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The complete integral solution is found for the convectively unstable and oscillatory-forced linear Klein–Gordon equation as a function of spatial variable, , and time, . A comparison of the integral solution with series solutions of the Klein–Gordon equation elucidates salient features of both the transient and long-time spatially growing solutions. A rigorous method is developed for identifying the key rays associated with saddle points that can be used to characterize the transition between transient temporally growing and long-term spatially growing waves. This method effectively combines the procedure given by Gordillo & Pérez-Saborid (Phys. Fluids, vol. 14, 2002, pp. 4329–4343) for determining the ray at which the forced spatial growth response affects the observed waveform and competes with the transient response, with an established methodology for identifying the leading and trailing edge rays of an impulse response. The method is applied to a linearized system describing an oscillatory-forced liquid sheet and asymptotic predictions are obtained. Series solutions are used to validate these predictions. We establish that the portion of the solution responsible for spatial growth in the signalling problem is correctly identified by Gordillo & Pérez-Saborid (Phys. Fluids, vol. 14, 2002, pp. 4329–4343), and that this interpretation is in contrast with the classical literature. The approach provided here can be applied in multiple ways to study a convectively unstable oscillatory-forced medium. In cases where numerical or series solutions are readily available, the proposed method is used to extract key features of the solution. In cases where only the forced long time behaviour is needed, the dispersion relation is used to extract: (i) the time required to see the forced solution; (ii) the amplitude, phase and spatial growth of the forced solution; and (iii) the breadth of the transient.
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J. Fluid Mech. (2012), vol. 699, pp. 115–152. c
Cambridge University Press 2012 115
doi:10.1017/jfm.2012.86
On the response of convectively unstable flows to
oscillatory forcing with application
to liquid sheets
N. S. Barlow1, S. J. Weinstein2and B. T. Helenbrook3
1Department of Chemical and Biological Engineering, University at Buffalo,
State University of New York, Buffalo, NY 14260, USA
2Department of Chemical and Biomedical Engineering, Rochester Institute of Technology,
Rochester, NY 14623, USA
3Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam,
NY 13699, USA
(Received 1 June 2011; revised 20 November 2011; accepted 9 February 2012;
first published online 19 April 2012)
The complete integral solution is found for the convectively unstable and oscillatory-
forced linear Klein–Gordon equation as a function of spatial variable, x, and time, t.
A comparison of the integral solution with series solutions of the Klein–Gordon
equation elucidates salient features of both the transient and long-time spatially
growing solutions. A rigorous method is developed for identifying the key x/trays
associated with saddle points that can be used to characterize the transition between
transient temporally growing and long-term spatially growing waves. This method
effectively combines the procedure given by Gordillo & P´
erez-Saborid (Phys. Fluids,
vol. 14, 2002, pp. 4329–4343) for determining the x/tray at which the forced spatial
growth response affects the observed waveform and competes with the transient
response, with an established methodology for identifying the leading and trailing
edge rays of an impulse response. The method is applied to a linearized system
describing an oscillatory-forced liquid sheet and asymptotic predictions are obtained.
Series solutions are used to validate these predictions. We establish that the portion
of the solution responsible for spatial growth in the signalling problem is correctly
identified by Gordillo & P´
erez-Saborid (Phys. Fluids, vol. 14, 2002, pp. 4329–4343),
and that this interpretation is in contrast with the classical literature. The approach
provided here can be applied in multiple ways to study a convectively unstable
oscillatory-forced medium. In cases where numerical or series solutions are readily
available, the proposed method is used to extract key features of the solution. In cases
where only the forced long time behaviour is needed, the dispersion relation is used to
extract: (i) the time required to see the forced solution; (ii) the amplitude, phase and
spatial growth of the forced solution; and (iii) the breadth of the transient.
Key words: absolute/convective instability, instability control, thin films
1. Introduction
The response of a potentially unstable medium to an oscillating point or line source
is known as ‘signalling’. In the context of a one-dimensional propagation problem,
Email address for correspondence: barlow.nate@gmail.com
116 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
h
Oscillating source
x
Transient region
Signalling
region
(x t)S
(x t)T
(x t)F
FIGURE 1. A schematic of signalling behaviour at a fixed time. Key regions of the response
are delimited by x/trays that move downstream with the wave packet.
this corresponds to the response induced by a forcing term f(x,t)=Aδ(x)eiωftwhere
ωfis the forcing frequency, Ais the forcing amplitude, xis the coordinate along the
medium and δis the Dirac delta function. The response to such forcing is directly
applicable to liquid sheets, particularly in the context of thin locally uniform film
formation used in coating flows (Weinstein & Ruschak 2004). As shown in Clarke
et al. (1997) and Weinstein et al. (1997), a pressure disturbance in the surrounding
air can lead to undesirable deformations of liquid sheets (curtains) in the process of
curtain coating. To better understand the stability of forced mediums in general, a
small subset of problems termed ‘signalling problems’ have appeared in the spatio-
temporal stability literature over the past three decades.
Bers (1983) studied the signalling problem applied to convectively unstable
waves and discussed the competition between a temporally growing transient and a
temporally stable yet spatially growing oscillatory response downstream of the forcing.
Bers correctly concluded that the spatial growth is directly related to poles in the
Fourier integral solution, but attributed the spatial growth to a residue contribution
that, in fact, spatially decays (see § 2.2). A similar interpretation is given in Huerre
(1987,2000) and Huerre & Monkewitz (1990). Gordillo & P´
erez-Saborid (2002)
identified the correct integral responsible for spatial growth and, through analysis,
introduced the idea of a ‘limiting ray’ x/tthat determines the space–time ratio below
which the magnitude of the spatially growing response and the transient response are
comparable.
A schematic of signalling behaviour is shown in figure 1, where wave amplitude h
is plotted against xfor a fixed time. In the figure, key regions of the response are
delimited by x/trays that move downstream with the wave packet. The limiting ray
of Gordillo & P´
erez-Saborid (2002) is labelled (x/t)T, beyond which the total response
is dominated by the transient. The rays (x/t)Sand (x/t)Findicate the breadth of the
transient. These rays are identified using the criteria given in Gaster (1968) (see § 2.3)
and are denoted respectively as Vand V+in Huerre (2000) for a non-oscillatory
‘impulse response’, where the forcing is given by Aδ(x)δ(t). The rays (x/t)Sand (x/t)F,
The response of convectively unstable flows to oscillatory forcing 117
respectively, define the trailing and leading edges of the temporally growing
wave packet. The interpretation of these rays in Gaster (1968), Monkewitz (1990)
and Huerre (2000) is identical to that given here for the oscillatory-forced problem,
with one exception. For the impulse response, the flow is calm upstream of the
ray (x/t)S(V). However, in an oscillatory-forced medium, the flow can exhibit
downstream spatial amplification in this region, as shown in figure 1. The ‘signalling
region’ labelled in figure 1indicates the region of flow that is affected by varying the
input forcing frequency, ωf, and forcing amplitude, A. A complete description of these
rays and regions is provided in this paper.
The objective of this work is to construct long-time asymptotic predictions for
each region of figure 1, and to provide an exact solution for the spatially growing
oscillatory response, both of which can be used to interpret an existing numerical or
series solution. Using the dispersion relation alone, we develop a method for extracting
certain x/trays that characterize the response of the system. The ray (x/t)S, indicated
in figure 1, is inherently important, as it provides the time beyond which the transient
response has moved out of a given region; beyond this time, the system resides in the
purely spatial growth regime (i.e. the signalling region of figure 1), where the response
is controllable by varying the amplitude and frequency of the forcing. Our method
leads to an exact solution for this regime, which explicitly gives the amplitude and
frequency of the induced wave response as a function of the amplitude and frequency
of an input disturbance. This transfer function has practical utility, as the source of the
induced response can often be isolated and controlled in a process.
The paper is organized as follows. In § 2, we solve the oscillatory-forced linear
Klein–Gordon equation in order to develop the asymptotic technique. In § 2.1, the
equation is solved via a series solution, and both temporal and spatial stability
analyses are used to interpret the simulated wave dynamics. In § 2.2, we find the
asymptotic behaviour of the integral solution and identify errors in the previous
literature. In this section, we provide a complete spatio-temporal description of
convectively unstable flow for the oscillatory-forced linear Klein–Gordon equation.
This leads to a procedure, described in § 2.3, for extracting the key x/trays that
characterize both the transient and continually forced solution in a convectively
unstable system. To illustrate the generality of the method, we apply the analysis
to a planar liquid sheet. A description of the physical model, governing equations
and boundary conditions for a forced liquid sheet is given in § 3.1. The integral and
series solutions of a linearized system describing the forced sheet are also given in
this section. In § 3.2, we evaluate the series solution for sinuous and varicose waves in
a sheet and compare the behaviour with asymptotic predictions obtained through the
procedure described in § 2.3.
2. Signalling in the Klein–Gordon equation
With the goal of developing a general procedure for extracting the important
features of spatially developing forced flows, we start with a tractable model
problem: the forced linear Klein–Gordon equation. We wish to find the response to
an oscillating line source located at x=0 in an infinite medium, where the wave
height h(x,t)is governed by
t+U
x2
h2h
x2µh=Aδ(x)eiωft,t>0,
h(x,0)=h
t(x,0)=0 for all x
(2.1)
118 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
where Uand µare real parameters and h(x,t)=0 for t<0. These parameters
ultimately affect wave dynamics, but otherwise are not linked to any particular
physical properties; (2.1) is considered dimensionless in this exercise. The unforced
Klein–Gordon equation subject to an initial disturbance has been studied extensively
by Huerre (1987); therefore, we only focus on the forced response of an initially
undisturbed medium. Huerre showed that the problem is stable for µ < 0, absolutely
unstable when µ > 0 and |U|<1 and convectively unstable for
µ > 0 and |U|>1.(2.2)
The analyses herein assume the constraints given in (2.2), since we are only
interested in convective instability. In an absolutely unstable medium, an initial
disturbance grows and eventually contaminates the entire domain, causing continued
forcing to become subdominant and inconsequential as t→ ∞. In this limit, the
absolutely unstable forced case is equivalent to the unforced case studied by Huerre
(1987). Equation (2.1) is hyperbolic, and its characteristic velocities are given by
dx/dt=U±1. This indicates that convectively unstable waves either travel completely
downstream (U>1) or completely upstream (U<1). Regardless of direction, the
foregoing analysis of growth and propagation is identical. To organize ideas, we limit
our discussion to waves that convect and grow downstream (x>0) of the source by
choosing U>1.
