 U.S.A.

i ¢_

,_

@,

Table

of Contents

Abstract iv

Nomenclature Figures

vi

and Tables

ix Page

INTRODUCTION

lo

o

G

.

.

1

THE AERODYNAMICS

PROBLEM

2.1

Synopsis

2.2

Conservation

2.3

Turbulence,

2.4

Non-dimensionalizafion

2.5

Canonical

2.6

Well-posed

5 5

law

systems

5

Reynolds-averaging

8 9

form

10

boundary

APPROXIMATION,

conditions

ERROR

3.1

Overview

3.2

Approximation,

3.3

Error extremization,

3.4

Spatial

3.5

Fully

3.6

Summary

14

CONSTRAINT

15 15

measure

of error

the weak

semi-discrefization, discrete

form,

finite

algebraic

15

statement volume,

16 finite

statement

element

17 21 23

WELL-POSEDNESS,

STABILITY,

4.1

Overview

4.2

WeU-posedness,

boundary

4.3

Stability,

dissipation

4.4

Accuracy,

4.5

Stability,

4.6

Summary

THE

STATEMENT

CONVERGENCE

25 25

Taylor

asymptotic artificial

REM/AERO

5.1

Synopsis

5.2

Finite

element

5.3

REMI

algorithm

conditions

25 29

convergence

32

dissipation

33 39

CFD

ALGORITHM

43 43

TWS h algorithm matrix

statement

nomenclature illustrations

43 51

Page

5.4 The REMI 5.5

6.

algorithm

56

Summary

64

AUXILIARY 6.1

RaNS/E

PROCEDURES,

LINEAR

ALGEBRA

65

Synopsis

65

6.2

Initial

condition

generation

6.3

Implicit

6.4

Equilibrium

6.5

Tensor

6.6

Summary

7.

DISCUSSION

7.1

Synopsis

7.2

Subsonic

7.3

Transonic

7.4

Supersonic

inviscid

7.5

Hypersonic

Euler

7.6

Viscous

8.

SUMMARY

Runge-Kutta reacting

matrix

65

algorithm

67

air algorithm

70

factorization

71

product

77

AND

RESULTS

79 79

inviscid inviscid

transonic

AND

verifications,

d=2,3

verifications,

79

benchmarks,

verifications,

d=2

88

d=2

93

verification,

validation,

d=2 axisymmetric

100

benchmark,

validation,

d=2

104

CONCLUSIONS

111

References

117

Appendices A.

AKCESS.AERO

B. TWS h FE REMI C. AKCESS.AERO D.

AKCESS.AERO

REMI

template,

algorithm, template, REMI

d=2, Newton

122

d= 1,2,3

133 137

d=2, TP quasi-Newton template,

°°.

III

quasi-Newton,

d=3

151

Abstract

The

solicitation

evaluation

of

aerodynamics fitted to

for a

CFD

derivation

law

Phase

applicable

an "arbitrary

weak

possessing

completion of all theoretical calculus and vector field theory



intrinsic

embedding

of

methodologies

a continuum

Galerkin

approximate

solution

I project

(TWS)"

CFD

weak

in

the

capturing

statement

algorithm,

applicable

published

numerical

stability law

specification

useable



a fully discrete theory algebraic quasi-Newton iteration method stationary relaxation solvers

system eligible for any appropriate using sparse, block-banded and/or

Phase

the

I project

finite

conservation



any time

discretization,

results

provided

implicit,

element)

explicit

theoretical

spatial

Euler,

laminar

conservation



a diagonal construction



well-posed statement stability

flow

and

semi-

or multi-step

foundation

specific options for coding and verification in a Phase II project. theoretically independent option was selected as follows: •

for The

selection

level

Navier-Stokes

Reynolds-averaged

law systems

scalar

simplification

boundary

to the

conditions

for

Euler/Navier-Stokes methodology

TWS

numerical

continuum

constructions

iv

to

Navier-Stokes

employing

extremizing

volume,

contributed

amenable discretization

The

(finite

completion

body-

continuum

and

error for any approximation

any

a well-ordered,

attributes:

previously

for shock

critical

configuration

Reynolds-averaged

details

sixteen

requiring

Phase

and

and



with

to

The

design

three-dimensional

not

the following





grid

Euler

requested

to

statement

potential,

systems,

dissipation

I project

for robustness."

