v

N 87- 26017

Stability, Transition and TurbulenceM. Y. Hussaini

Institute for Computer Applications in Science and Engineering

NASA Langley Research Center

Hampton, VA 23665

1 Abstract

A glimpse is provided of the research program in

stability, transition and turbulence based on numer-

ical simulations. This program includes both the so-

called abrupt and the restrained transition processes.

Attention is confined to the prototype problems ofchannel flow and the parallel boundary layer in the

former category and the Taylor-Couette flow in the

latter category. It covers both incompressible flows

and supersonic flows. Some representative results are

presented.

2 Nomenclature

D :

F

M :

OD :

pPr :

Re :

t

T :

U :

"76

V :

e_ :

pS

O"

0

V

specific heat at constant pressure

specific heat at constant volume

average of diagonal stress components

forcing functionMuch number

average of off-dlagonal stress components

pressurePrandtl number

Reynolds numbertime

temperaturefluid velocity

ratio of specific heats (Cp/C,)Kronecker tensor

gradient operator

thermal conductivity

molecular viscosityviscous dissipation function

density

average of stress componentson the scalar level

viscous stress tensor

(T - T,o)/(T - Too)average of stress componentson the vector level

Superscripts

" : Favre average

Subscripts

k, l : Cartesian indicesw : wall value

co : freestream value

3 Introduction

The phenomena of transitionto turbulence are so

complex as to defy a unifiedtheory at the present

time. As such, directnumerical simulationsof these

phenomena within the framework of Navier-Stokes

equations have assumed a dimension equal in im-

portance to experimental and even theoreticalstud-

ies.An effectivesimulation not only mimics a phys-

icalexperiment but has the added advantages of of-

feringreadilyretrievableclean informationwherever

and whenever itisneeded, and alsoa precisecontrol

of parameters unachiewble in the physical experi-

ment. Theoreticaland mathematical approaches pro-

vide progress inunderstanding through the processes

of abstractionand idealization;the resultsof anal-

ysis then furnish,apart from specificpredictions,a

deeper comprehension ofthe underlyinggeneralprin-

ciples.These featuresare not intrinsicto a numerical

simulation.However, they can be builtintothe simu-

lationstosome extent.Carefulconsiderationsofsuch

factorsare the bases of the researchprogram in this

area.Here atLaRC/ICASE our researchprogram in-

cludesboth the so-calledabrupt and restrainedtran-

sitionprocesses.Attention isconfined to the proto-

type problems of the channel flow and the parallel

boundary layerin the former category,and the wake

and the Taylor-Couette flows in the lattercategory.

It covers both incompressible flows and supersonic

flOWS.

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4 Poiseuille and Blasius Flows

4.1 Background

Although our knowledge of laminar-turbulent tran-

sition is by no means complete, the broad featuresare now clear at least in low subsonic shear flows.

This is in a large measure due to the the clas-

sical experiments of Klebanoff, Tidstrom and Sar-gent [1], Kovasznay, Komoda and Vasudeva [2], Hama

and Nutant [3]. These experiments were conductedin a controlled, identifiable disturbance environmentwhere the nondeterministic disturbances were im-

measurably small. Briefly stated, the instability and

transition process involves the following stages: 1)a primary instability evolving in accordance with

the linear theory, 2) a secondary instability lead-ing to the emergence of a flow of a pronounced

three-dimensional nature, and the appearance of the

streamwise vortex system, 3) the development of de-

tached high-shear layers, 4) at least a tertiary in-stability (if not an extremely rapid succession of a

sequence of instabilities) resulting in turbulent spot

development and 5) convection and coalescence ofturbulent spots to form a fully developed turbulent

flow. Similar patterns have been observed in plane

Poiseuille flow by Nishioka, Iida and Ichikawa [4].More recently, the experiments of Kachanov and

Levchenko [5], and Saric and Thomas [6] have un-covered some new details in the secondary instability

stage of the transition process in boundary layers.The flow pattern outlined above is relevant to natu-

ral transition under some limited, but not all, condi-

tions. The reviews of Stuart [7] and Tani [8] providea detailed discussion of the classical experiments andthe then available theoretical explanations of the iso-

lated stages or events of the transition process. The

more recent review of Herbert and Morkovin [9] pro-vides an overview of applied problems in stability andtransition, and a discussion of the various theoreti-

cal approaches to the secondary instability problem.Craik's monograph [10] deals with the general prob-

hm of wave interactions in various fields, and includes

a discussion of the secondary instability pertinent towall-bounded shear flows.

