of Akron. Akron, OH 44325. SUMMARY. The numerical solution of exterior problems is typically accomplished by introducing an artificial, far field boundary.
....
NASA Technicaq ICOMP-88-2
Accurate Problems
...........
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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DENSITY MOMENTUM
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(D) TIME-GOO-METHOD ON THE
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TIME
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1.5
l
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2.0
2.5
(D) TIME-6OO-METHOD-TH. ON THE SHORT
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TIffE = _00, 600.
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(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
I
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 _
