PHYSICAL REVIEW LETTERS
PRL 96, 224504 (2006)
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Spatiotemporal Growing Wave Fronts in Spatially Stable Boundary Layers
T. K. Sengupta,* A. Kameswara Rao, and K. Venkatasubbaiah
Department of Aerospace Engineering, I. I. T., Kanpur 208 016, India
(Received 27 February 2006; published 8 June 2006)
In fluid dynamical systems, it is not known a priori whether disturbances grow either in space or in time
or as spatiotemporal structures. For a zero pressure gradient boundary layer (also known as the Blasius
boundary layer), it is customary to treat it as a spatial problem, and some limited comparison between
prediction and laboratory experiments exist. In the present work, the two-dimensional receptivity problem
of a Blasius boundary layer excited by a localized harmonic source is investigated under the general
spatiotemporal framework, by using the Bromwich contour integral method. While this approach is seen
to be equivalent to the spatial study for unstable systems, here we show for the first time how spatially
stable systems show spatiotemporally growing wave fronts.
DOI: 10.1103/PhysRevLett.96.224504
PACS numbers: 47.20.Pc, 47.15.Fe, 47.20.Ib
Hydrodynamic stability theory aims to link laminar and
turbulent flows. Despite significant advances made [1,2],
there are still issues of transition that remain incompletely
understood. Classical approaches to instability studies identify an equilibrium state, whose stability is studied by eigenvalue analysis by linearizing the governing equations.
This analysis seeks the least stable eigenmode, for parallel
flows, of the linearized Navier-Stokes equation in the spectral plane [giving the Orr-Sommerfeld equation, as given
by Eq. (2)] whose solution exhibits waves for particular parameter combinations, and these are known as the TollmienSchlichting (TS) waves. The Reynolds number (Re) at
which the equilibrium flow first becomes unstable for any
harmonic excitation is the critical Reynolds number (Recr ).
Results obtained by this approach match with laboratory
experiments for thermal and centrifugal instabilities.
Instabilities dictated by shear force do not match as well,
e.g., (i) Couette and pipe flows are found to be linearly
stable at all Re, while the former is known to suffer
transition at Re 350 and the latter at Re 1950 [3] in
laboratory experiments, with the exact value dependent
upon facilities and background noise level; (ii) plane
Poiseuille flow has a Recr 5772, whereas in experiments
transition was seen to occur at Re 1000 [4]. According
to Ref. [3], even for Blasius boundary layer, success of
eigenvalue analysis is of a lesser degree.
The above discrepancies of linear stability theory in not
being able to predict the subcritical transition have
prompted many to seek alternatives in nonlinear theories
[5,6], secondary instabilities [7], etc. In recent times, the
failure of eigenvalue analysis has been attributed to analysis methods, and, instead, subcritical transition is attributed
to nonorthogonal or non-normal eigenvectors that can
cause large transient energy growth [8,9] for stable systems. One of the features of this nonmodal amplification is
that it applies to three-dimensional perturbation only [3].
In the present work, an overlooked aspect of linearized
analysis is highlighted to account for subcritical transition.
Usual eigenvalue analysis treats it as either a spatial or a
0031-9007=06=96(22)=224504(4)
temporal problem. Specially, spatial stability is used for
velocity profiles without inflection points, as for the
Blasius boundary layer. This has been established by receptivity analysis [10], where the problem shown schematically in Fig. 1 was solved for the Blasius boundary
layer excited by a harmonic localized source. The nondimensional disturbance stream function ( ) was obtained
from the Bromwich contour integral [11,12],
x; y; t
ei0 t Z
y; ; 0 eix d;
2
Br
(1)
where 0 is the circular frequency of excitation and Br
indicates the Bromwich contour followed in evaluating the
above integral in the complex wave-number () plane.
