Revisiting Crossflow-Based Stabilization in Channel Flows

The control of fluid flows in channels is among the most complex open problems in classical theory, a precise understanding of which is yet to be established despite multiple decades of research. The shift from a benign laminar state to the complex chaos of turbulence is, to all appearances, highly non-linear, and incurs key structural and mechanistic changes in the processes driving mass and momentum transport. These modifications, depending on the exact setting, can be both desirable or undesirable, so that strategies to suppress – or even instigate – them are typically of significant interest. In this work, our aim is to reassess the efficacy of one such scheme: the superposition of a homogeneous, vertical (i.e., in the wall-normal direction) crossflow, such as that generated by a steady, spatially uniform transpiration of fluid through a porous boundary (or boundaries). Such systems have widespread utility, for example, in filtration processes, medical apparatus, and various geophysical and astrophysical phenomena, although, surprisingly, it is applications in external flows that are more frequently cited in the literature (e.g. modulating flow separation or delaying inflectional instabilities over swept wings; see [1] or [2]).

To preface with a broader perspective, the debate on transition and instability in fluid flows has undoubtedly been a long-standing one. Here, the standard analysis proceeds by deriving an eigenvalue problem from the equations of motion linearized around some base state of interest, usually laminar. Eigenmodes with positive growth rates suffer exponential amplification in time, and a critical value, $Re_{c}$, of the Reynolds number defines the threshold below which such an instability cannot survive. For simple parabolic profiles, such as those encountered in rectilinear pressure-driven flows, this approach predicts $Re_{c}\approx 5772$, with the Reynolds number defined based on the channel half-height $h$ and the center-line velocity $U_{p}$. As expected, the inclusion of crossflow, described by its own Reynolds number, say $R_{v}$, dramatically alters the picture, and the first few treatments may perhaps be attributed to [3] and [4]. Both authors resolved, at least for small to moderate $R_{v}$, a significant increase in $Re_{c}$ (again based on $U_{p}$), suggesting that the overarching influence of a net throughflow was very much stabilizing. Evidently, this remained the prevailing opinion until [5] proposed an alternative non-dimensionalization of the dynamical variables, setting the “actual” streamwise maximum as opposed to $U_{p}$ as the reference velocity unit. Because the injected wall-normal flux dampens the base streamwise component, this framework, they argued, preserved the distinction between the base velocity magnitude and the base velocity distribution when investigating stability. With this scaling, [5] demonstrated pockets of stabilization and destabilization, called “branches” in their terminology. To recount their example of choice, the wavenumber $\alpha=1$ at $Re=6000$, known to exhibit a positive growth rate for Poiseuille flow, becomes increasingly stable until a turning point of $R_{v}\approx 3.4$. Thereafter, the trend reverses and the instability is restored for $R_{v}\gtrapprox 42.5$, only to be suppressed once again for $R_{v}\approx\mathcal{O}(10^{2})$.

Apart from the initial study by [3], the work of [6] appears to be the first to provide a rigorous extension to the case of moving (upper) boundaries, the so-called Couette-Poiseuille flows, which present interesting stability characteristics in and of themselves. In particular, adopting the dimensionless ratio $\xi=U_{c}/U_{p}$ (the equivalent proxy in [6] is $k$), where $U_{c}$ is the wall speed, [7] showed that complete linear stability – $Re_{c}\to\infty$ – could be achieved for $\xi\gtrapprox 0.7$, termed the “cutoff”. This is, of course, not surprising since the Couette flow itself is known to be linearly stable to infinitesimal perturbations for various rheological models; see, for example, [8]. At any rate, in the presence of crossflow, [6] reported trends largely similar to the $\xi=0$ case, except that the cutoff threshold could now be reduced to, say, $\xi\approx 0.2$ using intermediate $R_{v}$ (a claim that, as they noted, was only valid for $Re\leq 10^{6}$, the upper limit of their numerical study). As special instances of this base flow, “generalized” Couette-like profiles have also been investigated by [9]; these are precisely linear and develop when the influence of the crossflow in the streamwise momentum budget is neutralized by a suitably chosen pressure gradient. In the presence of viscosity, these profiles exhibit complete linear stability up to $R_{v}\approx 24$, beyond which the critical Reynolds number can descend to as low as the relatively mild $Re_{c}\approx 725$ [9]. Even in the inviscid (Rayleigh) limit, excellently treated by [10], a perturbation has the potential to grow. Finally, we remark that the spatially developing (i.e., in the streamwise direction) analog has also been of interest and has been rigorously explored using Berman-type similarity solutions in the works of [11] and [12].