The primary instability has as its basis linear the-

ory and has long been well established. The the-

ory for the secondary instability is however more re-

cent, but is also fairly well established (Herbert [11],Nayfeh [12]). There have been no improvements in

the theoretical models for the other stages since the

survey article of Stuart [7] Full scale numerical sim-

ulations provide a unified basis for investigating the

relevance and relative importance of various events

right up to the formation of turbulent spots. In thepresent paper we shall dwell on numerical experi-

ments which have uncovered the secondary instability

associated with the center modes in plane Poiseuilleflow. We shall also discuss simulations which have

brought out the relative effectiveness of the various

laminar flow control (LFC) techniques such as heat-ing, favorable pressure gradient and suction. We will

present some representative results in each category.

The incompressible variable property Navier-

Stokes equations, in usual notation, are (Zang and

Hussaini [13])V u = 0 (1)

Ou I_.V--cgt+ u- Vu = -Vp + Re •(_Vu) + Fu (2)

00 1---_V. (gV0) + fo (3)_-_ + u. VO = RePr

where the lengths are scaled by the half channel-

width (or for the boundary layer by the displacement

thickness at the streamwise station of interest), ve-

locities by the corresponding center velocity for the

channel flow (or the corresponding mainstream ve-

locity for the boundary layer), and pressure by the

dynamic head; the forcing functions Fu and Fo are

designed to ensure parallel flow. The viscosity and

conductivity are given functions of temperature forthe heated wall case, and constants in other cases.

The applied boundary conditions assume periodic-

ity in the horizontal directions, and the no-slip ve-

locity on solid walls. For the heated water bound-

ary layer case, the no-slip condition for the velocityand a uniform temperature are enforced at the wall;

perturbations are assumed to be zero in the main-

stream. This permits a Fourier discretization in the

streamwise and spanwise directions, and a Cheby-shev discretization in the vertical direction. The ini-

tial condition consists of a triad of waves - one two-

dimensional wave and two skewed waves. The general

fractional step algorithm for efficiently solving the rel-

evant equations is given by Zang and Hussaini [14].

A survey of spectral algorithms for fluid dynamic

calculations is given in Hussaini and Zang [15], andCanuto, Hussaini, Quarteroni and Zang [16]. The lat-

ter is a monograph which deals with both theory and

applications of spectral methods.

4.2 Instability Due To A Triad Of

Center Modes

The so-called center modes of plane channel flows are

in fact temporal eigenfunctions of the Orr-Sommer-

212t-

feld equation which decay in time. However, a com-bination of a two-dimensional center mode with two

skewed modes is susceptible to an instability simi-lax to the secondary instability encountered in a sim-ilar non-resonant triad of wall modes. The essen-

tial qualitative difference between the center mode

and the wall mode {as is evident from Fig. la andlb) is that the maximum amplitude for the former

occurs away from wall towards the channel center,

while for the latter it occurs near the wall, and hence

the terminology. There are of course other quanti-

tative differences with regard to wavelengths, phasevelocities and decay rates. The Reynolds numberfor the simulation was 8000 based on the channel

half width, and both the streamwise and spanwise

wavenumbers were unity; the initial two-dimensional

and three-dimensional amplitudes were 10% and 3_

respectively of the channel center velocity. The reso-

lution requirements were monitored and the grid was

refined to resolve the fine structures as they emerged.

The finest grid was 96x128x192. Plotted in Fig. 2 isthe time history of the harmonic contents of the so-

lution. Note that the two-dimensional mode, labeled

{1,0), decays almost exponentially, but the three-

dimensional mode, labeled {1,1), grows after an initial

decay for about three periods 2_r being the charac-teristic period. The results axe presented at t _- 87

(a little less than 14 periods) at which time the finest

grid was just about sumcient to resolve the sharp gra-dients of the flow field. The vortex lines are displayed

in Fig. 3a, and for comparison purposes vortex lines

for an analogous case of wall modes are presented

in Fig. 3b. The streamwise and spanwise vorticity

contours are displayed in Fig. 4 at four equidistantstreamwlse planes over a wavelength. Fig. 5 dis-plays similar contours for the case of the wall modes.

