The theoretical basis for well posed marching of a Parabolic Navier-Stokes (PNS) computational technique for supersonic flow is discussed and examples given to verify the analysis. It is demonstrated that stable computations can be made even with very small steps in the marching direction. The method is applied to cones at large angle of attack in high Reynolds number, supersonic flow. Streamline trajectories generated from the numerical solutions demonstrate the development of vortex structures on the lee side of the cone

Barnett, M.

Davis, R. T.

Rakich, J. V.

English

NASA Technical Reports Server

3 1176 00508 8043 //_/_'/_:_ --"7--//_/}--'_:::)Z7/_'_
r
NASA Technical Memorandum 84262
NASA-TM-84262 19830014280
Simulationof LargeTurbulent
Structureswith the Parabolic
Navier-Stokes Equations
J. V. Rakich, R. T. Davisand M. Barnett
June 1982 [IB_.AR'+,I_OPY
,JUL8 1982
LANGLEY RESEARCH CENTER
LIBRARY, NASA
HAMPTON°VIRGINIA
NASA
NationalAeronautics and
Space Administration
https://ntrs.nasa.gov/search.jsp?R=19830014280 2020-03-21T04:35:42+00:00Z
D T _[_€-_l L_'- wi u° .__-_iK!t_;_l,..'-)__lJE t_.i_2/"-_
U,-il'l_. L D J , :- _ IgSIIE i _'
U_IO_H._:_IF.LED DOCUMENT
_-m9Jo HAg i ;5 -'4262 /llg/ 1...... 0 n P_QGES ,,- ....,c-c -
UTTL: Simulation o{ large turbulent strucf.-ures _u,:th the parabolic Navier-Stoi_e'_-_.
equaLions
it!_lTH" R/RAKICH. ,_1. V.: B/DAVIS, R. T • C!B{_RhETT H. P.qA" B/(Cincinnati
Llniv.): C/(Cinc,:nnat. 1 IJni,,.:, , _. •
C:ORP: E'lational _eronautics and SF::---_ a -_ _umlni--.traf_.i_-,n_ _ . _mes K_'_arch Center.
..... L TTC " He [_Mo£fe_t Field Ca{';{ AVAI! ,I ....... SAP 021HF ROI
Presented at L--;th Intern. (_-;on,_. on Numeric.at Hethods in Flbid Dyn.. Aachen.
Nest Germany. -o Jun. - i Jr,! I °°°
MAJS: /_F,!GLE OF _TI_Ct::i+-CONES/_REYNOLDS , , _,_ _.F ,_ rNUf,BER/._oL!r_RoOF.II,_ FLOtq!+:TIJRBI_ILENT FLOb;
I!IH_, / COMPUTERIZED _,IHULaTIOIi/ I'I_UTFR-_TnV_m EOU_TION/ PNR_BOLIC DTFFERENTINL
E0LI6"_TIONS
_iBA: S.L.
t- t_ 1-'__ i
,4E15: The theoretical basis {'or uell posed marching of- a F'araDolic ;'la,/ier-o,.o_es
(F>NS) computational technique {or-,upersonic £loi-, ,_s discussed and
e'_ampies given to verify the analysis, tt is demons'rated thai stable
,computations can be made even .,ith very small -steps it: t.he marching
,direction. The method is applied to cones at large ang'!e c,f atLacl< in high
Reynolds number, supersonic £1ou. c;treamline trajectoriF::=.._aer, erated {tom
the numerical solutions demonstrate the development ,:,f -'orte': sf.ructures
on the lee side of" the cone.
