NASA Contractor Report 178258 

ICASE REPORT NO. 87-9 

ICASE 

STABILITY, TRANSITION AND TURBULENCE 

(NASA-CR-178258) SIIAEILIIY, 'IEANSXTlON AID NS7-205GO 

1CfiBULENCE €inal Beport (NASL) 22 CSCL p 2QD Unclas 

63/34 45173 

M. Y. Hussaini 

NASA Contract No. NAS1-18107 

February 1987 

INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING 

NASA Langley Research Center, Hampton, Virginia 

Operated by the Universities Space Research Association 

23665 

National Aeronautics and 

Space Administram 

Lnoly-- 

HamptanVirginia 23665 
STABILITY,TRANSITION, AND TURBULENCE 

M. Y. Hussaini 

Institute €or Computer Applications in Science and Engineering 

ABSTRACT 

A glimpse is prov-ded of the research program -n stability, transition 

This program includes both the l and turbulence based on numerical simulations. 

so-called abrupt and the restrained transition processes. Attention is 

confined to the prototype problems of channel flow and the parallel boundary 

layer in the former category and the Taylor-Couette flow in the latter 

I category. It covers both incompressible flows and supersonic flows. Some 

I representative results are presented. 

Research was supported under the National Aeronautics and Space Administration 

under NASA Contract No. NAS1-18107 while the author was in residence at ICASE, 

NASA Langley Research Center, Hampton, VA 23665. 

i 
1 Abstract 

Superscripts 

A glimpse is provided of. the research program in 

stability, transition and turbulence based on numerical simulations. This program includes both the socalled abrupt and the restrained transition processes. 

Attention is confined to the prototype problems of 

channel 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. 

: Favre average 

Subscripts 

k, 1 : Cartesian indices 

w : wall value 

00 : freestream value 

2 

CP 

C" 

D 

F 

M 

OD 

P 

Pr 

Re 

t 

T 

7 

6 

V 

CL 

P 

S 

U 

IC 

U 

0 

V 

Nomenclature 

specific heat at constant pressure 

specific heat at constant volume 

average of diagonal stress components 

iorcing function 

Mach number 

average of off-diagonal stress components 

pressure 

Prandtl number 

Reynolds number 

time 

temperature 

fluid velocity 

ratio of specific heats (Cp/Cv) 

Kronecker tensor 

gradient operator 

thermal conductivity 

molecular viscosity 

viscous dissipation function 

density 

average of stress components 

on the scalar level 

viscous stress tensor 

average of stress components 

on the vector level 

(T - Tw)/(T - Tco) 

3 Introduction 

The phenomena of transition to turbulence are so 

complex as to defy a unified theory at the present 

time. As such, direct numerical simulations of these 

phenomena within the framework of Navier-Stokes 

equations have assumed a dimension equal in importance to experimental and even theoretical studies. An effective simulation not only mimics a physical experiment but has the added advantages of offering readily retrievable clean information wherever 

and whenever it is needed, and also a precise control 

of parameters unachievable in the physical experiment. Theoretical and mathematical approaches provide progress in understanding through the processes 

of abstraction and idealization; the results of analysis then furnish, apart from specific predictions, a 

deeper comprehension of the underlying general principles. These features are not intrinsic to a numerical 

simulation. However, they can be built into the simulations to some extent. Careful considerations of such 

factors are the bases of the research program in this 

area. Here at LaRC/ICASE our research program includes both the so-called abrupt and restrained transition processes. Attention is confined t.o the P_PO~.Otype problems of the channel flow and the parallel 

boundary layer in the former category, and the wake 

and the Taylor-Couette flows in the latter category. 

It covers both incompressible flows and supersonic 

flows. 

1 
4 Poiseuille and Blasius Flows 

4.1 Background 

Although our knowledge of laminar-turbulent transition is by no means complete, the broad features 

are now clear at least in low subsonic shear flows. 

This is in a large measure due to the the classical experiments of Klebanoff, Tidstrom and Sargent [I], Kovasznay, Komoda and Vasudeva [2], Hama 

and Nutant [3]. These experiments were conducted 

in a controlled, identifiable disturbance environment 

where the nondeterministic disturbances were immeasurably 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 leading to the emergence of a flow of a pronounced 

three-dimensional nature, and the appearance of the 

streamwise vortex system, 3) the development of detached high-shear layers, 4) at least a tertiary instability (if not an extremely rapid succession of a 

sequence of instabilities) resulting in turbulent spot 

development and 5) convection and coalescence of 

turbulent 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 uncovered some new details in the secondary instability 

stage of the transition process in boundary layers. 