a "Taylor

transonic

conservation

using

system of

SBIR

algorithm

analysis,

coordinate

unsteady

the

dissipation

Taylor using

weak

Lyapunov

of

for each

a finite

element

spatial

quadrilateral distortion



single step 0-implicit time discretizations



a block-banded, matrix

original

verification

proposed

achieving

this goal

details

and

years,

from

for

extensible

to

the

the given The

associated

with

(only)

research

code

details

and to establish

the

cost, new

(FEMNAS)

was

Newton

proved

the

product arbitrary

Runge-Kutta

iteration

verification,

algorithm

via

benchmark

prototype

Phase

project

and

three-dimensional

II project

a practical

flows evolved,

starting

point,

was

In

necessity,

difficult

to

accurately

out

a

to confirm

especially

apparent

develop

the

necessary

the

two

myriad

operational The restriction

tensor

for

for

two-dimensional

results.

since

that period,

continued

extension,

and validation

and

performance

to thrash

and

fighter

very

formally

this

developed

benchmark

became

needed

coding

a generic

in the two-year

time

algorithm.

anticipated

about it

therefore

to provide

was

verification,

to two-dimensions

of

project

at no added

jacobian

implicit

quasi-Newton

three-dimensional

As

impossibility.

additional

tensor

potentially

two-stage

strategy

scope

for turbulent configuration.

an

a B-stable,

linear

of

geometries.

lex-delta

was

mesh

problems,

aerodynamics

The

and

using

elements

products

block

validation

hexahedral)

mesh-sweeping

tensor

algebraic,

@

semi-discretization

(hence

product

quasi-

arbitrarily-distorted

meshings. The

results

of these

fundamental

hypersonic

inviscid

and

verifications

reported

herein.

Newton

tensor

product

verification

and

operational

readiness

and

2-D

simulations

laminar-viscous Additionally,

jacobian

benchmark in the

flow

tests

are

the

fully

results.

production

test

cases

and

presented

herein,

along

3-D

theory code,

supersonic

constitutes

algorithm

AKCESS.,,

V

transonic,

3-D

The

detailed.

for

is only which

the

associated with now

and major quasi-

very

basic

approaching

is briefly

described

a

NOMENCLATURE a

A Ak B C CpCv

d D e

eh E

scalar

convection

element

matrix

kinetic

flux vector matrix

Courant

number,

specific

heat

dimension

mass

volume

4

kinetic

mesh

weak

measure,

turbulent dissipation

Im

mixing

mi

momentum

P

pressure

Pe

Peclet

kinetic

energy,

degree

conductivity

length

scale

differential

mesh

finite

polynomial

length

m

N

turbulent

length

reference

normal

space

basis eddy

ID

(resolution)

enthalpy

conductivity,

I'1

error

statement

thermal

Mach

data

factor

function

mass

element

(resolution)

Sobolev

Mk

domain

flux vector

trial space

M

d=3

total energy

flux vector

Galerkin

partial

prefix,

number

amplification

Z(.)

matrix

number

GWS

L

element

total energy,

specific

g

kt

d=2

approximation

dissipative

k

prefix,

matrix

semi-discrete

Euler

jacobian

capacities

specific

Eckert

d=l

of problem

diffusion

Eu

HP

prefix,

element

Ec

h

speed,

measure

equation

function vector

matrix,

(resolution)

collision

reference

factor,

number

coordinate element

basis

number

vi

function

molecular

mass

Pr

R R

RQ Re

Prandtl

number

polytropic

gas law

universal

gas law

residual

weak

Reynolds

source term

St

Stanton

t

time

ui U V

constant

statement

number

$

T

constant

number

temperature velocity

vector

reference scalar

(resolution)

velocity

speed

V

convection

INS

weak

wi

expansion

coefficient

xi

coordinate

system

Y

wall normal

Yi

species

-PUiUj

Reynolds

stress

Uiuj

kinematic

Reynolds

matrix

statement set (resolution)

