Accurate Boundary ConditiOns for Exterior Problems in Gas Dynamics

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of Akron. Akron, OH 44325. SUMMARY. The numerical solution of exterior problems is typically accomplished by introducing an artificial, far field boundary.
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NASA Technicaq ICOMP-88-2

Accurate Problems

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MemOrandum

100807

Z

Boundary ConditiOns in Gas Dynamics

Thomas Hagstrom Institute for Computational Mechanics Lewis Research Center _ _ Cleveland, Ohio_ .............

for Exterior

in Propulsion .............................................

and S.I.

Hariharan

University of Akron Akron, Ohio (l_ASJl-'Itl1 C0 8C7) _C£ gli,lT_ ECI_I: CC§D.I.TICNS FO_ "£1_B_ZCB PFCEI_I_£ £11iABICS (_i A.5 ll) 2_ F

Jl._¥ .,T.I_GAS CSCL

N88-151E2 12k G3/6

March

1988

q

Oaclas C128853

Accurate

Boundary

Conditions

Problems

in Gas Dynamics

Thomas Institute

for Exterior

Hagstrom*

for Computational Mechanics Lewis Research Center Cleveland,

in Propulsion

OH 44135 and

S.I. Department

Hariharant

of Mathematical University of Akron Akron, OH 44325

Sciences

SUMMARY

The

numerical

by introducing

an artificial,

a truncated boundary waves

domain.

with

spherical

and

'natural' condition

based nonlinear

numerical

energy

experiments

estimates to validate

boundary

a principle

variables

satisfy

are coupled. We propose

solution for the

accomplished

the equations conditions

However,

equations This suggests

a reflecting

on

at this

of no reflection.

in gas dynamics

be important.

on an asymptotic

is typically

and solving

systems,

by imposing

B.iemann

may

problems

far field boundary

symmetry

outgoing

reflections

of exterior

For hyperbolic

are often derived

incoming

obtain

solution

where that

boundary

of the far field equations. truncated

problem

We

and present

our theory.

*Department of Applied Mathematics and Statistics, SUNY at Stony Brook, Stony Brook, New York 11794 (work funded under Space Act Agreement C99066G). tResearch supported by National Science Foundation Grant DMS-8604047.

INTRODUCTION Interestingandimportantproblemsin gasdynamicsareoftenposedin exteriordomains.Examplesincludethe explosionof gasbubblesin various mediaandflowsexternalto aircraft. An approachto the numericalsolution of suchproblemsis to restrict the computationaldomainto a finite region through the introduction of an artificial boundary.For largetime computations interactionsbetweenthe solutionand the artificial boundarycan stronglyinfluencethe results.The focusof this paperis the development of an accuratetreatmentof theseconditions. A variety of authorshaveinvokeda principleof no reflection.Notable amongtheseareEngqulstandMajda [2]whostudiedthe generallinearcase andHedstrom[8] andThompson[10]whoconsiderednonlinearhyperbolic systems.However,aspointedout by GustafssonandKreiss[3], conditions satisfiedby the exactsolutionmayinvolvereflections.Thecurrentstudyinvolessphericalwaveswhichexhibit couplingbetweenincomingandoutgoing P_iemann variables.Oneexpectsthis couplingto resultin naturalreflections whichshouldbe accountedfor in an efficientnumericaltreatment.Indeed, Thompson[10]documentsthe disappointingperformanceof nonreflecting conditionsin suchcases. An alternateapproachis to incorporatethe asymptoticbehaviorof the solutionin the far field. Conditionsfor linearproblemsbasedon far field asymptoticshavebeensucccessfully employedby BaylissandTurkel[1]and HariharanandBayliss[7]. tions to the appropriate the region tained

2

exterior

which

weakly

to the

includes

Our procedure nonlinear

computational

appropriate

is to develop initial domain.

reflections

boundary

approximate

solu-

value problem

A condition

at the computational

is thus

in ob-

bound-

ary.