The qualitativesimilaritiesand differencesare obvi-

ous. The center-mode vortex loop iscomparatively

away from the wall,and the pinching at the neck is

lessintensethan for the wall-mode vortexloop. Also,

the high-shearlayerassociatedwith the centermodes

islessintensethan the one associatedwith the wall

modes. The physical relevanceof the center mode

instabilityto the high-intensitybypass to transition

isas yet undetermined.

4.3 Effect of Heating on the Second-

ary Instability of Blasius Flow

Laminar flow control (LFC) vehicles might operate

in a finite-amplitude disturbance environment, so it

is of interest to examine the relative performance of

the various LFC techniques in the nonlinear regime.

While some experimental studies have investigatedthe influenceofnonlineardisturbancesourcessuch as

roughness,vibrations,free-streamturbulence,etc.on

the basictransitionprocess,therehave been no com-

parative studieson the sensitivityof the LFC tech-

niques such as heating,pressuregradientand suction

in the nonlinearregime. In the absence of any rele-

vant nonlinearstabilitytheoriesdue to the complex-

ity of the problem, numerical simulationscan play

a crucialrole in thisfield.As an example of such a

use ofsimulation,we have performed a Navier-Stokes

calculationinvolvingconditionssimilarto those used

in the experiment of Kovasznay, Komoda and Va-

sudeva [2]except that in our simulationthe wallwas

heated. The idealizationof a parallelboundary layer

was used mainly in order to meet the resolutionre-

quirements while keeping the computing time within

reasonablelimits.The Reynolds number ofthe simu-

lationwas 1100 based on the displacement thickness,

and the initialamplitudes ofthe two-dimensionaland

three-dimensionalTollmien-Schlichtingwaves (wall

modes) were respectively2.7_ and 0.8_ of the free

stream velocity.The finestgrid was 72x162x192.

Three differentsituationswere studied: 1) the un-

controlledcase,2) the heated fixedtemperature case,

and 3) the heated active temperature case. In the

heated fixedtemperature case,the temperature was

kept fixedat the initialvalue pertinentto the mean

flow conditions,and the temperature evolutionwas

totallyneglected. In the heated activetemperature

case, the temperature evolution was taken into ac-

count by solvingthe temperature equation along with

the momentum equations.In both the casesthe wall

temperature was 2.75% above the free stream tem-

perature. Fig. 6 shows the harmonic history of

the perturbationenergy. Note that the {1,0)mode

which grows in the uncontrolledcase decays in the

heated almost up to 4 periods. Heating damps the

(1,1)mode initially,but itstartsgrowing within the

firstperiod. Itappears that when the energy in thethree-dimensionalwave overtakes that in the two-

dimensionalwave, itfeedsenergy partlyintothe two-

dimensional wave. Fig. 7 shows the spanwise vortic-

itycontours on the peak plane. In the uncontrolled

case (Fig.7 top left)a kink developsinthe high-shear

layer at time t equal to three Tollmien-Schlichting

periods. It isgenerallyaccepted that a irrevocably

quick successionof events followsthereafterleading

to a turbulent spot formation. Heating the wall to

2.75% above the freestream temperature diffusesthe

high-shearlayer as isobvious from Fig. 7 (bottom

left).However, within the subsequent one and one

fourth period,turbulent spot formation appears to

become imminent (Fig. 7 top right). In the fixed

213

temperature case, it is clear from Fig. 7 (bottomright) that the high-shear layer formation is mellowed

clown even up to four and one fourth periods. The

fixed temperature case overpredicts the weakening ef-

fect of heating on the secondary instability. In other

words, the effect of temperature evolution is signifi-

cant and deleterious in the nonlinear regime whereasitisquitenegligiblein the linearregime.

5 Taylor-Couette Flow

The instabilityand abrupt transitionprocessare typ-

icalof aerodynamic flows. On the other hand, the

transitionto turbulenceby spectralevolutionare typ-

icalof geophysicalfluiddynamics where there is a

destabilizingforce field. The Taylor-Couette flow 6

in the annulus of concentricrotatingcircularcylin-

ders typifiesa classof geophysicalproblems involv-

ing instabilitiesand turbulence due to the presence

ofa destabilizingcentrifugalforcefield.DiPrima and

Swinney [17]providean excellentreview of instabil-

itiesand transitionsin Taylor-Couette flows. Our

interestliesin finite-lengthTaylor-Couette flow as

it offersthe situationof weak turbulence which is

amenable to directsimulation.The prime objective

of our research effortto examine the variousquali-

tativemathematical theories(Benjamin [18]).Some

of the routesto mathematical chaos appears to havebeen observed in theseflows.The chaos theoriesare

based on model equations. Our investigationswill

establishthe relevanceof these theoriesto Navier-

Stokes equations.

