# Absolute and “upstream” convective instabilities in plane Couette-Poiseuille
flow

Kirthy K. Srinivas1  Sourabh S. Diwan1<EMAIL_ADDRESS>1Department of
Aerospace Engineering, Indian Institute of Science, Bengaluru 560012, India

###### Abstract

Here we report some interesting new features of the spatio-temporal
instability of the incompressible plane Couette-Poiseuille flow (CPF). First
of all, this flow represents the first instance of a “non-inflectional”
absolute instability, within constant-viscosity formulation, which is
triggered when one of the plates moves opposite to the bulk motion. More
strikingly, with further increase in the negative plate motion, the absolute
instability (AI) transitions to an “upstream” convective instability
($\textrm{CI}^{-}$), wherein an unstable wave packet moves opposite to the
direction of the bulk flow. Thus, the CPF exhibits a unique
$\textrm{CI}^{+}\to\textrm{AI}\to\textrm{CI}^{-}$ transition, for a given
Reynolds number ($Re$), where $\textrm{CI}^{+}$ denotes the commonly-observed
case of a “downstream” convective instability. This type of transition has not
been reported for other known examples of absolutely unstable flows. We
compute the leading and trailing edge velocities for an amplifying wave packet
and find that, for the plane Poiseuille flow, both these velocities approach
zero as $Re\to\infty$. As a result, at high $Re$, even the slightest of
negative plate motions is sufficient to trigger AI and subsequently
$\textrm{CI}^{-}$, as observed for the CPF. The wave-packet dispersion first
increases with $Re$, followed by a decrease, which points to a peculiar “dual”
role of viscosity in sustaining AI in the CPF, namely, viscosity promotes
sustenance of AI at moderate Reynolds numbers but suppresses it at low and
high Reynolds numbers. These results can be well understood within the
Ginzburg-Landau framework, and therefore can be expected to have a wider
applicability.

##  1 Introduction

One of the important developments in the modern theory of hydrodynamic
stability has been the classification of instabilities into convective and
absolute instability (Huerre & Monkewitz, 1990; Chomaz, 2005). The
convectively unstable flows behave as noise amplifiers, whereas the absolutely
unstable flows exhibit an intrinsic oscillator behaviour; the latter can also
lead to a “global” instability affecting the entire flow domain (Chomaz,
2005).

The prototypical examples of absolutely unstable flows include a mixing layer,
a near-wake and a heated jet (Huerre & Monkewitz, 1990); there have also been
several other types of flow, such as separated flows (Alam & Sandham, 2000),
the rotating-disk boundary layer (Lingwood, 1995), the Batchelor vortex
(Olendraru & Sellier, 2002) etc., in which absolute instability has been
observed. It is worth noting that these and other known cases of absolutely
unstable flows contain an inflection point in the base velocity profiles and
are therefore governed primarily by the inviscid instability mechanism (Huerre
& Monkewitz, 1990; Drazin & Reid, 2004). On the other hand, the non-
inflectional flows such as the Blasius boundary layer, the channel flow and
the plane Couette flow, which are governed by the viscous instability
mechanism (Drazin & Reid, 2004), have been found to be convectively unstable
(Huerre & Monkewitz, 1990). There exist more complex situations involving
absolute instability, such as a channel flow with viscosity stratification
(Govindarajan & Sahu, 2014), but the velocity profile for this case is found
to be inflectional. To the best of our knowledge, the onset of absolute
instability in a non-inflectional flow has not been reported so far. In this
work, we show that the plane Couette-Poiseuille flow, within the
incompressible constant-viscosity framework, provides the first instance of
such a “viscous non-inflectional” absolute instability. Interestingly, this
flow also exhibits an “upstream” convective instability.

The plane Couette-Poiseuille flow (CPF) is obtained by applying favourable
pressure gradient to a fluid between two plane parallel plates (Poiseuille
component of the flow) and setting one of the plates in motion (Couette
component of the flow), and is given by

$U(y)=\frac{\lambda}{2}(1+y)+(1-y^{2})\quad\mathrm{with}\quad\begin{aligned}
&y=-1,&&U=0\\\ &y=+1,&&U=\lambda\end{aligned}$ (1)

where $y$ is the wall normal coordinate, $U$ is the streamwise velocity and
$\lambda=\nicefrac{{U_{c}}}{{U_{p}}}$ is ratio of the upper plate speed,
$U_{c}$ and the Poiseuille component of centreline velocity, $U_{p}$; see
figure 1(a). Here $y$ is non-dimensionalized by the half-width of the channel,
$b$.