This is the signal problem where the transient part of the
solution is not considered and time dependence is harmonic. In the present formulation, all the lengths have
been nondimensionalized by the displacement thickness
( ) of the boundary layer that characterizes the effect of
mass defect by viscous action at the wall. All velocities are
nondimensionalized by the free-stream velocity (U1 ) and
time by =U1 . The bilateral Laplace transform defined
in (1) is obtained as a solution of the Orr-Sommerfeld
equation [1],
Uy 0 00 2
iv 22 00 4
U00 y
; (2)
iRe
where U is the parallel equilibrium flow taken at the
location of the exciter, which is equivalent to taking the
shear layer to be parallel, as indicated in Fig. 1. In the
above, primes indicate derivatives with respect to y and
Re U1 =, with as the kinematic viscosity. In
Ref. [10], the Bromwich contour was taken as parallel
and below the real axis, such that all the downstream
propagating modes are above this line. In this receptivity
approach, effects of all the modes are incorporated, which
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2006 The American Physical Society
PRL 96, 224504 (2006)
FIG. 1. Harmonic excitation of a parallel boundary layer corresponding to the location of the exciter.
is in contrast to normal mode eigenvalue analysis that tries
to explain everything in terms of the leading eigenmode
only.
In the signal problem [10], the implicit assumption that
the response is at the frequency 0 of the exciter may not
seem as drastic. But for absolute unstable systems, initial
transient associated with startup can fix the response at the
absolute frequency that is, in general, different from 0 . In
Ref. [13], an attempt was made to obtain a criterion
whereby flows could be analyzed by either the spatial or
the temporal theory —overlooking the possibility of simultaneous spatiotemporal response. We show here that the
latter is of prime importance for spatially stable systems.
In Ref. [14], spatiotemporal theory was used to ascertain
the inviscid instability of jets and wakes and to estimate the
characteristic frequency of the response. Similarly, in
Refs. [15,16], the receptivity problem of wave-packet
propagation created by a single pulse was studied in the
context of a stationary phase asymptotic solution obtained
by the saddle point method. In contrast to these spatiotemporal approaches, in Ref. [17] a complete spatiotemporal
problem was studied, without the assumption of the signal
problem, where was obtained from
x; y; t
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PHYSICAL REVIEW LETTERS
1 ZZ
; ; yeixt dd
22 Br
(3)
and obtained from the solution of Eq. (2) (with 0
replaced by ) along the Bromwich contours in the complex and planes. The choice of Bromwich contours in
the plane is restricted by causality, and the contour in the
plane is similar to that used in Refs. [10,17]. In the
following, Cartesian disturbance velocity components are
denoted by u and v, respectively. Therefore, for the excitation shown in Fig. 1, boundary conditions at y 0 are
u 0, x; 0; t Htxei0 t and for y ! 1: u; v !
0, where Ht is the Heaviside function and x represents
the Dirac delta excitation at the origin of the frame. The
conclusion [17] was that, for a spatially unstable case
(Re 1000 and 0 0:1), the evolving solution has the
same parameters as that obtained in Ref. [10]; i.e., the
space-time dependent solution is determined mainly by
the unstable eigenvalue, with the damped modes playing
no role. The results presented in Ref. [17] were for a
spatially unstable case only and not undertaken for stable
cases. This is investigated here, specifically to look for
spatiotemporal growing solutions that are otherwise spatially stable.
To discuss the spatiotemporal growth of waves for the
Blasius boundary layer, a few cases are considered marked
as A, B, C, and D in Fig. 2, with respect to the neutral curve
shown in the Re-0 plane, for the leading eigenmode. A
neutral curve demarcates stable and unstable regions of the
Re-0 plane —as indicated in this figure. The Bromwich
contour for point A is chosen in the plane on a line
extending from 20 to 20 that is below and parallel to
the real axis at a distance of 0.009, and in the plane it
extends from 1 to 1 above and parallel to the real axes
at a distance of 0.02. For the other points, the Bromwich
contour in the plane is located at a distance of 0:001
from the real axis. The choice of Bromwich contour in the
plane is such that all the downstream propagating eigenvalues lie above it. Equation (2) has been solved along
these contours with 8192 equidistant points in the plane
and 512 points in the plane. In Ref. [17], results were
reported for point A, with only half the number of points in
the and planes and 1200 points were taken in 0 y
6:97, as compared to 2400 points used here. An increase in
the number of points is mandatory to obtain solutions valid
over a longer domain and times. All other numerical details
are the same as in Ref. [17]. Spatial stability analysis
produces waves for the four points of Fig. 2 with the
properties as given in Table I. For point A, receptivity
analysis produces streamwise perturbation velocity (u) as
shown in the bottom frame in Fig. 3 at t 801:1. In Fig. 3,
the top two frames show solutions for the case of point B.