This problem, unlikethose consideredin the pre-

ceding sections,istruly inhomogeneous in two di-

rections. A spectralalgorithm necessarilyinvolves

Chebyshev polynomial expansions inradialand axial

directions,and Fourierexpansion inthe azimuthal di-

rection.The fractionalstep algorithm used to study

the channel flows and the parallelboundary layers

iseasilyextended totreatthisproblem (StreettandHussaini [19]).

The initial phase of study focussed on the steady-

state bifurcations in axisymmetric Taylor-Couette

flow (with the inner cylinder rotating, and the outer

cylinder and end walls remaining stationary} for as-

pect ratios of order unity. These problems provide wherea stringent test for a tlme-accurate method designed

to simulate the delicate unsteady processes leadingto transition. Among the numerous simulations car-

ried out to compare with the available experiments,

we present some typical results for the geometry of

Benjamin and Mullin [20]which had an aspect ratio

of1.05and radiusratioof0.615.The symmetric two-

cellmode isestablishedat a relativelylow Reynolds

number of 62, and then the Reynolds number isim-

pulsivelyraisedto 150. The development ofthe asym-

metric single-cellflowrepresentedby the streamlines

ina cross-sectionalplane isshown inFig. 8,and Fig.

9 displaysthe order parameter (which isa measure

of the asymmetry of the flow) and energy as func-tionsof time in unitsof characteristicdiffusiontime

scale. The continuing simulationsare on the verge

of capturing bifurcationsintotime-dependent states.

The finalobjectiveisto calculatethe dimension of

the strange attractorwhich issupposed to representthe weak turbulence.

Compressible Transition and

Turbulence

Transition to turbulence in supersonic and hyper-

sonic flows is a gray area. There are differing experi-

mental results, and unexplained visual observations.The picture is quite piecemeal compared to low Mach

number flows. The purpose of the research program

is to answer questions such as: a) do secondary in-

stability mechanisms observed in low subsonic flows

persist in supersonic flows, b) is the turbulent spot

formation the usual way transition occurs, and c) is

the transition process abrupt or restrained. The ini-

tial focus of our research program is on question Ca),i.e., whether the known prototypes of instabilities in

incompressible flows persist at high Mach numbers.

A spectral simulation of a three-dimensional insta-

bility process in a parallel boundary layer at Machnumber 4.5 was carried out.

The full compressible Navier-Stokes equations are,

in dimensionless form, (Erlebacher and Hussaini [21])

apa--f+ v = 0 (4)

1 Va(p,.,)+ v = .. (5)at Re

ap ..7

+u-Vp+'Tp_"u -- RePrM2 _7. (,VT)+ (',/- 1)(I)

(6)

r ,aUk au,r I_ 2= + ax, " 5.(v (7)

2" axl + _-_xk]aki (8)

214

and the equation of state is

_M£p-- pT (9)

with _ = Cv/C, , the ratio of specific heats. Athigh Much number, temperature variations across

the boundary layer become important and the tem-

perature dependence of viscosity must be taken into

account. The Prandtl number was assumed equal

to 0.7, and the Sutherland's law was prescribed for

viscosity variation with the temperature. The free

stream Math number was 4.5, and the Reynolds num-ber was 10000 based on the displacement thickness.

The boundary conditions were identical to those used

in the incompressible boundary layer simulations dis-cussed in the earlier section. The initial conditionconsisted of an unstable two-dimensional wave and a

pair of three-dimensional oblique waves superimposed

on a prescribed parallel mean flow. These waves were

the eigenfunctions of an eighth order eigenvalue prob-

lem, the compressible analogue of the incompressible

Orr-Sommerfeld equation. The initial amplitudes ofthe two-dimensional and three-dimensional perturba-

tions were taken to be respectively 5.4_ and 1.2_

of the free stream velocity. The temporal evolu-tion of this triad of waves was followed for about

eight periods of the initial two-dimensional Tollmien-

Schlichting wave (TS wave). The time history of sev-eral Fourier harmonics of the perturbation kinetic en-