The plane Poiseuille flow (PPF), obtained by setting $\lambda=0$ in (1), has
been studied extensively as a prototype of linear viscous instability (Drazin
& Reid, 2004; Nishioka et al., 1975), as it is a strictly parallel flow that
is an exact solution of the Navier-Stokes equations (as is the CPF). Deissler
(1987) carried out a spatio-temporal analysis of the PPF (which is the only
reported study on the topic) and showed that this flow is convectively
unstable at all Reynolds numbers. Interestingly, he reported that the trailing
edge velocity of the wave packet decreases with increasing Reynolds number,
but the implications of this finding was not explored further (Deissler,
1987). This observation, which sets the stage for the present investigation,
is found to have far-reaching consequences for the stability of the CPF.

The linear and nonlinear stability characteristics of the CPF have been
investigated in the past, both theoretically/numerically (Potter, 1966; Cowley
& Smith, 1985; Kumar & Walton, 2019; Kirthy K. & Diwan, 2021) as well as
experimentally (Klotz et al., 2017). An important result from the linear
stability analysis (Potter, 1966) is that the motion of the plate (either in
the positive or negative direction) makes the CPF more stable as compared to
PPF and for $|\lambda|>0.7$, there is a complete stabilization of the flow. A
physical explanation for this observation was given by Kirthy K. & Diwan
(2021) in terms of emergence of a region of negative production with an
increase in $|\lambda|$; they also carried out a detailed characterization of
eigenvalues for $\lambda>0$ and $\lambda<0$. Despite these studies, the effect
of plate motion on the spatio-temporal behaviour of the CPF has not been
reported so far. Here we show that the negative plate speed (i.e. opposite to
the Poiseuille component of the flow), although having a slight stabilizing
effect on the spatial growth rates, has a profound influence on the
convective/absolute character of the CPF - even the slightest reverse flow
induced by the moving plate is sufficient to trigger absolute instability,
provided the Reynolds number $Re$ is large enough. Here $Re=U_{p}b/\nu$, $\nu$
being the kinematic viscosity. As the negative plate speed increases, the
absolute instability is realized at increasingly lower values of $Re$. Another
striking feature we observe is that the absolute instability in the CPF is
sustained only for a certain range of $\lambda(<0)$ and $Re$. For $\lambda$
exceeding this range (for a given $Re$), the flow reverts back to being
convectively unstable; the direction of the wave-packet propagation, however,
is opposite to the bulk fluid motion, i.e., in the upstream direction.

##  2 Spatio-temporal stability analysis of Couette-Poiseuille flow

The viscous spatio-temporal problem for the evolution of infinitesimal
disturbances on a parallel base flow is governed by the Orr-Sommerfeld (O-S)
equation (Drazin & Reid, 2004). For the present base flow (1) the O-S equation
is solved using the Chebyshev pseudo-spectral method. The disturbances are
represented as Fourier modes - $v(x,y,t)=\hat{v}(y)e^{i(\alpha x-\omega t)}$,
where $x$ is the streamwise coordinate, $t$ is time, $v$ is the perturbed wall
normal velocity, $\hat{v}$ is the complex eigenfunction, and $\alpha$ and
$\omega$, respectively, are the complex (streamwise) wavenumber and frequency.
Solving the discrete form of the O-S equation (Kirthy K. & Diwan, 2021)
results in a dispersion relation of the form $\omega=\omega(\alpha,Re)$, from
which the convective/absolute nature of instability is determined using the
Briggs-Bers criterion (Briggs, 1964; Huerre & Monkewitz, 1985). This
necessitates locating the saddle point of the dispersion relation,
$\omega_{s}$, which is done numerically using the procedure suggested by
Deissler (1987). Our algorithm involves starting with an initial guess
$(\alpha_{o},\omega_{o})$ for the saddle point and fitting a complex quadratic
to the function $\omega(\alpha)$ using the points
$[\omega(\alpha_{o}),\omega(\alpha_{o}+\Delta\alpha_{o}),\omega(\alpha_{o}-\Delta\alpha_{o})]$.
By enforcing $\left(\frac{d\omega}{d\alpha}\right)=0$, we obtain a new value
for $\alpha_{o}$ which serves as the next guess for the saddle point. This
iteration is carried out till the location of the saddle point is calculated
to a desired degree of accuracy. For the present work, the eigenvalue
computations have been validated against existing results for the PPF (Schmid
& Henningson, 2012) and the CPF (Potter, 1966); see Kirthy K. & Diwan (2021).
The results from the saddle-point computation have been validated against the
data of Deissler (1987); see supplementary figure S1.

