0g10 

m 1 UNIVERSITY 1 

Interaction Between 

Tollmien-Schlichting Waves 

And Free-Stream Disturbances 

In Boundary-Layer Flows 

R. J. Bodonyi 

Department of Aeronautical and Astronautical Engineering 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 441 35 

Grant No. NAG 3-743 

Final Report 

RF Project 765803/719039 

August 1989 
. 

TABLE OF CONTENTS 

Preface 

Summary 

Introduction 

Problem Formulation 

Numerical Method 

Steady-state Solution 

Unsteady Solution 

Numerical Results 

Acknowledgement 

References 

Figures 

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PREFACE 

The work described in this report was carried out under NASA 

Grant No. NAG 3-743 l*Interaction Between Tollmien-Schlichting Waves 

and Free-stream Disturbances in Boundary-layer The program 

was monitored by Dr. M.E. Goldstein, Office of the Chief Scientist, 

NASA Lewis Research Center. 

Richard J. Bodonyi, Department of Aeronautical and Astronautical 

Engineering, The Ohio State University. 

will appear in a paper co-authored with J.W.C. Welch, P.W. Duck and 

M. Tadjfar in the Journal of Fluid Mechanics. This report 

constitutes the final technical report for this project. 

The principal investigator was Professor 

The results of this study 

iii 
SUMMARY 

A numerical study of the generation of Tollmien-Schlichting waves 

due to the interaction between a small freestream disturbance and a 

small localized variation of the surface geometry has been carried 

out using finite difference methods. 

of the viscous-inviscid interactive type while the unsteady disturbed 

flow is assumed to be governed by the Navier-Stokes equations 

linearized about this flow. Numerical solutions illustrate the 

growth or decay of the T-S waves generated by the interaction between 

the freestream disturbance and the surface distortion, depending on 

the value of the scaled Strouhal number. 

receptivity problem is the numerical determination of the amplitude 

of the Tollmien-Schlichting waves. 

The nonlinear steady flow is 

An important result of this 
1. Introduction 

The steady and unsteady effects of small surface-mounted 

obstacles on the boundary-layer flow over a surface have been of 

concern for many years. These effects include most notably 

separation and instability, often leading to transition to 

turbulence. 

suggested: either the surface distortion produces, in effect, a 

locally separated shear flow which is susceptible to inviscid 

instabilities associated with the inflectional velocity profile [see 

Smith & Bodonyi 1982, 1985, 1987 and Bodonyi, Smith & Gajjar 1983 for 

a discussion of these inviscid type instabilities], or there is a 

sensitive interaction between the surface distortion and the basic 

flow, possibly with unsteadiness/turbulence in the free stream, which 

can readily accentuate the viscous-inviscid growth of the TollmienTwo main reasons possible for this transition have been 

Schlichting instabilities 

case 

The major steady-flow 

well understood, at least 

usually present in boundary layers in any 

phenomenon observed, separation, is now 

in two-dimensional flows. It is generally 

of an interactive viscous-inviscid type in which the flows inside and 

outside the boundary layer affect each other significantly within a 

relatively short length scale. 

separating flow or other locally distorted steady or unsteady motions 

is always present, however, and this has started to receive increased 

The question of the stability of the 

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attention, in part because of modern developments in boundary-layer 

methods. 

In this study our concern is with the possibility of the 

generation of Tollmien-Schlichting (T-S) waves due to the interaction 

of small free-stream disturbances and localized variations in the 

surface geometry. In general terms, the sequence of events that 

begins with the excitation of spatially growing T-S waves in a 

boundary layer by free-stream disturbances is known as the 

receptivity problem [Morkovin 19691. 

of authors over the years and the area has been reviewed by Reshotko 

(1976). More recently, Murdock (1980), Goldstein (1983, 1984, 1985), 

Goldstein, Sockol & Sanz (1983) and Goldstein, Leib & Cowley (1987) 

have theoretically investigated the role that small free-stream 

disturbances play in generating T-S waves in boundary-layer flows in 

a variety of situations. In particular we note that Goldstein (1985) 

studied the effect that small variations in surface geometry have on 

scattering weak unsteady free-stream disturbances into TollmienSchlichting waves. Using the triple-deck scalings of Stewartson 

(1969), Goldstein concluded that relatively small surface variations 

which provoke correspondingly small pressure changes can produce a 

large coupling between the T-S waves and the imposed disturbance when 

these variations are sufficiently rapid, i.e., when they occur on the 

scale of a T-S wave. Goldstein's analysis provided a qualitative 

explanation of the Leehey & Shapiro (1979) boundary-layer receptivity 

measurements. 

It has been studied by a number 

Further comparisons with the Leehey & Shapiro 

2 
experiments using a solution of the local Orr-Sommerfeld equation 

have been made by Goldstein & Hultgren (1987). 

However, Goldstein’s analysis is limited in that he took for the 

steady flow the linearized solution of Stewartson (1970, 1971) for 

the interactive flow in the vicinity of a sharp corner in an 

incompressible flow. Thus Goldstein could not consider the effects 

of surface variations of sufficient size to provoke boundary-layer 

separation or even a nonlinear response in the steady flow, although 

his analysis does account for nonparallel flow effects on the 

stability of the flow. 

