Linear Stability Analysis of Oblique Couette-Poiseuille flows

We use the standard equations of motion for an incompressible Newtonian fluid. In the absence of body forces, these can be expressed in dimensional format as follows

where $\widetilde{\boldsymbol{u}}=\begin{pmatrix}\widetilde{u}&\widetilde{v}&\widetilde{w}\end{pmatrix}^{\intercal}$ is the Eulerian velocity field, $\widetilde{p}$ the hydrodynamic pressure, $\rho$ the fluid density, and $\mu$ the dynamic viscosity. The flow of interest in this study is illustrated in the schematic presented in Figure 1. Two rigid surfaces, infinite in the wall-parallel directions and located at $\widetilde{y}=\pm h$, confine an incompressible fluid subject to a fixed streamwise pressure gradient $-\mathrm{d}p/\mathrm{d}x>0$. A crossflow is established by additionally translating the top wall with a constant velocity $U_{w}$ at an angle $\theta$ with respect to the positive $\widetilde{x}$-axis. The steady laminar velocities in the streamwise and spanwise directions, respectively, are given by

The resulting system is, therefore, a viscous-induced three-dimensional boundary layer, for which the flow angle, defined as $\phi(y)=\tan^{-1}(\widetilde{W}/\widetilde{U})$, changes with the wall-normal direction. These configurations are herein referred to as oblique Couette-Poiseuille flows (OCPfs) and, to our knowledge, have not received prior treatment in the stability literature, despite being among the simplest three-dimensional flows capable of retaining homogeneity in the streamwise and spanwise directions. Respectively, $\widetilde{U}$ and $\widetilde{W}$ are Couette-Poiseuille and Couette profiles, their relative strengths modulated by the direction of wall movement. In the limit $U_{w}\to 0$, standard Poiseuille flow is recovered. On the other hand, for $\theta\to 0$ and $U_{w}\neq 0$, the crossflow vanishes and the system reduces to the well-known aligned Couette-Poiseuille flow (ACPf), in which the pressure gradient and wall motion coincide exactly.

The parameter space characterizing OCPfs is rather complex, and, as is the case for ACPf, there exist multiple routes to rendering the governing equations non-dimensional. An obvious candidate is $U_{p}$, the so-called Poiseuille velocity scale, which is the streamwise maximum computed in the absence of wall motion. The other option is $U_{\max}$, the “actual” streamwise maximum, and is preferred if non-equilibrium effects are expected to significantly distort the streamwise profile away from $U_{p}$. However, in all possible realizations of OCPf, the boundedness of $\cos\theta$ and $\sin\theta$ ensures that $\widetilde{U}$ is $O\left(U_{p}\right)$. Therefore, to facilitate comparison with the previous literature, we choose to scale with $U_{p}$. More specifically, the following non-dimensionalization scheme is adopted

which yields the dimensionless form of the momentum equations

Here, Equation (5) represents the incompressibility constraint, and $Re=\rho U_{p}h/\mu=U_{p}h/\nu$ is a Reynolds number, with $\nu$ being the kinematic viscosity. The base velocity profiles become

where by defining $Re_{w}=U_{w}h/\nu$, we can interpret $\xi=Re_{w}/Re=U_{w}/U_{p}$ as the non-dimensional wall-speed. In this setting, the influence of the shear angle on the base profiles becomes more apparent. Suppose that $\xi$ is fixed and $\theta$ is varied; while $W$ maintains its Couette nature, $U$ evolves continuously as a one-parameter homotopy between ACPf and the plane-Poiseuille flow ($\theta=n\upi/2$ for odd $n$). Therefore, it is reasonable to limit attention to pairs $\left(\xi,\theta\right)\in\left[0,1\right]\times\left[0,2\upi\right]$, the former due to its physical relevance and the latter due to the periodicity of the base profiles that can be expected to permeate the forthcoming calculations. For select values of the flow parameters, the associated non-dimensional profiles are offered in Figure 2.

This section follows standard monologues on hydrodynamic stability, and we refer the reader to the works of Schmid & Henningson (2001) or Drazin & Reid (2004), for example. In operator format, the Navier-Stokes equations can be rewritten as

where $\mathcal{N}$ is a non-linear function of the state vector $\boldsymbol{u}^{*}=\begin{pmatrix}\boldsymbol{u}&p\end{pmatrix}^{\intercal}$. We decompose $\boldsymbol{u}^{*}$ as $\boldsymbol{u}^{*}=\boldsymbol{U}^{*}+{\boldsymbol{u}^{\prime}}^{*}$, where $\boldsymbol{U}^{*}$ is a time-independent base state superposed by a set of infinitesimal fluctuations ${\boldsymbol{u}^{\prime}}^{*}=\begin{pmatrix}\boldsymbol{u}^{\prime}&p^{\prime}\end{pmatrix}^{\intercal}$. In particular, we have