coordinate

mass

fraction tensor

vii

stress

tensor

Ct

A

8V 82 V2 Vh E

artificial

dissipation

parameter

artificial

dissipation

parameter

central

difference

Kronecker central

difference

laplacian discrete

dissipation

alternator

0

implicitness

tt

absolute

¢a

potential

vi

operator level

wavelength

viscosity,

artificial

viscosity

density function

test

space

function

set

trial

space

function

set

domain boundary (t}

II

vorticity, column

i} T

row

[] FJ

wave matrix

number

matrix

square diagonal

operator

parameter

mode

P

derivative

tensor

kinematic eddy

operator

viscosity

Fourier

vt

second

divergence

error,

derivative

operator

Eijk

V

first

delta

matrix matrix

viii

dissipation

parameter

List

of

Figures

Figure 3.1

Domain

partitioning

into

Page 2O

a discretization,

a) hexahedra on R 3, b) composite hexahedra and subdivision into five tetrahedra 3.2

Tensor

product

dispositions, 4.1

Amplification

element

and phase

various weak from Chaffin 5.1

finite

domains

a) two-dimensional,

with

and node

eight

nodes

coordinate

20

b) three-dimensional

velocity

error distributions,

41

statement algorithms, and Baker(1994)

Finite

element

space

(_e)

5.2

Gauss

symmetric

5.3

AKCESS.AERO

REMI

template

for {FR}e

54

5.4

AKCESS.AERO

REMI

template

for {FM1}e

57

6.1

AKCESS.AERO

REMI

template

jacobian

6.2

AKCESS.AERO

REMI

template,

7.1

Converging subsonic b) d=3, 4:1 area ratio

7.2

AKCESS.,

domains

for tensor

in physical product

basis

quadrature

template

duct

space

46

for d=2

47

form

coordinates

[RE,E]e,

TP jacobian

verification,

for REMI

(D.e) and transform

76

d=2

[RE, ETP],

a) d=2, 2:1 area

d=2

77

ratio,

80

81

d=2 IC generation,

a) nodal density, element-averaged metric data, b) average density, Gauss quadrature element matrices. 7.3

AKCESS.,, template for REMI d=3 IC generation, density, Gauss quadrature element matrices.

7.4

Converging

duct

a) modestly

non-cartesian

7.5a

IC algorithm Main

validation

TWS h density

-- 0.2, averaged

metric

check mesh

case,

d=2,

A, b) highly

solution,

84 distorted

d=2 converging

template,

ix

82

averaged

a) mesh

A,

mesh, duct,

b) mesh

B. 85

B

7.5b

IC algorithm Main

7.6

TWS h density

=_0.2, Gauss

REMI

FE

7.8

duct,

Total

pressure

duct,

Main

duct

nozzle

verification

REMI

deLaval

nodal

nozzle,

algorithm,

REMI

B, d=2,

at=O.O05,

B.

solutions,

87

B, d=2, a) [5=0.3, b) _=0.2 89

converging

a) _=0.3,

b) _=0.2.

--- 0.2, 4:1 area

ratio,

solution

velocity

with

a) cross-section

for axial

REMI

pressure

solutions,

pressure

solution

A,

9O

distributions,

91

momentum.

TWS h solution a) t=0.4,

number,

92

15% parabolic

94

for Mach

b) t=l.0,

c) t=1.2,

e) t=2.8

algorithm

arc, scalar

TWS h Euler

_=0.2

c) entropy, REMI

Euler

problem,

unsteady

d) mesh

d=3, Main

b) REMI

deLaval

d) d=1.8,

7.12

solution, [5=0.3.

