NASA Contractor Report 178025

lCASE REPORT NO. 85-56

NASA-CR-] 78025

19860010083

leASE

WAVE PHENOMENA IN A HIGH REYNOLDS NUMBER

COMPRESSIBLE BOUNDARY LAYER

Alvin Bayliss

Lucio Maestrello

Paresh Parikh

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Contract Nos. NASI-I 7070, NASI-18l07

November 1985

INSTITUTE FOR CO~WUTER APPLICATIONS IN SCIENCE AND ENGINEERING

NASA Langley Research Center, Hampton, Virginia 23665

Operated by the Universities Space Research Association

NI\SI\

National Aeronautics and

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WAVEPHENOMENAIN A HIGH

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A. Bayliss

Exxon Corporate Research Science Laboratories

L. Maestrello

NASA Langley Research Center

P. Parikh

Vigyan Research Associates, Inc.

E. Turkel

Institute for Computer Applications in Science and Engineering

and

Tel-Aviv University, Tel-Aviv, Israel

ABSTRACT

Growth of unstable disturbances in a high Reynolds number compressible

boundary layer is numerically simulated. Localized periodic surface heating

and cooling as a means of active control of these disturbances is studied. It

is shown that compressibility in itself stabilizes the flow but at a lower

Mach number, significant nonlinear distortions are produced. Phase

cancellation is shown to be an effective mechanism for active boundary layer

control.

Partial support for the first and fourth authors was provided by the

National Aeronautics and Space Administration under NASA Contracts No. NASI17070 and NASI-18107 while they were in residence at ICASE, NASA Langley

Research Center, Hampton, VA 23665-5225.

Partial support was provided for the third author under NASA Contract No.

NASI-17919.

Introduction

This paper is a numerical study of the behavior of spatially unstable

waves in a high Reynolds number, compressible boundary layer. The numerical

simulations are conducted by solving the laminar, two-dimensional,

compressible Navier-Stokes equations over a flat plate with a fluctuating

disturbance generated at the inflow. The primary objectives of this work are

to study the nonlinear growth and distortion of the unstable waves and also to

study techniques for the active control of these disturbances by time-periodic

surface heating and cooling. The results presented here are an extension of

the results presented in [I], [2].

An extensive experimental investigation of the evolution of linearly

unstable waves in a boundary layer is described in [3]. The numerical

simulations closely parallel the conditions of this experiment except for the

Mach number of the mean flow which is considerably higher than in the

experiment. The authors in [3] investigated the initial stage of the

development of the disturbance, in particular the growth of the higher

harmonics. It was shown that in this region three-dimenslonal effects were

not important. Our results are in qualitative agreement with these

experimental observations. Murdock [4] studied the growth of spatially

unstable disturbances in an incompressible flow. The results presented here

are similar to those obtained in [4]. However, our results are obtained for a

much smaller initial disturbance level and bring out some additional features

of the wave development. Furthermore, we account for compressibility effects.

The active control of unstable waves by surface heating was introduced in

[5] and [6] for disturbances in water. The idea is to introduce a temperature

disturbance by surface heating which is out of phase with the propagating
-2-

disturbance. The growth of the unstable wave is reduced because the two waves

cancel. In [5] and [6] a feedback control mechanism was used to generate the

control signal. A feedback mechanism is needed since by an appropriate choice

of phase the signals can be made to amplify rather than cancel. In the

experiments of Maestrello [7], instantaneous transition in air was achieved by

localized surface heating.

The use of active control techniques which attempt to modify the unstable

wave, rather than the basic mean flow, offers considerable promise as an

efficient method of delaying transition. It was shown in [8] that this

technique should be considerably more difficult to apply in air than in

water. There are three reasons for this. First, steady heating tends to

destabilize the mean flow. Second, a much larger temperature disturbance is

required to generate an equivalent change in the viscosity. Finally, there is

the possibility that temperature disturbances can be transformed into acoustic

disturbances, thereby losing effective phase control. Nevertheless, the

numerical simulations presented in [2] and in the present paper demonstrate

that this technique is feasible.