The flow pattern outlined above is relevant to natural transition under some limited, but not all, conditions. The reviews of Stuart (71 and Tani [8] provide 

a detailed discussion of the classical experiments and 

the then available theoretical explanations of the isolated stages or events of the transition process. The 

more recent review of Herbert and Morkovin [9] provides an overview of applied problems in stability and 

transition, and a discussion of the various theoretical approaches to the secondary instability problem. 

Craik's monograph [lo] deals with the general problem of wave interactions in various fields, and includes 

a discussion of the secondary instability pertinent to 

wall-bounded shear flows. 

The primary instability has as its basis linear theory and has long been well established. The theory for the secondary instability is however more recent, but is also fairly well established (Herbert [Ill, 

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 simulations provide a unified basis for investigating the 

relevance and relative importance of various events 

right up to the formation of turbulent spots. In the 

present paper we shall dwell on numerical experiments which have uncovered the secondary instability 

associated with the center modes in plane Poiseuille 

flow. We shall also discuss simulations which have 

brought out the relative effectiveness of the various 

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

present some representative results in each category. 

The incompressible variable property NavierStokes equations, in usual notation, are (Zang and 

Hussaini [ 131) 

v.u=o (1) 

v . (~vo) -t Fo (3) 

where the lengths'are scaled by the half channelwidth (or for the boundary layer by the displacement 

thickness at the streamwise station of interest), velocities by the corresponding center velocity for the 

channel flow (or the corresponding mainstream velocity for the boundary layer), and pressure by the 

dynamic head; the forcing functions FU and Fe are 

designed to ensure parallel flow. The viscosity and 

conductivity are given functions of temperature for 

the heated wall case, and constants in other cases. 

The applied boundary conditions assume periodicity in the horizontal directions, and the no-slip velocity on solid walls. For the heated water boundary layer case, the no-slip condition for the velocity 

and a uniform temperature are enforced at the wall; 

perturbations are assumed to be zero in the mainstream. This permits a Fourier discretization in the 

streamwise and spanwise directions, and a Chebyshev discretization in the vertical direction. The initial condition consists of a triad of waves - one twodimensional wave and two skewed waves. The general 

fractional step algorithm for efficiently solving the relevant equations is given by Zang and Hussaini [14]. 

A survey of spectral algorithms for fluid dynamic 

calculations is given in Hussaini and Zang [15], and 

Canuto, Hussaini, Quarteroni and Zang [16]. The latter is a monograph which deals with both theory and 

applications of spectral methods. 

ao 1 

at RePr -+u.ve=- 

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-Sommer2 
feld equation which decay in time. However, a combination of a two-dimensional center mode with two 

skewed modes is susceptible to an instability similar to the secondary instability encountered in a similar non-resonant triad of wall modes. The essential qualitative difference between the center mode 

and the wall mode (as is evident from Fig. la and 

lb) 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 quantitative differences with regard to wavelengths, phase 

velocities and decay rates. The Reynolds number 

for 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 resolution requirements were monitored and the grid was 

refined to resolve the fine structures as they emerged. 

The finest grid was 96x128~192. Plotted in Fig. 2 is 

the time history of the harmonic contents of the solution. Note that the two-dimensional mode, labeled 

(l,O), decays almost exponentially, but the threedimensional mode, labeled (l,l), grows after an initial 

decay for about three periods 27r being the characteristic period. The results are presented at t = 87 

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

grid was just about sufficient to resolve the sharp gradients 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 equidistant 

streamwise planes over a wavelength. Fig. 5 displays similar contours for the case of the wall modes. 

The qualitative similarities and differences are obvious. The center-mode vortex loop is comparatively 

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

less intense than for the wall-mode vortex loop. Also, 

the high-shear layer associated with the center modes 

is less intense than the one associated with the wall 

modes. The physical relevance of the center mode 

instability to the high-intensity bypass to transition 

is as yet undetermined. 