Since our interest is in the stability of nonparallel flows and 

especially those with strong local streamwise variations in surface 

geometry, it is appropriate to take the steady nonlinear viscousinviscid interactive solutions, of the triple-deck and similar kinds, 

for the basic steady motion. This is because, as is now well-known, 

flow reversal for small or large scale separations occuring in such 

flows is not necessarily a catastrophic event: the solution at the 

separation point is regular due to the presence of interaction, 

unlike that in steady classical, i.e. noninteractive, boundary layers 

for instance. Hence a steady nonparallel basic flow with a small 

localized region of reversed flow can be described fully by the 

classical boundary-layer equations, subject to an unknown pressure 

which must also be computed as part of the solution. 

we shall assume that the unsteady flow is governed by the linearized 

Navier-Stokes equations, as discussed in the following section. 

Additionally, 

3 
2. Problem Formulation 

We wish to study the interaction between an unsteady freestream 

and a small surface perturbation on a flat plate, such as a hump or 

trough, for an incompressible two-dimensional viscous flow. Thus 

following Goldstein (1985) we take the upstream motion to consist of 

a uniform flow with velocity UQ plus a small harmonic perturbation of 

frequency n and constant amplitude uQ << UQ, so that the unsteady 

motion can be analyzed as a linear perturbation of the uniform steady 

* 

* * 

* * * * -int* flow, U-, i.e., UQ(l + u-/UQe 1. 

** * Consider a Cartesian coordinate system (x ,y ) with x tangent to 

* 

and y 

edge. Further, define the Reynolds number Re = UQL /Y, where L is 

the distance of the surface perturbation from the leading edge of the 

flat plate and Y is the kinematic viscosity of the fluid. 

convenience, we introduce the small parameter c = Reo1I8 and consider 

solutions of the Navier-Stokes equations when Re >> 1. Specifically, 

normal to the flat plate with the origin taken at the leading 

** * 

For 

we wish to consider the problem of flow over a small hump of length 

O(L*c3) and height O(L*c5) with a profile of the form 

* 

y*/L* = c5F((x*-L*)/c3L ) , (2.1) * with F = 0(1), positioned at a distance L from the leading edge. The 

4 
are 

, 

where the steady velocity components,Uo, Vo, and pressure, 

normalized by U, and pUz2, respectively and u,v,p the unsteady 

velocity and pressure terms normalized by u, and 

respectively. Furthermore, we define 

* --- 

* ** 

pu,U,, 

** * * ** 

X = (X -L )/L , y = y*/L*, t = nt , 6 = UaD/U,. (2 5) 

Substituting (2.2) - (2.4) into the Navier-Stokes equations and 

neglecting terms of 0(62), we get the linearized perturbation 

equations 

- - - 

SUt + uoux + uuox + V OY i + ;U OY - -i X + Re'l[Gxx + ;,I, (2 6) 

S;, + UoGx + ;Vox + V OY + GV OY = -; Y + Re-l[Gxx + ;,I, (2.7) 

where 

- - 

ux + v - 0, 

** Y 

s = QL /u, , 

is the Strouhal number. 

Finally, we note that the physical interaction between the 

oncoming boundary layer, freestream disturbance, and hump is governed 

by a triple-deck structure, centered near the surface distortion. 

The details of the structure as applied to this problem have been 

given by Goldstein (1985) and we,therefore, only summarize the 

relevant portions here. As is usually the case the viscous 

interaction problem essentially reduces to a study of the lower-deck 

equations. Thus the appropriately scaled variables in the lower deck 

for the steady flow are 

where 

5 

(2.10) 

(2.11) 

(2.12) 

(2.13) 
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U, V, and P are found from the solution of the lower-deck equations: 

ux + vy = 0, 

wx + wy = -Px + uyy I 

subject to the boundary conditions 

(2.14) 

(2.15) 

U-V-0 onY=F(X), (2.16) 

(u,v,p) * (y,o,o) as Ixl * 8 

U + [Y + A(X)J , Y + , all X, 

(2.17) 

(2.18) 

where we have assumed that a simple renormalization of the variables 

has been carried out in order to set the value of the wall shear of 

the oncoming, undisturbed boundary layer to unity. 

Finally, the interaction condition for incompressible flow is given 

by the Cauchy Hilbert integral 

P(X) = 1/rJ -(D o(X-<)'l(dA/d<)d< . (2.19) 

This steady problem was originally formulated by Smith (1973). 

For the unsteady flow we introduce the following lowerdeck variables 

(2.20) 

(2.21) 

In these expressions we have utilized the fact that since the 

unsteady flow is governed by the linearized Navier-Stokes equations 

we can seek solutions which have a harmonic time dependence. 