Consequently, crossflow-laden channel flows demonstrate rich stability portraits and form an interesting testbed for studying transition and disturbance amplification. From a purely application-oriented point of view, previous work, particularly on spatially non-developing cases, can be summarized in the statement by [6], who posited that modest wall velocities $\xi$ coupled with weak-to-moderate crossflows strike the optimal balance between stability and practicality. We argue here – and this is our main contribution – that this is not necessarily the case. In particular, we cite the non-normality of the linearized operator, typical of inertia-dominated flows, as the main culprit. Since the unstable eigenvalues are only really relevant at sufficiently large time horizons, when all other modal contributions have more or less decayed, one can rarely accurately judge the stability of a system from its spectrum alone. In shear flows, non-modal growth mechanisms are generally more dominant and therefore more informative, which is a well-established result in hydrodynamic stability theory [13, 14, 15, 16]. Surprisingly, despite their apparent simplicity, a comprehensive analysis on non-modal perturbations in (internal) crossflow-laden flows is yet to be performed, at least to the best of our knowledge. One relevant work in this regard seems to be that of [17], who investigated transient, unforced, energy amplification using the base flow of [5]. However, the scope of their study was rather narrow, and, as we discuss below, leaves much room for further investigation. On the other hand, the single-plate, zero-pressure-gradient, analog – the so-called asymptotic suction boundary layer (ASBL) – has received much more attention [18, 19, 20], and we make comparisons where appropriate.

We structure this paper as follows.  Section II introduces the base flow and describes our analysis frameworks. Section III augments previous discussion on eigenvalues, while Section IV presents a systematic summary of our non-modal calculations, with a particular focus on the purely pressure-driven case. Section V investigates the effects of wall motion, while Section VI offers conclusions.

We begin by supplementing classical perspectives on the modal stability of this flow. We define the critical Reynolds number, $Re_{c}$, as the smallest value of the Reynolds number below which the flow is linearly stable. At this threshold, a disturbance, described by the critical wavenumber $\left(\alpha_{c},\beta_{c}\right)$ must achieve neutral stability. Despite a non-zero mean velocity component in the wall-normal direction, Squire’s transformation [30] remains applicable, allowing us to set $\beta_{c}=0$ a priori and restrict attention to transversal modes. In contrast to the purely streamwise ($V=0$) case, however, the Squire operator (see Equations (15) and (17)) cannot be ignored, since the associated eigenmodes need not be damped. To confirm this, we follow [15] by converting to a formulation involving the complex phase speed, $c=\omega/\alpha=c_{r}+ic_{i}$, and multiplying the homogeneous Squire problem by $\eta$. Integrating across $y$, leveraging the associated Dirichlet conditions, and isolating imaginary components, we find

Assuming $\alpha>0$, the second term on the right-hand side of Equation (27) is necessarily negative, although the first may or may not be. The latter, of course, vanishes when $V=0$, ensuring $c_{i}<0$. Thus, for our purposes, we consider the complete Orr-Sommerfeld-Squire system when investigating modal stability.

To preface the ensuing discussion, we cite a key finding from [5], referring in the process to Figure 2, which plots at $Re=6000$ the growth rate $\omega_{i}$ corresponding to the least stable eigenmode for various representative choices of the streamwise wavenumber $\alpha$. [5] focused specifically on the case $\alpha=1$, which is unstable for the Poiseuille flow ($R_{v}=0$), and showed that it experienced stabilization followed by destabilization for small to intermediate $R_{v}$, see Figure 2. Subsequently, at the so-called “Branch-I” $R_{v}$ ($\approx 42.91$), exponential growth could be re-established (cf. [5], the mode became unstable “again”) and sustained until the “Branch-II” $R_{v}$ ($\approx 636.16$), which initiated yet another region of stability.