ENTER:
NASA Technical Memorandum84262
Simulationof LargeTurbulent
Structureswiththe Parabolic
Navier-Stokes Equations
J. V. Rakich, Ames ResearchCenter, Moffett Field, California
R. T. Davis
M. Barnett, University of Cincinatti, Cincinatti, Ohio
N/K
NationalAeronautics and
SpaceAdministration
Ames Research Center
Moffett Field,California 94035
SIMULATION OF LARGE TURBULENT VORTEX STRUCTURES WITH THE
PARABOLIC NAVIER-STOKES EQUATIONS
J. V. Rakich, I R. T. Davis, 2 and M. Barnett 3
ABSTRACT
The theoretical basis for well-posed marching of a Parabolic Navier-Stokes (PNS)
computational technique for supersonic flow is discussedand examples given to verify
the analysis. It is demonstrated that stable computations can be made even with very
small steps in the marching direction. The method is applied to cones at large
angles of attack in high Reynolds-number, supersonic flow. Streamline trajectories
generated from the numerical solutions demonstrate the development of vortex
structures on the lee side of the cone.
INTRODUCTION
The computation of steady supersonic viscous flows with one set of equations,
valid in both the boundary layer and the inviscid supersonic region, has long been
attempted to avoid the difficulties with matching conditions for separate inviscid
and boundary-layer equations. Use of the unsteady form of the NS equations has been
successful but ineffieient_ in terms of computer time and memory requirements.
Therefore, many authors [I-8] have used steady marching or iterative methods to solve
numerically the time-invariant NS equations. Single-pass methods have been used
for supersonic flows, where the inviscid region is a well-posed initial-value
problem, and where the viscous region is a boundary layer, which is itself amenable
to a marching solution. The difficulty with solving the complete inviscid/viscous
domain at once lies in the thin subsonic layer near the wall, where the pressure
is determined by the solution. The absence of a downstream boundary condition makes
the problem ill-posed. This combined problem is basically an interaction of a
boundary layer with an inviscid, supersonic stream. Lighthill [9] analyzed this
problem and found solutions for the boundary-layer displacement thickness of the
form 6 = a exp(kx). One can interpret this exponential growth as the onset of
streamwise expansion or compression, which may lead to separation; these solutions
have become known as "departure" solutions corresponding to some undetermined
downstream boundary condition. This paper describes an approach for avoiding the
exponentially growing solution, and presents results for turbulent flow over a cone
at large angles of attack.
THE PNS EQUATIONS
The governing equations are written in general curvilinear conservation-law
form as follows [8]:
_-_ {J-l[$xE* + SyF* + SzG*]} + _ {J-l[_x(E - Ev) + ny(F - Fv) + nz(G - Gz)]}
(l)
+ _ {J-l[_x(E - mv) + _y(F - Fv) + _z(G - Gv)]} = _ {-J-l[_xpI + _yP2 + _zP3 ]}
iResearch Scientist, NASA Ames Research Center, Moffett Field, California, U.S.A.
2professor of Aeronautical Engineering and Applied Mechanics, University of
Cincinnati, Ohio, U.S.A.
3Graduate Research Assistant, University of Cincinnati, Ohio, U.S.A.
Here J is the Jacobian of the coordinate transormation; E, F, and G are the
inviscid flux vectors; and Fv and Gv are the viscous flux vectors--all written in
terms of Cartesian velocity components. Note that the viscous terms are omitted
from the _ derivative term; this is the fundamental parabolizing approxim@tion.
The ( )* terms are central to the discussion of parabolic marching, and are given
by the column vectors
* 2 }TE = {0u,pu + rap,puv, 0uw,(0et + p)u ,
F* = {pv, puv, pv2 + up, pvw, (pet + p)v}T , (2)
G* = {pw, puw, pvw, pw2 + up , (pet + p)w}T ,
where
pet = p/(y - i) + 0.5p(u2 + v2 + w2)
Here u, v, and w are Cartesian velocity components, p is the pressure, p the
density, and y the ratio of specific heats. The P terms on the right side are
given by
PI = {0, (i- re)p, O, O, 0}T , ]
P2 = {0, 0, (i- re)p, O, O}T , (3)
P3 = {0, O, O, (1- _)p, O}r .