Substituting (2.13), (2.20) - (2.22) into (2.6) - (2.8) yields, to 

leading order in C, 

-ic2SU + uux + uux + vuy + vuy = -px + uyy , (2.23) 

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ux+vy=o. (2.24) 

As noted in the Introduction, our interest in this paper is in 

the relatively high frequency case where we choose Q to be of the 

same order as the Tollmien-Schlichting wave frequency at and upstream 

of the lower branch of the neutral stability curve. For this reason 

we require that Q = 0(c2) i.e., the Strouhal number S = O(C-*). 

we define a scaled Strouhal number So such that 

Thus 

so = A, so = O(1). (2.25) 

Furthermore, in this case the Stokes-layer thickness is found to be 

of O(C) also, and, therefore, it will be of O(t5) in terms of the 

lower-deck scalings. Hence the Stokes-layer thickness is the same as 

that of the lower deck. Upstream of the triple-deck region, where 

the mean flow changes on the scale of x, the unsteady flow in the 

boundary layer is given by the Stokes solution, which can be written 

in terms of the lower-deck scalings, as 

(2.26) 

p = ixso . (2.27) 

The lower-deck problem is completed by solving (2.23), (2.24) 

using (2.25) subject to the no slip condition at the wall 

u=v=O onY=F(X) , (2.28) 

matching with the main-deck solution 

u + 1 + a(X), Y Q# all X, (2.29) 

and matching with the upstream Stokes layer solution given by (2.26) 

and (2.27) for X - --. Finally, the relationship between a(X) and 

dp/dX is given by the interaction condition 

d2a/dX2 = 1/11 -Q =[dp/d<-iSo]/(<-X)d( . (2.30) 

7 
It was found that numerical solutions of the disturbance 

equations were not obtainable for supercritical disturbances using 

the interaction law (2.30). Thus an alternative method used by 

Bodonyi 61 Duck (1985) in treating three-dimensional interacting flows 

has been employed here. In this approach, the relationship between 

the pressure p(X) and displacement thickness -a(X) is found through a 

numerical solution of the upper-deck equations as opposed to the 

Hilbert integral representation (2.30). Specifically, it can be 

shown that the appropriate boundary-value problem in the upper-deck 

for the disturbance pressure is given by 

A A 

Pn + P$$ = 0 , 

with boundary conditions 

c;(X?O) = d2a/dX2 , 

fi(x,$) + 0, y" + -, all x , 

(2.31) 

(2.32) 

(2.33) 

$(x,?) + isox -. p(x) as ? + 0, 

+ o as as x + -=, 

ex - ikc + 0, as X + Q, 

where we've written 

P - P, = + 8 y/r3 (2.37) 

Note that (2.36) defines a radiation condition applied on the 

disturbance pressure at the downstream boundary to simulate the 

outward propagating pressure disturbances there. The wave number, k, 

which depends on Sot is found from the solution of the classical OrrSommerfeld eigenvalue problem for the Stokes' layer flow. 

(2.34) 

(2.35) 

(2.36) 

a 
Alternatively, k can be computed iteratively from the numerical 

computations, as will be discussed below. 

The entire unsteady solution is thus known once the solution of 

this boundary-value problem has been obtained. 

Goldstein (1985) gives an analytical solution to the problem using a 

linearized solution of the steady-flow problem for a slightly cranked 

flap on a flat plate due to Stewartson (1970,1971). 

results for U(X,Y), V(X,Y) and P(X) Goldstein was able to solve the 

unsteady problem using a Fourier transform technique. 

As a special case 

Utilizing these 

In this paper we solve the corresponding problem for surface 

distortions such that the steady flow is fully nonlinear in 

character. 

problems must be solved numerically. 

approach used in this study it is appropriate to note that the 

problem formulated above differs in a significant way from the usual 

hydrodynamic stability problem which generally involves the solution 

of the Orr-Sommerfeld equation. 

problem when applied to boundary-layer flows leads to the solution 

for "free" disturbances which are, in fact, the normal modes of the 

boundary layer and they are usually referred to as the TollmienSchlichting waves. 

mode representation of a small disturbance spectrum cannot be 

conveniently extended to finite-amplitude, i.e., nonlinear 

disturbances, nor are they useful by themselves in understanding how 

external disturbances, such as free-stream turbulence or surface 

However, in this case both the steady and the unsteady 

Before discussing the numerical 

The Orr-Sommerfeld eigenvalue 

While important in their own right, the normal 

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roughness, etc, feed energy into the boundary layer, thereby exciting 

spatially growing T-S waves. 

One approach to understanding both the linear and nonlinear 

evolution of T-S waves in a boundary layer is to consider an initialvalue problem, using the unsteady triple-deck equations. Such 

studies have been carried out recently by Smith (1985), Smith & 

Burggraf (1985) , and Duck (1985) 

Alternatively, one can study the nature of the coupling between 

an imposed free-stream disturbance and the growth of T-S waves in the 

boundary layer. 

hydrodynamic stability problem both physically and mathematically. 

Physically, it is the response in the boundary layer to some 

externally imposed disturbance. Mathematically, it is no longer an 

eigenvalue problem. 

seen from equations (2.23) - (2.24) and (2.26) - (2.30), wherein the 

boundary layer is driven by some external forced oscillation with its 

response being a solution of the linearized disturbance equations 

having the same frequency and phase speed as the particular forcing 

disturbance being studied. The primary objective of this work is a 

detailed numerical study of this receptivity problem for a range of 

values of So, which represents the nature of the freestream 

disturbance, and a representative surface distortion which we take to 

be 

This receptivity problem differs from the 

Instead it is a boundary-value problem as can be 

F(X) = h(1 + x 2 ) -1 # (2.38) 

where h is an order one factor which gives the height of the 

distortion relative to the lower-deck scalings. 

10 
3. Numerical Method 

3.1 Steady-State Solution 

First we consider the steady flow problem defined by equations 

(2.14) - (2.19) along with (2.38). Numerical solutions have been 

found using a finite-difference procedure developed by Smith &I 

Bodonyi (1985). Briefly, the governing equations are replaced by 

difference representations for $, 

steps in x, Y. The computational 

x = x2 (>O) and from Y = 0 to Y = 

specified, in effect, at X = X1. 

