A dual reciprocal boundary element formulation for viscous flows

N94-23666 A DUAL

RECIPROCAL

BOUNDARY

FORMULATION

FOR Olu OLTech

ELEMENT

VISCOUS

FLOWS

Lafe' Corporation

Innovative 11795

Computing Sherwood

('hesterland,

Ohio

Group Trail

44026-1730

SUMMARY The

advantages

exploited

inherent

to solve

ciprocal

technique

unknown

coefficients converts

via

distribution

These

iutegral

kernel

is the

convective the

delermined

consist

flow problem. portion

of the

by using

boundary

integral

1)y collocal

ion.

differential

of two of the

governing Dual

The

parts.

The

PDEs.

aud

is a _iomain

The

domaiu

(DR)

which

(I'DEs)

into

strength

is a boundary

sources

part

by a global

of unknown first

for potential

ion of a so-called

apl)roach,

equations

fictitious

Reciprocal

(BEM)

iul ro(lucl

represented

or dil)o[es

-I'he second

the

is the are

sources

strength

,Method

trick

terms

partial

of fictitious

equations

boundary

pure

are

unknown

Element The

the convective

the governing

convection-free

Boundary

flow prol_lems.

in which

erative, the

in tile

viscous

the

integl'atiou

concept.

The

whose

is necessarily

it-

equations

on

boundary.

the

term,

whose

solution

whose

can

Re-

integral

integral te_'m

are

Dual

function

fuudamental

integral

flows

kernel

of a is the

be transformed

resulting

to

formulation

is a

process.

computational

INTRODUCTION The

major

of the of the BEM

advantage

computation problem. has

1. the

to the The

received

the reduced advantages

tlw

e|fici¢,ncv

2. a much

to halldh,

reduced

3.

the

ease

with

4.

the

restriction

aml

with the

large

t,,chniques

problems

domain

is the

iu the

in continua

li1_'ral urn' during

for no special

confinement

effective can

the past

dimension

1)e solved

decade.

discretization,

using

Apart other

from

derived

(Iomain._:

matrix:

siugularities

are

handled: errors

good

as the

5.

the

rol)ustness

6.

the ability to find solutions a l)osteriori 1)y the domain (liscrel izal ion:

wheu

other

is 1he rethwtion

linear

in the

need

over

resuh

which

of llw (lis('retizalion descril)t

enjoys

The

meution

infinitely

coefficient which

al)l)roach

l)(mn¢lary.

COllsiderable

dimensionalilv include: ability

BEM

ion of the

boundary

COml)lex

geometries

t()tile geomet are

477

so that

the

solution

is as

rv:

involve(l:

at desired

t President

boundary,

points,

m)t

at

nodes

predetermined

"

7. tile great latitude in solving transient dent

hmdamental

domain Efforts

a decade

e.g..

Brebbia

Lafe

et al..

[1981]:

La&

et al..

some the

tim

The first

integral

focus

Lafe

was

£

('ahan The

of BE3I credit

suggested

for

£

litmar

on

prol)lems

Reciprocal

Dual

Reciprocal

an innovative

approach

problems

such

as

recent.

potential

continua

methods

which

a major

path

were

problems.

utilized. This

set of complete PDEs. door

In this Method

made

and

coordinate

Excellent to the

This

author

(e.g..

Cheng

still

require

for exploiting

flows.

No

domain

is followed. goes

COiiC_'I;{

for tral|sl}irming

results

application papeL

difficult (see

funcl ions which

have

have

been

of BE._I

we l)rtment

(DRBEM).

convergence

his co-workers

to .Vardini

domain

£ Brebbia

integraIs

obtained

Io a wide

the

[1982]

to tile bound-

spect

Let the flow region are: •

('ontinuitv

is represt'ntecl

by 9. and

have

on a fanlily

COml)lete

of con_l)lex

of the

convective

GOVERNING

the

rum

for a class

_! rd.. [199:i])

l)oeu tested with

full formulation

for incoml)ressible

or impossible

('heng

l)ual

set.

of strongly This

et al., func-

of nonlinear

recently work

derived

a

nonlinear opens

the

flow prol)lems. Reciprocal

Boundary

Element

flows.

EQUATIONS the Imundary

is F. File

pertinent

flow equations

Equation

V.v=0 • ('onservation

where

(see (e.g.,

ary. However, until recently, prior investigations (see r..q.. Brebbia [1991]; Partridge [1992]) did not make use of a complete Set ot_gloimi ftmctions. A series Of local radial tions

For

flows

for nonlinear

as convective

approach

BE3!

such

creates

depen-

in a transform

are quite

t'ornmlatious

iterative

techl|ique

time

procedure.

in heterogeneous

heavily

Reciprocal

Dual

marching

BE3I

or those

relied

lhe appropriate the technique

problems

[1981]).

nonlinear

the

a time

to nonlinear

on

Liu

[1990])

Dual

to solve when

or c) using

largely

have

by a) using

iou: b) a pl,lying

techniques

Liggett

[1987-1992]

is involved original

or Fourier):

[1.)_4].

integration.

advantages

integration

main

et al..

domain

who

Laplace boundary

over

[1984]:

solut ion in tile fonmdat

(e.g..

at applying

probh'nls

of Moment

v is the

viscous stress the form:

velocity, tensor.

