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NASA Technical Memorandum 82835 

Generation of Instability Waves 
at a Leading Edge 

(.NA3A-;\1-d23j5) GEN EriATIC N OF TNSI ABlLli* N82-22453 


nC A 02/ At AO 1 CSCE 20D 


Gj/3 4 05637 

Marvin E. Goldstein 
Lewis Research Center 
Cleveland Ohio 

Prepared for the 
Third Joint Thermophysics Fluids, Plasma and Heat Tram 
cosponsored by the American Institute of Aeronautics an 
Astronautics and the American Society of Mechanical Ei 
St. Louis, Missouri, June 7-11, 1982 

cgf ^ 



invited paper 
Marvin E. Goldstein 

National Aeronautics and Space Administration 
Lewis Research Center 
Cleveland, Ohio 4413S 


This paper describes the generation of 
instability waves downstream of a leading edge by 
an imposed upstream disturbance. Two cases are 
considered. The first is concerned with mean 
flews of the 81asius type wherein the instabil- 
ities are represented by Tol lmien-Schl ichting 
waves. It is shown that the latter are generated 
fairly far downstream of the edge and are the 
result of a wave length reduction process that 
tunes the free stream disturbances to the Tol lmien- 
Schl ichting wave length. The other case is con- 
cerned with inflectional, uni-directional, trans- 
versely sheared mean flows. Such idealized flows 
provide a fairly good local representation to the 
nearly parallel flows in jets. They can support 
inviscid instabilities of the Kelvin-Helmholtz 
type. The various mathematically permissible 
mechanisms that can couple these instabilities to 
the upstream disturbances are discussed. The 
re> ults are compared to some acoustic measurements 
and con- elusions are drawn about the generation 
of the instabilities in these flows. 



A. Background 

It is now well established that there are many 
flows wherein the boundary layer turbulence is a 
direct result of the amplification of linear 
spatially growing instability waves (i.e., 
Tollmien-Schlichting waves) in the laminar portion 
of the boundary layer. These waves grow as they 
propagate downstream and, at least initially, the 
two dimensional waves exhibit the most rapid growth 
rates. However, once the Tollmien-Schlichting 
waves reach a certain amplitude, nonlinear effects 
rapidly set in and produce significant lateral 
energy transfer, which ultimately distorts the two- 
dimensional character of the flow. This stretches 
the vortex filaments and thereby produces further 
increases in the unsteady velocity until the flow 
breaks down into bursts of turbulent like motion. 

At this point, the boundary layer is well on its 
way to becoming turbulent. The length of laminar 
boundary layer over which these nonlinear phenomena 
occur is often significantly shorter than the 
length over which the instability waves are gov- 
erned by a linear equation (namely the Orr- 
Sommerfeld equation). The transition point (or, 
more precisely, the transition Reynolds number 
based on the distance from the leading edge) can, 
therefore, be predicted from linear theory in these 

It is also well known (Schubauer & 

Skramstad , and Spanqler & Wells 2 ) that the 
transition Reynolds number is strongly affected by 
the level of turbulence in the free stream. 
Schubauer a Skramstad* showed that the transition 

Reynolds number of a flat plate boundary layer 
increases with decreasing free stream turbulence 
until the intensity drops below about 0.1%. At 
lower intensities, the transition Reynolds number 
remained relatively constant at 2.8x10°. How- 
ever, Spangler and Wells-* found that they could 
increase the transition Reynolds number to about 
5.2x10°. They attributed this increase to the 
fact that background acoustic disturbances (noise) 
represented only a small fraction of the measured 
'turbulence' level in their experiment which 
implies that even random acoustic disturbances may 
be more efficient in generating turbulence than 
free stream turbulence. 

After it was discovered that free stream 
turbulence and random background acoustic distur- 
bances can have an important effect on transition, 
it was natural to study the effect of a regular two 
dimensional small amplitude free stream oscillation 
of a single frequency, say «, imposed on a two 
dimensional steady flow with uniform upstream 
velocity, say U». In such a flow, the stream- 
wise component u»(x) of the free stream veloc- 
ity at the outer edge of the boundary layer is of 
the form 

u.(x) - U 0 (x) + uj(x) e- 1 "* (1) 