U = $*, T = Uy, and P with uniform 

domain extends from X = X1 (<O) to 

YoD, with starting conditions (2.17) 

The nonlinear difference equations at a given streamwise location 

X are solved to within a tolerance of 

Newtonian iteration using Gaussian elimination and back substitution. 

The solution is then advanced to the next streamwise location and the 

in absolute value by 

process repeated until the entire domain is covered. Since the 

problem is interactive, multiple forward-marching sweeps are 

necessary until a tolerance of lo-’ between successive values 

obtained for P(X) is satisfied for all X. At this point the solution 

is said to have converged in the global sense. 

dominant nature of the finite-difference form of the interaction law 

The diagonally 

(2.19) makes this multi-sweeping process both fast and stable. 

Whenever flow reversal occurs, i.e., U < 0, windward differencing is 

used to represent Wx in finite-difference form. 

The numerical solution has been found using (2.38) for the 

surface shape with h = 0.1, 1.0 and 5.0. Representative 

11 
distributions of the wall shear r(X,O), and pressure, P(X), of the 

steady flow are given in Figure 2 for h = 5.0. 

With the steady solution known, we now proceed to consider the 

numerical solution of the complex unsteady linearized boundary-layer 

equations (2.23) - (2.24). In our approach, the governing equations 

were solved in the physical plane using a finite difference method. 

3.2 Unsteady Solution 

The unsteady equations (2.23) - (2.24) were replaced, after 

subtracting out the Stokes shear-wave solution, by a system of 

difference equations of second-order accuracy to be consistent with 

the numerical method used for the steady flow problem. 

governing equations are linear no iteration in the normal direction 

is necessary at a fixed streamwise location. A single sweep across 

the boundary-layer region was sufficient to determine the solution 

there. 

accomplished quickly. Multiple sweeps of the entire domain are still 

necessary to obtain the global solution, however, due to the elliptic 

nature of the interaction law (2.30) or equations (2.31) - (2.36). 

Since the 

Thus one complete sweep of the computational domain could be 

Initial attempts to solve the problem were made using the 

pressure-displacement interaction law (2.30) in a form utilizing the 

ideas first put forward by Veldman (1979) and fully discussed by 

Smith C Bodonyi (1985). 1 acceptable 

solutions could be found. However, as the scaled frequency, 

increased towards its critical (i.e. neutral) value of socrit = 2.296 

acceptable numerical solutions became increasingly more difficult to 

obtain. 

Indeed, for values of So 

was 

so , 

It appears that these difficulties are related to the use of 

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the interaction condition (2.30). Numerically, the Hilbert integral 

is truncated to the finite range X1 5 ( < X2, thus it is implicitly 

assumed that the tails of the integral over -Q < ( < X1 and ( > X2 

are negligibly small. 

the upstream Vail", but it is questionable for the downstream 

behavior of the integral. Furthermore, in addition to the 

difficulties associated with the finite integration domain, there is 

some question as to the convergence properties of the Cauchy-Hilbert 

integral itself, at least when the disturbances are supercritical, 

due to the exponentially growing form of the disturbance quantities 

downstream. 

- 

This assumption appears to be satisfactory for 

c 

To overcome these difficulties, the method developed by Bodonyi & 

Duck (1988) for solving flows with viscous-inviscid interaction has 

been succesfully applied to this problem. 

relationship between the disturbance pressure and displacement 

thickness is found through a numerical solution of the appropriate 

upper-deck equations (2.31) - (2.36) as opposed to the Hilbert 

integral representation of the solution, equation (2.30). The 

crucial feature of the scheme is the inherent numerical coupling 

between the viscous boundary-layer solution and the inviscid outer 

flow solution which is carried out simultaneously in the spirit of 

the scheme proposed by Veldman (1979). Using this approach, the 

difficulties associated with the convergence of the Hilbert integral 

for supercritical disturbances can be avoided and the proper 

downstream boundary condition (radiation condition, see 2.36) can be 

applied to the disturbance pressure. 

In this method the 

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For numerical convenience we apply the Prandtl transposition law 

and we also subtract out the Stokes solution. Thus consider the 

following change of variables 

v(X,Y) = vo(X,Y) I dp/dX - iso = dpddX (3.2) 

Then the disturbance equations can be written as 

'0, (3.3) uox + voY 

- (uX - iSo)uo + Uuox + U v + Vuoy + dpddX - uoYy - Yo 

(3.4) 

with boundary conditions 

u0 - vo = 0 on Y = 0, all X 8 (3.5) 

+ 0 as X+ -- , all Y , (3.6) 

as Y -* -, all X. (3.7) 

- a(X) + e i3/2S&/2Y 

uO 

Given a guess or an update for uo, vo, po(X) and $(X,$) 

everywhere, equations (3.3) - (3.7) are marched forward in X, while 

simultaneously equations (2.31) - (2.36) are solved along a line of 

varying 9. 

p and $ (and hence a(X)) at a given streamwise location X. 

through all X stations constitutes one global iteration, Convergence 

is finally attained when a global convergence test on the disturbance 

displacement thickness -a(X) is satisfied. 