Conditions

the discarded

involving region

The particular spherically

incoming

have also been

equations

symmetric,

isentropic

0z

conditions

density,

We also assume r0) and that

by Gustafsson

consideration

in

[4,5].

are the Euler equations

Oz Or

2 z r'

-

(1.1)

0 [z2 7 + f(p)] =

+

for

fluid flow;

z is the momentum

= p0(r)

that

proper

conditions

and

(1.2) f(p)

field and

derive

is the pressure.

established contains

a discussion

In section

5 numerical

problem.

for values conditions.

and

Our

Initial

domain

In the final section dimensional

r>

is shown

at r = L (L > we assume

L.

in section

in the far

estimates

extensions

4

conditions.

for an idealized the correct

are

3. Section

of the boundary

to yield

that (1.4)

solutions energy

those required

we propose

(1.3)

2 we follow the construc-

problem

are presented

than

0,

Nonlinear

experiments

smaller

Finally

asymptotic

treatment

technique

r >_ r0.

is located

In section

conditions. finite

=

of the numerical

of L significantly

three

z0(r)

[11] to obtain

boundary

for the resulting

zo(r),

boundary

is as follows:

by Whitham the

z(r, 0) =

at r0 are specified.

= p_

The plan of this paper tion presented

and

the computational

p0(r)

the truly

by inhomogeneities

are

p(r, 0)

plosion

proposed

under

Op 0--_ +

Here p is the

waves generated

weak ex-

steady

state

by the nonreflecting of our conditions

for

case.

3_

DERIVATION

OF ASYMPTOTIC

We find it convenient ables.

They

BOUNDARY

to work with equations

CONDITIONS

involving

RJemann

vari-

are

= s

:

z

- + c(p),

(2.1)

P z

G(p).

P

(2.2)

Here

c(p) Then equations

=

f V_dp. P

(2.3)

(1.1) and (1.2) take the form

cgt +

+

o-7 + Here we assume

that

Or

-

=

_

pr

'

=

for r > L z

That is, the flow is subsonic

in the far field.

the

l_iemann

outgoing

terms

couple

outgoing

problem

boundary

incoming

the equations

wave generates

To derive value

and

the

variables.

for the RJemann an incoming

boundary

(2.4),

(2.5)

conditions, and

Note that

variables.

the lower order

This means

that

an

wave.

(1.4)

we consider on the

exterior

the

initial

domain

boundary

r >_ L with

condition R(Z,t)

Solving

Then R and S are, respectively,

this problem

:

g(t).

(2.6)

yields

s(z,t)

=

J:[g(.)].

(2.7)

That

is the incoming Equations

(2.6) and

L. However, Therefore,

variable

(2.7) represent

the explicit we construct

for L sufficiently

large.

is a functional

form

(Similar

in Hagstrom

and

behavior

of solutions

of the

an exact

boundary

of the functional

an asymptotic

be found

of the outgoing

Keller

of the exterior for steady

[6].) Consistent

linearized

condition

9v is not known

solution

constructions

variable.

in general.

problem

state problems

with the known

equations

at r =

we expand

valid can

far field

R and

S as

follows:

R(r,t)

= Ro + Rl(r,t) P

s(_,t)

=

+

So + s_(r,t) + P

_2 S2(r,t)

_

+ ..., + ....

(2.8) (2.9)

We note that

We further

for some

Ro = -so = c(poo).

(2.10)

g(t) = Ro + _(t) L

(2.11)

assume

function

Equations

for R1 and $1 are given by:

OR___A ot +\ ( R1 _+ &

ot +_

H(t).

_

+ p(po_)(Rx ,,- sl) ) oR_o,.-

+ _ _--

• - P-P°°-(R_ r

o, (2.12)

Or

Here P(u) Following speed

Whitham

to suppress

-

[11] we have retained nonuniformities

_,f"(_,) 4f'(u)

"

! corrections 7,

in the expansion

to the characteristic

as r approaches

infinity.

5. !

Note

that

mann

the source

invariants

terms

are absent

of the approximate

characteristics

corresponding

at this order, equations

That

is, we have

R1 using

a simple

themethod

characteristics

wave.

=

and (2.13).

0.

(2.14) (2.12)

can be solved

The differential

equations

for

for the

-

the fact that

R1 is constant

along the characteristics

we find that

Rl(r,t(r;7-)) = H(7-), t(r;7-) where

Since the

at t = 0, we deduce

Now equation

of characteristics.

$1 are RJe-

are given by

_rr Using

(2.12)

to $1 all originate S1(r,t)

so R1 and

we have

=

(2.16)

r - L - B(-',')_[_+ B¢-,-)_ + B(r)_ + _ '

7-

(2.17)

introduced H(7-)[_

+ e(poo)]

B(7-) We remark

that

sect, in which nonvartishing

this solution

case a shock correction

may break

must

down where

be fitted

to the incoming

characteristics

in. In order variable

to compute

we consider

interthe first

the equation

for $2;

Ot To solve pressed

Or equation

(2.18)

as a function

we make

of 7- and

a change

of variables

r . This yields

the

in which

above

S_ is ex-

equation

in the

R1(7-),

(2.19)

form

( _1

07-0t

D,7-,r,ff_r, ) 07-) 0S207-

D(v,r)_--r2

-

which we write as

os2

#(_,r)D(T,r)0S2

8T

Or

_ #(T,r)al(_).