The present results obtained for point A are indistinguishable from the growing asymptotic solution obtained by
signal problem analysis in Ref. [10]. This type of receptivity analysis provides additionally the local solution (in
the neighborhood of the exciter), and a frontrunner (for B
and not for A) preceding the asymptotic solution. It can be
shown analytically that the local solution is due to the
essential singularity of the plane and numerically identified in Ref. [10]. For point A, the receptivity solution is
determined by the first mode alone, without any effects
coming from the second and third modes of Table I. In
contrast, for point B, the asymptotic solution is due to the
first mode of Table I (in terms of wavelength and decay
rate) and the growing wave front corresponds to the second
0.2
0. 15
0. 1
0.05
0
C
B
Stable
A
Unstable
D
Stable
250 500 750 1000 1250 1500 1750 2000
Re
FIG. 2. Neutral curve for the Blasius boundary layer identifying stable and unstable regions.
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PHYSICAL REVIEW LETTERS
PRL 96, 224504 (2006)
TABLE I. Wave properties of selected points in Fig. 2.
Mode
r
i
Vg
Vs
Ve
A1
A2
A3
B1
B2
B3
C1
D1
D2
0.279 826
0.138 037
0.122 020
0.394 003
0.272 870
0.189 425
0.246 666
0.160 767
0.062 141
0:007 287
0.109 912
0.173 933
0.010 493
0.167 558
0.322 635
0.013 668
0.001 520
0.069 659
0.4202
0.4174
0.8534
0.4267
0.2912
0.1159
0.5026
0.3908
0.2762
0.42
0.42
u
0.001
C: β 0 = 0.1
Re = 300
t = 801.1
D: β 0 = 0.05
Re = 1000
t = 801.1
0
-0.001
0.002
0.352
u
0.352
0.50
0.33
0.50
0.33
mode, in terms of the wavelength. The effects of the third
mode are not seen to contribute to the overall solution. It is
noted that the leading edge of the asymptotic solution
continues to decay at the same rate predicted by the spatial
eigenvalue analysis, while the frontrunner continues to
grow spatiotemporally, although the spatial theory identifies this as a damped mode.
The necessary condition for the creation of a frontrunner
can be found by looking at the receptivity solutions for
points C and D, with the former having a single stable
mode and the latter with two damped modes. Results for u
are shown in Fig. 4 at the indicated time. The essential
difference between these and previous cases in Fig. 3 is that
the latter have three modes, while C possesses a single
mode and D possesses two modes. The frontrunner in
Fig. 3 is due to interactions of multiple stable modes. In
the absence of multiple modes—as for point C—no such
frontrunner is seen in Fig. 4. Again, for point D, there are
only two stable modes that create a spatiotemporally growing frontrunner. Thus, for fluid dynamical systems, the
presence of a minimum of two stable modes is necessary
to produce a spatiotemporally growing wave front, when
the least stable mode is spatially damped.
0
-0.002
0
100
200
x
300
400
500
FIG. 4. u plotted as a function of x at indicated times for Re
300 and 1000 for the indicated 0 at y 0:278.