This then determines the complex-valued functions uo, vo, 

Sweeping 

The main features of the numerical scheme are the following. Two 

and three-point differencing in Y is used for equations (3.3) and 

(3.4), respectively, wih equation (3.7) applied at Y - Yo. Threepoint central differencing is used to approximate (2.31) in both 

dimensions, while condition (2.33) is applied at 9 = y,. 

A Equation 

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(2.32) is approximated by one-sided differencing in $ and a secondorder scheme for X derivatives. Finally, the radiation condition is 

applied in the following form to estimate the disturbance pressure at 

the downstream boundary 

A -AX,?) J/C~~ICAX-~ J 6 (Xmax , 9) = [6 (Xmax-2AX, $1 -4P Pmax 

The value for k is either prescribed as discussed earlier, or by 

estimating its value from the relation (a$/aX)/iG from values of X 

reasonably far downstream, and then feeding this value back into the 

numerical computations. 

slight differences in estimating k in these two ways. This isn't too 

Numerically, the results indicate only 

surprising since the viscous-inviscid interaction is a local 

phenomenon and the behavior far downstream should approach that of 

the classical stability theory. 

Supposing we have n points in Y and m points in 9, then at each X 

station, the difference approximation of equations (3.3), (3.4) and 

(2.31), together with the interface conditions (2.34) and (3.7) can 

be written conveniently in matrix form as discussed by Bodonyi & Duck 

(1988). The overall scheme is nominally second-order accurate in the 

grid spacings AX, AY and A?. 

solved using standard Gaussian elimination procedures and back 

substitution. 

The resulting matrix equation is then 

15 
4. Numerical Results 

In this section the results of the numerical computations will be 

presented. As part of a collaborative effort to confirm the finitedifference numerical solutions, Dr. P.W. Duck and Mr. J.W.C. Welch of 

the University of Manchester, Manchester, England, have solved the 

same receptivity problem discussed in the previous section using both 

time-marching and time-periodic spectral methods. 

computations were not supported by this NASA grant. 

figures that follow, the finite-difference results are represented by 

solid curves and the spectral results by the symbol o for comparison 

purposes. 

applied to this problem can be found in Bodonyi, Welch, Duck and 

Tadjfar (1989) . 

These spectral 

However,in the 

A detailed discussion concerning the spectral method as 

We first consider the effect of the Strouhal number, So, on the 

interaction between the unsteady flow and the surface distortion. To 

minimize any nonlinear effects, a small value of the hump height, h = 

0.1, for which separation does not occur in the mean flow, was 

chosen. 

ranging from 0.5 to 3.5. In most cases 200 points were taken in the 

streamwise direction over the range -10 s X s 10 (AX=O.lo). To solve 

the lower-deck equations, 50 points were taken across the lower-deck 

region over the range 0 s Y s 7.5. Also, an- additional 50 points 

were used in the upper-deck 9 scaling over the range 0 s 9 s 5, to 

solve Laplace's equation for the pressure in the upper-deck region. 

To test the sensitivity of the thickness of the upper-deck region, in 

The numerical solution was found for several values of So, 

16 
some cases the upper-deck range was extended to 9 = 10 (100 points), 

but no significant differences in the results were noticed. 

Using the finite difference method of solution, convergence of 

the numerical computations was achieved when the absolute value of 

the difference in the displacement thickness, -a(X), between two 

successive iterates was less than 10'' 

iterations required for convergence was found to be quite sensitive 

on the value of So under consideration and also on the initial 

for all X. The number of 

guesses taken for the disturbance profiles. For a subcritical 

disturbance of So = 0.5, less than one hundred iterations were 

sufficient for convergence. However, for a supercritical disturbance 

with So = 3.0 approximately 3400 iterations were necessary to obtain 

converged results. 

using the converged solution for the previous value of So as the 

initial guess, we could reduce the number of iterations required and 

also maintain a small maximum fluctuation between successive 

iterative values of the displacement thickness. 

Also, by gradually increasing the value of So and 

The disturbance produced by the interaction with the hump should 

ultimately decay sufficiently far downstream of the hump if the 

scaled freestream Strouhal number is below its critical value, Socrit 

m 2.296. To illustrate this eventual damping of the unsteady 

disturbances, an extended range of -10 s X s 40 (AX=0.25) was 

considered. The real parts of the complex-valued disturbance 

pressure and wall shear distributions for a representative 

subcritical case, So - 2, are presented in Figure 3 from both the 

finite difference and spectral methods. The decay in the disturbance 

17 
amplitude for all quantities is clearly seen for X L 10. 

the disturbances should amplify downstream of the surface distortion 

if So is supercritical. 

considered for a typical supercritical case, So = 2.5 and Figure 4 

shows the amplification of the disturbances in the streamwise 

direction in this case. 

Similarily, 

The same extended X range was thus 

The supercritical results given in Figure 4 were calculated 

using a linearized time-marching scheme and are shown for t = 14~. 

For both cases, excellent agreement exists between the finite 

difference and spectral method results for the positioning of the T-S 

waves. 

amplitudes. This difference can be reduced by increasing the number 

of grid points in the X direction of the finite difference method as 

discussed below. 