2.1. Series solution and basic stability analysis
The solution of (2.1) on either a finite or infinite domain can be thought of as a
superposition of modes, each given by
hk(x,t)=Ckei(kxωt),(2.3)
where Ckis the amplitude associated with a particular mode, k=kr+ikiis the possibly
complex wavenumber, and ω=ωr+iωiis the possibly complex wave frequency. The
relationship between the wavenumber and wave frequency for a given medium is
obtained by substituting (2.3) into the homogenous version of the constant coefficient
linear partial differential equation of interest. This transforms the differential equation
in h(x,t)into the algebraic equation, D(k, ω) =0, which is referred to as the
dispersion relation. Substituting (2.3) into (2.1) (neglecting initial conditions) with
A=0 leads to
D(ω, k)= − Uk)2+k2µ(2.4)
where D(ω, k)=0 describes the relationship between kand ωin the Klein–Gordon
equation.
2.1.1. Series solution
To validate the following stability analysis, we use the Fourier series solution
of (2.1) for periodic boundary conditions h(L,t)=h(L,t), given by
h(x,t)=A
2L
×
n=N
X
n=−N
cosh n
2t+(2Uikn2iωf)sinh n
2tnet/2eiωft
D(kn,iωf)eiknx(2.5)
where n=2pµ+(12U2)k2
n,kn=nπ/L, and D(kn,iωf)is obtained using (2.4).
The series solution (2.5) is constructed using real wavenumbers kn. Note that we could
The response of convectively unstable flows to oscillatory forcing 119
have also constructed the same solution using complex wavenumbers, as outlined
in Barlow et al. (2010). Solutions are valid provided that the leading edge of the
downstream response does not reach the boundaries of the domain at x= ±L. For the
figures and comparisons to follow, we use a domain length of L=200, take 401 terms
in the series (N=200) and use parameter values of µ=1 and U=2.
2.1.2. Stability analysis
If we assume that the dominant spatial or temporal growth is exponential in form,
the maximum growth rates can be inferred from (2.3) by applying D(ω, k)=0 in (2.4).
In classical temporal analysis, the maximum temporal growth rate is given by the
largest positive value of ωifor a real-valued wavenumber kr. Conversely, the spatial
growth rate associated with a particular real-valued forcing frequency ωfis given by
the largest negative value of kifor waves moving in the x>0 direction. Waves studied
in this analysis travel downstream only. However, the ideas developed herein also
apply to waves that travel upstream away from the source, except the spatial growth
rate would then be given by the most positive value of kifor a particular ωf. The
D(k, ω) =0 solutions of ωand kin (2.4) are given respectively by
ω±(k)=Uk ±pk2µ(2.6)
and
kf1,f2= − ωfU
1U2±1
1U2qω2
f+µ(1U2)(2.7)
where kf1and kf2are the respective +and roots associated with real-valued
frequencies ω=ωf. A temporal amplification plot is given in figure 2, where the
imaginary part of (2.6) is plotted versus real k. Since ωi|k=0=µis a maximum
in figure 2, the group velocity ∂ω/∂k=ωr/∂kr+i(∂ ωi/∂kr)takes on a real value,
and ω/∂ k|k=0=Uspecifies the speed at which the peak of the transient wave
packet travels away from the source, growing like eµt. This unique real-valued mode
(where ωi/∂kr=0, ωi>0) can be used to describe the motion of the transient
solution to (2.1). This can be seen in figure 3, where the series solution is plotted
for µ=1,U=2, ωf=1.632. The peak of the transient moves exactly x=Ut units
away from the source; the Gaussian-like waveform results from the culmination of
both slower and faster moving waves, which also grow exponentially with smaller
growth rates than the maximum. After the transient has passed a certain region, an
oscillatory-forced spatially growing region (a tail) is left behind. The solution is shown
at a later time (t=40) in figure 4, where both the Gaussian-like transient (figure 4a)
and the spatially growing tail (figure 4c) are clearly observed. Here, the transient has
moved exactly x=Ut =80 units away from the source and grows like eµtas t→ ∞;
this is shown in figure 4(b) where the maximum amplitude of the series solution is
plotted against time.
The variation of spatial growth rate with forcing frequency is given in the bottom
half-plane of figure 5, provided x>0. The frequency range that allows for x>0
spatial growth (ki<0) is ω2
f<|µ(1U2)|. Forcing in this range is a necessary but
not sufficient condition to have spatially growing waves. We will show in §2.2 that
it is also necessary for temporally unstable saddle points to exist in the dispersion
relation (2.4), ensuring that a growing transient leads the spatially growing waves.
For the parameters chosen, ωf=1.6321 falls within the range of potential spatial
instability. The spatial growth rate is given by Im(kf1)= −0.1933 from (2.7), which is
consistent with a region that follows the transient and grows like eIm(kf1)x, shown in
120 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
k
r
FIGURE 2. Temporal amplification plot of the Klein–Gordon equation; ω(k)is given by (2.6).
FIGURE 3. Convectively unstable evolution of the forced Klein–Gordon equation (2.1). The
series solution is given by (2.5) with µ=1,U=2 and ωf=1.632. On the scale of the
transient response, the influence of the sinusoidal forcing is not clearly seen in this figure.
Figure 4(c) provides a close-up of the spatially growing region that precedes the transient
response.
figure 4(c). This can be considered a steady spatially growing envelope and a solution
of (2.1) within a certain distance from the source, since the wave envelope remains
at the same amplitude for all time at a given location, although the phase varies as
The response of convectively unstable flows to oscillatory forcing 121
Region of (c)
–200
–3
–2
–1
0
h
1
2
3
10–5
100
105
hmax
1010
1015
1020
(a)(b)
–2000
–1500
–1000
–500
h0
500
1000
1500
2000
–150 –100 –50 0
x
50 100 150 200 0 5 10 15 20
t
25 30 35 40
0 5 10 15 20
x
25 30 35 4540
t = 40
(c)
(× 1015)
FIGURE 4. Series solution of (2.1) with µ=1,U=2 and ωf=1.632. (a) Full domain of
the solution at t=40. (b) Maximum growth of the solution () compared with the curve
Aeµt(—–). (c) Magnification of the solution at t=40, showing the steady spatially growing
envelope, which aligns with the curve AeIm(kf1)x(− −), where Im(kf1)= −0.1933 from (2.7).
the system is continually forced. Note that the other root (kf2) predicts spatial growth
upstream (x<0) of the source with a spatial growth rate given by Im(kf2)=0.1933.
In the following section, we show that only downstream spatial growth is relevant for
U>1.
The classical stability analysis provides us with a qualitative yet incomplete
description of the solution, as it cannot be used to sufficiently anticipate the Fourier
series results. We now continue to extract the key features of the series solution by
examining the spatio-temporal behaviour of the relevant integral equations governing
the system. We will find that this evaluation leads to a quantitative description of
the flow.
2.2. Integral solution
To obtain the double integral solution of a constant coefficient linear partial differential
equation, the Fourier and Laplace transforms are taken, the transformed dependent
variable is then solved for and successive inverse transforms are taken to return
to the original variable; detailed examples are given in Barlow et al. (2010) for
similar wave evolution equations. Following this procedure, the integral solution
122 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
Im[kf]
U2 – 1)]
U2 – 1)]
U2 – 1)]
U2 – 1)]
[
[
[
[
FIGURE 5. Spatial amplification plot of the Klein–Gordon equation; kff)is given by (2.7).
to (2.1) is
h(x,t)=1
4π2Z
−∞ Z∞+iτ0
−∞+iτ0
Ai ei(kxωt)
ωf)D(k, ω) dωdk(2.8)
where the parameter τ0is taken such that the horizontal integration path in the ωrωi
plane passes above all singularities in the integrand; this comes from the definition of
the inverse Laplace transform (Morse & Feshbach 1953). In addition, the path of the
Fourier inversion is along the real axis in (2.8). For simple ωpoles, the inner (Laplace)
integral of (2.8) can be evaluated first by the method of residues, leaving us with the
Fourier integral solution,
h(x,t)=
h+
A(x,t)
z }| {
A
2πZ
−∞
ei[kxω+(k)t]
[ω+(k)ωf]D
ω [k, ω+(k)]
dk
+
h
A(x,t)
z }| {
A
2πZ
−∞
ei[kxω(k)t]
[ω(k)ωf]D
ω [k, ω(k)]
dk
+Aeiωft
2πZ
−∞
eikx
D(k, ωf)dk
| {z }
hB(x,t)
(2.9)
where D/∂ω[k, ω±(k)] = ∓2pk2µ, revealing branch point singularities k= ±µ.
When complex contour integration is used, the poles of all integrals in (2.9) are given
The response of convectively unstable flows to oscillatory forcing 123
by (2.7); in h+
A(x,t)of (2.9), the pole kf1satisfies ω+(k)ωf=0; in h
A(x,t)of (2.9),
the pole kf2satisfies ω(k)ωf=0.
When contour integration is used to evaluate hB(x,t)of (2.9), the poles kf1and kf2
both satisfy D(k, ωf)=0. This integral can easily be evaluated by taking the residue
with respect to k. Depending on the sign of x, the integration path is closed by a
semicircle in either the upper or lower half of the krkiplane. The real part of the
exponential argument along this deformed contour must be negative as k→ ±∞, to
ensure convergence. After the residue of hB(x,t)is evaluated and the expression for
D/∂ω[k, ω±(k)]is inserted into h+
A(x,t)and h
A(x,t), (2.9) becomes
h(x,t)=h+
A+h
A+hB(2.10)
where
h+
A(x,t)+h
A(x,t)=A
2πZ
−∞
ei[kxω+(k)t]
2pk2µ[ω+(k)ωf]dk
+A
2πZ
−∞
ei[kxω(k)t]
2pk2µ[ω(k)ωf]dk(2.11)
and
hB(x,t)=Aieiωft
H(x)eikf2x
D
k(kf2, ωf)H(x)eikf1x
D
k(kf1, ωf)
.(2.12)
In (2.12), His the Heaviside step function used to combine the evaluation of the
integral hB(in (2.9)) when x>0 and when x<0. For this problem, kf2is enclosed
in the upper half (x>0) k-plane and kf1is enclosed in the lower half (x<0) k-plane.