IRK ODE

7.11

verification,

scalar

c) mesh

_, Mesh

_ d=2, mesh

overlay,

duct,

steady-state

= 0.2, scalar

---0.2, scalar

Converging

b) steady 7.10

Main

Euler

86

d=2 converging

template,

loss error for REMI

a) IC density

7.9

quadrature

TWS h algorithm

converging 7.7

solution,

d)

solution,

{1} T a) 65x35 Mach

algorithm

mesh,

number,

e) axial

TWS h Euler

solution,

arc, Main

= 0.68,

scalar

of a) axial

momentum,

_=0.2,

steady-state, b) velocity

steady-state,

{1} T perspective

b) transverse

vector

field,

momentum. 15% parabolic

and contour

graphs

c) Mach

number,

momentum,

95

d) pressure. 7.13

REMI

algorithm

TWS h Euler

solution,

0 = 20 °, [5 = 0.3 {1}T, a) initial isoclines, 7.14

REMI

c)

1st adapted

algorithm

mesh,

TWS h Euler

0=20 °, [5 = 0.3 {1} T, a) final contour 7.15

REMI

and perspective algorithm

65x35

[5 = 0.3 {1}T, solution-adapted isocline distributions.

uniform

d) resultant solution,

adapted

graphs,

TWS h Euler

supersonic

mesh,

65x35

X

mesh,

b) density

supersonic meshing,

96

isoclines. wedge

number,

flow,

b) density

density

supersonic

c) Mach

solution,

wedge

flow,

98

isoclines;

d) pressure. shock

resultant

reflection, density

99

7.16

REMI

algorithm

TWS h Euler

_a = 0.3 {1}T final mesh, a) pressure, b) entropy. 7.17

REMI

algorithm

blunt-body

surface 7.18

algorithm

REMI

arc,

7.21 7.22

algorithm

_=0.2,

boundary

algorithm

shock-boundary separation

Sub-grid wave, k=l;

region

viscous

Burgers

shoch

trailing

4% parabolic

of a) axial

107

shock

a) density,

laminar

Main

resolution;

comparisons

symbols

are data from

problems,

= 2.15,

inviscid

k=l or 2, b) p-embedded Re=105,

xi

108

b) Mach

velocity

simulation,

momentum,

problem.

to 103, d) p-embedded

106

edge.

interaction,

FE verification

105

momentum,

layer

WS n solution,

or 2, solution, converged verged to 10 -9 .

4% parabolic

b) axial

laminar,

solution,

c) skin friction,

p-embeddin_ a) standard

mesh,

mass

space.

validation

TWS h Navier-Stokes

b) surface pressures et al (1987) 8.1

layer

surface

presentations

near

layer, Re=105, 13= 0.3,{1}T c) axial momentum.

Supersonic a) REMI

closeup

103

species

viscous,

solution

perspective

mesh,

Ma_=8,

real-air

laminar,

in nodal

a)

102

density.

streamline/body

b) companion

101

perspective

steady-state,

TWS h Navier-Stokes

c) pressure

Shock-laminar

boundary number, 7.23

plotted

Re=4.0xl06,

REMI

stagnation

d)

_=0.2{ 1 }T, a) non-uniform

momentum

b) pressure,

solution,

hypersonic

Ma_=6.5, and

number,

TWS h Navier-Stokes,

arc, Re=4.0xl06,

REMI

meshes,

reflection,

graphs,

Euler solutions,

quad

Mach

of temperature,

algorithm

c) axial

of c)

real-air

state

shock

perspective

= 8.0, contour

TWS h Euler

and

distributions fractions

7.20

Ma_

supersonic

and

65x35

distribution;

a) ideal-air

7.19

adapted

distributions

REMI

contour

TWS h steady

flow

b) density

solutions,

c) standard solution,

Re=105,

109

on Degrez

square solution, WS h, k=l k=l,

con-

116

List of Tables Table Page

2.1 4.1

Euler-admissable Dirichlet boundary conditions Summary of CFD algorithms from Baker and Kim (1987) Gauss

quadrature

FE k=l basis

within

coordinates

interpolation

Taylor

and weights,

matrix

[M200]

xii

(BC)

weak

statement,

14 40

d=2

47

for d=1,2,3

48

I. INTRODUCTION

The work and

plan

broad-range

efficient

validation

on

regions, with

bounded

the

of

appeared

development

projects

of the

iteration

Both

was

with

relative

the MacCormack

diffusion,

viewed

genuine

viscous

splitting

methods

dissipative

as

combination

of Riemann

solvers,

However,

differencing,

and

order

These

spatial

nominally-uniform

aerodynamics

accuracy

various

volume

following

a variety

a

CFD

detraction This

boundary-fitted geometry.

law

was

were

meshings

CFD

algorithm,

to

to

the

Euler which

meshing-induced

total

vectorizability.)