###  2.1 Absolute and “upstream” convective instabilities

Figure 1: (a) The base velocity profile for the CPF. CF indicates the Couette
flow component. (b) The absolute growth rate $\Im(\omega_{s})$ as a function
$Re$ for $\lambda>0$, and (c) $\lambda<0$.

Figures 1(b) and 1(c) plot the imaginary part of the saddle point,
$\Im(\omega_{s})$ (which is the temporal growth rate associated with
$\omega_{s}$), as a function of $Re$, respectively, for $\lambda>0$ and
$\lambda<0$. We find that for $\lambda\in(0,0.7)$ and $Re>Re_{c}$ ($Re_{c}$
being the critical Reynolds number), the CPF is convectively unstable, i.e.,
$\Im(\omega_{s})<0$ (figure 1b), which is consistent with Deissler’s result
for $\lambda=0$ (Deissler, 1987). On the other hand, for $\lambda<0$ (and
$>-0.2924$; see figure 2) and for a certain range of $Re$, the flow is found
to be absolutely unstable, i.e., $\Im(\omega_{s})>0$, as seen from figure
1(c). Another interesting observation from figure 1(c) is that, for a given
$\lambda$, the absolute instability (AI) turns back into convective
instability (CI) beyond a particular $Re$. Thus, the CPF is seen to undergo
$\textrm{CI}\rightarrow\textrm{AI}\rightarrow\textrm{CI}$ transitions with
increase in $Re$; the first transition happens at $Re_{l}$ and the second at
$Re_{u}$; figure 1(c). As $\lambda$ decreases from $-0.1$ to $-0.25$, both
$Re_{l}$ and $Re_{u}$ go on decreasing (figure 1c). For a given $\lambda<0$,
the absolute growth rate exhibits a maximum, denoted as
$\Im(\omega_{s})_{max}$, at the Reynolds number denoted as $Re_{max}$. Figure
2(a) shows that as $\lambda$ becomes more negative, the (maximum) absolute
growth rate first increases and then decreases, reaching zero at
$\lambda\approx-0.3$ ($\lambda=-0.2924$ to be precise). For a more negative
$\lambda$, $\Im(\omega_{s})_{max}<0$, implying transition to a convective
instability (Huerre & Monkewitz, 1985). Thus absolute instability is realized
for the CPF only for $-0.2924<\lambda<0$. Interestingly, at $\lambda=-0.2924$,
$Re_{max}=Re_{c}$ and as $\lambda\to 0$, $Re_{max}\to\infty$ (figure 2b). More
comments on this behaviour will be made in relation to figure 4.

To explore these characteristics further, we plot the spatial branches
($\omega$ real) for $\lambda=-0.25$ in figure 3 for four different values of
$Re$. Here $\alpha^{+}$ represents the downstream travelling branch and
$\alpha^{-}$ the upstream branch. The identity of upstream and downstream
travelling branches has been confirmed by taking a contour in the complex
$\omega$ plane above all the unstable temporal modes. This results in a
separation of spatial branches into the lower and upper half $\alpha$ planes
(Huerre & Monkewitz, 1985). For $x>0$ (i.e., the direction of bulk motion),
the contour is closed in the upper half $\alpha$ plane and, for $x<0$, in the
lower half of $\alpha$ plane (supplementary figure S2). For $Re<Re_{l}$, the
amplifying spatial branch is $\alpha_{I}^{+}$ (figure 3) which advects in the
direction of the bulk flow. This is the well-understood case of “downstream”
convective instability ($\textrm{CI}^{+}$), exhibited by a variety of other
flows (Huerre & Rossi, 1998). For $Re>Re_{u}$, however, the spatial branches
move into the upper half of the $\alpha$-plane and for this case,
$\alpha_{II}^{-}$ becomes the amplifying branch (figure 3). Thus, for
$Re>Re_{u}$, the unstable wave packet travels upstream, i.e., opposite to the
bulk motion, exhibiting an “upstream” convective instability
($\textrm{CI}^{-}$). These features seen for $\lambda=-0.25$ are also observed
for the other values of $\lambda$ included in figure 1(c). This is shown in
supplementary figure S3, in which the movement of the saddle point
$(\alpha_{s},\omega_{s})$ is tracked in the complex $\alpha$ and $\omega$
planes for $\lambda=\pm 0.1,\pm 0.15,\pm 0.2,\textrm{and}\pm 0.25$. These
results imply that the transitions between CI and AI seen in figure 1(c) are
of the form $\textrm{CI}^{+}\rightarrow\textrm{AI}\rightarrow\textrm{CI}^{-}$,
and these are observed for the the entire range of $\lambda$ for which the CPF
is absolutely unstable, i.e., $\lambda\in(-0.2924,0)$. This is evident from
the parametric plot presented in figure 4(a).