Consequently, hBis neutral in time and represents a spatially decaying waveform
determined by the forcing frequency, ωf. It is important to point out that, even when
forced at an ‘unstable’ forcing frequency (ω2
f<|µ(1U2)|), hBcannot be responsible
for spatial growth, as the step functions that arise to assure convergence preclude such
growth.
In Bers (1983), Huerre (1987,2000), Huerre & Monkewitz (1990) and possibly
other publications, spatial growth is incorrectly attributed to hB. In these publications,
the double integral (2.8) is evaluated in the same order as above, with the ω
integration performed first, leading to a kintegral given by (2.9). Also, in each
of these publications, the ωpath of integration is originally taken to be above
all singularities in the ωiωr-plane and the kpath of integration is originally taken
along the kr-axis; these are also the paths used here. Note, that the Fourier inversion
integral can also be taken anywhere in the strip of absolute convergence (i.e. strip
of analyticity) in the complex k-plane, defined by Im(1) < ki<Im(2), where 1
and 2are the closest singularities (of the integrand) respectively below and above the
kr-axis (Morse & Feshbach 1953). The necessary inclusion of the kr-axis within this
strip is central to the argument that hBis evaluated incorrectly in the literature. This
inclusion comes from the classical derivation of a Fourier transform, which starts with
a discrete Fourier series that arises from a self-adjoint eigenvalue problem on a finite
domain. By definition these eigenvalues are real. The transform arises in the limit of
the domain becoming infinite in extent. The eigenvalues congeal into a continuous
spectrum, corresponding to the locus of real-valued wavenumbers (kr) in the complex
k-plane (Morse & Feshbach 1953; Lighthill 1962). Cauchy’s theorem then allows one
124 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
to use any path that lies within the strip of absolute convergence to evaluate the
Fourier inversion integral (Morse & Feshbach 1953; Bers 1983).
When evaluating the Fourier inversion integral of (2.8), Huerre (2000) reasoned
that the k-integration path can be moved below a fixed kfpole that lies below the
kr-axis (outside the strip of absolute convergence). Cauchy’s theorem was then applied
to this path, resulting in an expression with the correct growth rate. However, the
spatially growing hB(of (2.9)) described in Huerre (2000) is unbounded at x= ±∞.
Having obtained the correct spatial growth rate, Huerre did not continue the analysis
to determine whether the remaining portion of the solution (hAin (2.9)) cancels this
unbounded growth. In § 2.2, we show that hAonly cancels pieces of the correctly
evaluated hBthat spatially decay towards x= ±∞. Using a path that lies outside the
strip of absolute convergence in the complex k-plane, Huerre (somewhat fortuitously)
obtained the correct spatial growth rate from the wrong integral. Any path deformation
away from the original Fourier inversion path of hB(in our case, the kr-axis) must
be closed back to the original inversion path, and this will inevitably lead to a
spatially decaying result for hB(equation (2.12)). When Gordillo & P´
erez-Saborid
(2002) compared their (correct) evaluation of hBwith that of Huerre (2000), they
incorrectly concluded that the solutions are equivalent.
Although the problem is left non-specific in Bers (1983) and Huerre (1987), the
procedures given lead to conclusions that attribute spatial growth to the integral hB.
In Huerre & Monkewitz (1990) and Huerre (2000) these procedures were applied
to the oscillatory-forced linear Ginzburg–Landau equation. We note that hB, given
here by (2.12), is identical in form for both the forced Klein–Gordon equation and
the forced Ginzburg–Landau equation; they both spatially damp. A perhaps less
demanding (but not general) argument that substantiates the spatial decay of hB, is
the absolute integrability (with respect to k) of 1/D(k, ωf)for both the Klein–Gordon
and Ginzburg–Landau equations. This ensures, through the Riemann–Lebesgue
lemma (Lighthill 1962; Weinberger 1965), that the integral hBgiven in (2.9) spatially
decays. In the current work, we demonstrate that it is, in fact, the integrals
corresponding to hAin (2.11) that are responsible for spatial growth due to forcing, as
correctly identified by Gordillo & P´
erez-Saborid (2002).
We now focus on the evaluation of the integrals in (2.11). It is impractical to
evaluate hAusing contour integration on the same complex semicircular paths used
to evaluate hB. The sign of [kx ω(k)t]cannot be readily determined, and so one
cannot close the contour in any simple way. Instead, we look for an integration
path that enables asymptotic analysis as t→ ∞. It is natural, then, to utilize the
method of steepest descent. In this method, a path is chosen that passes through
saddle points of the argument of the exponential in such a way that the real part
of [k(x/t)ω(k)]tis constant. This enables the dominant exponential behaviour to
be identified along the path as tbecomes large, using Laplace’s method or Watson’s
lemma (Bender & Orszag 1978). The complex k-plane may contain an infinite number
of saddle points, ks, which satisfy (∂ω±/∂k)|ks=x/t, each for some real-valued ray
x/t. In the context of a linear wave problem, each ksx/tpair can be thought of in
the following way: an observer travelling at a velocity x/twill see waves governed
by D[ks, ω(ks)] = 0, each moving with a phase of [ksxω(ks)t](Whitham 1974). For
the Klein–Gordon equation, a fixed x/tray provides two saddle points, ks±, which
satisfy Re((∂ω±/∂k)|ks)=x/tand Im(∂ω±/∂ k|ks)=0, where ω±/∂ kis determined
The response of convectively unstable flows to oscillatory forcing 125
–1.5 –0.5
kr
0.5 1.5
–1
ki0
1
–1.5 – 0.5
kr
0.5 1.5
–1
0
1
EDB
A
C
FG EDB
A
C
FG
(a)(b)
, ks
, ks
, ks
, ks
FIGURE 6. Location of saddles (×) for (a)h+
Aand (b)h
A, given by (2.13). Here µ=1,
U=2. The saddles are densely populated within each line segment, shown separated here
for equal 1(x/t). (a) AB, U<x/t<U+1; BC, U1<x/t<U; ED, −∞ <x/t<U1;
FG, U+1<x/t<. (b) AB, U1<x/t<U; BC, U<x/t<U+1; ED, U+1<x/t<;
FG, −∞ <x/t<U1. There are no saddles between D and B or between B and F.
from (2.6). For U>1 and µ > 0, they are given by
ks±= ± µ(x/tU)
p(x/tU)21
.(2.13)
For unstable waves (µ > 0), the numerator of (2.13) is always real. The denominator is
either purely real (|x/tU|>1) or purely imaginary (|x/tU|<1). This indicates that
all saddles either lie on the real axis or the imaginary axis in the kikrplane, leading
to sets of saddles, shown in figure 6(a) for h+
Aand figure 6(b) for h
A. The saddles
along ED, AC and FG, are densely populated. The separation distances between any
two saddles in figure 6are shown for equal 1(x/t); this is done only to illustrate the
variation of x/t, governed by (2.13). As mentioned previously, the saddles ks= ±µ
at D and F are also branch points, making DF the appropriate location to make a
branch cut to enable further analysis (Morse & Feshbach 1953). The only other saddle
to lie along the line DF is ks=0, which is also the wavenumber responsible for
maximum temporal growth, ωi=µ. This saddle is paired with x/t=U, which is
consistent with the classical analysis in § 2.1. This maximum growth saddle–ray pair
belongs to the locus of saddles along the imaginary axis, which is shown in figure 6.
These purely imaginary saddles (line ABC) must be responsible for temporal growth,
because all of the real saddles (lines ED and FG) lie outside the temporally unstable
region of figure 2. If one wished to track the transient (i.e. temporally growing)
solution to this problem, one could focus on the ray x/t=U, as done in § 2.1. This
ray lies in the centre of the transient region illustrated in figure 1. Based on the
saddle locations shown in figure 6, we can now identify the limits of the temporally
unstable wave packet as U16x/t<U+1. For problems where the saddles are
not purely real or imaginary, determination of the temporally unstable saddles is not
as mathematically straightforward. However, the procedure is identical conceptually,
as we shall see for the liquid sheet problem in § 3.2. For fully complex saddles, the
temporally unstable saddles that define the transient region are characterized as having
a non-zero ‘aggregate growth’ σ, which is the real part of i[ks(x/t)ω(ks)](Gaster
126 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
1968; Monkewitz 1990). The value of σcan also be interpreted as the temporal growth
rate of a wave taken in a frame of reference moving at a velocity given by the
corresponding x/tray. The characterization of x/trays based on their corresponding σ
value is outlined in Huerre (2000) for an impulse response, and is put into the context
of the signalling problem in § 2.3.
Evaluation of the steepest descent path through the saddle of maximum aggregate
growth usually provides the asymptotic description of waves in the unforced problem
as t→ ∞ (Barlow et al. 2010). However, this saddle does not provide any information
about the envelope of steady spatial growth induced by forcing; for that, we will need
to consider the spectrum of all possibly growing saddles and identify x/trays that are
critical to determining the ultimate evolution of the forced solution.
The contour used to join the steepest descent path to the original path can often
be chosen in a number of ways, provided the contribution along this connecting
contour is subdominant and the coefficient of the time-varying exponential is
analytic (Lighthill 1978). When (2.11) is evaluated via contour integration, the
resultant complex integral is not necessarily analytic everywhere, because there are
poles at ω±(k)=ωf, given by kf1,f2in (2.7) and branch points at k= ±µ. If a
pole is enclosed between a steepest descent path and the original integration path (the
kr-axis) of the integrals in (2.11), the resultant residue can dominate the asymptotic
behaviour of the solution. These poles are shown in figure 7, each for a different
positive forcing frequency. The trends of complex kcorresponding to increasing ωfare
also shown in figure 7. As ωfis increased, the poles become purely real and spatial
growth ceases. This occurs when ωf= [µ(U21)]1/2, shown as point Bin figure 7.
As ωfis increased further, kf2increases along the kr-axis, while kf1decreases along the
kr-axis until reaching the branch point (kf1=µ,ωf=µU), shown as point Cin
figure 7. As ωfis increased even further, both kf1and kf2increase along the kr-axis.