Evidently, as verified in Figure 2, this classification is not appropriate for all combinations of $\left(\alpha,Re\right)$. As a simple example, the wavenumber $\alpha=0.5$ does not exhibit instability at $Re=6000$ for the Poiseuille flow, rendering the notion of a Branch-I $R_{v}$ inherently ill-defined. Indeed, since the upper and lower branches of the Poiseuille neutral curve decay as $Re^{-1/11}$ and $Re^{-1/7}$ at large $Re$ (see, for example, [31, 32]), such a $R_{v}$ cannot be demarcated for any $Re$ at even smaller wavelengths, say $\alpha\geq 1.5$. [6] recognized this and circumvented the problem by introducing a third branch (which marked the transition from unstable to stable when $R_{v}$ initially increases from zero) whose existence was predicated on the stability of $\left(\alpha,Re,R_{v}=0\right)$. In summary, a more refined characterization was needed, and here we attempt to add to the discussion.

Figure 3 summarizes a numerical search for the critical parameters $\left(\alpha_{c},Re_{c}\right)$ for the range $0\leq R_{v}\leq 250$. Similarly to previous studies, we ignore $R_{v}<0$ due to the invariance of the eigenproblem under the transformation $\left(y,R_{v}\right)\to\left(-y,-R_{v}\right)$. Consistent with the narrative suggested by Figure 2, we observe a rather sharp, monotonic increase in the critical Reynolds number $Re_{c}$ through $R_{v}\lessapprox 1$. This trend persists beyond this interval, adjusting, however, to an approximate power-law behavior, $Re_{c}\sim R_{v}^{1.26}$. Throughout the latter range, the value of $Re_{c}$ increases rapidly to $Re_{c}\approx\mathcal{O}\left(10^{6}\right)$, before abruptly descending to $Re_{c}\approx\mathcal{O}\left(10^{3}\right)$ at $R_{v}=R_{v}^{d}\approx 22.175$. Shortly thereafter, the minimum of $Re_{c}\approx 667.48730$ is achieved at the turning point $R_{v}\approx 38.75$, which is in good agreement with the findings of [5]. Meanwhile, the critical streamwise wavenumber $\alpha_{c}$ initially experiences a brief drop from its value at $R_{v}=0$ ($\alpha_{c}\approx 1.02$), before recovering at $R_{v}\approx 1$ – note that such a discontinuity does not exist for $Re_{c}$. Following this, with stronger crossflows, we observe an increasing preference for short-wavelength instabilities up to the critical value of $R_{v}=R_{v}^{d}$. Here, in conjunction with $Re_{c}$, another discontinuity is encountered and $\alpha_{c}$ rapidly descends to near-zero ($\approx\mathcal{O}\left(10^{-2}\right)$ with the extent of our computation) before rising to what appears to be an asymptote. We remark here that for $20.8\lessapprox R_{v}\lessapprox 22$, [6] report an unconditional linear stability for the crossflow-laden Poiseuille flow (in fact, even for the Couette-Poiseuille flow $\xi\neq 0$ – see Section V), which is in contrast to Figure 3. According to our understanding, this discrepancy appears to be due to a combination of insufficient numerical resolution (they used $N=120$ collocation points) and a restriction of their search space to $Re\leq 10^{6}$. We verified that our calculations remained robust even when the resolution was doubled, indicating genuine instability.

To unravel the discontinuous nature of the critical parameters, Figure 4 details the movement of the neutral stability curves (NSCs) in the $\left(\alpha,Re\right)$-plane, particularly before and after $R_{v}=R_{v}^{d}$. Focusing first on $R_{v}<R_{v}^{d}$, Figure 3$(a)$ indicates that the presence of crossflow is exclusively stabilizing, and this is verified in Figure 4$(a)$ by a net displacement of the NSCs in the direction of increasing $Re$. A secondary minimum develops at intermediate $R_{v}$ in this range, which quickly coalesces with the primary one and appears to signal a small jump in the NSC towards larger wavenumbers, consistent with the first discontinuity for $\alpha_{c}$ observed in Figure 3$(b)$. Henceforth, the NSCs continue to shift deeper into the upper-right corner of the $\left(\alpha,Re\right)$-plane.