In Section 2 we discuss the numerical model. The major feature of the

numerical scheme is the use of fourth-order accurate finite differences for

the inviscid terms of the Navier-Stokes equations. It is well known (see, for

example, [9], [I0]) that fourth-order accuracy is essential in preventing

numerical dispersion and dissipation from significantly degrading the accuracy

of wave propagation problems. Fourth-order accuracy is even more essential

for the present problem, in order to prevent numerical errors on the inviscid

terms, which act as an additional source of viscosity, from lowering the

effective Reynolds number of the mean flow and hence, incorrectly stabilizing

the flow.
-3-

In Section 3 we present numerical results. The section is divided into

two parts. In the first part the nonlinear evolution of the uncontrolled wave

is described and compared with experimental observations. In the second part

the simulation of localized tlme-periodic surface heating and cooling is

discussed. Generally, the active control would be expected to be most

effective in the linear regime before nonlinear growth and distortion becomes

significant. When nonlinear effects predominate then periodic control is no

longer possible as harmonics develop and the waves will not have a welldefined phase. One needs to develop control techniques which account for

random amplitudes and phases. In Section 4 we discuss our conclusions.

2. Numerical Model

In this section we describe the numerical model. An extensive discussion

of the model is presented in [i]. The discussion here will be brief and the

reader is referred to [I] for further details.

The laminar, compressible Navier-Stokes equations can be written in the

form

w = F + G . (2.1)

t x y

Here, w is the vector (p, pu, pv, E)T, P is the density, u and v are

the x and y velocity components respectively, and E is the total

energy. The functional forms of the functions F and G are standard and

will be omitted for brevity.

The computational domain is the rectangle shown in figure I. First, a

basic steady flow, in this case a spreading boundary layer, is computed. Then
-4-

an unsteady disturbance is specified at inflow and the development of the

disturbance as it propagates through the steady flow is simulated by solving

the system (2.1). Since the disturbance must be followed over a large number

of wavelengths, it is essential to use a higher order accurate scheme. We

therefore use a scheme which is second-order accurate in time and fourth-order

accurate in space.

For the one-dimensional equation

Wt +Fx = 0,

we have

_i+l W n At n

n] n- n

= i _ _ 7-Fi+l - FI_ - _Fi+2

f

Fi+ I )

wn+l l(_n+l + n At {7_n+l _---n+ I ._--n+I _---n+l)

i = 7 i Wi 6Ax - i - _i-l) - [Fi-I - _i-2) " (2.2)

The scheme (2.2) becomes fourth-order when F = F(u) if it is alternated with

a symmetric variant in which there are backward differences in the predictor

and forward differences in the corrector. It has a greatly-reduced truncation

error compared with the second-order MacCormack scheme. Our experience has

been that fourth-order accuracy is necessary to efficiently compute the class

of problems considered here. Operator splitting is used so that the twodimensional system (2. I) is solved by successive applications of onedimensional solution operators of the form (2.2). This scheme is fourth-order

accurate on the inviscid terms. The scheme is fourth-order on the viscous

terms for a constant viscosity but is only second-order accurate on the

viscous terms when the viscosity is spatially dependent. For this problem,
-5-

due to the high Reynolds number, the primary source of error is due to the

inviscid terms and the scheme is very accurate. A sixth-order, in space,

algorithm is presented in [2].

A detailed description of the boundary conditions is given in [I].

Radiation conditions are used at the outflow and upper boundaries. At the

inflow we must specify three boundary conditions. These are the three

incoming characteristic variables based on linearizing the function F in

(2.1) and ignoring variations in the y direction. Let Q denote an

incoming variable. We specify Q at inflow by

imt

Q = Qsteady + _ e Qos(Y)

where Qsteady is the steady state solution, and Qos(Y) is obtained from

the linearized Orr-Sommerfeld equation (linearized about the inflow steady

profile) for the (unstable) frequency m. The Orr-Sommerfeld solutions were

obtained from a program developed by J. R. Dagenhart at NASA Langley Research

Center. This program neglects compressibility effects. Thus, there exists a

discrepancy when comparing with linear theory at large flow velocities.

However, for an inflow Mach number of 0.4 close agreement with linear theory

is obtained.

3. Numerical Results

In this section we present numerical results for the model described in

Section 2. The section is divided into two parts. Part A is concerned with

wave propagation through the mean flow without external active control. In

Part B the effect of active surface heating and cooling is discussed.
-6-

A. Uncontrolled Wave Propagation

We first consider a boundary layer with a free stream Mach number of

0.4. The unit Reynolds number is 3.0 × 105 . The computational domain is

chosen so that at inflow Re6, (Reynolds number based on displacement

thickness) is 990 and at outflow Re_, is 1730. Based on the inflow profile,

frequency F = I2_fv/g_) of .8 x 10-4 is unstable.

the nondimensional

(Here f is the frequency in Hertz, v the kinematic viscosity and U the

free stream velocity.)