The existence of these poles requires us to completely determine each path of
steepest decent, as it is only then that the relative contributions of the poles and
saddle points can be correctly assessed as tbecomes large. This is in contrast with
the unforced problem, where a single saddle point contribution governs the asymptotic
behaviour, and the long time asymptotic behaviour can be deduced solely via a portion
of the path in the immediate vicinity of the saddle.
The steepest paths through the saddle points are given by the equation
Re hkx
tω±(k)i=Re hks
x
tω±(ks)i(2.14)
where Re denotes the real part of the bracketed argument. For the saddles along the
line ABC in figure 6, Re[ks(x/t)ω±(ks)] = 0. Substituting this and (2.6) into (2.14)
leads to a family of curves describing the steepest paths, given by
kr"kr
a2
+ki
Im(ks)2
1#=0 (2.15a)
where
a=pµ/[(1+x/tU)(1x/t+U)](2.15b)
and ksis given by the purely imaginary saddles (line ABC) shown in figure 6and
determined by (2.13). The above equation describes two paths: kr=0 (the ki-axis)
and an ellipse, shown in figure 8for saddles lying along lines AB (SDAB) and
BC (SDBC). The path of steepest descent passes perpendicular to isocontours of
The response of convectively unstable flows to oscillatory forcing 127
00.2
–1.0
–0.8
–0.6
–0.4
–0.2
0
ki
0.2
0.4
0.6
0.8
1.0
0.4 0.6 0.8
kr
1.0 1.2 1.4 1.6
A1kf2
A2kf1
C
BD
FIGURE 7. Location of poles for h+
A(kf1) and h
A(kf2), given by (2.7). The real positive
forcing frequency increases from ωf=0 (points A1and A2) in the direction of the arrows.
When the frequency is increased beyond ωf= [µ(U21)]1/2(point B), the spatial poles
are purely real; no spatial growth occurs above this frequency. As ωfis increased further,
kf2increases along the kr-axis towards point D, while kf1decreases along the kr-axis until
ωf=µUand kf1=µ(point C, branch point). As ωfis increased even further, kf1
increases along the kr-axis towards point D. In this figure, µ=1 and U=2.
Im[k(x/t)ω(k)](not shown) that decrease away from the saddle, decreasing the
magnitude of the exponential argument in hAfor a given time. For a large time, the
main contribution then comes from the saddle point, which is a maximum along this
path, and allows for the asymptotic behaviour to be deduced. The other steepest path
(that of ‘ascent’) is not useful for the asymptotic analysis, since the magnitude of the
exponential argument increases away from the saddle and leads to a non-converging
integral (Morse & Feshbach 1953). For saddles lying along the line ABC in figure 6
for both h+
Aand h
A, the steepest descent path is the ellipse and the steepest ascent
path is the kiaxis; a proof is given in appendix A. For this problem, the steepest
descent paths intersect the original path of integration (the real axis) at kr= ±a
(2.15b), eliminating the need for a ‘connecting contour’ to close the path. If this were
not the case, one would need to justify that a subdominant connecting contour exists;
examples are given in Bleistein (1984) using Jordan’s lemma. We partition the Fourier
integral h±
A=h±
A1+h±
A2+h±
A3such that the integration path of each term is, respectively,
Ra
−∞,Ra
aand R
a, shown in figure 8. The inner piece, h±
A2, is evaluated by forming a
closed path with both the top and bottom semi-ellipses of steepest descent, also shown
in the figure. The choice of ellipse is determined by the x/tray of interest, and its
association with either the saddles lying along the lines AB or BC of figure 6. The
128 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
A
B
C
SDAB
SDBC
ki
kr
32
(–a, 0) (a, 0)
Im
Im
FIGURE 8. Paths of steepest descent (ellipse) and ascent (vertical axis) through saddles (×)
lying along the lines AB (SDAB) and BC (SDBC ) for a particular x/tray. The ellipse intercepts
±aand ±|Im(ks)|are given by (2.15b) and (2.13), respectively. The markers ,+and
indicate possible pole locations (2.7) in relation to the integration path described by (2.17).
closed path formed by h±
A2and either the upper or lower semi-ellipse equates h±
A2to
the sum of the steepest descent path (taken in the opposite direction) and any pole
contributions that are enclosed by the semi-ellipse. For a given x/tray, the solution
then becomes
h±
A|x/t=h±
A1ZSD±+2πiXresidues +h±
A3(2.16)
where RSD is the steepest descent path through a saddle associated with a particular x/t.
By fixing x/twithin the range of the saddles lying along the lines AB and BC in
figure 6, integration by parts can be directly applied to h±
A1and h±
A3to show that they
decay like 1/tas t→ ∞ (Bender & Orszag 1978). The solution is then dominated by
the saddle point and pole contributions, leading to the asymptotic result
h±
A|x/t∼ − ZSD±+2πiXresidues,t→ ∞.(2.17)
The individual residues are multiplied by 1, 1 or 0, depending on whether a kf
pole is enclosed by the ellipse in the upper half-plane (), enclosed in the lower
half-plane () or not enclosed at all (+), as indicated in figure 8. The integration paths
for hAare shown in figure 9(a–d) for h+
A|U16x/t<U,h
A|U16x/t<U,h+
A|U<x/t6U+1and
h
A|U<x/t6U+1, respectively. For all plots, a single unstable forcing frequency is chosen
and an x/tvalue is assumed such that the ellipse of steepest descent is large enough to
enclose the kfpoles, if they exist in the relevant quadrant. If a stable forcing frequency
The response of convectively unstable flows to oscillatory forcing 129
–2 –1 0 1 2
–1.5
–1.0
–0.5
0
ki
0.5
1.0
1.5
–2 –1 0 1 2
–2 –1 0 1 2 –2 –1 0 1 2
–1.5
–1.0
–0.5
0
0.5
1.0
1.5
–1.5
–1.0
–0.5
0
ki
0.5
1.0
1.5
–1.5
–1.0
–0.5
0
0.5
1.0
1.5
SD+SD+
SD+SD+
SDSD
SDSD
kf1
kf1
kf2
kf2
ks+
ks+
ks
ks
(a)(b)
(c)(d)
krkr
FIGURE 9. Integration paths for U1<x/t<U, (a)h+
Aand (b)h
A, and U<x/t<U+1,
(c)h+
Aand (d)h
A: (×) saddles; () poles; (− −) branch cut; µ=1,U=2, ωf=1.632.
is chosen (one that leads to real-valued kfpoles), the residue contribution is replaced
by a principal value with half the magnitude. Recall, that it is also possible that a pole
is not enclosed by the ellipse, for certain x/trays (not shown); this ‘limiting’ process
of enclosing poles is important to our problem and is discussed later. The branch cut
from kr= −µto kr=µis drawn in figure 9as a dashed line. From (2.15), it
is seen that the ellipse of steepest descent intersects the branch cut only for the ray
x/t=U, that of the fastest growing mode. The classical analysis (§ 2.1) sufficiently
describes the behaviour of this ray. Therefore, to avoid passing through the branch cut,
we are careful not to directly include the ray x/t=Uin our analysis. On the other
hand, we do include the rays x/t=U1 and x/t=U+1 in the scope of figure 9.
Equation (2.15) shows that these particular rays lead to circular paths of steepest
descent that pass through two saddles for each part of the solution (h+
Aand h
A): one
saddle at ks= ±iand another at ks= ±∞. This effectively bridges the saddles lying
along the lines AB and BC, respectively, with the saddles lying along the lines FG and
ED in figure 6. It should be noted that (2.17) is exact for x/t=U1 and x/t=U+1
for all t, since the entire kr-axis closes the circular path. This is discussed further in
appendix B, where the contributions of the saddles lying along the lines ED and FG in
figure 6are examined.
130 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
Finally, we can evaluate (2.16) and write the integral solution for hAin the range
U16x/t<Uas
[hA(x,t)]U16x/t<U
t→∞ ∼ −ZSD+x/tZSDx/t
iAei(kf1(x/t)ωf)t
D
ω
ω
kkf1f
+iAei(kf2(x/t)ωf)t
D
ω
ω
kkf2f
(2.18)
where the third term is zero if kf1is not enclosed by RSD and R
−∞ and the fourth
term is zero if kf2is not enclosed. The integral solution for hAin the range
U<x/t6U+1 is
[hA(x,t)]U<x/t6U+1
t→∞ ∼ − ZSD+x/tZSDx/t
(2.19)
since no poles are enclosed by the paths in figure 9(c,d). In (2.18) and (2.19),
[RSD±]x/tt1/2ei[ks±x/tω±(ks)]tfor simple saddles (2ω/∂ k2|ks6= 0) (Bender & Orszag
1978). Note that the residues in (2.18) have no restriction on the sign of Im(kf),
in contrast with those of (2.12) where the Heaviside function, used to guarantee
convergence, suppresses any spatial growth for hB. The integral responsible for spatial
growth is, in fact, hA. From the form of (2.18) and noting that x/t>0, we can
now state the necessary and sufficient conditions for downstream spatial growth: (i)
Im(kf) < 0 (signalled by ωf); (ii) kfis enclosed by the kraxis and a steepest descent
path through a saddle of the dispersion relation. As indicated by figure 9, poles are
enclosed for U16x/t<U, but not for U<x/t6U+1, indicating spatial growth
(and decay) to the left of the peak (x/t=U) of the transient waveform, but not
necessarily to the right. As it happens, the solution is identically zero to the right of
the leading edge (x/t=U+1) of the transient; this is shown in the following analysis.