ergy (integrated in the vertical direction) is shownin Fig. 10. After a period of slow growth of the

three-dimensional wave, labeled (1,1) mode, the non-

linearinteractionsbetween the (0,I)spanwise mode

and the primary (1,0)mode triggerthe secondary in-

stabilityafterapproximately 5 TS periods. Contour

plotsofthe streamwise velocity(Fig.11) at the time

of5 TS periods illustratethe incipientstagesofa K-

type breakdown (Erlebacher and Hussaini [21]). In

the supersonic simulations, the critical layer is an or-

der of magnitude further from the wall than in the

incompressible cases. One consequence is that thestructures are farther removed from the walls. Ini-

tlal results indicate that compressibility retards theonset of the secondary instability. However, a more

extensive parameter study is required before any firmconclusion can be drawn.

Our research program has recently focused its at-tention on the development of subgrid-scale mod-

els for compressible turbulence. A compressible ex-

tension of Bardina's linear combination model [22]

has been developed. Favre-averagingwas employed

to produce spatiallyaveraged compressible Navier-

Stokes equations which closelyresemble the incom-

pressibleones. For compressible as opposed to in-

compressible flow, the trace of the subgrid stresses

cannot be incorporated with the pressure sincethe

pressure is a true thermodynamic variable. These

isotropic stresses must now be modeled, and a newmodel constant, termed the isotropic constant, mustbe included.

This new model has been testedagainstdirectsim-

ulationsat Much numbers ranging from 0.0to0.6 and

on gridsof 643 to 1283. The turbulence model was

compared against this direct simulation (DS) data

suitably injected onto a coarser grid. The coarse

grid would normally be used in a largeeddy simula-

tion.A leastsquare fitbetween the totalstresscalcu-

lated from the DS and the modeled stresscomputed

from spatiallyfilteredvelocitiesallows the constants

to be determined. The constant determination was

performed on the vector level(Erlebacher,Hussaini,

Speziale,Zang [23]).Table 1 summarizes the correla-tion coefficientsbetween the modeled and the exact

D

OD

V

S

M0=0.0 M0=0.682 81

85 84

72 71

73 74

Table 1:

totalstressesat Much 0 and Much 0.6. Results are

presented on the tensor,vectorand scalarlevels.The

vector levelisthe divergence ofthe tensor,while the

scalarlevelisdefinedby the product ofV •_ and 6.

The Leonard stressisomitted from the totalsubgrid

stressbecause itcan be calculatedexactly. Corre-

lationcoefficientsof over 80_0 are obtained on the

tensor level,and above 70_ on both the vector and

scalarleveis.A more thorough study can be found in

Erlebacher et al. [23].

7 Concluding Remarks

Numerical algorithms for studying the physics of

transitionand turbulence in simple geometries have

been developed. Detailed studiesofthe highlynonlin-

ear stagesoftransitionpriorto turbulentspot forma-

tionhave been made. The use ofsimulationstostudy

the sensitivityof laminar flow controltechniques in

the nonlinear regime has been demonstrated. A

new instabilitymechanism associatedwith the center

modes inchannel flowshas been uncovered, although

215

its physical significance is as yet undetermined. The

first numerical simulations of their kind are being car-

ried out for supersonic boundary layer flows. The pre-

liminary results revealed the existence of a secondaryinstability similar to the one in incompressible flows.

References

[11]

[12]

[1] Klebanoff, P. S.; Tidstrom, K. D.; and Sar-gent, L. M.: The Three-Dimensional Nature

of Boundary-Layer Instability, J. Fluid Mech., [13]Vol. 12, 1962, pp. 1-34.

[2] Kovasznay, L. S.; Komoda, H.; and Vasudeva,

B. R.: Detailed Flow Field in Transition, Proc.1962 Heat Transfer and Fluid Mechanics Insti- [14]

tute, pp. 1-16, Stanford Univ. Press, Palo Alto,1962.

[3] Hama, F. R.; and Nutant, J.: Detailed Flow-Field Observations in the Transition Process in [15]

a Thick Boundary Layer, Proc. of the 1963 Heat

Transfer and Fluid Mechanics Institute, pp. 77-93, Stanford Univ. Press, Palo Alto, 1963.

[16]

[4] Nishioka, M.; Asai, M.; and Iida, S.: An Exper-

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(eds.), pp. 37-46, Springer-Verlag, Berlin, 1980.