(1)

um

_)v

!

0"-7 +(v'V)v=-

Vl,+-V'r+g p /9

p is pressure, I[" 1_ is viscosity,

i

(2)

g is tho _raxitalional th¢'t_ [or a Newtonial|

accolevator

vector,

r is the

fluid,

r is expressible

in

r = pVv Dimensionless

Equations

Let L = characteristic length scale, i: = mean (x). We can define the following dimensionless

x. = x/I. 478

velocity, variables:

and

q is tlw

elevation

of tile

point

(3)

v.

= v/_

(4)

i,. = (v+ r_,l)/(S) t. = _UL With

these,

tile above

conservalion

statements

can

V.v. t)v. t)t--"[ + (v..

(5) (6)

I)e made

dimensionless:

= 0

V)v.

(7)

= -Vp. +

V2v.

(8)

where R, The

governing

equations

= p'fL/tl

= Reynolds

call l)e rearranged

and

V'"(I,(x..

t.l

Numl.)er

writ tell

= F(x..

(9)

in tile

pseudo-Poissou

form: (10)

t.)

where _ _ v. t p.

X'elocity Pre.,csure

(11)

and F =

The

pressure

of the

IL [/)v./t)/. + (v. -V. [(v. g')v]

equation

momentum

is obt aiu(,d

equation.

• X-)v.

by introducillg

Note

that

in the

two (for 2-D and axi-synltlletric problems) now drop the • prefix in I},, dimensionh,ss For most -

flow problems

• Diriehlet

Boundary

+ _1,.]

Xelocity Pressure

the c(mt inuity velocity

equation

equation.

conditions

into

4) an(I

or t lifo(' (for 3-D prol)lenls) variables, for ('onvenieuce.

th(" Imun(tary

(12)

Equation Equation

will generally

tile divergence

F are

vectors

COml)onents.

consist

of three

with We will

types:

(I'.) ¢=¢,,

• Neumann

Boundary

(I'_)

Q=--=@, /)n where

?)¢b//)n

• Mixed

(r._l)

= X-qb. n. and

n is the

tu,it

w'ctor

C(O. V'(I,. x.t) where ( is some sl)ecilicd fuu('tion. A free-surface iterative schemes it is usual to recast tile 3Iixrd the

Dirichlet

or the

X('tmtanl_

types.

479

normal

to the

boundary.

= 0

will I)o an example I,outldarv condition

of the in the

third. form

In most of either

BOUNDARY We

will use

If fictitious into

the

equation sources

integral

(10)

INTEGRAL

as tile

of strength expression

representative

u' are (see

q>(x) = fr w(x')g(x, where

g is the

free-space

Greeu's

PDE

distributed

.lasu'on

EQUATIONS ill developing

around

,(: S vmn)

[1!)77]):

x')dx'+

.fn F(x')g(x.

function

which

must

the

F. equation

integral

(10)

can

equations. be converted

(13)

x")dx"

satisfy:

V2g(x.x ') = ,6(x.x') where

(5 is the

solution

Dirac

to equation

delta (l-l)

fimction

is ( (;re_'_d_el_g

g(x.

in which convert

r = this

(Chenget DUAL

Ix -x'

into

al..

I.

applied

x')

The

-

term

on the

a point

x' and

Mt

at

x.

The

closed

form

[1971]):

Inr/2r: l/(4r, r)

last

all integration

at

(14)

(15)

in two-,linlensions in Ihree-(limensions

in equation boundary

t t3)

rel)resents

we intro(luce

a donlain

t]w D,M

integral.

Reciprocal

To

concept

[1993]).

RECIPROCAL

TECHNIQUE

Consider nr points on l" and (j = 1.2...-nT) such that:

in ft..

We

introd,lce

a family

of coordinate

functions

M./(x)

F(x)

(16) j= I

where

.:t are

Mj(x).

there

coefficients exists

to I),, determined

an asso('iale,

V(e assume

i)v ,'olI,,,'ation.

I function

tPs(x)

V_2q.i(x)

slwll

for

each

function.

thai:

= 3/,{x)

(17) =--

It can

be shown

.lid = r'"g"

(('hetJ_

tile function

& Ouazm' ip, is given

[19!13])

t}ml

5,ra

two-

dimensional

problem

for which

by:

(is) m

where

the

square

the argument. Table 1.

brackets

Sohttious

in lhe Ul)per for other

possible

limit

of tile

families

48O

sum,lation of coordinate