We now examine the relationship between saddle and pole location in the complex
k-plane, in order to understand the extent of the transient region in figure 1, and to
develop a closed-form solution in the signalling region. As noted above, a residue is
generated in (2.18) when a pole resides between a steepest descent path and the real
axis. The integration paths for h+
A(—–) and h
A(−−) for the rays U16x/t<U
are shown in figure 10. As x/tdecreases, the enclosure of poles (,) occurs for some
‘limiting ray’, (x/t)T, and for all rays U1<x/t< (x/t)T. It should be clarified that
this is not the limiting ray at which the transition between steady-state spatial growth
and unsteady temporal growth occurs, since contributions from temporally unstable
saddle points and spatially growing residues both exist for U1<x/t< (x/t)T. We
refer to (x/t)Tas the ‘transient limiting ray’, since all rays (x/t)T<x/t<Uare
limited to temporally growing saddle point contributions (no residues are accrued
through hA). In the method introduced by Gordillo & P´
erez-Saborid (2002), this type
of limiting ray is evaluated for simple saddles by first locating the path of steepest
descent that passes through both ksand kf, and then finding the x/tvalue associated
with this path. Applying this procedure to (2.14), one obtains
Re(kf) (x/t)Tωf=Re(ks) (x/t)TRe[ω±(ks)],(2.20)
which is exactly the first part of (13) in Gordillo & P´
erez-Saborid (2002). Further
details concerning (x/t)T(denoted by (x/t)lim in Gordillo & P´
erez-Saborid (2002)) are
given in this reference. Substituting (2.13) into (2.20) for the U16x/t<Uset of
The response of convectively unstable flows to oscillatory forcing 131
–1.5 – 1.0 –0.5 0
kr
0.5 1.0 1.5
–1.5
–1.0
–0.5
ki0
0.5
1.0
1.5 Decreasing
Decreasing
FIGURE 10. Paths of steepest descent through the saddles (×) of h+
A(—–) and h
A(− −) for
U1<x/t<U. From innermost to outermost ellipse: x/t=1.8,1.5,1.4,1.3 and 1.2. Here
kf1() and kf2() poles, given by (2.7), are shown for range of forcing frequencies, ωf>0;
U=2, µ =1.
saddles leads to
(x/t)T=U1
U(2.21)
which is independent of the forcing frequency ωf. The ellipse associated with this
ray is easily determined by substituting (2.21) into (2.15). As shown in figure 10, the
limiting ellipse for h+
Afalls along the path of all poles that lead to spatial growth (kf1,
denoted by in the figure). For U16x/t< (x/t)Tthere is competition between
the transient saddle point contribution and the steady spatial-growth pole contribution.
The transient contribution becomes smaller as x/tdecreases from Uto U1. For x/t
slightly above U1, ksis still within the temporally unstable regime of saddles (i.e.
along line BC in figure 6aand along line AB in figure 6b). For x/tslightly below
U1, kslies in the temporally stable regime of saddles (i.e. along line ED in figure 6a,
and along line FG in figure 6b). This suggests that the ‘spatial-growth limiting ray’ is
given by (x/t)S=U1.
Even though there is no temporal growth associated with the real-valued saddles, in
appendix Bwe show that spatial growth is present for 0 <x/t<U1, when forced
at particular forcing frequencies. By applying Cauchy’s integral theorem and using
our knowledge of the placement of real-valued saddles, we show that h
A=0 and h+
Ais
given by (B 4) for x/t(U1). The solution, h=h+
A+hB, for the spatially growing
forced region is then given by the superposition of (B 4) and (2.12):
[h(x,t)]x/t(U1)= −iAei(kf1(x/t)ωf)t
D
ω
ω
kkf1f
+iAei(kf2(x/t)ωf)t
D
kkf2f
.(2.22)
132 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
–200 –150 –100 –50 0 50 100 150
100
105
1010
1015
Region of (b)
0 5 10 15 20
x
25 30 35 40 45
–2000
–1500
–1000
–500
h0
500
1000
1500
2000
(a)
(b)
FIGURE 11. Series solution of (2.1) at t=40 with µ=1,U=2 and ωf=1.632. (a) Rays
(− −) from left to right: (x/t)S=U1, (x/t)T=1.5,x/t=Uand (x/t)F=U+1. (b)
Magnification showing the steady-state spatially growing region. The series solution (—–)
exactly matches the residue prediction () given by (2.22).
Although (2.22) is given in the limit as smaller x/tvalues approach x/t=U1, we
expect that it is valid for the entire range 0 <x/t<U1. The series solution verifies
the result (2.22), shown in figure 11. In figure 11(a), the wave height is provided on
a logarithmic scale and negative values are shaded in grey. Apart from the logarithmic
scale, figures 11 and 4(a) are identical. The ray of maximum growth (x/t=U) and the
The response of convectively unstable flows to oscillatory forcing 133
limiting rays which separate important regions of the domain are shown as vertical
dashed lines. Figure 11(b) shows that the residue prediction () of (2.22) is an
exact match with the series solution (—–) in the steady spatially growing region.
This provides us with (at least) two physical interpretations: (i) for a fixed time,
the oscillatory-forced spatially growing region is entered as one moves closer to the
source (x=0); (ii) for a fixed location, the oscillatory-forced spatially growing region
eventually arrives and remains for all time. These interpretations are entirely consistent
with figure 1, and are expected, considering the nature of convectively unstable flows,
which indicate that transients eventually leave any finite region. While this insight
is significant by itself, it is useful to know the exact location (interpretation i) or
time (interpretation ii) at which the spatially growing region is observed, which also
indicates when the ‘unsteady’ transient has left the (x,t)coordinate of interest.
For the range of saddles that never enclose a pole (U<x/t6U+1), it can be seen
that x/t=U+1 is also a limiting ray, (x/t)F, that tracks the ‘front’ of the transient.
An expression for hAbeyond this region is obtained by applying Cauchy’s integral
theorem and again using our knowledge of the placement of real-valued saddles. This
is done in appendix B, where we show that h+
A=0 and h
Ais given by (B 9) for
x/t(U+1)+. The solution, h=h
A+hB, for waves beyond the transient is then
given by the superposition of (B 9) and (2.12):
[h(x,t)]x/t(U+1)+=iAei(kf2(x/t)ωf)t
D
ω
ω
kkf2f
+iAei(kf2(x/t)ωf)t
D
kkf2f
=0.(2.23)
Although (2.23) is given in the limit as larger x/tvalues approach x/t=U+1, we
expect that it is valid for the entire range U+1<x/t. The two terms in (2.23) are
equal and opposite, since [(∂D/∂ω)(∂ω/∂ k)]kf2f= −D/∂ k|kf2f. Thus, the sum is
equal to zero, indicating that beyond (x/t)F, the medium is calm.
For the linear forced Klein–Gordon equation, the limiting rays (x/t)Sand (x/t)F
that enclose the transient are exactly the characteristic velocities of (2.1), which is
expected because the operator is second-order hyperbolic. Dispersive non-hyperbolic
wave problems, in which characteristic velocities do not exist, are examined in § 3.
2.3. Ray extraction method
In the previous section, the key saddles, integration paths and limiting rays are
explicitly obtained for the forced linear Klein–Gordon equation (2.1). For most
wave problems, this is an arduous task and often unnecessary for examining the full
behaviour, since series solutions are fairly straightforward to generate for many linear
wave systems (Barlow et al. 2010; Barlow, Helenbrook & Lin 2011). Although one
may certainly estimate the region of spatial growth using numerical or series solutions,
the limiting rays provide an exact prediction of the location of this region at any given
time. In this section, we provide a method that allows one to extract the limiting rays
of the signalling problem using only the dispersion relation.
Before giving the full details of the technique, let us first summarize some crucial
concepts. Limiting rays are paired with limiting saddles through the growth/phase
argument i[ks(x/t)ω(ks)], which is used to evaluate the asymptotic behaviour. To
extract these rays, a map of saddles can be constructed in the complex k-plane,
defined by Im(∂ω/∂ k|ks)=0. Once the relevant sets of saddles ksare available,
the corresponding rays are simply given by x/t=Re(∂ω/∂k|ks). As mentioned
in § 2.2, the aggregate growth of the solution along each ray is determined
134 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
by σ=Re{i[ks(x/t)ω(ks)]}. The rays (x/t)Sand (x/t)Fare defined as follows:
σ6= 0 if |x/t|> (x/t)S(2.24a)
σ=0 if |x/t|< (x/t)S(2.24b)
σ=0 if |x/t|> (x/t)F(2.25a)
σ6= 0 if |x/t|< (x/t)F.(2.25b)
The above definitions also apply to the response of a non-oscillatory impulsively
forced medium (i.e. the Green’s function) (Gaster 1968; Monkewitz 1990; Huerre
2000). For example, when the forcing is given by Aδ(x)δ(t), the rays (x/t)Sand (x/t)F
represent the trailing and leading edges of the full response. In the cases studied
herein, where the forcing is given by Aδ(x)eiωft, the rays (x/t)Sand (x/t)Frepresent
the trailing and leading edges of the transient region (cf. figure 1). In addition, the
ray (x/t)Tpredicts where the saddle and pole contributions cease to compete. For
x/t> (x/t)T, the solution is dominated by the transient response (cf. figure 1). The
saddle ksassociated with (x/t)Thas a steepest descent path that passes through the
pole kfassociated with a given ωf. This is the ray moving slowest away from the
source, for which the steepest descent path does not enclose a kfpole. As proposed
by Gordillo & P´
erez-Saborid (2002), (x/t)Tis calculated by equating the steepest
descent curve through kswith the steepest decent curve through kf; this is given
by (2.20).
In the following procedure, we combine definition (2.24) and (2.25) with the ray
extraction technique of Gordillo & P´
erez-Saborid (2002), so that one may predict the
full behaviour of an oscillatory-forced convectively unstable flow.
(a)Step 1. Determine the real wavenumber kmax associated with the maximum
temporal growth rate using the conditions ∂ωi/∂ kr|kmax =0 and ωi>0. If the
dispersion relation can be solved explicitly for ω(k), kmax is easily obtained from
a temporal amplification plot. This is given in figure 2for the Klein–Gordon
equation, where kmax =0. Note that kmax is, by definition, a saddle point.
(b)Step 2. Construct a grid of (kr,ki) values that include kmax as an interior point;
it is expected that other saddles will lie nearby. Calculate the ω(k)roots of
D(k, ω) =0 for each grid point. The goal here is to choose the range of kiand kr
so that the saddles associated with (x/t)S, (x/t)Fand (x/t)Tare either within the
domain or are resolved to a desired accuracy. Since we have not yet determined
these rays, this step is iterative.