[5] Kachanov, Yu. S.; and Levchenko, V. Ya.: The

resonant interaction of disturbances at laminar-

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[6] Saric, W. S.; and Thomas, A. S. W.: Experi-

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Phenomena in Fluids, T. Tatsumu (ed.} NorthHolland, 1984.

[2o][7] Stuart, J. T.; Hydrodynamic Stability, Appl.

Mech. Rev., Vol. 18, 1965, pp. 523-531.

[8] Tani, I.: Boundary-Layer Transition, Ann.Rev. of Fluid Mech., Vol. 1, 1969, pp. 169-196. [21]

[9] Herbert, T.; and Morkovin, M. V.: Dialogue

on bridging some gaps in stability and tran-sition research, Laminar-Turbulent Transition, [22]

R. Eppler and H. Fasel (eds.), pp. 47-70,

Springer Verlag, Berlin, 1980.

[10] Craik, A. D. D.:Wave Interactions and Fluid

Flows, Cambridge University Press, 1985.

Herbert, T.: Floquet analysis of secondary

instability in shear flows, Stability of TimeDependent and Spatially Varying Flows, D. L.

Dwoyer and M. Y. Hussaini (eds.), pp. 43-57,Springer Verlag, New York, 1987.

Nayfeh, A. H.: On secondary instabilities in

boundary layers, Stability of Time Dependent

and Spatially Varying Flows, D. L. Dwoyer and

M. Y. Hussaini (eds.}, pp. 43-57, Springer Ver-lag, New York, 1987.

Zang, T. A.; and Hussaini, M. Y.: Numeri-

cal Experiments on the Stability of Controlled

Shear Flows, AIAA Paper No. 85-1698, 1985.

Zang, T. A.; and Hussaini, M. Y.: On Spectral

Multigrid Methods for the Time-DependentNavier-Stokes Equations, Appl. Math. Comp.,

Vol. 19, 1986, pp. 359-372.

Hussaini, M. Y.; and Zang, T. A.: Spectral

Methods in Fluid Dynamics, Ann. Rev. Fluid

Mech., Vol. 19, 1987.

Canuto, C.; Hussaini, M. Y.; Quarteroni,

A.; and Zang, T. A.: Spectral Methods in

Fluid Dynamics, Springer-Verlag, Berlin, 1987.

DiPrima, R. C.; and Swinney, H. L.: Transition

in Flow between Rotating Concentric Cylin-

ders, Transition and Turbulence, R.E. Meyer(ed.}, Academic Press, 1981.

Benjamin, T. B.: Bifurcation Phenomena in

Steady Flow of a Viscous Fluid I. Theory, Proc.

Roy. Soc. London, Vol. A 359, 1978, pp. 1-26.

Streett, C. L.; and Hussaini, M. Y.: Finite

Length Effects in Taylor-Couette Flow, ICASE

Report No. 86-59, 1986.

Benjamin, T. B.; and Mullin, T.: Anamalous

modes in the Taylor Experiment, Proc. Roy.Soc. London, Vol. A377, 1981, pp. 221-249.

Erlebacher, G.; and Hussaini, M. Y.: Incipient

Transition Phenomena in Compressible Flows

over a Flat Plate, ICASE Report 86-39, 1986.

Bardina, J.; Ferziger, J. H.; and Reynolds,

W. C.: Improved Turbulence Models Based on

Large Eddy Simulation of Homogeneous, In-

compressible, Turbulent FLows, Report TF-19,

Department of Mechanical Engineering, Stan-ford University, Palo Alto, CA, 1983.

216

[23] Erlebacher, G.; Hussaini, M. Y.; Speziale, C.

G.; and Zang, T. A.: Subgrid-Scale Models

for Compressible Turbulence, ICASE Report,1987.

217

ORIGTNAI; PAGE IS

I .o

ORR-SOMMERFELD EIGENl

I I I

3-D osc

-UNC

-2

OF TON

I I I I

POOR

-1 .o - .5 0 .5 I .O Y

Figure la. C e n t e r mode.

ORR-SOMMERFELD EIGENFUNCTON

I I I I 1 .o -1 .o

-I .o - .5 0 .5 Y

F i g u r e l b . Wall mode.