Beyond $R_{v}=R_{v}^{d}$, more dynamic behavior can be resolved. Specifically, two different NSCs begin to co-exist. The first set of unstable modes, relegated to $Re\geq O\left(10^{6}\right)$, can effectively be traced back to the (sole) NSC that occurs for $R_{v}<R_{v}^{d}$ and, as such, can be interpreted as its continuation into this $R_{v}$ regime. Neither [5] nor [6] reported this, instead implying the presence of a single NSC for all unstable $R_{v}$; we label these (and their counterparts in $R_{v}<R_{v}^{d}$) as “Type-I” instabilities and note that in both $R_{v}$ regimes, they generally exhibit a monotonic displacement toward increasing $Re$ as the crossflow becomes stronger. The second group of instabilities, deemed “Type-II” and active only in $R_{v}\geq R_{v}^{d}$, manifest in the form of an NSC emerging from $\alpha\approx 0$ and are, of course, precisely those reported in previous work since they formally define criticality for this $R_{v}$ regime. As highlighted in the inset of Figure 4$(b)$ these modes temporarily destabilize before stabilizing, defining in the process the turning point observed for $Re_{c}$ in Figure 3$(a)$.

To dissect the physical mechanism driving the instability, one can turn to the perturbation energy budget. By writing the energy density as $u_{k}u_{k}/2$, where a repeated index implies Einstein summation, the total disturbance energy can be written as

where $V\equiv\left[0,2\pi/\alpha\right]\times\left[-1,1\right]\times\left[0,2\pi/\beta\right]$ is taken as one full disturbance wavelength. Using the normal mode ansatz, Equation (13), evolution equations for $\mathsf{E}$ then take the form

where

Here, $\mathsf{PR}$ represents production against the background shear (contributed to only by $\mathsf{D}U$) and is responsible for the transfer of energy from the base flow to the disturbance through the action of the Reynolds stress. The second term $\mathsf{VD}\geq 0$ instead denotes viscous dissipation. In general, a (positive) production destabilizes, whereas dissipation stabilizes the disturbance field, although for a marginally stable mode, as we will discuss here, these contributions must exactly cancel.

Figure 5 explores for Type-I and Type-II instabilities the spatial variation of terms that contribute to the energy budget at criticality. For Type-I modes below $R_{v}^{d}$, energy production $\mathsf{PR}$ operates primarily near the suction boundary, its peak becoming sharper in tandem with $R_{v}$. For the weakest crossflow strengths in this range, an additional small positive hump (not shown here) can also be resolved near the lower wall, although it decays very rapidly as $R_{v}$ increases. Similarly, viscous dissipation $\mathsf{VD}$ remains confined to the upper wall and also increases in conjunction with $\mathsf{PR}$, as required to ensure neutral growth. As highlighted in the second row of Figure 5, these trends appear to translate well to a Type-I criticality in $R_{v}\geq R_{v}^{d}$, which provides further evidence that they are, in fact, a continuation of Type-I modes from $R_{v}<R_{v}^{d}$. However, for Type-II instabilities, the picture changes dramatically. Here, $\mathsf{PR}$ develops noticeable regions of energy negation and, more importantly, a wide positive peak in the lower (injection) half of the channel. We find the latter behavior quite intriguing, particularly in light of recent work showing that it is, in fact, the boundary inflow (outflow) that supports destabilization (stabilization); see [10]. Equally important to mention here is that for moderate to large crossflows (such as those found in $R_{v}\geq R_{v}^{d}$), variations in the streamwise shear $\mathsf{D}U$ are primarily located near the upper wall (with $\mathsf{D}U\to 1/2$, the Couette value, elsewhere). However, since the amplitude of positive production instead peaks specifically near the lower wall, we conclude that the crossflow influences Type-II modes predominantly through modification of the Reynolds stress distribution.

Formally, the last row in Figure 5 corresponds to the regime explored in Section 5 of [6]. In their Figure 17, for example, they reported that the dissipation at criticality for $\left(\xi,R_{v}\right)=\left(0.5,25\right)$ was completely overshadowed by the energy production throughout the channel width. Therefore, any removal of energy from the perturbation field arose directly from energy negation, rendering the movement of the critical layers, as they say, “irrelevant”. Clearly, as highlighted in Figure 5, this is not an entirely robust mechanism, at least for $\xi=0$, since, despite neutrality, $\mathsf{VD}$ for $R_{v}\geq R_{v}^{d}$ operates at least one order of magnitude higher than $\mathsf{PR}$ (though only very close to the wall) and generally intensifies with $R_{v}$. In other words, investigating the critical layers, the set of points where $U\left(y\right)=c_{r}$, could be instructive here. These points constitute regular singularities for the stability equations in the Rayleigh limit, but are smoothed over with the addition of viscosity in the Orr-Sommerfeld-Squire formulation. More importantly, it is well known that peaks in the amplitude of energy production are usually located within these layers [33, 15, 6, 34].