(c)Step 3. Calculate ω/∂ kfor each kvalue, make a contour plot of Im(∂ω/∂ k|ks)
on the krkigrid, and identify the zero contours; these are the locus of saddle
points. For problems such as the Klein–Gordon equation, where branch points are
present, it is useful to write ω/∂ k|ksin complex polar form, as done in (A 6)
(appendix A). This allows one to draw continuous contours of Im(∂ω/∂k|ks)
separately for each Riemann surface. The contours for one Riemann surface of
Im(∂ω+/∂k|ks)for the Klein–Gordon equation is shown in figure 12(a); the zero
contours (locus of saddles) are shown in bold. Although it is not shown here,
both Riemann surfaces share the same zero contours of Im(∂ω+/∂k|ks). The x/tks
distribution for the first and second Riemann surfaces are different; however, the
limiting rays of each surface are the same. Note that the plot in figure 12(a) is
identical to figure 6(b), which was obtained explicitly through (2.13) for h
A.
(d)Step 4. The Im(∂ω/∂ k)=0 curves from step 3 provide one with an array
of complex wavenumbers. For a given krvalue, its corresponding kivalue is
The response of convectively unstable flows to oscillatory forcing 135
0 0.5 1.0 1.5 2.0 3.02.5 3.5 4.0
–1.5
–1.0
0.5
0.5
0
1.0
1.5
S
Max growth at ( 2
–3 –2
–2
–1
–1
0
0000
000 000ki
1
1
2
2
3
3
kr
2.95
2.93
2.81
2.5
3.28
1.05
1.06
1.11
20.72
0.94
3.07
0.25
0.25
–0.25
–0.25
–0.25
–0.25
–0.25
–0.25
–0.25
–0.5 –0.5
–0.5
–0.5
–0.5
–0.5
–0.5
–0.75
–0.75 –0.75 –0.75
–0.75
–0.75
–0.75
–1
–1
–1
–1
0.25
0.25
0.75
0.75
0.75
1
1
1
1
0.5
0.5
0.5
(a)
(b)
FIGURE 12. The ksx/tdistribution for the forced Klein–Gordon equation with U=2 and
µ=1, determined from D(k, ω) =0 in (2.4). (a) Labelled contours of Im(∂ω/∂ k); zero
contours (shown in bold) indicate the locus of saddles. Relevant saddles (×) are labelled with
their x/tvalue. Here (x/t)Tis obtained by solving (2.20). (b) Aggregate growth (σ) versus x/t,
indicating the values of (x/t)Sand (x/t)F.
established along one of these curves, and the saddle ks=kr+ikiis identified. This
array of saddles can be inserted into the additional definition of a saddle point,
x/t=Re(∂ω/∂ k|ks), which provides an array of x/tvalues. Using both the saddle
point and x/tarrays, one can then obtain an array of σ=Re{i[ks(x/t)ω(ks)]}
values for each Riemann surface.
(e)Step 5. Finally, one can plot σversus x/tand immediately determine the values
of (x/t)Sand (x/t)Fby using definition (2.24) and (2.25). This is shown for
the Klein–Gordon equation in figure 12(b). Here, we are only interested in the
136 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
positive values of σ; negative values are shown in figure 12(b) to remind us that
σis multi-valued. The limiting ray values shown in figure 12(b) agree with those
found in § 2.2. Note that, for this particular problem, the wavenumbers associated
with the saddles for (x/t)Sand (x/t)Fdo not have finite values (cf. figure 6), and
so they can only be approached in a limiting sense. We shall see that this is
not the case for the liquid sheet problem in § 3. The ray (x/t)Tis determined by
evaluating (2.20), using the saddle point and x/tarrays constructed in step 4, and
akfvalue determined from the dispersion relation with a chosen ωf.
In addition, for a given integral solution
h(x,t)=1
4π2Z
−∞ Z∞+iτ0
−∞+iτ0
f(k, ω)ei(kxωt)
ωf)D(k, ω) dωdk,(2.26)
where f(k, ω) is an arbitrary function that contains no singularities and the path is
taken along the real k-axis, the steady spatial growth envelope can be approximated by
the leading-order contributions of the residue series
h(x,t)|x/t|<(x/t)SX
kf
if(kf, ωf)ei(kf(x/t)ωf)t
D
kkff
,(2.27)
which is a generalization of (2.22), combining the relevant contributions hA(spatial
growth) and hB(spatial decay). For a particular forcing frequency ωf, one should first
determine the location of the kfpoles in the complex k-plane. This indicates the sign
of the residue associated with each pole (cf. figure 9). The complete prediction for the
region x/t< (x/t)Sfor the forced linear Klein–Gordon equation is given by (2.22), and
is shown in figure 11(b) by . This approximation provides a simple transfer function
between the input forcing and the output waveform. If one is not concerned with the
initial transient, the only information needed to describe the asymptotic behaviour of
the solution is provided by (2.27), where x/t< (x/t)Sgives the region of validity. Note
that this is only ensured for a convectively unstable case, where we are confident
that the transient response convects away from the source, leaving only the forced
oscillatory response behind.
3. Signalling in a liquid sheet
A brief review of the stability of liquid sheets and its applications is given by
Lin & Jiang (2003) and Barlow et al. (2011). Here we mention only the work
that is directly relevant to this study. As shown in the analysis of Rayleigh (1896),
there are two linearly independent wave modes of a liquid sheet. The sinuous mode
moves the two free surfaces of a sheet in phase. The varicose mode symmetrically
moves the free surfaces in opposite directions. These modes were later confirmed
in the experiments of Taylor (1959). The onset of wave instability was analysed
by Squire (1953) through the use of classical temporal stability theory. The classical
theory predicts, for finite Q=ρgl(ρgand ρlbeing, respectively, the gas and liquid
densities), that varicose waves are temporally unstable. In addition, sinuous waves
are temporally unstable if the Weber number is greater than one. The Weber number
is defined as We =ρlU2h0/S, where Uis the liquid velocity in the sheet, h0is the
half-sheet thickness, and Sis the interfacial tension. This instability has since been
classified as convective; the spatio-temporal analysis can be found in Lin (2003).
A sheet subject to an impulsive initial velocity disturbance can also become unstable
The response of convectively unstable flows to oscillatory forcing 137
h0
x
y
U
h0
FIGURE 13. Definition sketch, a sheet with uniform thickness.
when We <1, due to an algebraic absolute instability (Barlow et al. 2011). However,
here we limit our study to We >1, since we are studying prediction mechanisms for
forced oscillatory flow, which are only relevant in a convectively unstable medium.
3.1. Formulation
Consider an inviscid liquid sheet of uniform thickness, 2h0, in an inviscid ambient gas
of density, ρg. The gas is stationary and the liquid is flowing at a constant velocity,
U. The fluids are assumed incompressible and the perturbed flow is assumed to be
irrotational, and thus the flow potential, φ, is governed by the Laplace equation, given
respectively for the liquid and the gas as
(∂xx +yyl=0 (3.1)
and
(∂xx +yyj=0(j=1,2)(3.2)
where (x,y)are the Cartesian coordinates in the unit of half-sheet thickness, xbeing
the flow direction and yperpendicular to the flow. The indices j=1 and 2 refer to
the gas above and below the liquid sheet, respectively (cf. figure 13). An oscillating
pressure disturbance, Aδ(x)eiωft, is introduced at x=0. To induce sinuous waves
at the outset, the pressure disturbance is applied in the same direction to both
the top and bottom surfaces of the sheet. To induce varicose waves, the pressure
disturbance is applied in opposing directions to the top and bottom surfaces. Typical
disturbances may be viewed as a linear combination of these motions. Here, we
examine sinuous and varicose modes separately. Through the application of appropriate
boundary conditions to the liquid, we allow the sign of Ato be arbitrary and derive
the dispersion relations for each mode separately. The linearized balance of pressure
difference at each interface with the surface tension force provides the dynamic
boundary condition,
Qtφj(∂t+xl+Aδ(x)eiωfty=(1)j+1=(1)jWe1xx hj(j=1,2)(3.3)
where h1and h2are the distances measured in the unit of h0from the x-axis to the
upper and lower liquid–gas interfaces, respectively, and time is normalized by h0/U.
The linearized kinematic boundary conditions are
[yφl]y=(1)j+1=(∂t+x)hj(3.4)
[yφj]y=(1)j+1=thj(j=1,2). (3.5)
138 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
To ensure that the solution is bounded in the gas above the liquid sheet where y>1
and below the liquid sheet where y<1, a boundedness condition is given by
φj(x,y,t)0 as y(1)j+1(j=1,2). (3.6)
For the sinuous mode of solution, φlis an odd function and the two free surfaces
move in unison; this ‘odd’ boundary condition is given by
yyφl(x,y=0,t)=0.(3.7)
For the varicose mode of solution, φlis an even function and the two free surfaces
move in opposite directions; this ‘even’ boundary condition is given by
yφl(x,y=0,t)=0.(3.8)
3.1.1. Integral solution
The double integral solution to this problem is obtained by first Fourier transforming
the system (3.2)–(3.5) and applying the boundary conditions (3.6)–(3.8) in Fourier
space to solve for the coefficients. The Laplace transform is then applied to the
resultant characteristic equation and the doubly transformed interface height is solved
for explicitly. Finally, the original variable his recovered through successive inverse
Fourier and Laplace transforms. This process is carried out in Barlow et al. (2011) for
the unforced sheet. For the forced sheet, the integral solution of the linearized system
is given by
h(x,t)=1
4π2Z
−∞ Z∞+iτ0
−∞+iτ0H0(k, ω) +Aik
ωf)D(k, ω ) ei(kxωt)dωdk(3.9)
for t>0; h(x,t)=0 for t<0. The dispersion relations for the sinuous and varicose
modes (Barlow et al. 2011) are given respectively by
D(k, ω) =sgn(kr)Qω2+tanh(k) (kω)2We1k3(3.10)
and
D(k, ω) =sinh(k)sgn(kr)Qω2+cosh(k) (kω)2sinh(k)We1k3.(3.11)
In (3.9), the unforced contribution H0(k, ω) is determined by the initial conditions,
and is given by (2.23) in Barlow et al. (2011). The second term in (3.9) is the
forcing contribution. Since the full solution is the superposition of the forced and
unforced solution, there is no need to repeat the analysis of Barlow et al. (2011)
for the unforced problem. We consider a forced sheet that is initially undisturbed,
reducing (3.9) to
h(x,t)=1
4π2Z
−∞ Z∞+iτ0
−∞+iτ0
Aikei(kxωt)
ωf)D(k, ω) dωdk,t>0 (3.12)
where the integration path is taken along the real k-axis. Note that (3.12) has a form
identical to (2.8), excluding the factor of kin the numerator of the integrand.