There is a discrepancy of approximately 10% in the wave 

Numerical experiments using the finite difference method indicate 

that we can enhance the convergence rate in subcritical cases by 

extending the X-domain. 

boundary, the more the disturbance is damped near the boundary and 

this, in turn, speeds up the global convergence properties of the 

scheme. 

the downstream boundary is placed, the larger the value of the 

disturbance amplitude becomes, and this results in a further increase 

in the number of iterations required for convergence. 

The further downstream we place the 

Conversely, for supercritical disturbances the further away 

We next consider the nonlinear problem with a hump height of h = 

1.0. In this case the mean flow is still attached everywhere in the 

flowfield. The disturbance solution for this case has the same shape 

18 
and follows the 

over the entire 

same pattern as the previous solutions for h = 0.1 

range of Strouhal numbers considered in this study. 

However, now the amplitudes of the disturbances are increased by a 

factor of approximately 10, which indicates a linear mechanism for 

the amplification of the disturbances due to increasing the height, 

at least for this range of values of h. 

The growth rate, ki, and the wavelength, A, of the T-S waves can 

be calculated from the numerical results and compared with the values 

given by the analytical theory (Duck 1985). 

Rea1(pl(X)eoit), where pl(X) is calculated numerically. Now pl(X) = 

Ceim + a function of X which decays algebraically as X + 8 and C is 

a constant. 

where k = kr + iki and x = 2x/kr. 

calculating k should become more accurate when k is calculated from 

higher order derivatives of the pressure. 

values obtained at each X station for the linearized time-periodic 

spectral results for the case So = 1.0 calculated from the second 

derivative of the pressure. 

difference results for k calculated using the first derivative 

approximation for the pressure are also shown. 

substantially the same results for X > 10. In the region 12 < X < 

40 the values of kr are within 3% of the analytical value of kr = - 

0.522 and the values of ki within about 10% of ki = 0.121. 

reason for the larger Z error in ki compared to kr is that lkil < 

lkrl. 

algebraic terms become significant as the influence of the hump 

The pressure p(X,t) = 

Therefore, (dil/dX)/il = ik and (d2pl/dX2)/p, = -k 2 , 

This approximate method for 

Figures 5a,5b show the 

For comparison purposes, the finite 

The two methods give 

The 

As is to be expected, as we move upstream of this region the 

19 
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increases and the calculated values of kr, ki show greater variation 

from their analytical values. 

Figure 6 shows the results obtained for h = 1.0, So = 3. The 

finite difference results were calculated using an X grid -10 s X 5 

15 with AX = 0.125. The time-marching spectral method results here 

are for t = 5% (with the negative of the results plotted for 

comparison with the finite difference solution). 

time-periodic solution is established in the region approximately X < 

13 and here there is excellent agreement with the finite difference 

results. 

By this time the 

Next, results for h = 5, for which the mean flow has a small 

region of separated flow, are presented. 

corresponding mean flow distributions for the pressure and wall shear 

are presented. We note that the solutions obtained by the finite 

difference method and the steady spectral method are in excellent 

agreement. There are some small oscillations in the wall shear 

results in the reversed flow region suggesting the need for a 

somewhat finer grid for the finite difference method. 

was confinned by running the finite-difference code on a somewhat 

finer grid (-10 s X s 10, 400 points). In this case the oscillations 

noted above did indeed disappear. 

In Figure 2, the 

This effect 

For this value of h, the finite difference method was used to 

obtain converged solutions for So = 1 and 1.2. Pressure and wall 

shear distributions for the unsteady flow over the X-domain, -10 s X 

s 10 is given in Figure 7 for So =l. 

20 
Converged solutions using the finite difference method for 

larger values of the Strouhal number are possible in principle; 

however, the computer time necessary to achieve them becomes 

prohibitive. Furthermore, the accuracy of the numerical solutions 

deteriorates as the streamwise step size, AX, is increased. This 

deterioration can best be seen in the wall shear results. 

Maintaining the same number of mesh points (200) and increasing AX we 

have been able to find three other %umerically converged" solutions 

for h =5 and So = 1. 

solutions are given in Figure 8. Note that for all cases the general 

shape of the solutions do not change. However, by increasing AX the 

magnitude of the disturbance amplitude was reduced while maintaining 

its same form and location. For the largest value of AX, which 

corresponds .to an X domain of -10 s X s 90 (AX=0.50), the numerical 

breakdown is clearly seen in Figure 8c even though the general shape 

of the distribution has not changed appreciably. 

domain, another "numerical solution" was obtained for half the step 

size by doubling the number of points to 400 and the results are 

shown in Figure 8d. Note that the numerical oscillations have 

disappeared and the same general shape of the curve remains. 

However, the magnitude of the amplitude is the same as that in Figure 

8b. 

help on the convergence rate and a smaller step size to maintain an 

accurate solution. 

The wall shear distributions for these 

For this same XThere thus seems to be a trade-off between a larger X-domain to 

For h = 5, So = 1, the finite difference results calculated on 

the three grids: -10 s X s 40, AX = 0.25; -10 s X s 20, AX = 0.15; 

21 
and -10 5 X 5 10, AX = 0.1; giving discrepancies in the amplitude of 

the T-S wave of up to 30%, 10% and 5%, respectively, when compared 

with the time-periodic spectral method results which were calculated 

on a grid sufficiently fine and extensive to give grid independent 

results to graphical accuracy. A comparison of these results for the 

spectral method and the finite difference method using the third grid 

is also given in Figure 7. 