(2.20)

Here

,,r ,,+m_,, +s_

r

(,-L)s(_)11

N(v,r) ,+B(_-)J _//,(p_)] and D(%r) Again grate

we solve equation

=

(2.20)

1 +

by the method

from the first characteristic,

to know

$2 at the boundary

(P-_)H(T) _r

_" = 0, where

r = L . Thus

S_(L,_) =

_0

of characteristics.

T

We inte-

$2 = 0. We will only need

we obtain

(2.21)

Rl(s)lV(s,_(_;_,L))d_,

where

d_

-

d8 To put

equation

respect

to T:

(2.21)

-lVD,

in a more

_(_;T,L) = Z. convenient

form,

(2.22)

we differentiate

8S2(L,T)

with

(2.23)

_T

Now the integral

term

we neglect

it along

S0 + _$2

and RI(T)

is of order

with

Ot

-_ contributions

= L(R(L,t)-

sl,=

_1 due to the presence to N.

Ro) we finally

L = v/T(P°°)(a(z't)2L

Using

of -_('sNTherefore

_(L,_)

= 1, S

=

obtain

a0)

(2.:4)

7.

Equation that

(2.24)

is the

boundary

condition

we propose.

We also note

the relationship

zP can be used to derive

G(p) = -C(poo) + o(_), a number

of asymptotically

(2.2s)

equivalent

conditions.

For

example

Os

z_

Ot

(2.26)

Lp

and

(c(p) - v(p=))v_p=)

c_S i

Ot

ESTIMATES

ENERGY

We now

study

the field equations

the

problem

on the

truncated

region

(1.1) and (1.2) in a convenient

r0 = 0 for definiteness.

Thus

the problem

0--[ + Oz

(2.27)

L

under

r_cgr

=

1 0 . 2z2.

[r0, L], rewriting

form.

Moreover,

consideration

we take

is

0 ,

(3.1)

0 f

o-7 + _z_ (_ 7 ) + _[ (p)] = o.

(3.2)

Initial conditionsare

p(r, 0) Our boundary

£ :

=

condition

p0(r),

=

z0(r)

at r = L has the integrated

G(p) - G(poo)+ _

P Corresponding,

z(r, 0)

-_d_+ Z

r > 0 .

(3.3)

form

(G(p)-G(poo))d_. (3.4)

P respectively,

to (2.24),

(2.26) and (2.27)

= Z _ _

we have:

(3.5) 2L

'

a

_

S'vq_-=) L

a In addition

we need

the singularity

=

0,#

-

to introduce

of the

# = o, L

(3.7)

a finiteness

equations

(3.1)

(3.6)

and

condition

(3.2).

This

at r = 0 due is accomplished

to by

demanding z(r,t) It is difficult

to establish

0

_ the

(3.8)

weU-posedness

problems

for nonlinear

hyperbolic

derivation

of bounds

on the growth

(physical)

energy

density

as I"-,0

systems.

of initial-boundary

We content

of the total

is defined

energy

ourselves

value with the

of the system.

The

by: 1 z 2

Zwhere

the internal

energy

2 p

+

pe(p),

(3.9)

e satisfies:

f(P) p--T"

e'(p) =

(3.10)

We also define q Here

the gradient

is with respect

q Taking

the inner

product

:

=

(3.11)

VE.

to the variables

2 p2

-{" 7-1

p and z. Then

,

(3.12)

of (3.1) and (3.2) with q we obtain OE Ot

+

1 0 ----_ r 2 Or

=

O,

(3.13)

where

[_2z(e(p)+ --;--

+

2p 2

Now integrating

(3.13)

over [0, L] with the weight

r 2 we obtain

• '(t) + _io _ = o,

(3.14)

_(t) =

(3.15)

where

which

is the total

energy

of the system. =

Clearly,

(3.8) implies

that

Theorem

=

pe + f

+

exists

_:

.

(3.16)

us with (3.17)

of this section.

a bounded/unction

P=o, so that the total energy

to rewrite

-9(L,z(L,t),p(L,t)).

the main theorem

1 There

It is useful

@ = 0 at r = 0 leaving

¢'(t) We now state

P

_2Edr,

fo L

F(t),

of any generalized

depending

solution

only on f and

of (3.1)-(3._),

(3.8)

satisfies:

,(t)

_< 3(0)

+ r(t).