In figure 2 we plot the growth rates of the unstable disturbance as a

function of Re6,. The growth rates are computed by computing the RMS of

pu(t,y), integrating the result in y and normalizing by the value at

inflow. The results in figure 2 are plotted for two different values of

(amplitude of the inflow perturbation) and compared with results obtained from

linear (incompressible) stability theory.

It is apparent that for small g, the growth rates are very close to

those predicted by linear theory. Differences can be attributed to

nonparallel and possibly compressibility effects. In particular, the solution

decays at roughly the same position as predicted by linear theory. For larger

values of _, the solution does not decay and exhibits a nonlinear growth.

This behavior is similar to that observed by Thomas [11] using a vibrating

ribbon in air.

In order to analyze the solution as a function of y we plot in figure 3

the RMS of pu as a function of y at four different x locations. The

results in figure 3 are obtained with _ = 0.02. In this case the profiles

follow the basic shape of the inflow Tollmein-Schlichting profile. However,

the amplitude increases as the disturbance grows downstream. The shape of
-7-

Tollmein-Schlichting profile is preserved even though the solution is

exhibiting a growth which is not predicted by linear theory (which, in fact,

predicts decay for Re_, > 1500).

In figure 4 the amplitude of the fundamental (FI) and the first harmonic

(F2) are plotted as a function of y/_ where 6 is the local boundary layer

thickness for Re_, = 1579. The data is normalized so that the peak of the

fundamental is 1.0. It can be seen that the amplitude of the first harmonic

has grown to roughly 30% of the peak of the fundamental. It is also apparent

that the harmonic has maximum near the wall and thus the nonlinearity is most

pronounced there. This is in agreement with the experimental measurements of

[3]. In figure 5 we reproduce a figure from [3] illustrating the experimental

fluctuation level.

The experimental data clearly shows a much larger degree of nonlinearity

than the computations. However, the flow velocity in [3] is much less than

the present computation. The value of Re_, can not be determined from the

information presented in [3]. The computation does not produce a double peak

for the fundamental although there is some indication of a double peak for the

harmonic.

It is apparent from both figures that nonlinearity is expected to be most

noticeable in the solution near the wall and near the turning point where the

fluctuation goes through zero. This can also be seen in figure 6 where pu

is plotted as a function of nondimensional time at several y locations for

Re6, = 1579. We observe that at this location the disturbance has grown so

large that there is a cylic separation and reattachment near the wall.

The compressibility effects for this case are not large. In order to

examine the effect of compressibility we compare in figure 7 the growth of the
-8-

disturbance for free stream Mach numbers 0.4 and 0.7. The data is taken so

that Red, at inflow is 998.0 in both cases. The parameter s is also the

same in both cases. Both inflow Tollmien-Schlichting profiles are taken from

an incompressible program and would therefore be expected to be slightly less

accurate for M = 0.7.

The results demonstrate a very strong stabilizing effect due to increasing

compressibility. In fact the M = 0.7 decays downstream following the

stability curve while the M = 0.4 case does not decay due to nonlinear

effects. Examination of the solution indicates that there is much less

nonlinearity in the M = 0.7 case.

B. Active Control by Surface Heating and Cooling

We next consider the effect of active surface heating and cooling. All

results are obtained for the M = 0.4 case described previously. The surface

heating and cooling is effected by imposing a boundary condition of the form

T _ Tw ± (c_+ B sin(_ + i))2 (3.1) Tref Tref

where the + sign is for heating and the - sign is for cooling. Tw is the

temperature at the wall (520°R) and Tre f is a reference temperature. The

form of (3.1) is chosen to model a combination of a DC current and an AC

current. No attempt was made to optimize the coefficients = and _, however,

the effect of varying the phase i was studied.

It is well known that static heating is destabilizing in air and static

cooling is stabilizing. The numerical results with _ >> B confirm this. In

order to demonstrate the effect of phase we plot the growth rates in figure 8
-9-

for cooling with different values of i and in figure 9 for heating. In all

cases the control strip extends over roughly 20% of a wavelength and is

centered at Re_, = 1263.