Figure 2: (a) Variation of the maximum absolute growth rate,
$\Im(\omega_{s})_{max}$, (for a given $\lambda$) as a function of $\lambda$
for the CPF. (b) Variation of the Reynolds number corresponding to the maximum
absolute growth rate, $Re_{max}$, as a function of $\lambda$. Figure 3:
Spatial branches for $\lambda=-0.25$ exhibiting the
$\textrm{CI}^{+}\rightarrow\textrm{AI}\rightarrow\textrm{CI}^{-}$ transitions.
The four Reynolds numbers used are $2.2\times 10^{4}$ ($Re<Re_{l}$),
$2.3\times 10^{4}$, $5\times 10^{4}$ ($Re_{l}<Re<Re_{u}$) and $7\times 10^{4}$
($Re>Re_{u}$). Note that the branches corresponding to AI do not represent
physically meaningful spatial branches as they do not satisfy causality
(Huerre & Monkewitz, 1990); they are included here to track the movement of
the saddle point from the lower to upper $\alpha$ planes.

As seen earlier (figure 1c), for a given $\lambda$, $Re_{l}$ and $Re_{u}$
represent the $\textrm{CI}-\textrm{AI}$ transition points and are marked in
figure 4(a). For $\lambda=-0.2914$, below which AI is not present, $Re_{l}$
and $Re_{u}$ coincide with each other and are equal $Re_{c}$ (figure 2b). Note
that the $\textrm{CI}^{+}\rightarrow\textrm{AI}\rightarrow\textrm{CI}^{-}$
transitions are also observed at a given $Re$ as $\lambda$ becomes more
negative (figure 1c), with $\lambda_{l}$ and $\lambda_{u}$ representing the
transition points, as shown in figure 4(a). Thus, for the CPF, the AI can be
sustained only for a certain range in $\lambda$ and $Re$, namely,
$\lambda\in(-0.2924,0)$ and $Re\in(Re_{l},Re_{u})$, resulting in a “pocket” of
absolute instability; the ranges in $\lambda$ and $Re$ for which AI is
realized are plotted in figures 4(b) and (c) respectively. For a given
$\lambda$, the pocket of AI divides the convectively unstable region into two
parts - $\textrm{CI}^{+}$ region for $Re\in(Re_{c},Re_{l})$, and
$\textrm{CI}^{-}$ region for $Re\in(Re_{u},\infty)$ (figure 4a). A pocket of
AI surrounded by a region of CI was also observed for the instability of the
Batchelor vortex (Olendraru & Sellier, 2002); however the direction of
propagation of CI for this flow was always in the direction of the bulk
(axial) flow. The “upstream” convective instability, therefore, seems to be a
unique feature of the CPF. Interestingly, this upstream convective instability
is observed for any $\lambda<0$ (and $>-0.7$), provided $Re>Re_{u}$; for
$\lambda\in(-0.7,-0.2924)$, there exists one continuous $\textrm{CI}^{-}$
region for $Re>Re_{c}$ (figure 4a). Thus, for $\lambda\in(-0.7,-0.2924)$, the
only modal instability possible for the CPF is $\textrm{CI}^{-}$, with the
wave packet moving opposite to the bulk motion. (Note that the average
velocity for the CPF is $\left(\frac{\lambda}{2}+\frac{2}{3}\right)$ (1),
which is greater than zero for the entire range of unstable
$\lambda\in(-0.7,0)$.)