4.3 Effect of Heating on the Secondary Instability of Blasius Fiow 

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 investigated 

the influence of nonlinear disturbance sources such as 

roughness, vibrations, free-stream turbulence, etc. on 

the basic transition process, there have been no comparative studies on the sensitivity of the LFC techniques such as heating, pressure gradient and suction 

in the nonlinear regime. In the absence of any relevant nonlinear stability theories due to the complexity of the problem, numerical simulations can play 

a crucial role in this field. As an example of such a 

use of simulation, we have performed a Navier-Stokes 

calculation involving conditions similar to those used 

in the experiment of Kovasznay, Komoda and Vasudeva (21 except that in our simulation the wall was 

heated. The idealization of a parallel boundary layer 

was used mainly in order to meet the resolution requirements while keeping the computing time within 

reasonable limits. The Reynolds number of the simulation was 1100 based on the displacement thickness, 

and the initial amplitudes of the two-dimensional and 

three-dimensional Tollmien-Schlichting waves (wall 

modes) were respectively 2.7% and 0.8% of the free 

stream velocity. The finest grid was 72x162~192. 

Three different situations were studied: 1) the uncontrolled case, 2) the heated fixed temperature case, 

and 3) the heated active temperature case. In the 

heated fixed temperature case, the temperature was 

kept fixed at the initial value pertinent to the mean 

flow conditions, and the temperature evolution was 

totally neglected. In the heated active temperature 

case, the temperature evolution was taken into acthe momentum equations. In both the cases the wall 

temperature was 2.75% above the free stream temperature. Fig. 6 shows the harmonic history of 

the perturbation energy. Note that the (1,O) mode 

which grows in the uncontrolled case decays in the 

heated almost up to 4 periods. Heating damps the 

(1,l) mode initially, but it starts growing within the 

first period. It appears that when the energy in the 

three-dimensional wave overtakes that in the twG 

dimensional wave, it feeds energy partly into the twodimensional wave. Fig. 7 shows the spanwise vorticity contours on the peak plane. In the uncontrolled 

case (Fig. 7 top left) a kink develops in the high-shear 

layer at time t equal to three Tollmien-Schlichting 

periods. It is generally accepted that a irrevocably 

quick succession of events follows thereafter leading 

to a turbuient spot formation. Eeating the wall to 

2.75% above the free stream temperature diffuses the 

high-shear layer as is obvious 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 

ccnnt hy s&i:g the temnomfiire r -- -- - -- - rynat.ion alone with 

3 
temperature case, it is clear from Fig. 7 (bottom 

right) that the high-shear layer formation is mellowed 

down even up to four and one fourth periods. The 

fixed temperature case overpredicts the weakening effect of heating on the secondary instability. In other 

words, the effect of temperature evolution is significant and deleterious in the nonlinear regime whereas 

it is quite negligible in the linear regime. 

5 Taylor-Couette Flow 

The instability and abrupt transition process are typical of aerodynamic flows. On the other hand, the 

transition to turbulence by spectral evolution are typical of geophysical fluid dynamics where there is a 

destabilizing force field. The Taylor-Couette flow 

in the annulus of concentric rotating circular cylinders typifies a class of geophysical problems involving instabilities and turbulence due to the presence 

of a destabilizing centrifugal force field. DiPrima and 

Swinney [17] provide an excellent review of instabilities and transitions in Taylor-Couette flows. Our 

interest lies in finite-length Taylor-Couette flow as 

it offers the situation of weak turbulence which is 

amenable to direct simulation. The prime objective 

of our research effort is to examine the various qualitative mathematical theories (Benjamin [ 181). Some 

of the routes to mathematical chaos appears to have 

been observed in these flows. The chaos theories are 

based on model equations. Our investigations will 

establish the relevance of these theories to NavierStokes equations. 

This problem, unlike those considered in the preceding sections, is truly inhomogeneous in two directions. A spectral algorithm necessarily involves 

Chebyshev polynomial expansions in radial and axial 

directions, and Fourier expansion in the azimuthal direction. The fractional step algorithm used to study 

the channel flows and the parallel boundary layers 

is easily extended to treat this problem (Streett and 

Hussaini [ 191). 

The initial phase of study focussed on the steadystate bifurcations in axisymmetric Taylor-Couette 

flow (with the inner cylinder rotating, and the outer 

cylinder and end walls remaining stationary) for aspect ratios of order unity. These problems provide 

a stringent test for a time-accurate method designed 

to simulate the delicate unsteady processes leading 

to transition. Among the numerous simulations carried out to compare with the available experiments, 

we present some typical results for the geometry of 

Benjamin and Mullin [ZO] which had an aspect ratio 

of 1.05 and radius ratio of 0.615. The symmetric twocell mode is established at a relatively low Reynolds 

number of 62, and then the Reynolds number is impulsively raised to 150. The development of the asymmetric single-cell flow represented by the streamlines 

in a cross-sectional plane is shown in Fig. 8, and Fig. 