It is apparent from figures 8 and 9 that the response is very sensitive to

the phase. In particular, cooling can destabilize with an appropriate choice

of phase and heating can stabilize. These results clearly indicate that phase

cancellation can be a viable mechanism for the control of _nstable

disturbances. By appropriate choice of phase the active heating and cooling

can have effects exactly opposite from the static case.

In figures i0 and II we examine time traces for 0u(t) at different y

locations. The traces are all taken at x = 1.0 ft., significantly downstream

of the heating (or cooling) strip which was placed at 0.6 ft. The figures

show that the residual effects of the active heating and cooling are an

amplitude and phase change in the propagating wave. Very little distortion in

the wave form is introduced by the active heating and cooling.

4. Conclusions

The behavior of unstable disturbances in a high Reynolds number flow can

be effectively computed provided a fourth-order accurate finite difference

scheme is used. The results show that significant nonlinear distortion is

produced which is in qualitative agreement with experiment.

It is shown via a full Navier-Stokes solution that increasing

compressibility can significantly stabilize the flow over a flat plate. In

addition, it is shown that the mechanism of phase cancellation is a viable

mechanism for the control of growing disturbances. The authors are currently
-i0-

extending these results to flows over curved surfaces thus accounting for nonzero pressure gradients.
-iiReferences

[i] Bayliss, A., Maestrello, L., Parikh, P., and Turkel, E., "Numerical

Simulation of Boundary Layer Excitation by Surface Heating/Cooling,"

AIAA-85-0565, March 1985.

[2] Bayliss, A., Parikh, P., Maestrello, L., and Turkel, E., "A Fourth-Order

Method for the Unsteady Compressible Navier-Stokes Equations," AIAA-85-

1694, July 1985.

[3] Kachanov, Yu. S., Kozlov, V. V., and Levchenko, V. Ya., "Nonlinear

Development of Wave in a Boundary Layer," Fluid Dynamics, Translation,

Vol. 12, No. I, January-February, 1977.

[4] Murdock, J., "A Numerical Study of Nonlinear Effects on Boundary Layer

Stability," AIAA J., Vol. 15, No. 8, August 1977, pp. 1167-1173.

[5] Liepmann, H. W., Brown, G. L., and Nosenchuck, D. M., "Control of

Laminar Instability Waves Using a New Technique," J. Fluid Mech., Vol.

118, 1982, pp. 187-200.

[6] Liepmann, H. W. and Nosenchuck, D. M., "Active Control of LaminarTurbulent Transition," J. Fluid Mech., Vol. 118, 1982, pp. 201-204.

[7] Maestrello, L., "Active Transition Fixing and Control of the Boundary

Layer in Air," AIAA-85-0564, March 1985.
-12-

[8] Maestrello, L. and Ting, L., "Analysis of Active Control by Surface

Heating," AIAA J., Vol. 23, No. 7, July 1985, pp. 1038-1045.

[9] Kreiss, H. O. and 01iger, J., "Comparison of Accurate Methods for the

Integration of Hyperbolic Systems, Tellus, Vol. 24, 1980, pp. 119-225.

[i0] Turkel, E., "On the Practical Use of Higher Order Methods for Hyperbolic

Systems," J. Comput. Phys., Vol. 35, 1980, pp. 319-340.

[Ii] Thomas, A. S. W., "The Control of Boundary Layer Transition Using a

Wave-Superposition Principle," J. Fluid Mech., Vol. 137, 1983, pp. 233-

250.
-13-

Y

r I

Inflow! Computational] Outflow.

" I " domain

Figure I. Schematic of computational domain.
-14-

1.4 1.5

I

--0- Linear theory

--0- Computed, 4th order

--6-- Computed, 2nd order

1.01.1

2

6

3

5

4

J(pU); IJ(puJ; 0

7

Figure 2. Comparison of amplitude growth with linear theory.
-15-

J(PU)2

.125

filii' .....---- ....... _

.100

.075

.050

.025

o 2 4 6 8 10

Figure 3. RMS pu versus y at different x locations.
-16-

Amplitude

1.00

.25 _///-2f

I I I

0 .3 ..6 .9 1.2

y/&

Figure 4. F 1 and F2 versus y/6 at Re_, = 1579.
-17-

#

U

0 .5 1.0 • 1.5

Figure 5. Experimental analysis of harmonic content of fluctuating

disturbance (from [3]).
-18-

pu(t)

1.1

y16= 1.0.5 (pu)2 i 03_8

1.0

.9

.93 (.716)

1.00381

•91

.9 _ (.556)

.8

(.0465)

.7

.8 (.359)

.6

(.0473)

.4

.3

.1

1.i011

-.i I I I

6.1 6.4 6.7 7.0

tu /L o0

Figure 6. pu versus t for selected values of y; Red. = 1579.
-19-

RMS

amplitude

ratio

8.