where we suppose that the unsteady streamwise 
velocity amplitude ui(x) is much smaller than 
the mean streamwise velocity Ug, x denotes the 
streamwise distance measured along the surface of 
the body and nondimensionalized by lL/«, and t 
denotes the time. Such a flow was studied by 
Obremski and Fejer 3 and Miller & Fejer 4 . How- 
ever, they produced the unsteady motion with a 
variable speed rotating shutter valve downstream of 
the test section of their wind tunnel and their 
unsteacy motion could probably not be considered to 
be two dimensional. They measured transition 
Reynolds numbers and showed that they were signi- 
ficantly affected by the amplitude |u^| of the 
imposed free stream disturbance, but they did not 
make any measurements that would allow them to 
determine how the imposed disturbance affected the 
Tollmien-Schlichting waves. This disturbance may 
have generated the Tollmien-Schlichting waves 
directly or it may only have affected their growth 
rates by changing the stability characteristics of 
the boundary layer. It could even happen that the 
Tollmien-Schlichting waves were bypassed in these 
experiments and the free stream disturbance was 
able to generate turbulence directly by some non- 
linear process (Morkovin 6 ). 

It is, therefore, important to measure the 
effect of the imposed disturbance on the Tollmien- 
Schlichting waves themselves. This was done by 
Shapiro 6 whose unsteady disturbance was quite 
two-dimensional. We will discuss his results 

We listed three mechanisms by which free 
stream disturbances might affect transition. 
However, only the first of these is truly linear in 
the sense that it can be described by equations 
which are linear in the unsteady flow perturba- 
tion. The other mechanisms would invoke terms that 
are quadratic in the unsteady motion. We, there- 
fore, anticipate that the first mechanism will 
dominate when the amplitude of the imposed un- 
steadiness i s sufficiently small. 

B. General Linear Theory 

The physics of this linear interaction will 
now be described. 

The relevant mathematical problem has been 
solved numerically for a flat plate by Murdock* 8 
and analytically in the general case by Gold- 
stein*’. The following discussion is mainly 
based on the analysis of ref. 12. 

When we say that the Tollmien-Schlichting 
waves are generated by the free stream disturb- 
ances, we imply that they are solutions to a well 
defined boundary value problem. But, the Tollmien- 
Schlichting waves are eigensolutions of the Orr- 
Somnerfeld equation (which is obtained by linear- 
izing the Navier-Stokes equations about the mean 
flow and assuming that the latter is nearly paral- 
lel, which is usually a good approximation in the 
boundary layer). Moreover, it is well known that 
one can always ado 3n arbitrary multiple of an 
eigensolution to the solution of a boundary value 
problem and still satisfy all the imposed boundary 
conditions. This raises the question of how 
Tollmien-Schlichting waves can be coupled with the 
imposed free stream disturbance. But, the 
spatially growing Tollmien-Schlichting waves will 
only be eigensolutions of the Orr-Sommerfeld equa- 
tion when no upstream (initial) conditions are 
imposed (i.e., they are eigensolutions when the 
mean parallel flow extends from -» to+w). The 
coupling comes about when upstream boundary condi- 
tions (i.e., initial conditions) are imposed. 

However, the initial conditions cannot be 
applied directly to the solution of the Orr- 
Somnerfeld equation. Near the leading edge of the 
boundary layer (actually within a region that 
occupies the first few wavelengths of the boundary 
layer) the wave length of the disturbance is very 
long compared to the boundary layer thickness, and 
the streamwise derivatives are small. The diver- 
gence of the mean flow has a fist order effect on 
the unsteady motion rather than being a higher 
order effect that can be treated as a 'slowly 
varying' correction to classical parallel flow 
stability theory. In this region, inertia terms 
involving the cross stream component of the mean 
flow velocity have to be included in the lowest 
order equation for the unsteady flow. However, one 
can neglect unsteady pressure fluctuations across 
the mean boundary layer, which is still relatively 
thin (on a wave leigth scale). The flow is then 
governed by the linearized unsteady boundary layer 
equation rather than by an Orr-Sommerfeld equation 
with slowly varying coefficients. 

ihis latter equation, whose eigensolutions are 
the Tollmien-Schlichting waves, is only valid 
further downstream. The upstream initial condition 
for the solution to this equation should, there - 
fore, be that it 'match', preferably in" the 
'matched asymptotic expansion' sense, onto a 

solution of the unsteady boundary layer equation in 
some intenwedi ate region that overlaps the unsteady 
boundary layer and Orr-Sommerfeld regions . 

For definiteness, we restrict the discussion 
to flat plates whose ‘nose radii' are 0(lL/<*>). 