Another interesting observation from figure 4(a) is that even for arbitrarily
small negative values of $\lambda$, the CPF exhibits regions of AI and
$\textrm{CI}^{-}$, although at very high values of $Re$. Note that the limit
of $Re_{l}$ and $Re_{u}$ as $\lambda\to 0$ does not exist, since for
$\lambda=0$, the flow (i.e., the PPF) exhibits only $\textrm{CI}^{+}$
(Deissler, 1987). As $\lambda$ becomes more negative for the CPF, $Re_{l}$ and
$Re_{u}$ drop significantly and approach $Re_{c}$ as
$\lambda\rightarrow-0.2924$ (figure 4a). Therefore, it should be possible to
realize AI and $\textrm{CI}^{-}$ in the CPF in physical situations at moderate
plate speeds, provided the background disturbance levels are sufficiently low
to enable modal growth. No such instance has been reported in the literature
to our knowledge.

Figure 4: (a) Parametric representation ($Re$, $\lambda$) of the AI/CI
instability regions for the CPF. Here $S$ indicates linear stability. (b) The
range of $\lambda$, ($\lambda_{l}-\lambda_{u}$), supporting AI, as a function
of $Re$. (c) The range of $Re$, $(Re_{u}-Re_{l})/Re_{m}$, supporting AI as a
function of $\lambda$; here $Re_{m}=(Re_{u}+Re_{l})/2$. Note that $Re_{u}$ and
$Re_{l}$ are not relevant for $\lambda=0$, which exhibits $\textrm{CI}^{+}$
for all $Re$ (Deissler, 1987).

###  2.2 Wave-packet propagation and dispersion

The surprisingly rich behaviour exhibited by the CPF, with regard to the AI-CI
transitions, warrants a further investigation. It is well known that, the
primary instability mechanism in the CPF, which has a non-inflectional
profile, is viscous in character (Drazin & Reid, 2004). However, the role
viscosity plays in promoting and sustaining absolute instability for this flow
has not received attention in the literature. To understand this aspect
better, we make a Galilean transformation of the observer frame of reference
to $x=x_{0}-Vt$ (Deissler, 1987), where $x_{0}$ represents the laboratory
reference frame, and $V$ is the observer speed; see section 2.3 for a
discussion on an “objective” frame of reference for the CPF. We then proceed
to determine the range of $V$ ($=x/t$) for which the wave packet shows
temporal growth. The two rays where this growth rate is zero correspond to
leading and trailing edges of the amplifying wave packet; see also Huerre &
Rossi (1998).

Figure 5: (a) Leading ($V_{LE}$) and trailing edge ($V_{TE}$) velocities of
the disturbance wave packet as a function of $Re$. (b) Wave-packet dispersion,
$V_{LE}-V_{TE}$, as a function of $Re$.

The variation with $Re$ of leading and trailing edge velocities ($V_{LE}$ and
$V_{TE}$ respectively) for the CPF is plotted in figure 5(a). We see that
$V_{TE}$ decreases monotonically with $Re$ for all $\lambda(\leq 0)$, implying
that viscosity has a stabilizing effect on the onset of AI, i.e., a decrease
in $Re$ makes it harder for the flow to reach the critical condition for
absolute instability. A similar observation has been reported earlier in the
context of a wake (Monkewitz, 1988) and a separated boundary layer (Avanci et
al., 2019). Interestingly, $V_{TE}\rightarrow 0$ as $Re\rightarrow\infty$ for
$\lambda=0$, implying that the PPF is on the verge of becoming absolutely
unstable for asymptotically large $Re$ (although it is never realized) and
therefore, at these values of $Re$, a slight negative plate speed can make the
flow absolutely unstable, as seen in figure 4(a). Furthermore, $V_{LE}$ also
decreases monotonically with $Re$. (This was observed by Deissler (1987) for
PPF at lower values of $Re$ but he did not calculate $V_{LE}$ for large $Re$
as done presently.) As a result, a further small increase in the negative
plate speed can make $V_{LE}$ negative for sufficiently large $Re$. This would
make the wave packet switch the direction of propagation and move opposite to
the bulk motion. As $Re$ decreases, higher values of negative $\lambda$ are
necessary to bring $V_{TE}$ and $V_{LE}$ down to zero and make them negative
(figure 5a). Thus the main reason for the switch from AI back to CI (or to
$\textrm{CI}^{-}$) is the significant decrease in $V_{LE}$ with $Re$, which
facilitates an upstream propagation of the wave packet for a certain range of
negative plate speeds. The peculiar behaviour of $V_{TE}$ and $V_{LE}$ seen in
figure 5(a) is what is responsible for the
$\textrm{CI}^{+}\rightarrow\textrm{AI}\rightarrow\textrm{CI}^{-}$ transitions
observed for the CPF for $-0.2924<\lambda<0$.