Figure 6$(a)$ presents the variation of $c_{r}$ at criticality versus $R_{v}$; we immediately observe discontinuities synonymous with those of $\alpha_{c}$, Figure 3$(b)$. The exact location of the associated critical layers can be explicitly determined as the solution(s) to the following transcendental equation

where we have elected to set

Equation (31) is satisfied by

where $W_{n}$ is the $n^{\mathrm{th}}$-branch of Lambert’s $W$-function and $\Pi_{3}=-R_{v}\operatorname{csch}R_{v}e^{\Pi_{2}R_{v}}$. Here, since $-1/e\leq\Pi_{3}<0$, there exist two distinct critical layers (which can alternatively be concluded by noting that $c_{r}<1$ and inspecting the velocity profiles in Figure 1$(b)$), and we only need to consider $n\in\left\{-1,0\right\}$. We observe that, as $R_{v}\to 0$, the symmetry of the resulting profile requires the absolute values of these roots to coalesce, despite being in opposite halves of the channel. Thus, the solutions, which we label $y_{s}$ and $y_{b}$, respectively, can be naturally identified with the suction (top) and blowing (lower) boundaries. On the other hand, due to the asymmetry inflicted upon $U$ by the crossflow, one can expect both critical layers to eventually shift toward the upper half of the channel, an intuition that is verified in Figures 6$(b$–$c)$. Specifically, commensurate with the movement of the peak in energy production for Type-I modes, $y_{s}$ generally increases, effectively approaching $y_{s}=1$ for sufficiently large $R_{v}$. Separately, beyond $R_{v}=R_{v}^{d}$, $y_{b}$ undergoes a sudden jump into the suction half of the channel, which is consistent with the wide production peaks (spanning almost the entire lower half-width) observed for Type-II modes in the last row of Figure 5.

We now direct our focus toward non-modal energy amplification. As shown in Section III, an increase in the crossflow component starting from $R_{v}=0$ generally has a stabilizing effect, at least in the sense of the critical Reynolds number. Following a paradigm shift at $R_{v}=R_{v}^{d}$, however, positive growth rates can be achieved for Reynolds numbers as low as $Re\approx 700$. When operating solely on this information, one would conclude that the most optimal (and practical) stabilization is granted by weak to, at best, modest $R_{v}$. In this section, we show that non-normality indicates an entirely different story.

We begin with the resolvent $\mathsf{R}$. In particular, the following bounds on the norm $\left\lVert\mathsf{R}\right\rVert_{E}$ are standard

where $\mathrm{dist}\left(\upsilon,\Lambda\left(\mathsf{S}\right)\right)$ is the shortest distance between $\upsilon$ and the spectrum $\Lambda\left(\mathsf{S}\right)$ (equivalent to the spectral radius of $\mathsf{R}$) and $\kappa\left(\mathsf{W}\right)$ is the 2-norm condition number of $\mathsf{W}$ [13, 26, 15]. For systems governed by normal operators, $\kappa=1$, so that the bounds in Equation (35) coalesce into equality. On the other hand, for non-normal linear dynamics, $\kappa\gg 1$ and the eigenfunctions of the underlying operator can be highly oblique, so that even pseudoresonant $\upsilon$ far from an eigen-frequency are capable of generating a substantial response. Thus, to probe the potential for this energy growth, we define the following quantity

which, if the resolvent is interpreted as a transfer function between the excitation and its response, represents an $H_{\infty}$-norm. In Figure 7, we visualize $\mathcal{R}$, along with the maximizing frequency $\upsilon=\upsilon_{\max}$, for some sample wavenumbers at $Re=500$, which is sub-critical and, therefore, admits no modal instability for all $R_{v}$ treated here. The dashed lines denote the lower bound in Equation (35), plotted for $\upsilon_{\max}$; the significant discrepancy observed with $\mathcal{R}$ is entirely a consequence of non-normality.