9 displays the order parameter (which is a measure 

of the asymmetry of the flow) and energy as functions of time in units of characteristic diffusion time 

scale. The continuing simulations are on the verge 

of capturing bifurcations into time-dependent states. 

The final objective is to calculate the dimension of 

the strange attractor which is supposed to represent 

the weak turbulence. 

6 Compressible Transition and 

Turbulence 

Transition to turbulence in supersonic and hypersonic flows is a gray area. There are differing experimental 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 instability 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 initial focus of our research program is on question (a), 

i.e., whether the known prototypes of instabilities in 

incompressible flows persist at high Mach numbers. 

A spectral simulation of a three-dimensional instability process in a parallel boundary layer at Mach 

number 4.5 was carried out. 

The full compressible Navier-Stokes equations are, 

in dimensionless form, (Erlebacher and Hussaini (211) 

aP 

- + v. (pu) = 0 at (4) 

1 + v. (puu) = -v . u a(PU) 

at Re 

V.(pVT)+(7-1)@ aP -+u.vp+ypv.u = at RePrMZ 

(5) 

where 

1 auk aul 

2 as[ ask @= -(-+ -) ukl 

4 
and the equation of state is 

with 7 = Cp/Cv , the ratio of specific heats. At 

high Mach number, temperature variations across 

the boundary layer become important and the temperature 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 Mach number was 4.5, and the Reynolds number was 10000 based on the displacement thickness. 

The boundary conditions were identical to those used 

in the incompressible boundary layer simulations discussed in the earlier section. The initial condition 

consisted 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 problem, the compressible analogue of the incompressible 

Orr-Sommerfeld equation. The initial amplitudes of 

the two-dimensional and three-dimensional perturbations were taken to be respectively 5.4% and 1.2% 

of the free stream velocity. The temporal evolution of this triad of waves was followed for about 

eight periods of the initial two-dimensional TollmienSchlichting wave (TS wave). The time history of several Fourier harmonics of the perturbation kinetic energy (integrated in the vertical direction) is shown in 

Fig. 10. After a period of slow growth of the threedimensional wave, labeled (1,l) mode, the non-linear 

interactions between the [O,l) spanwise mode and the 

primary (1,O) mode trigger the secondary instability after approximately 5 TS periods. Contour plots 

of the streamwise vorticity (Fig. 11) at the time of 

5 TS periods illustrate the incipient stages of a Ktype breakdown (Erlebacher and Hussaini [21]). In 

the supersonic simulations, the critical layer is an order of magnitude further from the wall than in the 

incompressible cases. One consequence is that the 

structures are farther removed from the walls. Initial results indicate that compressibility retards the 

onset of the secondary instability. However, a more 

extensive parameter study is required before any firm 

conclusion can be drawn. 

Our research program has recently focused its attention on the development of subgrid-scale models for compressible turbulence. A compressible extension of Bardina’s linear combination model 1221 

has been developed. Favre-averaging was employed 

to produce spatially averaged compressible NavierStokes equations which closely resemble the incompressible ones. For compressible as opposed to incompressible flow, the trace of the subgrid stresses 

cannot be incorporated with the pressure since the 

pressure is a true thermodynamic variable. These 

isotropic stresses must now be modeled, and a new 

model constant, termed the isotropic constant, must 

be included. 

This new model has been tested against direct simulations at Mach numbers ranging from 0.0 to 0.6 and 

on grids of 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 large eddy simulation. A least square fit between the total stress calculated from the DS and the modeled stress computed 

from spatially filtered velocities allows the constants 

to be determined. The constant determination was 

performed on the vector level (Erlebacher, Hussaini, 

Speziale, Zang 1231). Table 1 summarizes the correlation coefficients between the modeled and the exact 

Mo = 0.0 Mo = 0.6 

Is1 73 I 74 

Table 1: 

total stresses at Mach 0 and Mach 0.6. Results are 

presented on the tensor, vector and scalar levels. The 

vector level is the divergence of the tensor, while the 

scalar level is defined by the product of V . fi and fi. 

The Leonard stress is omitted from the total subgrid 

stress because it can be calculated exactly. Correlation coefficients of over 80% are obtained on the 

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

scalar levels. A more thorough study can be found in 

Erlebacher et al. [23]. 

7 Concluding Remarks 

Numerical algorithms for studying the physics of 

transition and turbulence in simple geometries have 

been developed. Detailed studies of the highly nonlinear stages of transition prior to turbulent spot formation have been made. The use of simulations to study 

the sensitivity of laminar flow control techniques in 

the nonlinear regime has been demonstrated. A 

new instability mechanism associated with the center 

modes in channel flows has been uncovered, although 

5 
its physical significance is as yet undetermined. The 

first numerical simulations of their kind are being carried out for supersonic boundary layer flows. The preliminary results revealed the existence of a secondary 

instability similar to the one in incompressible flows. 