(3.18)

Proof: Using Fubini's

z( L, t ) p(L,t)

theorem

it may easily be verified

(3.4) solved for _ yields:

- a(p(Z, t))- a(p_) + (,_+ _) foot e°('-')(a(p(L, _))- a(;_))d_.

Since

the

bracketed

(3.17)

can be positive

nondecreasing

quantity

in (3.16)

is positive,

only if _ is negative.

function,

a positive G(p) p

10

that


L. Following

the

2, we work with the symmetric

S, as well as the angular

momenta.

The formal

and

Riemann

variables,

expansion

R

we postulate

has the form:

Rl(r, 0, ¢, t)

R(r,e,¢,t)

= Ro +

s(r,e,¢,t)

= So + sl(r,e,¢,t) r

m(_, 0,¢, t)

p

R2(_,0,¢, t)

+

m3(_,0, ¢, t) r3

-{-

+ ...,

p2

+ s2(,,e,¢,t) r2

m_(_,0,¢, t) r2

--

+

(6.5)

...,

(6.6)

-{- ...,

(6.7)

q(_,0, ¢, t) _ q2(r,0,¢,t) 0,¢, t) + ..., r2 + q3(_,r3 where,

borrowing

menta,

m and

$2 are taken order

terms

results q, are O(_)

unchanged involving

with our inclusion

as r --* _.

from

section

that

the primary

radial

Expressions

equations

Although involving

direction

o,

of propagation corrections

the angular

mo-

for R1, R_, $1 and

2. This involves

terms

for the first

case, we assume The

0 and ¢ derivatives.

of lower order

expectation one.

from the linear

(6.8)

the neglect

not entirely

of lower consistent

this is justified

by the

in the far field is the

may then

be copied

from

(2.14),(2.16)and (2.17):

sl(_,0,¢,t) R_(_,e,¢,t(_,e,¢,_))

= 0, :

(6.9)

_(e,¢,_),

(6.10)

[,+e(o,_,_)l

t(_,e,¢,_)

r - L - B(0, ¢,v)in

LL+B(o,¢,_.)j

(6.11)

I7

More importantly, the boundaryconditions(2.24), (2.26)and (2.27) are unchanged.Equationsfor m2 and q2 0m2

ot

zl

are:

0m2

8pl

+ _:_ o_

Oq2

zl

ot +

+ :'(:oo) oe - o,

Oq2

rpoo cgr

(6.12)

f'(Poo) OF1

+

sin6

- o.

69¢

(6.13)

Here, Zl

pooR1 2

m

m

pooH 2

and p_R1 Pl

We include ular,

the

at a point

is required. derivative

t term I=

2_"

characteristics

can be computed.

zl > 0, no boundary

we may

to update

pooh

2J-f(p_)

so that

of outflow,

At inflow terms

--

simply

condition

use (6.12)

and

momenta.

In summary

the angular

(6.13)

In partic-

for m2 and without

q2

the r

we have:

_(a(L,O,¢:)-ao) oT

2L

o_

L p( L,8 ,dp,t )

(_(,(L,0,¢,t)-

(6.14)

CC._ ))_ L

and, if z(L,8,¢,t)

< O,

Om'L_-[ ( ,O,¢,t)

1 8 r2 _0 [f(p(L,O,¢,t))],

(6.15)

1 _(L,8,¢,t))

18

=

r2sinO O¢ [f(p(L,O,¢,t))].

(6.16)

REFERENCES

1. Bayliss,

A. ; and Turkel,

Comm.

Pure Appl.

2. Engquist,

of Waves.

3. Gustafsson,

4. Gustafsson,

Math.

Absorbing Comp.,

Boundary.

Appl.

5. Gustafsson,

Numer.

B.: Far SIAM

6. Hagstrom,

Conditions 1988,

Boundary

139,

for the Numerical

1977,

pp. 629-651.

for Time Dependent

at Open Boundaries

Problems

for Wave Propagation

to appear.

1988,

for Time-Dependent

Hyperbolic

to appear.

H.B.: Exact Boundary

Equations

Equations.

vol. 30, no. 3, 1979, pp. 333-351.

Conditions

J. Sci. Stat. Comp.,

Conditions

Conditions

Phys.,

for Wave-Like

pp. 707-723.

vol. 31, no.

J. Comput.

Field

Differential

Condition

Boundary

Boundary

Math.,

T.; and Keller,

for Partial

in Cylinders.