We also suppose that the characteristic wave number 
of uj is 0(w/UL)- The asymptotic expansion 
(alluded to above) is carried out in terms of the 
inverse Reynolds number based on the 'convective' 
wave length U,/w of the disturbance raised to 
the l/6th power, i.e., in terms of 

t * Uv/l £)*/ 6 (2) 

Allowing t * 0 in the nondimensionalized, incom- 
pressible, Navier-Stokes equations while assuming 
that x is order one, one obtains the unsteady 
boundary layer equation to lowest order of approxi- 
mation. The linearized unsteady boundary layer 
equation has been extensively studied in the liter- 
ature (Moore 7 , Lighthill 8 , Lam and Rott®, 

Ackerberg and Phillips*®). At small distances 
from the leading edge, the unsteady boundary layer 
is quasi-steady and grows at the same rate as the 
steady boundary layer. At large distances from the 
leading edge, the unsteady boundary layer is con- 
trolled by the frequency and, to lowest order of 
approximation, behaves somewhat like a Stokes layer 
whose thickness remains constant, independent of x. 
The Stokes layer-like solution is independent of 
the upstream conditions and of the mean boundary 
layer. This type of asymptotic behavior occurs 
because the unsteady boundary layer equations are 
invariant under the Galilean transform 

f * ■> 

x = x - J u (t) dt 

t = t 

r (3) 

u = u - u ( t ) 

V * V ^ 

into an accelerated reference frame. Here, u and 
v are the streamwise ana transverse velocity 
components in the boundary layer. 

Then, when uj becomes constant far down- 
stream we can take u* = Re uj e - ’^ in this r egion 
and transform the fluctuating stream problem in the 
problem of an oscillating wall. But, in this 
region, the steady boundary layer is thick relative 
to the Stokes layer penetration distance (v/w)*'^ 
and we might expect the unsteady flow to be the 
same as that produced by an oscillating wall bound- 
ed by a fluid that is at rest at infinity - which 
is precisely the Stokes layer problem. 

Ackerberg and Phillips*® and Lam and Rott® 
point out that the Stokes-like solution is essen- 
tially incomplete because it is uniquely determined 
Independently of the upstream conditions that must 
always be imposed when solving a parabolic partial 
differential equation. The remaining portion of 
the solution is represented mathematically by an 
infinite set of 'asymptotic elgersolutions ' of the 
unsteady boundary layer equation. They were 
originally discovered by Lam and Rott*. In the 
downstream region the unsteady boundary layer 
solution, therefore, consists of a Stohes-like 
solution plus the asymptotic eigensolutions, whose 
undetermined constants are found from the upstream 
condit’ons, as was actually done numerically by 


Ackerberg and Phillips. One can siy then that the 
asymptotic eigensolutions describe the approach of 
the full unsteady boundary layer solution to the 
Stokes-type solution. 

The asymptotic eigensolutions only exist for 
x > 0(1). They are physically and mathematically 
independent both of each other and of thi Stokes- 
type solution. Their amplitudes are determined by 
the behavior of the full unsteady boundary layer 
solution in the region 0 < x < 0(1). They decay 
exponentially as they propagate downstream. In 
fact, they behave like 

where x is a complex constant with Re x > 0, so 
that the ‘wave length' of their oscillation de- 
creases like x~t/2. This occurs because the 
asymptotic eigensolutions can produce no pressure 
fluctuations and must, therefore, behave somewhat 
like convected disturbances propagating into a 
region of decreasing streamwise velocity. Since a 
convected disturbance is one with zero convective 
derivative, (j/»t) + U(s/ax), where U is 
the mean velocity, its phase • must be ut - J 
dx/U) and its wavelength must, therefore, decrease 
in the streamwise direction if U does. Near the 
wal 1 

U « c « yhfi 

so that 

* - ut - x iU 

Thus, the wavelength of this disturbance de- 
creases like x"J'2 because it must penetrate 
into a region where the mean velocity decreases 

like x-1'2 and not produce any pressure fluctua- 
tions. The importance of explaining this wave- 
length reduction mechanism was emphasized by 

The asymptotic eigensolutions oscillate with a 
wavelength that decreases with increasing x while 
the mean boundary layer thickness increases. The 
cross stream pressure fluctuations, which are 
neglected in the unsteady boundary layer approxi- 
mation, must, therefore, eventually become impor- 
tant and the asymptotic eigensolutions, which are 
based on this approximation, must then become 

Goldstein^ showed that one can obtain a new 
solution whlcn applies further downstream than the 
unsteady boundary layer solution, by considering 
the limiting form of the governing equation as c * 
0 with xt =t'x (rather than x) held fixed. 