A comparison between the numerical computations and Goldstein's 

(1985) analytical theory for h << 1 can also be made. The 

disturbance amplitude computed is shown in Figures 9a,b for h = 1, So 

= 3 and h = 5, So = 1.2, respectively. For h = 1, the analytical 

theory and the numerical results are in good agreement over most of 

the region of interest. For smaller values of h and other values of 

' So, not shown here, indicate an even better agreement between the two 

approaches. 

applied for h < 1. Conversely, for h = 5, the disturbance amplitude 

predicted by Goldstein's theory does not agree well with that 

predicted by the numerical computations. The nonlinear base flow 

results in substantial differences of the receptivity problem from 

that computed by the analytical theory. 

Thus we can conclude that the analytical theory can be 

We next consider the physical implications of the numerical 

results, and in particular the effect of the hump height, h, on the 

disturbance wave amplitude. 

variation with h of several peaks in the disturbance wall shear 

solution. These peak values are scaled by the corresponding peak 

values of the linear results: Note that the location of the peaks 

We choose to follow the amplitude 

22 
does vary slightly with 

studied for subcritical 

h. The peak amplitude dependence 

values of So, namely So = 1,2 and 

on h was 

for the 

supercritical value So = 3. 

3.5, calculations for larger values of h were affected by a grid 

dependent rapid growth in the spectral plane solution for lkl >> 1. 

This possible Rayleigh instability (see Smith & Bodonyi 1985, Tutty & 

The So = 3 solutions were found for h s 

Cowley 1986) is suppressed in the time-periodic spectral method and 

the finite-difference method which treats the "steady-state" 

equations to find solutions which are periodic in time. 

Consequently, solutions can be found for 1 s h s 5. 

In Figures 10a,b we show the results for So = 1 and So = 3, 

- 

giving the scaled peak amplitude behavior for the wall shear r(x,O) 

with h for two different peaks: the results for So = 2, not presented 

here, follow a similar pattern. 

For h < 1, the scaled disturbance amplitude depends approximately 

linearly on the hump height. Experimentally, Azin & Polyakov (1979) 

found a linear dependence of the disturbance amplitude on h, for the 

interaction of upstream propagating sound waves with thin mylar 

strips fixed on a flat plate near the lower branch of the neutral 

stability curve. 

increasingly nonlinear enhancing effect on the disturbance amplitude. 

These results are confirmed by the finite difference results for So = 

1. For example, for h = 3 the disturbance amplitude is approximately 

twice that of the linear results for the same h. The results for the 

subcritical So calculated using the time-periodic spectral method and 

finite difference method show that this increasingly rapid 

For larger values of h, our results show an 

23 
enhancement of the llreceptivityll continues for h = 4 and 5. The 

results for h = 4,5 are approximately three and six times, 

respectively, the corresponding linear results for the same h. 

In conclusion we find a linear dependence of the disturbance 

amplitude on the hump height for sufficiently small values of h. 

moderate h ( 1 s h s 3) we find an enhancement of the receptivity by 

the nonlinear effect of hump height. 

separation can occur in the steady flow, we find a possible short 

wavelength instability in our time marching calculations and a 

rapidly increasing enhanced receptivity in our "steady statet1 

calculations. 

For 

For large h where local flow 

24 
Acknowledgement 

The author is grateful to the Ohio Supercomp-er Center for making 

computer time available on the Cray X-MP computer during this 

research program. I 

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REFERENCES 

Aizin, L.B. & Polyakov, N.F. 1979, (in Russian) Preprint 17, Akad. 

Nauk USSR, Siberian Div., Inst. Theor. Appl. Mech., Novosibirsk. 

(See also Nishioka, M. & Morkovin, M.V. 1986, J. Fluid Mech., 

V. 171, 219). 

Bodonyi, R.J., Smith, F.T. & Gajjar, J. 1983, IMA J. Appl. Math., 

V. 30, 1. 

Bodonyi, R.J. & Duck, P.W. 1988, Computers & Fluids, V. 16, 279. 

Bodonyi, R.J., Welch, J.W.C., Duck, P.W. 61 Tadjfar, M. 1989, J. Fluid 

Hech., to appear. 

Burggraf, O.R. & Duck, P.W. 1981, Symp. on Physical Numer. Aspects 

Aerodyn. Flows, Calif. State Univ., Long Beach, Calif. 

Cooley, J.W. & Tukey, J.W. 1965, Math. Comp., V. 19, 297. 

Duck, P.W. 1984, Q. J. Mech. Appl. Math., V. 37, 57. 

Duck, P.W. 1985, J. Fluid Mech., V. 160, 465. 

Duck, P.W. 1988, J. Fluid Mech., V. 197, 254. 

Duck, P.W. 1989, Computers &'Fluids, to appear. 

Duck, P.W. & Burggraf, O.R., 1986, J. Fluid Mech., V. 162, 1. 

Gedney, C.J. 1983, Phys. Fluids, V. 26, 1158. 

Goldstein, M.E. 1983, J. Fluid Mech., V. 127, 59. 

Goldstein, M.E. 1984, J. Fluid Mech., V. 145, 71. 

Goldstein, M.E. 1985, J. Fluid Mech., V. 154, 509. 