. . . "

i i"

iI

" I I

1.0 1. ] 1.2 1.3 1.4 1.5 1.6 1.7

Re.s* X10-a

Figure 7. Growth rates for M = 0.7 and M = 0.4.
-20-

(pu)2/f (PU)x= 0

5 _--oo-__

4

3

2 _ o . "P

1 .

I I I I I

0 .1 .2 ..3 " .4 .5

X/L

Figure 8. Effect of phase - cooling.
-21-

o

5 - ¢.= 180°-_

.4-

_=0 °

3-

q)= 90°

2 --

1 _ Locationofstrip

I I " I I )

0 .1 .2 .3 .4 .5

X/L

Figure 9. Effect of phase - heating.
-22-

Uncontrolled

pu(t) @@ = O180° °I Heated

1.05 _ ., r_

/

/

/ y

1.00 / - 1.898

81

.95

.92

\

\

/ \ = 0.963

\ I \ \/I

\ I \ I

\ I \

\1

.88

.88 _,

/

\

\

\

84 _ \ = 0.867 • \ \

\ \

I \ I

I \ I

/

i \J

.80

/\

I

.2

\

\ = 0.115

•1 \I1 \\

\ \

\

\

o [ I 1

2.9 3.1 3.3 3.5

tUco/L

Figure I0. pu versus t for different phases at x = 1.0 ft;

heating at x = 0.6 ft.
-23-

pu(t) Uncor_trolled

1.05 _-- I_== O° 180 ° ! Cooled

Y

1.00 \ w= 1.898

\ &l

.95

.92 T

.91 / = 0.963

.90

.84 = 0.867

J

.2

= 0.115

I

I

.1 !

2.9 3.1 3.3 3.5

tUQo/L

Figure II. pu versus t for different phases at x = l.O ft;

cooling at x = 0.6 ft.


1. Report No. NASA CR-178025 I2. Government Acceuion No. 3. Recipient" CltoilOll No.

ICASE Report No. 85-56

4. T.t1e and Subt.t1e 5. Report Olt.

November 1985

WAVE PHENOMENA IN A HIGH REYNOLDS NUMBER 6. Performing OrganiZltion Code

COMPRESSIBLE BOUNDARY LAYER

7. Author(sl 8. Performing Organization Report No.

Alvin Bayliss, Lucio Maestrello, Paresh Parikh, 85-56

and Eli 1'11rKPl 10. Work Unit No.

9. Performing Organization Name and Address

Institute for Computer Applications in Science 11. Contract or Grant No. and Engineering

Mail Stop 132C, NASA Langley Research Center NASi-l7070· NA~1-lRl()7

U<:>mnt-nn VA ?1fih'l-'ln'1 13. Type of Report and Period Covered

12. Sponsoring Agency Name and Address Contractor Report

National Aeronautics and Space Administration 14. Spon~ring Agency Code Washington, D.C. 20546

505-31-R1-01

15. Supplementary Notes

Langley Technical Monitor: Proc. of Workshop on Stability of

J. C. South Jr. Time-Dependent and Spatially

Final Report VaryinK Flows, Springer-Verla!!

16. Abstract

Growth of unstable disturbances in a high Reynolds number compressible

boundary layer is numerically simulated. Localized periodic surface heating

and cooling as a means of active control of these disturbances is studied. It

is shown that compressi bility in itself stabilizes the flow but at a lower

Mach number, significant nonlinear distortions are produced. Phase

cancellation is shown to be an effective mechanism for active boundary layer

control.

17. Key Words (Suggested by Authorlsll 18. Oinribution Statement

stability theory, 34 - Fluid Mechanics & Heat Transfer

compressible viscous flow 64 - Numerical Analysis

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