This, leads to a solution that applies when x ■ 

0 U~^). It is essentially the classical large 
Reynolds number - sma’l wave number approximation 
to the To 1 lmien-Schl icht ing wave solution of the 
Orr-Soumerf eld equation (Lin*- 3 Tollmien^, 
etc.), appropriately corrected for slow variation 
in boundary layer thickness. Thus, it decays ex- 
ponentially fast in the downstream direction when 
xj is relatively small and exhibits exponential 
growth when x^ is sufficiently large. 

Goldstein^ shows that this solution matches 
onto one of the asymptotic eigensolutions in some 
overlap domain and is, therefore, the natural 

continuation of this solution into the downstream 
region. The other asymptotic eigensolutions match 
with Tol lmien-Schl ichting waves that continue to 

The remaining portion of the asymptotic 
unsteady boundary layer solution, that is, the 
Stokes-type solution, remains uniformly valid in 
the downstream region and is, therefore, completely 
decoupled from the Tollmien-Schlichting waves. 

At large Reynolds numbers, the Tollmien- 
Schlichting wave solution of the Orr-Sommerfeld 
equation is basically inviscid except in a thin 
region near the wall and in a critical layer about 
the point where the inviscid equation becomes 
singular. It is well known *h*t the critical and 
wall layers coincide near th lower branch of the 
neutral stability curve. But., there are two 
inviscid regions outsiae this wall layer - a main 
inviscid region where the unsteady velocity is 
quasi-steady, and an outer region where the unsteady 
effects are important, but where the mean flow is 
nearly uniform. This 3-level structure is somewhat 
similar to the triple deck structui e found in steady 
boundary layers - but, the transverse scaling is 
quite different here. The complete structure of the 
unsteady boundary layer found in ref. (12) is sum- 
marized in figure 1. 

As we already indicated, the asymptotic eigen- 
solutions of the unsteady boundary layer equation 
and the Tollmien-Schlichting wave solutions of the 
Orr-Sommerfeld equation match in the overlap 
domain. There are infinitely many asymptotic 
eigensolutions and the characteristic equation 
which determines the eigenvalues of the Orr- 
Sommerfeld equation has one root for each. 

The progressive reduction in wave length of 
the asymptotic eigensolutions is a sort of ‘tuning 1 
mechanism which allows free stream disturbances to 
couple with Tollmien-Schlichting waves even when 
their streamwise wavelengths are vastly different. 
The Orr-Sommerfeld region acts like a high gain 
linear amplifier tuned to a very specific wave- 

C. Comparison with Experiment 

We now turn to the experiment of Shapiro 6 
that we alluded to above. His unsteady distur- 
bance was produced by an upstream acoustic speaker 
that generated a nearly plane acoustic wave, which 
propagated downstream parallel to the mean flow. 

The unsteady flow was, therefore, relatively two 
dimensional. The ratio of the acoustic wave length 
to the Tollmien-Schlichting wave length was about 
30 in this experiment - so the acoustic wave be- 
haved pretty much like a uniform oscillation of the 

Shapiro's 6 plate was relatively thick. But, 
it does correspond to the model described above 
since its noise radius was of the order of U./«. 

Shapiro took his data with a narrow band 
filter and measured transverse velocity profiles of 
the streamwise velocity fluctuation in the boundary 
layer. His measured profiles were In close 
agreement with the theoretical Tollmien- 
Schlichting wave profiles near the upper branch of 
the neutral stability curve where the instability 
wave would have presumably grown well beyond the 


level of the Stokes shear wave solution (whose 
amplitude does not change with streamwise distance). 
Near the lower branch of the neutral stability 
curve, where the Tol lmien-Schl ichting wave is just 
beginning to grow, the measured profiles appeared 
to be a composite of the Stokes shear wave and a 
Tol lmien-Schl ichting wave. Moreover, the measure- 
ments near the lower branch show that the mean 
amplitude of the unsteady disturbance remains 
relatively constant with streamwise distance when 
averaged over a wave length but, the amplitude 
itself oscillates about this mean with a wave 
length that is roughly equal to the Tollmien- 
Schlichting wave length. The data is shown in 
figure 2. Thomas and Lekoudas 18 and Murdock 18 
show that this behavior is precisely what one would 
expect if *he solution consisted of Stokes shear 
wave plus * relatively small amplitude Tollmien- 
Schlichtir., wave. As we already indicated, 
Goldstein's asymptotic solution is of this form in 
the vicinity of the neutral stability curve. 