Following Vigneron et al. [4], the parameter m is included with the pressure
term in the momentum equations. It was shown that the equations can be made formally
parabolic by utilizing an appropriate functional form for m. Considering the two-
dimensional viscous subset of equations (i), the mathematical character of the
equations is governed by the eigenvalues of the differential system, leading to the
condition
S YMx2/[l + (y - l)Mx2] (4)
If this condition is satisfied, then the equations remain parabolic even in
the viscous region near the wall where the Macb number is less than unity; a similar
form also governs the inviscid region, but is not pertinent to the present work.
The purpose of this paper is to further demonstrate the effectiveness of this
approach for interacting supersonic flows where the problem can be solved in a single
sweep of the PNS equations.
In previous work with the PNS method, a "sublayer" approach was used, in which
the pressure gradient term was completely removed in the subsonic region of flow.
This is equivalent to a step function for m, and this places greater significance
on the P terms on the right side (RHS) of the equation. Usually, backward-
difference approximations have been used for the pressure gradient term on the RHS.
However, it can be shown that a backward difference leads to numerical instability
for sufficiently small marching steps (which may be a manifestation of the departure
behavior). When the _ function is used, the RHS can be set to zero, to a good
approximation, making the numerical computation stable, even for small steps. Thus,
the _ function and a zero RHS together permit the investigation of departure
behavior, which requires very small streamwise steps.
NUMERICAL METHODS
The present computational results are obtained with the Beam-Warming factored
algorithm in delta form. Details can be found in reference [8]. Full second'order
accuracy can be maintained by use of central differences in _ and _, and with a
three-point backward difference in _. However, for the present results, the Euler
implicit method is used for the marching direction, making the method first-order
accurate in the _ direction.
TRIPLE-DECK ANALYSIS
For subsonic flows, that is, elliptic equations, a downstream boundary condition
must be specified. When attempting to march such a system without specification
of the downstream condition, there are an infinity of possible solutions that corre-
spond to various attached or separated flows. When the viscous layer is thin, the
departure solution is governed by the interaction of the viscous and inviscid regions
near the wall [9]. Stewartson [i0] further studied such flows using a triple-deck
analysis consisting of a Lighthill sublayer, an inviscid shear region, and the
inviscid external flow. This analysis establishes the order of magnitude of the
interaction region.
The pressure is constant across both the lower and middle decks, and the velocity
in the middle deck corresponds with the attached velocity profile upstream of the
interaction, but displaced by 6. Utilizing linear supersonic theory for the pressure
at the top edge of the middle deck, one obtains the following momentum equation for
the lower deck:
8u Du = 82 [_w(Me2u _xx+ v _y _ u 2/ - I)!i]--d26 (5)
_y---_-OeU e dx2 '
where the subscript w refers to wall, and e refers to the top edge of the middle
deck. Introducing the stream function in (5), and utilizing the Lighthill form for
the displacement thickness 6, yields an Airy equation. It is found from the
solution that the trlple-deck scale is given by
AT = 0.8272(M 2 -I)3/4(T=/Tw)I/2cf 5/4Re3/8 Ax = k Ax , (6)
where x is the nondimensional physical distance, Re is the Reynolds number, Cf
is the skin friction coefficient, T is the temperature, and M is the Mach number.
The pressure gradient that results is
dp/dx = C exp(kx) (7)
The importance of this analysis to computations with the PNS equations is that
it provides a test of the effectiveness of methods for damping the unwanted departure
solutions. To illustrate, we consider a flat-plate boundary layer. The solution
shown in Figure i at x/L = 0.5 is obtained with the m parameter turned on, and
then w is set to 1.O, so that the full PNS equations are marched for a short
distance. The solution immediately starts to "depart" until at x = 0.5055 the
factor is again turned on, and the solution decays toward the well-known boundary-
layer solution. If the triple-deck scale had not been resolved by the step size
Ax, the departure would have appeared as an oscillatory divergence. We note here
that other investigators report their methods are unable to suppress the departure
behavior for small marchingsteps, while the present approach is quite effective.