The difference between $V_{LE}$ and $V_{TE}$, which is a measure of the degree
of dispersion of the wave packet is shown in figure 5(b). For a given
$\lambda$, the dispersion first increases with $Re$, reaches a maximum and
thereafter decreases. As a result, for Reynolds numbers corresponding to large
dispersion, the AI can be expected to be present for a larger range in
$\lambda$, i.e., more negative values of $\lambda$ will be needed to transform
AI into $\textrm{CI}^{-}$. This is evident from figure 4(b), which shows that
the range $\lambda_{l}-\lambda_{u}$ for which AI exists first increases with
$Re$ and then decreases. These results are useful in understanding the role
played by viscosity in sustaining absolute instability in the CPF. It appears
that viscosity plays a “dual” role in this regard - for moderately large $Re$
(corresponding to large wave-packet dispersion; figure 5b), viscosity sustains
AI for a larger range of negative $\lambda$ (or “promotes” its sustenance;
figure 4b), whereas for low and high $Re$, AI is realized over a narrower
range of $\lambda$ (implying that viscosity “suppresses” the sustenance of
AI). This behaviour is qualitatively similar to the dual role of viscosity in
making the flow unstable in the first place, i.e., the viscous mechanism
promotes disturbance amplification only for intermediate values of
$Re(>Re_{c})$, and the instability gets progressively weaker as $Re\to\infty$
(Drazin & Reid, 2004). Note that the shapes of the edge velocity curves in
figure 5(a) are strikingly similar to those of the neutral curves for the
primary instability of CPF (Potter, 1966; Kirthy K. & Diwan, 2021).
Incidentally, for a fixed $Re$, the wave packet dispersion decreases
monotonically as $\lambda$ becomes more negative (figure 5b). This is
reflected in a monotonic decrease in the range of $Re$ (i.e., $Re_{u}-Re_{l}$)
for the realization of AI, as $\lambda$ is decreased from zero; see figure
4(c).

###  2.3 Choice of an “objective” frame of reference for
$\textrm{AI}/\textrm{CI}$ distinction

At this point, it is pertinent to ask whether there exists an “objective”
laboratory frame of reference for distinguishing between CI and AI for the
CPF, as it is a strictly parallel flow. In other words, would the distinction
between $\textrm{CI}^{+}$, AI and $\textrm{CI}^{-}$ disappear, if the observer
moved with a conveniently-chosen frame of reference? As discussed in Huerre &
Rossi (1998), for a weakly non-parallel flow or for a flow which is
continuously forced at a certain location, the laboratory frame of reference
is well defined as the streamwise invariance is broken. On the other hand,
strictly parallel flows such as the PPF and CPF are invariant with respect to
translation in $x$, but the introduction of viscosity requires the no-slip
condition to be satisfied, and therefore the laboratory frame of reference for
these flows is unambiguously defined. This may be interpreted as the PPF and
CPF not being Galilean invariant (Huerre & Rossi, 1998), in the sense that any
change in the observer frame of reference necessitates the plate velocities to
be appropriately changed to satisfy the no-slip condition in the new frame.