Goldstein, M.E., Sockol, P.M. h Sanz, J. 1983, J. Fluid Mech., 

V. 129, 443. 

Goldstein, M.E., Leib, S.?. & Cowley, S.J. 1987, J. Fluid Mech., 

v. 181, 485. I 

Goldstein, M.E. & Hultgren, L.S. 1987, J. Fluid Mech., V. 181, 519. 

Leehey, P. & Shapiro, P. 1979, In Laminar-Turbulent Transition (ed. 

Eppler, R. 61 Fasel), Springer, 321. 

Morkovin, M.V. 1969, Air Force Flight Dyn. Lab, Wright-Paterson AFB, 

Ohio, Rep. AFFDL-TR-68-149. 

26 
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1 

Murdock, J.W. 1980, Proc. Roy. SOC. Lnnd., A372, 517. 

Reshotko, E. 1976, In Ann. Rev. Fluid Mech., V. 8, 311. 

Smith, F.T. 1973, J. Fluid Mech., V. 57, 803. 

Smith, F.T. 1985, United Technologies Research Center, Rept. UTRC85- 

Smith, F.T. & Bodonyi, R.J. 1982, J. Fluid Mech., V. 118, 165. 

Smith, F.T. f Bodonyi, R.J. 1985, Aeronautical J., V. 89, 205. 

Smith, F.T. C Bodonyi, R.J. 1987, Studies Appl. Hath., V. 77, 129. 

Smith, F.T., Brighton, P.W.M., Jackson, P.S. & Hunt, J.C.R. 1981, 

Smith, F.T. & Burggraf, O.R. 1985, Proc. Roy. SOC. Lond., A399, 25. 

Stewartson, X. 1969, Mathematika, V. 16, 106. 

Stewartson, K. 1970, Q. J. Mech. Appl. Math., V. 23, 137. 

Stewartson, X. 1971, Q. J. Mech. Appl. Math., V. 24, 387. 

Tutty, O.R. C Cowley, S.J. 1986, J. Fluid Mech., V. 168, 431. 

Veldman, A.E.P. 1979, Netherlands Nat. Aerosp. Lab., NLR-TR-79023. 

36. 

J. Fluid Hech., V. 113, 123. 

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Figure 1. Schematic of the triple-deck flow structure. 
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Figure 2a. Pressure distribution for steady flow with h = 5. 

- : finite-difference method; o : spectral method. 
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Figure 2b. Wall shear distribution for steady flow with h = 5. 

-* finite-difference method; o : spectral method. 
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Figure 3a. Disturbance pressure for S 

finite-difference method; 8 : spectral method. 

- 2, h = 0.1. 0 . 
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0 

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Figure 3b. Disturbance wall shear-for S 

- : finite-difference metRod; o : spectral method. 

= 2, h = 0.10. 
1. 

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Figure 4a. Disturbance pressure for S = 2.5, h y 0.10. 

: finite-difference mzthod; o : spectral method. 
f 

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Figure 4b. Disturbance wall shear for S 

-* finite-difference metbd; o : spectral method. 

- 2.5, h - 0.10.. 
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Figure Sa. Tollmien-Schlichting wave growth rate. 

-- -: analytical value. 

: finite-difference method; o : spectral methbd; 
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Figure 5b. Tollmien-Schlichting wavelength 

- : finite-difference method; o : spectral method; .I- -: analytical value. 
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Figure 6a. Disturbance pressure for S = 3, h = 1. 

: finite-difference m?thod; o : spectral method. 
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Figure 6b. Disturbance wall .. . shear .. for S - 3, h - 1. 

: finite-difference metgod; o : spectral method. 
X 

Figure 7a. Disturbance pressure for S 

-- finite-difference &hod; o : spectral method. 

- 1, h - 5. 
x 

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Figure 7b. Disturbance wall shear for S -0 Zinite-difference metf?od; o : spectral method. 

= 1, h = 5. 
Figure 8a. Disturbance wall shear for So = 1, h - 5 and X 

varying between -10 s X s 20. 

- : finite-difference method; o : spectral method. 
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Figure 8b. Disturbance wall shear for So = 1, h = 5 and X 

varying between -10 s X s 40. 

- : finite-difference method; o : spectral method. 
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8 -- 

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Figure 8c. Disturbance wall shear for So - 1, h = 5 and X 

varying between -10 s X s 90 with AX - 0.50. 
Figure 8d. Disturbance wall shear for So - 1, h = 5 and X 

varying between -10 s X s 90 with AX = 0.25. 
- 

C 

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0 

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01 

A 

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A 

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in 0 b in 0 in 0 t, I L I I I I 

Figure 9a. Comparison of disturbance displacement distribution 

for h = 1, So - 3 computed by the finite difference 

method (-) and the analytical theory (- - -) . 
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Figure 9b. 

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Comparison of disturbance displacement distribution 

for h - 5, So = 1.2. - : finite difference method; ---. . analytical theory. 
0 

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3- 

w 

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Figure loa. Disturbance amplitude variation versus h for So - 1, 

normalized by the linearized results for h = 1, 

so - 1. 

near X = 13; ---. linear result. 

- : minimum near X = 18.5; - - - . peak 

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1 
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P 

F 1 'igure lob. 

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Disturbance amplitude variation versus h for So - 3, 

normalized by the linearized results for h = 1, 

so - 3. - : minimum near X - 9; - - - : peak near 

x - 7: ---- linear result.