Figure 2 demonstrates that the computed departure solutions (_ = i) agree with
Lighthill's linear analysis. This figure shows the magnitude of the computed
pressure gradient as a function of the triple-deck distance scale _ = kx. The
initial slope of the computed departure agrees with the linear analysis until
nonlinear effects take over.
COMPUTATION OF TURBULENT FLOWS
Turbulent computationsare performed with the Reynolds averaged form of the
PNS equations and an algebraic, or zero-equation, eddy viscosity model. Details are
given in reference [5] and will not be repeated here. We note, however, that no
special modifications were needed for the present calculations, even with a large
crossflow separation on the leeward side.
Figures 3 and 4 compare the circumferential variation of pressure and surface
stream angle from the present computations with the experimental results of Rainbird
[ii]. Generally, good agreement is obtained with experiment. The main differences
are caused by an incorrect prediction of the location of separation, believed to be
due to the inadequacy of the simple turbulence model used. These results were
obtained on a CDC 7600 computer with 50 grid points between the body and the shock,
and with 47 unequally spaced meridian planes.
FLOW-FIELD SIMULATION
The simulation of turbulent vortex structures is achieved by tracing particle
paths using the velocity field obtained from the PNS solution. Initial positions
for the tracer particles are specified at field locations near the cone surface,
and x/L = 0.i. The particle paths are then determined from a simple Euler predictor-
corrector, finite-difference scheme. The velocity field is taken at x = L and is
assumed invariant with x for the purposes of this simulation.
The flow simulation is shown in Figures 5a and 5b. Three distinct vortices are
observed--one emanating from the primary separation, the second from the secondary
separation of the flow coming down the leeward plane of symmetry, and a third below
the primary vortex and having the same rotation as the primary vortex. This flow
structure is suggestive of a global type of mixing in large turbulent structures,
which is significant to the development of turbulence models.
REFERENCES
[i] Davis, R.T., AlAN J., 8 (5), 843-851 (1970).
[2] Rubin, S.G. and Lin, T.C., PIBAL Rept. 72-8, Polytechnic Inst. of Brooklyn,
Farmingdale, N.Y. (1971).
[3] Lubard, S.C. and Heliwell, W.S., AlAN J., 8 (7) (1974).
[4] Vigneron, Y.C., Rakich, J.V., and Tannehill, J.C., AIAA Paper 78-1137 (1978).
[5] Rakich, J.V., Vigneron, Y.C., and Agarwal, R., AIAA Paper 79-131 (1979).
[6] Schiff, L.B. and Steger, J.L., AIAA Paper 79-130 (1979).
[7] Kovenya, V.M. and Chernyi, S.G., Chis2enne Method Mechaniki Spioshnoi Sred,
i0 (i) (1979).
[8] Tannehill, J.C., Venkatapathy, E., and Rakich, J.V., AIAA J., 20 (2) (1982).
4
[9] Lighthill, M.J., Proc. Roy. Soc. London (1953).
[i0] Stewartson,K., Adv. in Appl. Mech.,AcademicPress,14, 145-159 (1974).
[ii] Rainbird,W.J., AIAA J., 6 (12) (1968).
FLAT PLATE Moo= 3 Re = 50 X 106
= f(M) .8
-i_ _ i__/
_=f(M) r _ =1.0
I_uLL PNS EQUATIONS i_--_.
P _ D
_2 _ _p_Uoo
.8 __..._ .., _ rw'-_ .4
.d I I I
•500 .502 .504 .506
x/L
Fig. I. Influence of "Omega" parameter on
departure and recovery.
2.0
1.5 Tw/Tc_
0.75
1.0
• " "--X
,", 1.0 1.25
o 1.50&..
Om
i-:
Z .5
tu PRESENT PNS5
<
n-
0 t
LU
o0
o,)
UJ
n- -.5
a.