3.1.2. Series solution
In what follows, we examine the behaviour of (3.12) using the ray extraction
technique of § 2.3. First, however, we obtain the Fourier series solution of the
system (3.2)–(3.8) to assess the efficacy of the technique. As done in § 2.1, the series
The response of convectively unstable flows to oscillatory forcing 139
solution is found by solving the system as a boundary value problem, using periodic
boundary conditions: h(L,t)=h(L,t). The Fourier series solution of the linearized
system that describes an initially undisturbed forced liquid sheet is given by
h(x,t)=A
2L
×X
n∈[−N,1;1,N]
cosh n
2t+(bn2ianωf)sinh n
2t/∆net/2aneiωft
D(kn,iωf)eiknx
(3.13)
where kn=nπ/L, ∆n=pb2
n4ancn, and
an=Q/|kn| + f(kn)/kn(3.14a)
bn=2i f(kn)(3.14b)
cn= −knf(kn)+k2
n/We.(3.14c)
For sinuous waves, D(kn,iωf)is found using (3.10) and f(kn)=tanh(kn). For varicose
waves, D(kn,iωf)is found using (3.11) and f(kn)=coth(kn). The mode n=0 is
excluded from (3.13) because the eigenfunction takes a different form such that
h(x,t)|n=0=0.
3.2. Asymptotic predictions and forced solutions
In this section, the method outlined in § 2.3 is used to predict the key features of
convectively unstable waves (We >1,Q6= 0) in a liquid sheet. These features include
the limiting x/trays that bound the transient ((x/t)Sand (x/t)F), the ray that indicates
where the transient and forced solution compete for dominance ((x/t)T, which lies in
the transient region; cf. figure 1), and the resulting forced response at long time. The
predictions are validated by the series solution (3.13).
For small k(long waves) and Q=0, Lin (2003) derived a hyperbolic evolution
equation for sinuous waves in a sheet, with characteristic velocities x/t=1±We1/2.
For water sheets surrounded by air (at sea level), Q=0.0013, which is small enough
for long waves to be approximated along these characteristic rays. For We >1, the
characteristic rays indicate that sinuous waves propagate downstream from an initial
disturbance. In § 2.2 we determined that, for the hyperbolic Klein–Gordon equation,
the rays which enclose the transient ((x/t)Sand (x/t)F) are exactly equal to the
characteristic velocities of (2.1). To illustrate how limiting rays in a non-hyperbolic
dispersive system deviate from the characteristic rays of the hyperbolic limiting
behaviour (given above), we examine sinuous waves in a sheet for Q=0.0013 and
Q=0.13. We also examine the limiting rays for varicose waves with Q=0.13. The
results for the varicose case are of particular interest, since there are no characteristic
velocities in the long wavelength limit (Lin 2003). In all cases examined here,
We =20.
For the three cases to be examined, temporal amplification plots are shown in
figure 14; these are obtained through D(k, ω) =0 in (3.10) and (3.11). The maximum
growth saddles are indicated in each plot. Note that all plots in figure 14 are
symmetric about the ki-axis. This is a consequence of the sgn(kr)function appearing in
the dispersion relations, resulting in maximum growth at both ±krvalues.
Using the location of the maximum growth saddles in figure 14 as a starting point,
the temporal growth saddles are found using the method given in §2.3. These are
140 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
0
0.5 1.0 1.5 2.0
kr
2.5
0.01 0.02
kr
0.03
–2
–1
0
1
2
(× 10–3)
–0.2
–0.3
–0.1
0
0.1
0.2
0.3
0.5 1.0 1.5 2.0
kr
2.5
–0.2
–0.3
–0.1
0
0.1
0.2
0.3
0.2901
1.47
0.2758
1.6
(a)
(b)(c)
FIGURE 14. Temporal amplification curves for sinuous waves, given by D(k, ω) =0
in (3.10): (a)Q=0.0013,We =20 and (b)Q=0.13,We =20. (c) Temporal amplification
curve for varicose waves, given by D(k, ω) =0 in (3.11) with Q=0.13,We =20. All plots
are symmetric about the ki-axis. For each plot, the maximum growth saddles are indicated,
where ωi/∂kr=0.
shown for sinuous waves (Q=0.0013,Q=0.13) and varicose waves (Q=0.13) in
figures 15(a), 16(a) and 17(a), respectively. One striking difference between these
saddle maps and that of the Klein–Gordon equation (figure 12), is the ‘finite’ location
where the complex saddles meet the purely real saddles. As one might predict
from the analysis of the Klein–Gordon equation, these saddles correspond with the
limiting rays (x/t)Sand (x/t)F, which is confirmed in the aggregate growth plots of
figures 15(b), 16(b) and 17(b).
The saddle associated with the transient limiting ray (x/t)Tis shown in
figures 15(a), 16(a) and 17(a), and is computed by applying (2.20) to D(k, ω) =0
in (3.10) and (3.11) for sinuous and varicose waves, respectively. The approximated
characteristic rays x/t=1±We1/2(for sinuous waves) are also shown in figures 15(a)
and 16(a). For Q=0.0013 (figure 15a), the characteristic ray x/t=1We1/2=
0.7764 is a good approximation for the steady limiting ray, (x/t)S=0.7736. In
fact, any ray shown in the left half-plane of figure 15(a) for Q=0.0013, including
(x/t)T=0.775, is a good approximation for (x/t)S, as they are all close in value. In
contrast, 1 We1/2=0.7764 is more than double (x/t)S=0.3368 for sinuous waves
with Q=0.13 (figure 16a). The merit of the extraction method is clearly seen in
The response of convectively unstable flows to oscillatory forcing 141
–0.10 –0.05 0 0.05 0.10
kr
0.15
–0.08
–0.04
0
ki
0.04
0.08
0.7736
0.7736
0.9921
0.775
T1.226
F
1.226
F
0.40.2 0.6 0.8 1.0 1.41.2 1.6 1.8
0.5
0
1.0
1.5
2.0
2.5
(× 10–3)0.9921
Max growth at
(a)
(b)
FIGURE 15. The ksx/tdistribution for sinuous waves with We =20 and Q=0.0013,
determined from D(k, ω) =0 in (3.10). (a) Contours of Im(∂ω/∂k)=0 indicate the locus
of saddles. Relevant saddles (×) are labelled with their x/tvalue, shown here for one branch
of ω(k). Here (x/t)Sand (x/t)Fare determined from the aggregate growth plot in (b); (x/t)Tis
determined by applying (2.20) to D(k, ω) =0 in (3.10) with ωf=0.2.
figure 17 for varicose waves, where limiting ray values are clearly identified for a
system that has no approximate characteristic velocities.
To validate these limiting rays, the series solution (3.13) is evaluated using an ωf
that permits spatial growth. The ray that tracks the front of the response, (x/t)F, is
validated in figures 18(a), 19(a) and 20(a), where the series solution is plotted for a
142 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
–3 –2 –1 0 1 2
kr
3
–1.0
–0.8
–0.6
–0.4
–0.2
0
ki
0.2
0.4
0.6
0.8
0.9081
0.9081
0.874
T
1.438
F
0.3368
–12 01234
0.05
0
0.10
0.15
0.20
0.25
0.30
0.9081
Max growth at
0.3368
1.438
F
(a)
(b)
FIGURE 16. The ksx/tdistribution for sinuous waves with We =20 and Q=0.13,
determined from D(k, ω) =0 in (3.10). (a) Contours of Im(∂ω/∂k)=0 indicate the locus
of saddles. Relevant saddles (×) are labelled with their x/tvalue, shown here for one branch
of ω(k). Here (x/t)Sand (x/t)Fare determined from the aggregate growth plot in (b); (x/t)Tis
determined by applying (2.20) to D(k, ω) =0 in (3.10) with ωf=0.5.
fixed time. The ray that tracks the extent of the steady spatially growing envelope,
(x/t)S, is validated in figures 18(b), 19(b) and 20(b), where the residue predictions ()
are superimposed onto the series solution, identically matching for 0 <x/t< (x/t)Sin
The response of convectively unstable flows to oscillatory forcing 143
321012
kr
3
–0.8
–0.6
–0.4
–0.2
0
ki
0.2
0.4
0.6
0.8
(a)
(b)
0.3281
0.8678
0.51
T
1.437
F
–0.5–1.0 0 0.5 1.0 1.5 2.0 2.5 3.0
0.05
0
0.10
0.15
0.20
0.25
0.30
0.8678
Max growth at
0.3281
1.437
F
FIGURE 17. The ksx/tdistribution for varicose waves with We =20 and Q=0.13,
determined from D(k, ω) =0 in (3.11). (a) Contours of Im(∂ω/∂k)=0 indicate the locus
of saddles. Relevant saddles (×) are labelled with their x/tvalue, shown here for one branch
of ω(k). Here (x/t)Sand (x/t)Fare determined from the aggregate growth plot in (b); (x/t)Tis
determined by applying (2.20) to D(k, ω) =0 in (3.11) with ωf=1.
each figure. The residue prediction is obtained using the leading-order contributions
of (2.27), applied to (3.10) and (3.11). Note that no long-wavelength approximation
144 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
0 1000 2000 3000 4000 5000 6000
500 1000 1500 2000
x
2500 35003000
–6
–4
–2
0
h
2
4
6
(× 104)
(× 104)
(a)
(b)
0.7736
0.9921 1.226
F
–2
–1
0
h
1
2
FIGURE 18. Series solution (3.13) of sinuous waves at t=4000 with We =20,Q=
0.0013, ωf=0.2 and A=0.1. The solution is constructed on a domain of length L=10 000
with N=400. (a) Limiting rays shown as vertical dashed lines. (b) Magnification showing the
steady spatial growth envelope. Here (x/t)T=0.775 () is slightly ahead of (x/t)S=0.7736
(dashed line). The series solution (—–) exactly matches the residue prediction () for
0<x/t< (x/t)S, given by (2.27) applied to (3.10).
is used when finding the kf=kf)poles of (3.12). In order to calculate the residue
prediction (2.27), a Newton–Raphson iteration is used to obtain the leading-order
k-roots of (3.10) and (3.11).