To illustrate this point further, we carry out the following exercise. We
consider $\lambda<0$, with the upper plate moving in the negative $x$
direction (supplementary figure S4a), which exhibits $\textrm{CI}^{-}$ for a
range of $Re$ (figures 1c and 3). Suppose the observer moves with a reference
frame fixed to the upper plate. This makes the lower plate move in the
positive $x$ direction, resulting in $\lambda>0$ in the moving frame of
reference, with $|\lambda|$ remaining constant (supplementary figures S4b and
S4c). As a result of this change in reference frame, the wave-packet edge
velocities are shifted by an amount $+|\lambda|$ and the $\textrm{CI}^{-}$
appears to transform into $\textrm{CI}^{+}$; see supplementary figure S4(d)
for a specific case of $\lambda=-0.25$. (A similar shift in the edge
velocities in a moving frame is also seen for the PPF; supplementary figure
S5). However, it is important to note that the character of the base flow has
changed from $\lambda=-0.25$ in the stationary frame to $\lambda=+0.25$ in the
moving frame, and we do not expect $\textrm{CI}^{-}$ to be present for the
latter case since $\lambda>0$ (figure 1b). Furthermore, there are important
differences in the stability characteristics of the CPF between $\lambda<0$
and $\lambda>0$, with regard to the behaviour of the saddle point
(supplementary figure S3) as well as that of the temporal eigenvalues (Kirthy
K. & Diwan, 2021). In summary, an “objective” (or the laboratory) reference
frame for the CPF for $\lambda<0$ ($\lambda>0$), is any frame in which one of
the plates is stationary and the other moves in the opposite (same) direction
to the bulk flow. We impose the condition of ”zero group velocity” (for
determining the saddle point) in this frame of reference for making the
$\textrm{AI}/\textrm{CI}$ distinction. If $\lambda<0$, $\textrm{CI}^{-}$ (as
well as AI for $\lambda\in(-0.2924,0)$) is manifested, whereas, if
$\lambda>0$, only $\textrm{CI}^{+}$ is manifested.

##  3 Comparison with Ginzburg-Landau framework

The $\textrm{CI}^{+}\rightarrow\textrm{AI}\rightarrow\textrm{CI}^{-}$
transitions for the CPF can be well understood in terms of the Ginzburg-Landau
(G-L) framework. Towards this, we use a slightly modified form of the G-L
equation given by

$\frac{\partial\Psi}{\partial t}+\widetilde{U}\frac{\partial\Psi}{\partial
x}=\mu\Psi+\kappa\frac{\partial^{2}\Psi}{\partial x^{2}},$ (2)

where $\Psi(x,t)$ is the disturbance amplitude, $\widetilde{U}$ is velocity,
and $\kappa$ is a new parameter introduced to mimic kinematic viscosity in the
O-S equation; $\kappa=1$ results in the classical G-L equation (Huerre &
Monkewitz, 1990). Note that $\mu$ in (2) does not represent dynamic viscosity
but is a measure of disturbance growth. Introducing
$\Psi=\hat{\Psi}e^{i(\alpha x-\omega t)}$ into (2) and applying the Briggs-
Bers criterion gives us the condition for AI as
$\mu>\widetilde{U}^{2}/(4\>\kappa)$.

In (2), $\mu$ has the dimensions of velocity gradient and $\kappa$ that of
kinematic viscosity. We find it convenient to choose a velocity scale as $\mu
L$ and define a Reynolds number given by

$\widetilde{Re}=\frac{\mu L^{2}}{\kappa},$ (3)

where $L$ is the length scale used to non-dimensionalize $x$ in (2), assumed
fixed for the present purposes. Note that $\widetilde{U}$ could have also been
chosen as a velocity scale for defining $\widetilde{Re}$ but we prefer to use
it as a free parameter. We consider $\widetilde{Re}$ and $\widetilde{U}$ to be
respectively equivalent to $Re$ and $\lambda$ in the CPF analysis. For
enabling comparison, the AI/CI boundary for the CPF on the $Re-\lambda$ plane
in figure 4(a) is re-plotted in figure 6(a) in a different representation.

Figure 6: (a) The boundary dividing AI and CI in figure 4(a) represented as
$Re_{b}/Re_{c}$ as a function of $\lambda-\lambda_{m}$, where $Re_{b}$ denotes
the Reynolds number corresponding to the dividing boundary and can be either
$Re_{l}$ or $Re_{u}$, and $\lambda_{m}=(\lambda_{l}+\lambda_{u})/2$. (b) and
(c) The dividing boundary between AI and CI in the
$\widetilde{Re}-\widetilde{U}$ space for the G-L equation (2), with $\kappa$
and $\mu$ held constant respectively.