The growth of the frontrunner is due to the competing
groups associated with multiple stable modes, reinforcing
each other at the front. What we have called here the
frontrunner is also referred to as the forerunner in the
literature [18]. The propagation of a wave front and forerunner in a material medium has been of continuing interest. The propagation speed has been variously described as
the group velocity by Rayleigh, signal velocity by
Sommerfeld, and also the velocity of energy transfer in
Ref. [18]. It is noted [18] that the three definitions are
identical for nondissipative systems. But, in dissipative
systems, these can differ considerably. It is also shown
[18] that the forerunner is very weak and difficult to trace
for stable systems, and it can attain high amplitudes only
when the group velocity attains a minimum. In the context
of the present problem, group velocity (Vg ) is obtained
from eigenvalue analysis—as given in Table I. From
Figs. 3 and 4, one directly estimates the signal velocity
(Vs ) by tracking the crests, as shown in the second last
column of Table I. In the following, an estimate for the
energy propagation speed (Ve ) is obtained.
If the total mechanical energy of incompressible flow is
identified as E p= V 2 =2, then the evolution of disturbance energy can be studied following the method of
Ref. [19], where the disturbance energy (Ed ) is calculated
Ed
0.05
B : β 0 = 0.15
t = 801.1
A : β 0 = 0.1
t = 801.1
0
-0.05
Ed
0.25
0
-0.25
0
FIG. 3. Streamwise disturbance velocity (u) plotted as a function of x at indicated times for Re 1000, y 0:278.
100
200
x
300
400
500
FIG. 5. Disturbance energy (Ed ) plotted as a function of x at
indicated times for Re 1000, y 0:278.
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PHYSICAL REVIEW LETTERS
PRL 96, 224504 (2006)
Wave-front
u
0 .0 2 Exciter
t = 150
t = 300
0
- 0 .0 2
Asymptotic solution
100
x 50
0
FIG. 6. u vs x at y 0:648 obtained by solving the NavierStokes equation for simultaneous blowing-suction excitation at
the wall for Re 1000, 0 0:14.
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parallel flow approximation, which does not account for
the growth of the boundary layer.
We conclude by noting that a growing frontrunner is
created as spatiotemporal perturbations in a spatially stable
fluid dynamical system with multiple modes. For unstable
systems, no such frontrunner is seen. This is also verified
from the direct simulation of the 2D Navier-Stokes equation. This result could be of broad interest and contribute to
the understanding of the long-standing discrepancies observed in some systems between the analytical results
obtained using linear spatial stability theory and actual
experimental observations on flow transition.
from
r2 Ed r Vm
!d r Vd
!m ;
(4)
with V and ! representing velocity and vorticity field,
respectively, and the subscripts m and d identify equilibrium and disturbance quantities, respectively. If one represents Ed in terms of RR
its Fourier-Laplace transform via
2
ixt
^
dd, then the
Ed x; y; t 1=2
Br Ed ye
^
governing equation for Ed is given by
E^ 00d 2 E^ d 000 U 200 U0 0 U00 2 U
22 U0 :
(5)
We solved Eq. (5) for E^ d as a function of and and
reconstructed Ed as a function of x and t by performing
Bromwich integrals successively. Results for Ed are shown
as a function of x for points A and B in Fig. 5. Ed shows
smoothly decreasing variation, upstream of the exciter.
Also, Ed is less spiky as compared to u at the location of
the exciter. Once again, for point A, there is no frontrunner,
while point B displays the same as before. The rate at
which Ed propagates can be estimated roughly from this
figure. We note that the system dynamics is determined by
the least stable mode (A1) for the spatially unstable case,
with all three definitions of propagation speed producing
identical results—as seen in Table I. In contrast, for stable
systems with multiple modes, the frontrunner has identical
Vs and Ve , which lies between the group velocity values of
the leading two modes. For stable systems with a single
mode, all three definitions produce the same value. Thus,
for all systems the signal speed and energy propagation
speed are the same.
The appearance and propagation of a frontrunner is not a
transient phenomenon, and this can be further demonstrated by performing direct simulation of the 2D NavierStokes equation. The formulation and numerical method is
as noted in Ref. [20], and typical results of a run are shown
in Fig. 6, where the Blasius boundary layer is excited by a
simultaneous blowing-suction source at the wall—a similar problem was solved in Ref. [21]. Results shown for t
150 and 300 clearly identify the frontrunner. This verification is necessary, as the receptivity solutions are based on
*Electronic address: tksen@iitk.ac.in
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