We first comment on longitudinal ($\alpha=0$) and transverse ($\beta=0$) perturbations. The former class typically elicits the strongest resolvent response for purely streamwise base flows. Therefore, we expect this trend to persist at least for weak $R_{v}$, and this is verified in Figure 7$(a)$. In fact, we see that the amplification of streamwise-independent modes can even be intensified in the presence of small amounts of crossflow. As an example, for $R_{v}\approx 2$, for which Figure 3$(a)$ predicts $Re_{c}\approx 48500$, $\mathcal{R}\approx 2000$, so that a unit norm forcing can produce $\mathcal{O}(10^{3})$ energy growth (compare this with $\mathcal{R}\approx 1000$ at $R_{v}=0$). This is the first indication of the unreliability of eigenvalues; when the effects of non-normality are taken into account, weak crossflows are, in fact, the most prone to excitation.

Interestingly, for relatively large $R_{v}$, Figure 7$(a)$ indicates that $\mathcal{R}$ tends to decay for longitudinal perturbations; in contrast; an increasingly stronger response from spanwise-independent modes is seen in Figure 7$(b)$, a gap that appears to peak at $R_{v}\approx 40$. More generally, in intermediate $R_{v}$ regimes, oblique disturbances constitute the main preference. Compared to the crossflow-independent case, these trends are inherently distinct but not unexpected, since the addition of a third-order inertial term in the Orr-Sommerfeld-Squire system fundamentally alters its anatomy and, therefore, that of the underlying non-normality (see Appendix A). However, what is crucial to note here is that $\mathcal{R}$ is only truly attenuated when $R_{v}$ becomes very large, say $R_{v}\geq 70$. In other words, it is, in fact, the weakest crossflows that instigate the strongest linear growth, possibly transition arising from subsequent non-linear processes. Given that most practical applications are typically noise-heavy and that such (relatively) strong crossflows are often not even feasible, we begin to see that crossflow-based modulation in channels might not be as appropriate a choice as suggested in the previous literature.

Moving to the complex plane, $\upsilon\in\mathbb{C}$, in Figure 8, we illustrate, as is customary, the contours of $\lVert\mathsf{R}^{-1}\rVert_{-E}$, where

and $\sigma_{\min}$ represents the minimum singular value of the operator $\mathsf{W}\mathsf{R}^{-1}\mathsf{W}^{-1}$. To interpret this in the context of the $\epsilon$-pseudospectra, we note that $\left\lVert\mathsf{R}\right\rVert_{E}=\lVert\mathsf{R}^{-1}\rVert_{-E}^{-1}$, so that

Thus, within the level set $\lVert\mathsf{R}^{-1}\rVert_{-E}=\epsilon$, amplification of the order of $1/\epsilon$ can be induced as a consequence of harmonic forcing. The pseudospectra are extremely informative, revealing crucial insight on the sensitivity of the spectrum to operator-level perturbations (see [35] and [36]) or, as is more relevant here, energy growth in the unforced initial-value problem. More specifically, the Hille-Yosida Theorem states that $G\leq 1$ if and only if the pseudospectra lie sufficiently close to the lower (stable) half-plane [26]. This condition can be made explicit through either the pseudospectral abscissa or, alternatively, the Kreiss constant $\mathcal{K}$, defined as follows

which in turn provides the following lower bound

Hence, substantial transient growth can be achieved if the pseudospectra protrude significantly into the unstable half-plane. Returning to Figure 8, we immediately observe strong pseudo-resonance down to $\epsilon\approx 10^{-5}$, indicative yet again of the strong non-normality pervading the linear dynamics of this system. Reminiscent of the trends shown in Figure 7, this amplification evidently worsens in tandem with the crossflow strength, eventually decaying only in the intermediate to large $R_{v}$ range (which is not shown here). Variations in the Kreiss constant suggest that, at least for weak crossflows, streamwise-independent disturbances remain the most relevant for the unforced problem, although oblique modes appear to be not too far behind. This is indeed representative of the ground truth, as we now proceed to discuss.

In particular, for various $Re$, Figure 9 summarizes a numerical sweep for $G_{\max}$, defined as the maximum admissible transient gain in energy over time and across wavenumber space, i.e.

Here, to concentrate specifically on amplification that occurs in the absence of exponential instability, we have limited our analysis to Reynolds numbers that are globally sub-critical, $Re\lessapprox 650$. In doing so, we readily observe a familiar pattern: for all $Re$ considered here, peak energy growth is only enhanced by small levels of crossflow, with $R_{v}=R_{v}^{\max}\approx 4.5$ consistently characterizing the worst-case scenario. Furthermore, it is precisely at very large $R_{v}$ that $G_{\max}$ begins to appreciably weaken relative to the Poiseuille flow. Note that for the asymptotic suction boundary layer, [18] reported a similarly declining – but still significant – amplification relative to the Blasius (no suction) case, although this conclusion could very well be specific to the Reynolds number they opted to focus on.