Perhaps, most importantly, Shapiro's date •_ 
that the amplitude of the unsteady motion at any 
given point in the boundary layer increases 
linearly with the amplitude of the imposed free 
stream disturbance - indicating that the Tollmien- 
Schl ichting waves are indeed generated by the 
imposed disturbance through a mechanism that ’s 
entirely linear in the unsteady motion. 


A. General Background 

It is well known thac when small amplitude 
periodic flow occurs in the vicinity of a sharp 
trailing edge embedded in an otherwise steady 
flow, the pressure singularity that would other- 
wise occur in the infinite Reynplas number limit 
can often be relieved by the continuous shedding 
of vorticity downstream of the edge. One then 
says that a 'Kutta' condition is satisfied at the 
edge. Crighton 1 ' has shown that there are 
certain periodic trailing edge flows where the 
vortex shedding is represented mathematically by 
spatially growing instability waves of the Kelvin- 
Helmholtz type. 

Suppose that an infinitely thin flat plate is 
embedded in a uniform inviscid flow on which a 
small amplitude unsteady motion is imposed. Un- 
less the unsteady motion is a plane wave aligned 
with the plate, it will produce a square root 
singularity in the pressure at the leading edge. 

Now we have seen that the vi.rous mjtion near the 
edge is governed by the unsteady boundary layer 
equation which allows no transve-se pressure varia- 
tions. The viscous effects by themselves cannot, 
therefore, eliminate the pressure singularity in 
the inviscid solution. When there is no flow 
separation, the singularity does not appear in a 
real flow simply because all real plates have 
finite ‘noise radii'. The fluctuations in angle 
of attack produced by the unsteady flow must be 
small enough so that the laminar boundary layer on 
the rounded nose does not separate. Tol lmien- 
Schl ichting waves only make thei appearance far 
downstream in the flow and can produce no upstream 
influence that can affect the pressure at the 
leading edge to say nothing of eliminating the 
pressure singularity that would occur at an 
infinitely sharp edge. 

But, if the plate were embedded in a trans- 
versely sheared mean flow 18 with an inflectional 
velocity profile, as shown in figure 3, the Inci- 
dent disturbance could trigger a Kel vln-Helmholtz 
Instability at the edge which could then eliminate 
or relieve the pressure singularity that would 
otherwise occur at that edge. 

Such an instability wave is clearly detect- 
able in figure 4, which is comprised of photo- 
graphs of the flow over a wedge placed in a 
rectangular laminar jet. (The flow here is from 
left to right.) The photographs were taken during 
an edge tone experiment and the unsteady motion 
that triggered the instability wave could have 
been an acoustic wave reflected from the nozzle 
lip or a harmonic disturbance convected downstream 
by the mean fl<~w, or perhaps both. 

B. Theoretical Analysis 

Suppose that the flow is inviscid Since a 
transversely sheared mean flow is an exact solu- 
tion of the inviscid equations of motion 18 , it 
makes sense to calculate the small amplitude un- 
steady flow by linearizing these equations about a 
transversely sheared mean flow, As in the case of 
a completely uniform mean fir.., the resulting un- 
steady motion will, in general, possess a square 
root singularity at the sharp leading edge unless 
the imposed unsteady motion is a plane wave aligned 
with the plate. Moreover, as long as we are will- 
ing to allow such a singularity, we can always re- 
quire that the solution remain finite (i.e., that 
it does not 'blow up') at large distances from the 

However, it was shown by Goldstein 1 ^ that 
this problem possesses an eigensolution, which in- 
volves a Kelvin-Helmholtz instability wave propa- 
gating downstream from the edge (and is conse- 
quently unbounded at infinity). This eigensolution 
also possesses a square root singularity at the 
leading edge. Then, since one can always add an 
arbitrary multiple of an eigensolution of a given 
problem to any particular solution of that problem 
and still satisfy the imposed boundary conditions, 
we can add this eigensolution to the particular 
solution that is bounded at infinity and adjust 
the arbitrary constant to cancel out the singu- 
larity at the leading edge. 