For both sinuous and varicose waves, the rays (x/t)Sand (x/t)Fcompletely enclose
the transient, shown in the series solutions plotted in figures 18(a), 19(a) and 20(a).
The response of convectively unstable flows to oscillatory forcing 145
0 5 10 15 20
x
25 30
0 1020304050607080 10090
–1.0
–1.5
–0.5
0
h
1.0
0.5
Region of (b)
(× 106)0.874
T1.438
F
0.3368
0.9081
–200
–300
–100
0
h
100
200
300 0.3368
(a)
(b)
FIGURE 19. Series solution (3.13) of sinuous waves at t=60 with We =20,Q=
0.13, ωf=0.5 and A=0.1. The solution is constructed on a domain of length L=100
with N=400. (a) Limiting rays shown as vertical dashed lines. (b) Magnification showing
the steady spatial growth envelope. The series solution (—–) exactly matches the residue
prediction () for 0 <x/t< (x/t)S, given by (2.27) applied to (3.10).
The magnified regions of these solutions in figures 18(b), 19(b) and 20(b) indicate
that (x/t)Sand (x/t)Fconservatively bound the transient. As time increases, the wave
packet solution associated with the transient will continually grow in width to fill the
region between these two rays, since even small exponential growth, guaranteed to
reside between these two rays, will eventually manifest itself.
146 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
0 10203040506070
–200
– 400
0
h
–10
–5
0
h
5
10
200
400
Region of (b)
0.3281
0.3281
1.437
F
0.51
T
0.51
T
0.8678
0 5 10 15 20
x
25 30
(a)
(b)
FIGURE 20. Series solution (3.13) of varicose waves at t=50 with We =20,Q=0.13,
ωf=1 and A=0.001. The solution is constructed on a domain of length L=75 with
N=200. (a) Limiting rays shown as vertical dashed lines. (b) Magnification showing the
steady-state spatially growing region. The series solution (—–) exactly matches the residue
prediction () for 0 <x/t< (x/t)S, given by (2.27) applied to (3.11).
4. Conclusions
The analysis provided here clearly demonstrates the subtleties involved in
spatial–temporal stability analysis, particularly when studying an oscillatory-forced
medium. Steady spatial growth in the signalling problem is intimately related to the
The response of convectively unstable flows to oscillatory forcing 147
temporally growing transient response that precedes it. To witness spatial growth
downstream (upstream), one must first force the medium at a frequency that permits
negative (positive) imaginary wavenumbers, kf, determined from the dispersion relation.
In addition, one must be looking in a specific region, x, and at a specific time, t, such
that the x/tvalue corresponds with a saddle point of the Fourier integral solution. To
actually observe spatial growth, this saddle must lie along a steepest descent path in
the complex k-plane that encloses the pole kfwith the real k-axis. As first identified
by Gordillo & P´
erez-Saborid (2002) and established in this paper, the correct integral
responsible for spatial growth is necessarily the one which includes both a saddle point
and the pole responsible for spatial growth, given here as hA. This integral not only
allows one to determine the growth rate, but also provides the x/trays that mark
the transition between the spatially growing and temporally growing regions of the
solution.
The approach provided in this paper provides a tractable method for obtaining
practical information for convectively unstable flows that are continuously forced with
oscillatory disturbances. Such flows are, in fact, controllable because the transient
solution eventually propagates out of the domain. A spatially growing wave envelope
that is bounded in time lags the transient and eventually fills the domain. In this
work, we extract a closed-form transfer function that relates forcing amplitude and
forcing frequency with the amplitude and frequency of the resultant spatially growing
wave envelope. It is also important to determine the time at which the transient
leaves the domain, as control of uniformity is lacking while waves grow freely
in time. The precise x/tvalue that determines the time tat which the transient
leaves the location x, can be determined from the dispersion relation, following
the method provided in this work. In essence, we have combined the limiting ray
technique of Gordillo & P´
erez-Saborid (2002) with the established concepts found
in Gaster (1968), Monkewitz (1990) and Huerre (2000) for identifying leading and
trailing edge rays. The methodology developed here is shown to work for both a
hyperbolic equation system (Klein–Gordon) as well as a non-hyperbolic dispersive
system (liquid sheets); this is a demonstration of the generality of the procedure when
saddles arise in the dispersion relation and govern the transient response. The spatial
growth rate, wavelength and ultimately the phase of the steady spatially growing
envelope are determined by the purely real forcing frequency through the dispersion
relation. Although unstable flows are not desirable in processes that require flow
uniformity, such as film coating, those that are convectively unstable may be controlled
if disturbances to the system are process related and repetitive. Since the resultant
spatially growing waves are bounded in time, it is possible to control the magnitude of
these waves by controlling the magnitude of the offending disturbance.
Acknowledgements
The authors would like to thank Professor D. S. Ross of Rochester Institute
of Technology for useful mathematical discussions in support of this work. N.S.B.
acknowledges partial support from NSF Award No. 1048579.
Appendix A. Directions of steepest descent through convectively unstable
saddles in the Klein–Gordon equation
For (2.1), the saddles associated with the convectively unstable transient are purely
imaginary. The steepest paths away from these saddles are shown in figure 8as an
148 N. S. Barlow, S. J. Weinstein and B. T. Helenbrook
ellipse and the ki-axis. If we limit ourselves to a small region around the saddle,
kks= |kks|eiθ, the steepest directions θaway from the saddle are θ=nπ(ellipse)
and θ=nπ/2 (kiaxis), where nis an integer. If we Taylor expand the exponential
argument of (2.11) about the simple saddle ksand evaluate the denominator at k=ks,
we are then able to move the resultant coefficient of the exponential f±(ks)outside of
the integral, leading to
h±
Af±(ks)Z
−∞
e
γ2
z }| {
i(∂2ω±/∂k2)|ks(kks)2tdkF±(ks)Z
−∞
eγ2dγ(A 1)
where the variable of integration has been transformed into γand F±(ks)=
f±(ks)/(dγ /dk). The path of steepest descent follows the specific directions θthat
lead to sgn2)= +1, guaranteeing convergence of (A 1). If we define
i2ω±
k2ks=
2ω
k2ks
eiα,(A 2)
then the condition that restricts (A 1) to be a path of steepest descent is
2θ+α=mπ,modd (A 3)
which is an established result (Bleistein & Handelsman 1975). To determine which
steepest path is that of descent for h+
Aand h
A, one must first determine αfrom (A 2)
and then evaluate (A 3) to obtain the correct θ.
The branch cut between k= −µand k=µseparates the complex k-plane into
two Riemann surfaces, on which the integrands of h+
Aand h
Amust be appropriately
matched. In order to track the placement of kson these surfaces (and by extension
2ω/∂ k2|ks), it is useful to redefine the following quantity in polar form:
k2
sµ=ρeiπ+i2rπ.(A 4)
In (A 4), each integer rrepresents a different Riemann surface, rotated 2rπabout the
branch line. By applying (A 4) to the definition of a saddle point,
ω±
kks=U±ks(k2
sµ)1/2=x/t(A 5)
becomes
±ksρ1/2eiπ/2irπ=x/tU.(A 6)
The placement of saddles and choice of +or can be determined from (2.13) and
figure 6(line ABC) for two ranges of x/t:(x/tU) > 0 and (x/tU) < 0. To ensure
that the sign of ksin figure 6remains consistent with the sign of (x/tU)in (A 6),
the quantity ρ1/2eiπ/2irπmust be negative imaginary, which leads to an even rvalue.
Evaluating (A 2) and using r=0 gives
i2ω
k2ks=iµ
(k2
sµ)3/2
=iµρ3/2ei3π/2=µρ3/2eiα(A 7)
which leads to α=πfor saddles in both the lower and upper half-planes. Substituting
this into (A 3) reveals that θ=nπ; thus, the ellipse is the path of steepest descent,
leaving the ki-axis as the path of steepest ascent.
The response of convectively unstable flows to oscillatory forcing 149
Appendix B. Integration through stable saddles of the Klein–Gordon
equation
For (2.1), the temporally stable saddles that lag (x/t<U1) and lead (U+1<x/t)
the transient are purely real valued. The convectively unstable saddles associated with
the transient (U1<x/t<U+1) are purely imaginary. The transition between
real and imaginary saddles occurs at two locations in the complex k-plane: for the
rays x/t=U+1 and x/t=U1, as shown in figure 6. For h+
A, this transition
occurs between ks=iand ks= ∞ for x/t=U+1 and between ks= −iand
ks= −∞ for x/t=U1. For h
A, the transition occurs between ks=iand
ks= ∞ for x/t=U1 and between ks= −iand ks= −∞ for x/t=U+1.
In § 2.2, the asymptotic behaviour of the integral solution for purely imaginary saddles
(U1<x/t<U+1) is obtained by deforming the integration path along that of
steepest descent, which is a semi-ellipse, given by the bracketed argument of (2.15).
In the limits x/t(U+1)and x/t(U1)+, this semi-ellipse approaches a
semicircle of infinite radius. In § 2.2, we focus on the steepest descent paths associated
with the purely imaginary saddles, as it captures the limiting rays that bound the
transient. The real-valued saddles also contribute to the solution, and are necessary for
capturing the behaviour associated with x/tvalues that lead and lag the transient. In
practice, we need to determine the steepest descent paths associated with these saddles
to carefully examine their role. In this appendix, however, we focus on the limiting
case of the infinite semicircular path, where the contributions of real-valued saddles
may be elucidated easily and analytically.
To approach the rays x/t=U±1 from outside of the transient region, we approach
the infinite semicircle from either the saddles along the line ED or along the line FG,
shown in figure 6along the kr-axis. To preserve the direction of this limit on the
appropriate line of saddles, it is first useful to consider the following quantity in polar
form:
k2
sµ=ρeiθ.(B 1)
B.1. Temporally stable behaviour of h+
A
For the integral h+
A, given by the first term in (2.11), the real-valued saddles
representing x/t<U1 are shown along the line ED in figure 6(a). A solution
is constructed by closing the entire kr-axis with a semicircular path, leading to
h+
A|x/t(U1)= − A
2πZCR→∞
ei[kxω+(k)t]
2pk2µ[ω+(k)ωf]dk+2πiXresidues (B 2)
where RCR→∞ is the infinite semicircle. The individual residues are multiplied by 1,
1 or 0, depending on whether a