Regardless, as predicted by Figure 8, the optimal gain is indeed achieved for streamwise-elongated perturbations for small $R_{v}$ (say, $R_{v}\leq 2$), after which oblique modes become the norm (note that [17] apparently did not see the former for $Re=1000$). Interestingly, however, at the lower end of the Reynolds numbers treated here, sufficiently strong crossflows can once again establish a preference for streamwise-independent disturbances; we conjecture that expanding the range of $R_{v}$ investigated might lead to similar behavior at larger $Re$, although we did not attempt to verify this. Finally, the optimal time $t_{\max}$ also temporarily increases and then decreases monotonically in conjunction with $R_{v}$ (equivalently $G_{\max}$), although its peak was found instead at $R_{v}\approx 2\neq R_{v}^{\max}$. The immediate implication is that algebraic growth operates on much longer timescales in the presence of crossflow. [18] concluded similarly for ASBL but the effect is, in a sense, more pronounced here, since $G_{\max}$ also increases.

Figure 10 shows the initial condition and response pair attached to the optimal gain calculated for $R_{v}=R_{v}^{\max}$. Streaky structures, supported in both $x$ and $z$, develop at initial time and are then subsequently tilted and advected toward the suction boundary. A crucial question here is the precise mechanism driving the variation in the disturbance energy. It is well-known that longitudinal perturbations develop primarily via the lift-up effect, whereby strong streamwise streaks develop as a consequence of mean momentum transport in the wall-normal direction [37, 38, 39]. On the other hand, for perturbations with finite streamwise wavenumbers, the tilting mechanism of Orr dominates instead [40, 41]. In oblique disturbances, these processes operate simultaneously and [17] argued using a decomposition of the energy production proposed by [42] that at larger $R_{v}$, the Orr mechanism became increasingly relevant. When juxtaposed with Figure 9, this result should not be too surprising, since $\alpha_{\max}$ generally also increases with $R_{v}$. To refine this approach, however, we consider the time variation of the perturbation energy contained in each velocity component

where $\mathsf{PR}_{i}$ represents energy production, $\mathsf{PVC}_{i}=\mathsf{PS}_{i}+\mathsf{PD}_{i}$ the pressure-velocity correlation, $\mathsf{PS}_{i}$ the pressure-rate-of-strain, $\mathsf{PD}_{i}$ the pressure diffusion, and $\mathsf{VD}_{i}$ the viscous dissipation; see, for example, [43]. One can further relate the terms in Equations (42)–(44) to those in Equation (29) as follows

From here, we note that when $\alpha=0$, $\mathsf{PVC}_{u}$ vanishes, so that $\mathsf{E}_{v}$ and $\mathsf{E}_{w}$ decay simply due to a combination of the inter-variance pressure redistribution and viscous dissipation. As a result, we have $\mathsf{E}\approx\mathsf{E}_{u}$, whose growth in turn depends on the injection of energy from the mean shear through $\mathsf{PR}_{u}$. In other words, any changes in the amplification of streamwise-independent modes can be primarily attributed to modifications in the base shear and, by extension, to the Reynolds stresses. On the other hand, when $\beta=0$, the off-diagonal term in the Orr-Sommerfeld-Squire operator vanishes, so that the impact of non-normality is anyway diminished. Thus, we are left to deal primarily with oblique disturbances. In what follows, the pressure diffusion term $\mathsf{PD}_{i}$ was found to contribute negligibly to $\mathsf{PVC}_{i}$ and has therefore been omitted for the sake of clarity.