Conditions

at an Artificial

SIAM J. Math.

Anal.,

Boundary

vol. 17, no. 2,

pp. 322-341.

7. Hariharan, Ducts.

S.I.;

SIAM

8. Hedstrom, Systems.

9.

A.:

B.: Inhomogeneous

Problems.

Boundary

vol. 33, no. 6, 1980,

B.; and Kreiss, H.-O.:

with an Artificial

1986,

Math.,

B.; and Majda,

Simulation

Systems.

E.: Radiation

and Bayliss,

G.W.:

Non-Reflecting

J. Comput.

Phys.,

Numerical

Cambridge,

1985.

J. Comput.

11. Whitham

Radiation

J. Sci. Stat. Comput.,

Sod, G.A.:

10. Thompson

A.:

K.W.: Phys.,

, G.B.:

Time

Methods

Boundary

in Fluid

Dependent

from

Unflanged

Cylindrical

vol. 6, no. 2, 1985, pp. 285-296.

Conditions

vol. 30, no. 2, 1979,

vol. 68, no.

Linear

of Sound

Dynamics,

Boundary

1, 1987, pp.

and Nonlinear

for Nonlinear

Hyperbolic

pp. 222-237.

Cambridge

Conditions

University

for Hyperbolic

Press,

Systems.

1-24.

Waves.

Wiley,

New York,

1974.

19

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J

l

(B) TIME-8OO-METHOD-TH.

m

.8

D

.i 4

m

0

-

I

((

0

J

.5

1.0

I

J

J

2.0

2.5

1.5

I .5

I

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1.0

1,5

l

DISTANCE

2.0

(C) TI ME -2000-METHOD-HH3. FIGURE

22

6.

- SOLUTIONS

(D) TIME-2OOO-METHOD-TH. ON THE SHORT

INTERVAL,

TIME

= 800, 2000.

I 2.5

I ASA National Space

Aeronaotics

Report

and

Documentation

Page

Adm_n]stralion

I. Report No.NASA

2. Government

TM-100807

Accession

No.

3. Recipient's

Catalog

No.

ICOMP-88-2 4. Title and Subtitle

Accurate Problems

5. Report Date

Boundary Conditions in Gas Dynamics

March

for Exterlor

7. Author(s)

Thomas

Hagstrom

and S.l.

6. Performing

Organization

Code

8. Performing

Organization

Report No.

E-3988

Hariharan

10. Work

9. Performing

Organization

National Lewis

Research

Cleveland, 12. Sponsoring

and 44135-3191

New

Technical

Center

Department

I1794 (work

Sciences,

Administration

14. Sponsoring

Memorandum

Agency Code

Notes

Hagstrom,

York

or Grant No.

13. Type of Report and Period Covered

Agency Name and Address

National Aeronautics and Space Washington, D.C. 20546-000l

Thomas

11. Contract

Administration

Space

Center

Ohio

15. Supplementa_

Unit No.

505-62-21

Name and Address

Aeronautics

1988

and

the

funded

under

University

of

Applied

Institute

of

Space Akron,

for

Mathematics Computational

Act

Agreement

Akron,

Ohio

and

Statistics,

Mechanics C99066G);

44325

S.I.

(work

SUNY in

Hariharan,

funded

at

Stony

Propulsion,

by

National

NASA

Brook,

Stony

Lewis

Research

Department Science

of

Brook,

Mathematical

Foundation

Grant

DMS-8604047). 16. Abstract

The numerical solution of exterior problems ts typically accomplished by introducing an artificial, far field boundary and solving the equations on a truncated domain. For hyperbolic systems, boundary conditions at thls boundary are often derived by lmposlng a principle of no reflection. However, waves wfth spherical symmetry In gas dynamics satisfy equations where incoming and outgoing Riemann variables are coup]ed. This suggests that 'natura]' ref]ections may be important. He propose a reflecting boundary condition based on an asymptotic solution of the far field equations. We obtaln nonlinear energy estlmates for the truncated problem and present numerical experiments to validate our theory.

17. Key Words (Suggested

18. Distribution

by Author(s))

Fluid mechanics; Hyperbolic equations; Reflective boundary conditions; Finite differences; Asymptotics 19. Security

Classif. (of this report)

!20.

Security

Classif.

Unclassified NASA FORM 1626 OCT 86

Statement

Unclassified - Unlimited Subject Category 64

(of this page)

21. No of pages

Unclassified *For

sale

by the

National

Technical

Information

24 Service,

Springfield,

Virginia

22161

22. Price*

A02 _