References 

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

of Boundary-Layer Instability, J. Fluid Mech., 

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 Institute, pp. 1-16, Stanford Univ. Press, Palo Alto, 

1962. 

[3] Hama, F. R.; and Nutant, J.: Detailed FlowField Observations in the Transition Process in 

a Thick Boundary Layer, Proc. of the 1963 Heat 

Transfer and Fluid Mechanics Institute, pp. 77- 

93, Stanford Univ. Press, Palo Alto, 1963. 

[4] Nishioka, M.; Asai, M.; and Iida, S.: An Experimental Investigation of the Secondary Instability in Laminar-Turbulent Transition, LaminarTurbulent Transition, R. Eppler and H. Fasel 

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

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

resonant interaction of disturbances at laminarturbulent transition in a boundary layer, J. 

Fluid Mech., Vol. 138, 1984, pp. 209-247. 

[6] Saric, W. S.; and Thomas, A. S. W.: Experiments on the subharmonic route to turbulence 

in boundary layers, Turbulence and Chaotic 

Phenomena in Fluids, T. Tatsumu (ed.) North 

Holland, 1984. 

171 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. 

191 Herbert, T.; and Morkovin, M. V.: Dialogue 

on bridging some gaps in stability and transition research, Laminar-Turbulent Transition, 

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

Springer Verlag, Berlin, 1980. 

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

Flows, Cambridge University Press, 1985. 

6 

[ll] Herbert, T.: Floquet analysis of secondary 

instability in shear flows, Stability of Time 

Dependent and Spatially Varying Flows, D. L. 

Dwoyer and M. Y. Hussaini (eds.), pp. 43-57, 

Springer Verlag, New York, 1987 

[l2] Nayfeh, A. H.: On secondary instabilities in 

boundarv lavers. Stability of Time Dependent rll 

and Spatially Varying Flows, D. L. Dwoyer and 

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

[13] Zang, T. A.; and Hussaini, M. Y.: Numerical Experiments on the Stability of Controlled 

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

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

Multigrid Methods for the Time-Dependent 

Navier-Stokes Equations, Appl. Math. Comp., 

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

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

Methods in Fluid Dynamics, Ann. Rev. Fluid 

Mech., Vol. 19, 1987. 

[16] Canuto, C.; Hussaini, M. Y.; Quarteroni, 

A.; and Zang, T. A.: SpectralMethods in 

Fluid Dynamics, Springer-Verlag, Berlin, 1987. 

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

in Flow between Rotating Concentric Cylinders, Transition and Turbulence, R.E. Meyer 

(ed.), Academic Press, 1981. 

[18] 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. 

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

Length Effects in Taylor-Couette Flow, ICASE 

Report No. 8659, 1986. 

[20] Benjamin, T. B.; and Mullin, T.: Anamalous 

modes in the Taylor Experiment, Proc. Roy. 

SOC. London, Vol. A377, 1981, pp. 221-249. 

[21] Erlebacher, G.; and Hussaini, M. Y.: Incipient 

Transition Phenomena in Compressible Flows 

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

[22] Bardina, J.; Fersiger, J. H.; and Reynolds, 

W. C.: Improved Turbulence Models Based on 

Large Eddy Simulation of Homogeneous, Incompressible, Turbulent FLOWS, Report TF-19, 

Department of Mechanical Engineering, Stanford University, Palo Alto, CA, 1983. 
1231 Erlebacher, G.; Hussaini, M. Y.; Speziale, C. 

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

for Compressible Turbulence, ICASE Report, 

1987. 

7 
ORR-SOMHERFELD E I GENFUNCTON 

RE: 5000.0 ALPHA: 1.12 BETR: 2.00 

NC : 69. RE[ OMEGRl : 1 -066315 IMC OMEGRI : - -051005 

1 .o I I I 

3-D OSC 

-1 .o -1 I1 .o - .5 0 .. .5 1 .o 

Y 

Figure la. Center mode. 

ORR-SOMMERFELD EIGENFUNCTON 

RE: 5000.0 RLPHA: 1.12 BETR: 2.00 

NC : 69. RE[ OMEGA) : -36rlY73 IM( OMEGRI : - -076227 

-1 .o 

-1 .o - .S 0 .5 1 .o 

Y 

Figure lb. Wall mode. 