Using the energy-optimal initial conditions associated with $R_{v}=R_{v}^{\max}$ and $R_{v}=20$ (note that these disturbances are both oblique), Figure 11 compares the development of the energy budget over time for each component of the velocity. The primary motivator for the discrepancy in $G_{\max}$ can be immediately identified as the significantly attenuated production term, which evidently tapers around $t=45$ for $R_{v}=20$. In contrast, $\left\langle\mathsf{PR}_{u}\right\rangle$ for $R_{v}=R_{v}^{\max}$ operates longer and achieves much larger amplitudes. Furthermore, pressure redistribution remains active in both cases, although its influence is much more pronounced for the sub-optimal $R_{v}=20$. On inspection, this energy is transferred to both the wall-normal and the spanwise perturbations, and it is interesting to note that $\mathsf{E}_{v}$ ultimately achieves a slightly larger amplitude for $R_{v}=20$, so that a stronger amplification of the streaks observed in Figure 10 might be expected due to the lift-up process. However, we see that this effect is negated not only by a moderate increase in dissipation, but also by the rapid extraction of energy from $\mathsf{E}_{v}$ to $\mathsf{E}_{w}$ through a strongly negative pressure redistribution term starting from, say, $t\approx 12.5$. Consequently, any increase in the wall-normal perturbations is quickly eroded and energy production suffers. The net influx of energy into $\mathsf{E}_{w}$ appears to primarily dissipate away, in part, due to the lack of a feedback mechanism (e.g., a mean spanwise shear) to amplify it.

If either wall is translated in the streamwise direction with some finite velocity, say $U_{c}\neq 0$, we recover the base flow of [3] and [6]. More specifically, if this moving boundary is taken to be the upper (suction) one, the dimensionless streamwise velocity becomes

where $\xi=U_{c}/U_{p}$ and we have adopted a non-dimensionalization similar to that in Section II.1. Furthermore, we have denoted

We note two distinct scenarios in Equation (49), the need for which arises from the fact that both the crossflow and the Couette component skew the velocity profile towards the upper half of the channel [6]. Thus, for strong enough crossflows (alternatively, large enough wall speeds), the maximum of the streamwise velocity will necessarily occur at the moving boundary, that is, $U_{m}=U_{c}$ and $y\left(U=U_{m}\right)=h$. If the pair $\left(\xi,R_{v}\right)$ is chosen such that $\xi R_{v}=4>\Xi_{2}$, one recovers $U\left(y,\xi,R_{v}\right)=y$, which constitutes the “generalized” Couette profile of [9]. Throughout this section, we limit our attention to wall speeds $\xi\in\left[0,1\right]$, as in the prior literature.

We have performed a detailed re-assessment of stability in channel flows with crossflow. Previous work has attempted to determine parameter regimes supposedly ideal for delaying transition, and our primary contribution in this work has been to show that non-modal growth may preclude this apparent suitability.

The main focus of our study has been the crossflow-laden Poiseuille flow. Beginning with an eigenvalue (modal) analysis, we provide a global perspective on modal instability by tracking the trajectory of the neutral stability curves in the $\left(\alpha,Re\right)$-space. We show that beyond a discontinuity in the critical parameters, two individual neutral curves begin to co-exist, with distinct properties exemplified through consideration of the corresponding linear energetics. The movement of the critical layers is shown to be highly relevant in shaping the development of this budget.

From a non-modal perspective, which forms the crux of this paper, the resolvent is first inspected for real frequencies, and later the more general complex case through the $\epsilon$-pseudospectra. Substantial amplification is recovered even at relatively mild sub-critical $Re$, and is only damped for large, possibly unfeasible, crossflows. Similar patterns are observed when treating unforced algebraic growth, with non-modal interactions lasting longer (relative to the reference Poiseuille flow) and being more dangerous at weak $R_{v}$ – touted in the previous literature as the “ideal” stability configuration. The precise mechanism driving (and suppressing) energy growth is explored by considering a component-wise energy budget. The additional stability provided by large $R_{v}$ is shown to be due to a combination of decreased energy production and a more active velocity-pressure gradient term, which forces inter-component redistribution of energy in a way that significantly dampens the wall-normal fluctuations and, therefore, the lift-up effect.

This analysis has been extended to account for wall motion, cf. the flow of [6]. Here, we present novel instabilities for a region of parameter space previously thought to be unconditionally linearly stable. These instabilities occur at very high $Re$ but are physically genuine and of the short-wavelength type. Optimal growth is found to only worsen with the inclusion of wall motion, except beyond a regime change in the parameter space defined by $R_{v}=R_{v}^{\mathrm{shift}}\left(\xi\right)$, where the background shear decays as $1/(\xi R_{v})$ in the channel bulk. Here, the decrease in energy production is nonetheless counter-productive, since it is shown to be accompanied by an impractical cessation of mass transport brought on by a skewing of the velocity profile due to the combined effect of crossflow and wall motion. Collectively, our results suggest that crossflow-based stabilization schemes, at least in internal flows, are unlikely to be effective.