10% 2-0 CEN. 3.0% CEN

<'3,21,-- /--

- l o l l I I 20 qo 60 80 100

T I M E

F i g u r e 3a. Vor tex l i n e s a t t = 87 f o r t h e c e n t e r mode t r a n s i t i o n i n c h a n n e l f l o w a t a Reynolds number of 8000.

F i g u r e 3b. V o r t e x l i n e s a f t e r 3-7/8 TS p e r i o d s f o r a ( f u n d a m e n t a l ) K-type t r a n s i t i o n i n a boundary l a y e r at a Reynolds number of 1100.

F i g u r e 2. Harmonic h i s t o r y €or a Re = 8000 c e n t e r mode s i m u l a t i o n .

218

,1 - • _._ ..... ]

:I .... ' I

• /L, II+

;:ii

I

:i[++ +++++,it,. llZ

i'l i:+k ++++]

l]q LI_ +lq i

I:'i ++++:..... :: ::_:

!/_+::?+_+!++++++!

+/k

Figure 4. Streamwise (left) and spanwise (right)

vorticity contours at t = 87 for a channel center

mode at a Reynolds number of 8000. The contour

interval is 0.20. Dashed lines Indicate negativecontours.

-I14

Y -t/2

-314

-!

0

q14

y -I12

-314 +++:_ :" +" _l.ll.:

-| , • ,

0 I X/Lx = 5/8-1/4

Y -I/2 F z

LI;;-+/41 ...........?L-::'

x,+iI..........+....x,x,,81I-_:++:+.:::_=_.... - +.. :+,!_ii+++:+i:+!+,iii!;i-:!L+_++!

xltx=3/8 [ ............ _....... x/tx= 3Istt.............,..........................t

01 X/Lx7/8

-]/4

-U2

-314 :_ '

0 ]/4 ]/2 314

ZILz

L ................... ??. ...............

_: :)17x'4:,_]....... - +..+- .... ................... " ++o ..........

-+.. ..* ..- ,44 ,. --7-,.... - .....

....+_L :,,

................... K....... X/Lx= 7/8 ]• .L.... ....... .+.* lO '. ............ I

..... ,q_;t.., " " +..... o ++_,:'+:+ ;, \ ...."-- /{li_ +i o .+ -+

,, -_++ • :_+.<..... _;G+,.;-.." + _+ ,:++:.: +!+,+ ++:.+!:++::+:::+:++1/4 1/2 3/4 1

lit z

Figure 5. Streamwlse (left) and spanwise (right)

vorticity contours at t = I82 for a channel wall

mode at a Reynolds number of 8000. The contour

interval is 0.40. Dashed lines indicate negativecontours.

2.77. 2-D, 0.87. 3 B

,mE: tloo.o mLPlltx; .2s mtF,_my:

ii_: T2 J_: t_ relax;

-3-- -- • - [ q I

-4- //'"

)t_ ....

z_ s

/," I /," " /) I I-'7

I 2 3 4

Ti_E

+m

.,s2

i

5

Figure 6. Harmonic energy history of an

uncontrolled (solid lines) and a heated (dashed

lines) boundary layer undergoing a K-type

transition at a Reynolds number of ItO0.

i i " "_i_!" _) iiXlfO 11ixto 1_ ___1.1 ii i

o _

Figure 7. Vertical shear in th+ peak plane forRe = It00 boundary layer si_*utation.

ORIGINAL PAGE I8

OF POOR QUAI_

219

• = L_ 4

4) 5)

Figure 8. Cross-sectional streamline at 5 stages

of two-cell/one-cell exchange process.

HaFrTl. ConL. M=4.5 _te = 10000

--45V ......... .,:'].

""F ," !--5 38

v ooo oo i 60 3 2o 4 ao e 4,0 a oo

PERIODS

Figure i0. Harmonic energy history for a Mach 4.5

boundary layer undergoing transition at a Reynolds

number of I0,000.

R = 150

o.

'69V'/ .66

i I

O. 1. 2.

-2.

loglo

--4.

lOgl0 E '

--6.

-8.

-10.

tD (L2/u)

I

3.

Figure 9. Time history of order parameter and

perturbation energy.

........ l

• r ....... I _r T ...... ,

, -jr

U

Figure If. Streamwise vorticity contours for the

compressible boundary layer transition at Mach

4.5. The contour interval is 0.20. Dashed lines

indicate negative contours.

ORIGINAL PAGE IS

OF, POOR QUALITY

220