8 
-2 I I I 

-3 

1 

(1.0) 

Figure 2. Harmonic history for a Re = 8000 center mode simulation. 

9 
10 

-rt 

v1 

G 

(d 

$4 

U 

a 

a 

0 

I 

I 

I 

I a 
ORWNAL PAGE IS 

Of POOR QUALlrV 

aJ 

a 

h 

+I 

h 

h 

d 

a 

11 
J 

-114 

-112 

I 

n 

-114 

............. -112 

-514 

1 114 io 114 112 I O 112 514 

ZIlz dLz 114 

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 izdicatc negative contours. 

12 
Y -112 

-314 

-1 

__.. - .- . -114 ..................... ...................... 

......... .... ........... -'p.."' .. y -112 11 

-314 

-1 

.................. 

-114 .................. ....... ................. 

........ 

................. 

o 114 112 314 1 0 114 112 314 1 

ZlLZ Z/LZ 

Figure 5. Streamwise (left) and spanwise (right) vorticity contours at t = 

182 for a channel wall mode at a Reynolds number of 8000. The contour 

interval is 0.40. Dashed lines indicate negative contours. 

13 
0 1 2 3 5 

TIME 

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 1100. 

14 
I 

Y 

3, . , . , , . . , H(A110 IflXED 11 1.4114 

. 

Figure 7. Vertical shear in the peak plane for Re = 1100 boundary layer 

simulation. 

15 
n 

e 

16 

r( 

rl 

aJ 

u 

I 

a 

C 

0 

\ 

rl 

4 

a 

u 

I 

U 

U 

rn 

m 

U 

(0 

aJ 

C 

4 

.. 

U 

rn 

r( 

6" 

0 

4 

U 

u 

aJ 

m 

1 

I 
log10 

-8. 

-1 0. 

0. 

7-• 615 

R = 150 

Figure 9. Time 

1. 2. 

tD ( L2/Uj 

history of order parameter and perturbation 

3. 

energy. 

17 
1 

-4.30 

-4.57 

-4.84 

-5.11 

-5.38 

-5.65 

-5.92 

-6.19 

-6.46 

-6.73 

-7.00 

PERIODS 

Figure 10. Harmonic energy history for a Elach 4.5 boundary layer undergoing 

transition at a Reynolds number of 10,000. 

18 
XIL,= 0 

f i I 

I XIL,. 314 

1 1 

Y Y 

2 2 

Ut,= 114 

1 

Y Y 

WL,= 1/2 

1 0 1/11 112 3/4 0 1A 112 314 1 

LZ LZ 

Figure 11. Streamwise vorticity contours for the compressible boundary layer 

transition at Mach 4.5. The contour interval is 0.20. Dashed lines indicate 

negative contours. 

19 
Standard Bibliographic Page 

1. Report No. NASA CR- 178258 

ICASE Report No. 87-9 

2. Government Accession No. 3. Recipient’s Catalog No. 

~ 

6. Performing Organization Code 

7. Author(s) 8. Performing Organization Report No. M. Y. Hussaini 8 7-9 

4. Title and Subtitle 

STABILITY, TRANSITION AND TURBULENCE 

5. Report Date 

February 1987 

9. Performing Organization Name and Address 10. Work Unit No. 

Institute for Computer Applications in Science and 

Engineering 

Jerry C. South 

11. Contract or Grant No. 

Conf. on Supercomputing in 

Aerospace 

ton, 

Mail Stop 132C, NASA Langley Research Center 

VA 23665-5225 - 

12. Sponsoring Agency Name and Address 

Final Report 

16. Abstract 

NAS1-18107 

13. Type of Report and Period Covered 

A glimpse is provided of the research program in stability, transition 

Attention is 

and turbulence based on numerical simulations. 

the so-called abrupt and the restrained transition processes. 

confined to the prototype problems of channel 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. 

This program includes both 

National Aeronautics and Space Administration 

Washington, DC 20546 

Contractor Report 

14. Sponsoring Agency Code 

505-90-21-01 

17. Key Words (Suggested by Authors(s)) 18. Distribution Statement 

spectral simulation, Navier-Stokes 

equations, Poiseuille, Blasius and 

Taylor-Couette flows 

34 - Fluid Mechanics and Heat 

Transfer 

Unclassified - Unlimited 

19. Security Classif.(of this report) 

Unclassified 

20. Security Classif.(of this page) 21. No. of Pages 22. Price 

Unclassified 21 A02