Next we construct the AI/CI boundary in the $\widetilde{Re}-\widetilde{U}$
space for the modified G-L equation, which we do in two different ways. First,
we keep $\kappa$ constant so that $\widetilde{Re}\propto\mu$ (3) and the
dividing AI/CI boundary scales as $\widetilde{Re}\propto\widetilde{U}^{2}$
(figure 6b). Secondly, $\mu$ is held constant so that
$\widetilde{Re}\propto(1/\kappa)$ and the dividing boundary in this case is
given by $\widetilde{Re}\propto(1/\widetilde{U}^{2})$ (figure 6c). It is
evident that the first case corresponds to region “A” of the dividing boundary
for the CPF (figure 6a) where viscosity suppresses the sustenance of AI,
whereas the second case corresponds to region “B” where viscosity promotes the
sustenance of AI (by enhancing dispersion as $Re$ decreases; figure 5b).
Furthermore, the CI region for the G-L equation is divided into
$\textrm{CI}^{+}$($\widetilde{U}>0$) and $\textrm{CI}^{-}$($\widetilde{U}<0$).
This is evident from the sign of the imaginary part of the absolute wavenumber
for the G-L equation ($\alpha_{s}=-\frac{i\widetilde{U}}{2\kappa}$) (Huerre &
Rossi, 1998). When $\widetilde{U}$ changes from positive to negative,
$\alpha_{s}$ moves from lower half to upper half in the complex $\alpha$
plane, implying that the wave packet amplifies in the negative $x$ direction;
see supplementary figure S6. This analysis shows that the G-L equation
reproduces the CI-AI transitions for the CPF (i.e,
$\textrm{CI}^{+}\to\textrm{AI}\to\textrm{CI}^{-}$) more closely than it does
for other common examples, such as a mixing layer or a separated boundary
layer, which exhibit only the $\textrm{CI}^{+}\to\textrm{AI}$ transition; the
transition from AI to $\textrm{CI}^{-}$ for these flows has not been reported
to our knowledge (see, e.g., Huerre & Monkewitz, 1985). This provides a
further support to the utility of the G-L equation as a model problem for
studying the spatio-temporal instability of fluid flows.

##  4 Conclusion

We have reported a spatio-temporal stability analysis of the Couette-
Poiseuille flow using the Orr-Sommerfeld equation to reveal some remarkable
and previously unknown features with regard to the $\textrm{AI}/\textrm{CI}$
behaviour of the CPF, as summarized below.

  1. 1.

The CPF is shown to exhibit absolute instability for a range of negative plate
speeds, $\lambda\in(-0.2924,0)$. Even the slightest of negative plate motions
is sufficient to trigger AI, provided $Re$ is large enough.

  2. 2.

For a given $Re$, the CPF undergoes
$\textrm{CI}^{+}\to\textrm{AI}\to\textrm{CI}^{-}$ transitions, as $\lambda$ is
made progressively more negative. The presence of an “upstream” convective
instability ($\textrm{CI}^{-}$), i.e., an unstable wave packet moving opposite
to the bulk motion, seems to be a unique feature of the CPF not reported
previously.

  3. 3.

For $\lambda\in(-0.7,-0.2924)$, the only instability possible for the CPF is
the “upstream” convective instability. Neither $\textrm{CI}^{+}$ nor AI is
realized for these values of $\lambda$.

  4. 4.

For the PPF, the leading and trailing edge velocities of the unstable wave
packet decrease monotonically with $Re$, approaching zero as $Re\to\infty$. It
is this feature that enables realization of AI and $\textrm{CI}^{-}$ for the
CPF, even for small negative values of $\lambda$.

  5. 5.

For a given $\lambda$, the wave-packet dispersion first increases with $Re$,
following which it decreases. This reveals a peculiar “dual” role of viscosity
in sustaining absolute instability in the CPF; viscosity promotes sustenance
of AI at moderate Reynolds numbers but suppresses it for lower and higher
values of $Re$.

  6. 6.

The $\textrm{CI}^{+}\to\textrm{AI}\to\textrm{CI}^{-}$ transitions observed for
the CPF are consistent with the Ginzberg-Landau (G-L) framework, which also
admits similar transitions. In fact, the CPF exemplifies the G-L model better
than other known examples of linearly unstable flows.

We expect some of these results to be applicable to other non-inflectional
flows, governed by viscous mechanism, that might become absolutely unstable.
This can potentially open up new directions of research in understanding the
role of viscosity in triggering and sustaining absolute instability in shear
flows.

[Acknowledgements]The authors thank Dr. Abhijit Mitra from Israel Institute of
Technology, Technion, for lending his stability code. Thanks are due to Prof
O. N. Ramesh from IISc, Bengaluru, for useful discussions.

[Funding]SSD acknowledges financial support from IISc, Bengaluru as a start-up
grant (No. 1205010620).

[Declaration of interests]The authors report no conflict of interest.

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