1970s,

leading

algorithm,

to

designed

employed

for

development

of flux

with

Roe

extension were

(1981),

via e.g.,

some

differencing.

specifically-added

expansion

vector

developed,

upwind

to

to

to RaNS

all employing

directional

intrinsic

artificial

compromise

(1978),

of

added

a potential

variants

absence

artificial

characteristic-direction

were

developed

using

finite

to avoid

differencing. code-implemented The on

abiding

the

coordinate

An alternative

historic

middle

Many

was

The

to its

theories and

and

exceptional

stiffness.

systems,

for stencil

upwind

applied

due

the

prompted

Osher

semi-discretizations. cartesian

commensurate

computer,

factored

CFD

diffusion

algorithms

in

implicit

averaged-states

of switches

of direct

using (RaNS)

an

explicit

due

to parasitic

(1982),

numerical

to

witnessed

applications

initiated

(1976)

feature

and

(1969)

differencing.

vanLeer

versatility

has

a rebirth,

conservation

distinguishing

diffusion.

enjoyed

analysis.

central

application

of three-dimensional

requirement.

RaNS

a theoretical

(1978),

common

analysis

insensitivity

for hyperbolic

Steger-Warming

personnel

and Beam-Warming

both

and

Navier-Stokes

meshing

technical

thereby

aerodynamics

flux-vector

with

community

Beam-Warming

on

accurate

geometries

of discretizations

CFD

for

were

was

three-dimensional

the MacCormack

since

focus

stable,

coding

itself.

CFD

(It has

research

efficient

both

for derivation,

Reynolds-averaged

surfaces,

the

called

CFD algorithm, The

generation

inappropriate

stiffness.

Several

finite

decades

development

originally

low

two

aerodynamics

aerodynamics

parasitic

was

aerodynamics

resources,

configuration

and

of a CFD algorithm

past

expenditure

the

by

the capabilities In

The

to this

element

for general

laminar,

Adjunct

project

meshings.

analyses

(Euler),

simulations.

finite

arbitrary

aerodynamics flow

II contractual

of a new

absolutely

configuration inviscid

of this SBIR Phase

character

transformed

transformation spatial

semi-discretization

was

difference

use of structured,

computational from

the procedure

domain, transformed emerged

or

in

the

1980's,

using

in the

physical

codes

were

domain, time

specifically

"hp")

accuracy

with

This memory extra

solution

steps

Without

exception

cartesian

mesh

splitting

implicit

to compute the

triangle/tetrahedra

From

with

the

Phase

algorithm

that

mathematical

extension

to (meshing

hopefully

leading

weak

have

to attainment

statement

We the

goal,

moved and

to use

stiffness. to

as large and

structured

of operator-

Coincidentally,

finite

volume

CFD

to derive,

code

and validate

mesh

versatility

and

arbitrary

quality genuine

(Euler)

shock-capturing,

RaNS

applications.

of approximation

flux vector

linear basis, tensor of quads/hexahedra

product

tensor product iteration block

mesh,

implicit

manipulation

two step

finite

derived

meshing

the above

solution

and The

the

direct

decisions

error to produce

conservation law systems with intrinsic boundary conditions, suitable for Euler

i.e., an arbitrary

such

procedures

at least,

sought

exhibited

for extremization

and

considered

enhanced

of this goal were:

0-implicit one step discretizations

algebraic,

for

issues

has

amenable

enforceable via weak statement generated ideal and real-gas equations of state

matrix algebra

FE basis

refinement/de-refinement.

become

11 project

for)

Euler/RaNS

continuum well-posed

for mesh

parasitic

robustness,

requirements

Taylor-series

and

estimation.

to handle

since

Galerkin applications

meshing

computational

regions,

employed

et al (1991).

in 1987,

of a CFD

adaptability

methods

refined

promising

applications

layer

application

Taylor

generation/adaptation data

RaNS

in boundary

meshings

view

mesh

measure to

integration

c.f., Barth, this

ingredients

extension

embedding time

error

and

elements

and

the

Locally

error

by associated

stiffness,

from

appeared

via local

The

simulations

demonstrated,

efficiency

parasitic

Euler

method.