The time periodic solution to a problem can 
be obtained by finding the long time (i.e., steady 
state) beha-'-io, of the solution to an initial 
value problem. A solution to such a problem that 
is identically zero before the incident disturbance 
is 'switched on' is said to be causal. The causal 
solution to the present problem is singular at the 
leading edge and involves a Kelvin-Helmholtz insta- 
bility on the downstream flow (so that it is un- 
bounded at infinity). Consequently, neither the 
solution that is bounded at infinity nor the solu- 
tion that satisfies the leading edge 'kutta' con- 
dition is causal. However, it is not at all clear 
that the Steady state solution should be causal 
(Rienstra 20 ). But, neither is it clear that the 
solution should be finite at infinity since the 
linearization is, at best, only valid in a loca 1 
region near the edge aid one cannot, therefore, 
impose a condition on the solution at large dis- 
tances from that edge. Thus, at this point, it is 
not possible to establish which, if any, of these 
three solutions is correct. 


Unearned Invlscid theory of the type 
described above can be used to represent high 
Reynolds number turbulent flows when the turbu- 
lence Intensity is sufficiently small ano the un- 
steady Interaction being calculated is completed 
In a time that is short relative to thp decay, (or 
turnover) time of the turbulent eddies (Hunt” 1 ). 
This linear theory of turbulence is usually refer- 
red to as ’rapid distortion theory’. 

Then, in particular, we can use the inviscid 
flow model described above to represent the 
turbulent flow over a large flat plate placed 
downstream of the potential core in a turbulent 
airjet in the manner indicated in figure 5. 

The assumptions of rapid distortion theory 
(perhaps more appropriately called rapid inter- 
action theory in this case) are rather well 
satisfied in this flow. However, we must now use 
the 'gust' or 'hydrodynamic' solution of the 
inviscid equations (Goldstein? 2 - 23 ) to represent 
the incident turbulence. This solution is defined 
over the entire flow field even in the absence of 
the plate and (when the near flow is subsonic) it 
decays exponentially fast at infinity. It, there- 
fore, has no radiation field (i.e., it is non- 
acoustic) and can be used to represent the turbu- 
lence that would exist in the absence of the pi 
It has sufficient generality (i.e., it involves 
two arbitrary functions that can be specified as 
boundary conditions in any given problem) to 
represent an arbitrary incident turbulence field. 

Since this solution exists independently of 
the plate, it will not, in general, satisfy the 
physically required boundary condition that its 
normal velocity component vanish at the surface of 
the plate. He must, therefore, add to it another 
solution that exactly cancels this normal velocity 
component at ttie plate (this is permissible since 
we are dealing with linear theory and superposi- 
tion holds). However, the resulting solution will 
no longer exhibit exponential decay at infinity, 
but rather behave like an outgoing acoustic wave 
there. Thus, the plate 'scatters' the non- 
propagating motion associated with the gust or 
hydrodynamic solution into a propagating acoustic 

Much more interesting, however, is the fact 
that in both the causal ano Kutta condition solu- 
tions the incident turbulence generates downstream 
propagating instability waves which, in the real 
flow, roll up and break to form new turbulence. 

C. Comparison with Experiment 

Go’dstein 23 compared this analysis with 
Olsen'^ 4 measurements of the acoustic field 
produced by a large flat plate placed in the 
mi ing region of a turbulent jet in the manner 
indicated in figure 5. His solutions satisfy 
causality . ' 

Figure 6 is a comparison of Olsen's measure- 
ments of the sound radiated in the plane perpen- 
dicular to the plane of the plate in one third 
octave frequency bands as a function of angle 
measured from the nozzle inlet. 

The upper part of the figure corresponds to 
the high frequency limit of the solution. Here, 
the instability waves are 'cut off' and the issues 

of causality and leading edge 'Kutta' condition 
are irrelevant. 

The lower part of the figure corresponds to 
the low frequency limit. Here the instability 
waves have a large effect on the radiation field 
but, unfortunately, both the causal and Kutta 
condition solution lead to the same result. How- 
ever, it is worth noting that they both differ 
significantly from the low frequency limit of the 
solution that is bounded at infinity and the agree- 
ment between experiment and theory would have been 
quite poor if the latter solution had been used. 


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_j < 




Figure 4. - Vortex shedding downstream of a leading edge. 


NOZZLE DIAM, D, 10 cm (4 in.) 

R, 4.56 m <15 '11 

Figure 5. - Configuration of plate experiment 


Figure 6. - Goldstein (J. F. M. 1979) comparison of causal or L. E. Kutta condition 
solution with data of Olsen for Uj ■ 700 1/S.