THEORETICALLY PREDICTED
-1.0 SLOPE, LIGHTHILL (1953)
-1.5 I I I I I
0 .5 1.0 1.5 2.0 2.5
SCALED DISTANCE, kx
Fig. 2. Departure solution for a flat plate;
M_ = 3, Re = 50x106, Tw = i00 K.
12 --
8 _... EXPERIMENT,
'_).,J RAIN BIRD (1968)
o _ _ _PNSCOMPUTATION4
04
.E
o 0 III
o.
O
,-8
-12 I I I I I I
0 30 60 90 120 150 180
¢,deg
Fig. 3. Circumferential pressure distribu-
tion for a 12.5 ° cone; M_ = 1.8, e = 22.7 °,
Re = 25xi06 .
60
.EXPERIMENT,
4050 _RAINBIRD (1968)
.z:'_ . PNS
30 t_ COMPUTATION
€ 20
10
0 (3
-10
-20
-30 _ _ i _ t w
0 30 60 90 120 150 180
, deg
Fig. 4. Surface stream angle relative to
the generators; ec = 12.5 °, M = 1.8,
= 22.75 °, Re = 25xi0 G.
VORTEX SIMULATION VORTEX SIMULA
12. S DEG CONE 12.5 DEG
ALPHA 22.5 DEG ALPHA 22.5-/DEG
MACH 1.8 MACH 1.8
SIDE VIEW FRONT V_
(a) Side view. (b) Front view.
Fig. 5. Streamlines around a 12.5 ° cone; M = 1.8, _ = 22.75 °, Re = 25xi06.
1. Report No. I 2. GovernmentAccessionNo. 3. Recipient'sCatalogNo.INASA TM-84262"
4. Title and Subtitle 5. Report Date
SIMULATION OF LARGE TURBULENT VORTEX STRUCTURES WITH THE
PARABOLIC NAVIER-STOKES EQUATIONS* 6. PerformingOrganizationCode
7. Author(s) 8. PerformingOrganizationReport No.
J. V. Rakich, R. T. Davis,** and M. Barnett** A-8958
10. Work Unit No.
9. PerformingOrganizationNameandAddress T-4220
NASA Ames Research Center
Moffett Field, California 94035 11.ContractorGrantNo.
13. Type of Report and PeriodCovered
12. SponsoringAgencyNameand Address
National Aeronautics and Space Administration Technical Memorandum
_ashington, D.C. 20546 14. SponsoringAgencyCode
15. SupplementaryNotes *Presented at the Eighth International Conference on Numerical Methods in Fluid
Dynamics. June 28-July i, 1982, Aachen, West Germany.
**University of Cincinnati, Ohio,
Point of Contact: John V. Raklch, Ames Research Center, Moffett Field, CA, (_15)965-6192, FTS 448-619
16. Abstract
The theoretical basis for well-posed marching of a Parabolic Navier-Stokes (PNS) computational
technique for supersonic flow is dlscussedand examples given to verify the analysis. It is
demonstrated that stable computations can be made even with very small steps in the marching
direction. The method is applied to cones at large angles of attack in highReynolds-number,
supersonic flow. Streamline trajectories generated from the numerical solutions demonstrate the
development of vortex structures on the lee side of the cone.
17. Key Words(Suggestedby Author(s)) 18. DistributionStatement
Parabolic equations Unlimited
Boundary layer
Turbulent vortex
Finite difference
Subject Category - 34
19. S_urity Classif.(ofthisreport) 20. SecurityCla=if.(ofthi$ pe_) 21. No. of Pages 22. Price"
Unclassified Unclassified 9 A02
"ForsalebytheNationaITechnicallnformationService,Sprlngfield,Virginia 22161
!" i
• !
J
i
l 3 1176 00508 8043 '
.=l
0
i


Simulation of large turbulent structures with the parabolic Navier-Stokes equations

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server