methodology

of finite

et al (1986).

derived

were

is moderated

requirements,

constructions,

tetrahedra

adaptive

promise

to

dissipation

degree-of-freedom

bright

restricted

meshings"

Oden,

of the Lax-Wendroff

and

solution

non-structured

et al (1984),

generally

1984)

triangles

(termed

Loehner,

numerical

(Donea

nested

"absolutely

e.g.,

explicit,

added

generalization using

potentially

surface

combination,

element

spatial

to solution

if successful,

configuration

aerodynamics

2

integrals

Runge-Kutta

quasi-Newton

amenable

dissipation and RaNS,

time

semi-discretization

block-banded

linear

adaptivity

would CFD

lead

to attainment

algorithm,

applicable

of

f

to both

Euler

and

RaNS

circumvented

some

mathematical

robustness,

However, goal

was

the

conservation

detractions

severely

after

progress

included

practical

difficulties and

factors

also

spanning

in the

procedures

on "arbitrary"

additional

key areas

were

contributing of numerical

issues

contributed

years.

During

this period,

to the

both

and

Benchmarks

necessity.

The

Navier-Stokes generated

CFD

limited

approximation block

all

subsonic

tensor

this

slow

(6 months)

and

computer

(before

and code

practice

clean

theory

proved

product

matrix

algebra

lead

a no-cost

theoretical

effort

to resolution and

were

by Iannelli,

hypersonic

issues,

executed

form

as

Euler

and validations

over

algorithmic

to two-dimensional and

extension

pursued

of many

validations

supersonic

viable,

stiffly-stable Euler

stiffness inverse

family.

generated Reynolds

constitute

a comprehensive,

element

to two-dimensions,

and

in transonic,

by

his

a practical

and

reported

The

The

laminar

herein

and

second

It is useful via boundary number.

and definition

resultant

evaluations

order

were

for shock

The

meshing range

of the developed

the

replacement

of

with

transverse

these

results

arbitrary

3

as well

grid

finite

algorithm..

form

(ENO)

is achieved, for the

_term

non-rectangular

computational

weak

time-marching

capturing

flows

are highly

the

of the

non-oscillatory

diagonal

leave on

accurate

REMI

hypersonic

meshings

flows

Runge-Kutta

layer

and

assessment

CFD

essentially

a simplified

far-field

appropriate

implicit

quality

with

positive

statement

supersonic

remeshing

supersonic

via

weak

high

TWS h theory.

oscillation, integrals.

derivation

central

to request

for benchmark

finite

solution-adaptive

surface

of

results

to shocks

underlying

without

step

statements

need

limited

transonic,

arbitrary-grid,

Although

and

of

problem

reported

developed

of the

range

code,

to

project-

by this code.

The

using

computer

and

to the

workstation

elegantly

of the

two-dimensional

Theoretical

as the

a dedicated

(1991),

FEMNAS

available).

capability

accrued

Ames

for

meshings.

his dissertation practical.

potential

contributing

3300

selections

meshings.

a fledgling,

factors

dissipation

distorted)

exhibiting

meshings

NASA

role,

these

use of "arbitrary"

only

regular

on the

view,

the verification

of our SGI Model

speed

(highly

that

Practical

as he developed

developed

and

to achieve

years.

several

theoretical

while

efficiency

operation

a central

constructions,

on rather

T-1 communication

played

In our

to the extent

delivery

in remote

descriptions.

required

two

delayed

incomplete

These

of detail

capability

a period

NASNET

operating

under-estimated,

verification/benchmark end

of previous

code

volume

law

domain

statement-generated

algorithm

appears

for the

0-implicit

as handling resolution are

discussed

element

CFD

the

a truly single parasitic

on the

order

following algorithm.

As these operational

advances

in

our

became

achieved,

emerging

the

AKCESS.,

software

previously

established

"research"

codes.

algorithm

is thoroughly

detailed

herein,

quasi-Newton generated move to

jacobian. by AKCESS.*

rapidly

3-D,

as

to recovery

Only

modest

to date

are

of reported

AKCESS.*

moves

to

three-dimensional

The

platform,

resultant

including

the

and

FEMNAS

3-D

readiness

to

TWS h CFD

"REMr'

FE

product

its

results

we

as benchmark in

factorized

numerical

However,

as well

become

"code"

Euler

included.

has

successor

tensor

3-D

tests,

operational

the

3-D

verification-level available

algorithm

expect

to

extensions

parallel-processing

implementation. The

near-term

to greatly

shorten

we

warrants

hope

project required

completion. to convert

emergence the time

of this versatile

software

to implement/validate

the

significant

We

have

CFD theory

theoretical

government

certainly

learned

to genuine,

and

specifically

and/or

personal

a lasting

robust

4

platform,

lesson

and convergent

designed

practical

resources

committed

on estimating code

musings,

the

practice.

to effort

2. THE AERODYNAMICS

PROBLEM STATEMENT

2.1 Synopsis The goal and

is to establish

Navier-Stokes

a robust,

conservation

accurate

law

systems

configuration

aerodynamics

problem

statement.

mathematical

descriptions,

including

closure

and dissipation leads 2.2

mechanisms.

to mathematical

The

Following

law

basic

assumption

is

and

energy,

dissipative

and

modeled-turbulence

variable,

Denote as

q=q(x,t),

that

set

"q"

(PDE)

differential

and

in

equation,

spanning

a region

For where Therein, where

u i is termed Continuing

is the array,

included

state the

in

a

associated

thermodynamics

an eigenvalue

conservation

law

analysis

systems.

statements

the

variables,

the

expressed

as

desired usually

denote

i.e.,

the

state array

density,

L(-)

an

and is the

of

Euclidean

and/or

in (2.1), for laminar

variable

the

the

(matrix)

and

the

scalar

state

Then,

the

non-linear,

partial

x

corresponding

flow

specific

dependent

with

denotes

scalar

the

Finally,

is

vector

{p.m, EIT" transpose.

energy. f, while

s is a source

fjv term

Both

functional

tensor

real

m i = Pui,

modeling.

of state

cartesian

its

resolution

total internal

fv.

positive

q(x,t),=

kinetic flux

equation

heat transfer,

R + the

as

on closure

description,

a

t denotes

with

flux vector

of

9_d (1< d < 3), and

"T"

of the

form

system

selected

vector

resolution

Navier-Stokes viscous

the

coordinate

superscript

E=pe is the volume

as needed,

and

and

called

homogeneous

(global)

space

is usually

of the dissipative

for generality

mass,

mathematical

array.

governing

denotes

resolution

m is the momentum

"velocity,"

for

for thermodynamics,

yields

to

system,

following,

resolution

For the Euler

flow

for

domain of definition of (2.1) is R + x f2 with by I and Dc _d _ (x.lxl,,

+

_

_

_

#

=

I +

I

I

I

_

°

_

÷

I

.c:

I i ,4_

N I

A

O

.... I

_

=

I

_

=== I

.=,.¢

_3

L)

0

_

0

_

0

O ;>-.

33

¢=

.c

"_

.u L3

o

"

i O_GtNAL

PAGE

OF POO_ _m/UJl_ 4O

F_

0.1

I

-0.I

I

l

FD,All FE,Beta=0 Linear FE, Beta=. 1 Linear FE,Beta=.2 QUICK-5 FV QUICK-3 FV

I

..... - m--. ---x...... -_--.o-\

"i1:_

-0.2

C=I/2 -0.3

8=-I/2

-0.4 w

I

I

I

I

16

8

4

3

Modal

I

Wavelenmh

l

2

(_=nA x)

I

I

C=1/2 0=-112

o

.W •o ooO o° oO s

..4} .-

.-°"

._

°-

.-

! 16

8

i 3

4

Modal Wavelength Figure

4.1

Amplification statement

and

phase

algorithms,

velocity Chaffin 41

2

(X_-nA x)

error

distributions,

& Baker

(1994)

various

weak

5. 5.1

The

REMI

and

FE TWS h CFD

theoretically

algorithm

described.

corresponding

is coined Momentum

"REMI," the acronym _}T for 1< I _