By Taylor expanding $\mathcal{N}$ around $\boldsymbol{U}^{*}$ and neglecting terms that are $O(\left\lVert{\boldsymbol{u}^{\prime}}^{*}\right\rVert^{2})$, we obtain a linearized system of evolution equations for the perturbation variables. To reduce computational complexity and the size of the matrices dealt with, the usual procedure here is to eliminate the pressure. This yields a rephrased system based only on fluctuations in the wall-normal velocity/vorticity $\boldsymbol{q}=\begin{pmatrix}v^{\prime}&\eta^{\prime}\end{pmatrix}$

where $\nabla^{2}$ is the usual Laplacian in a Cartesian coordinate system and $\nabla^{4}\left\langle\cdot\right\rangle\equiv\nabla^{2}\left(\nabla^{2}\left\langle\cdot\right\rangle\right)$ is the bi-harmonic operator. Hereon, for notational brevity, we drop the prime notation. Note that, contrary to the case of a purely streamwise base flow for which $W=0$, the so-called Squire equation, Equation (10), is now forced by mean shear from both the streamwise and spanwise profiles, which are, in general, non-zero. The spatial homogeneity can be exploited via a Fourier Transform

to obtain the canonical form of the Orr-Sommerfeld-Squire (OSS) system. Here, $\alpha,\beta\in\mathbb{R}$ are the real-valued wavenumbers in the $x$ and $z$ directions and $\overline{\boldsymbol{q}}=\begin{pmatrix}\overline{v}&\overline{\eta}\end{pmatrix}$ is a block vector of Fourier coefficients. The transformed equations can be compactly written as

where, by denoting $\mathcal{D}\equiv\mathrm{d}/\mathrm{d}y$ and $k^{2}=\alpha^{2}+\beta^{2}$, we have defined

The Orr-Sommerfeld and Squire operators, $\mathcal{L}_{OS}$ and $\mathcal{L}_{SQ}$ respectively, are given by

Equation (12) forms an initial-value problem for the Fourier-Transformed state vector $\overline{\boldsymbol{q}}$ in wavenumber space, where the associated boundary conditions can be obtained by applying no-slip/impermeability at both walls. Whenever necessary, the velocity-vorticity formulation of the OSS problem can be recast into one for the primitive fluctuations using the transformation

For a modal or eigenvalue analysis, an additional Fourier Transform is conducted in time

where $\omega=\omega_{r}+i\omega_{i}\in\mathbb{C}$ is the complex wave frequency. Equation (12) then reduces to a generalized eigenvalue problem described by the linear operator pencil $\left(\mathsfbi{L},\mathsfbi{M}\right)$

with eigenvalues corresponding to $i\omega=i\omega_{r}-\omega_{i}$. Note that this is equivalent to solving for the eigensystem of $\mathsfbi{S}^{\prime}=\mathsfbi{M}^{-1}\mathsfbi{L}$. In general, the spectrum is a function of $\left\{\alpha,\beta,Re,\xi,\theta\right\}$, and exponential amplification occurs over time if $\omega_{i}>0$. Consequently, we seek the manifold of marginal stability, designated by

We note that the presence of a non-zero spanwise velocity in OCPfs prevents an application of Squire’s Theorem in its usual form. Although a two-dimensional problem may well be constructed (see, for example, Mack (1984) and Schmid & Henningson (2001)), the “effective” base velocity depends on both spatial wavenumbers and there is no a priori indication of the appropriate search space. Therefore, for a given configuration $\left(\xi,\theta\right)$, since a full stability portrait requires a sweep through the $\left(\alpha,\beta,Re\right)$-space, a numerical approach will inevitably be marred by a lack of resolution. While this is a valid criticism, we point out that most canonical shear flows only become linearly unstable at modest wavenumbers, if at all. Furthermore, in Section 3, we demonstrate that from the perspective of modal stability, oblique Couette-Poiseuille flows are essentially continuations of the aligned variant. Therefore, the results of a sufficiently broad numerical search, as conducted here, are likely global.

Before proceeding, we make some key observations. First, as is true for strictly streamwise base flows, the Squire modes remain damped. The proof proceeds in the usual way by converting to a formulation involving the $x$-phase speed, $c=\omega/\alpha$, multiplying the homogeneous Squire equation by the complex conjugate of the fluctuating normal vorticity, and integrating over $y$. Therefore, for a modal analysis, it suffices to consider only the Orr-Sommerfeld operator, Equation (14). Furthermore, since neither component of the base velocity is inflectional, OCPfs do not admit an inviscid crossflow-like instability as observed, for example, over swept wings or rotating disks. In particular, in the inviscid limit, Rayleigh’s criterion can be modified to require the following expression to hold at some wall-normal location

where $\gamma=\beta/\alpha$. Although Equation (20) will, for general flows, vary in wavenumber space, the linearity of $W$ implies that $\mathcal{D}^{2}W=0$ for OCPfs. Thus, since $\mathcal{D}^{2}U=-2$, the instability must be viscous in nature.

For most shear flows, a spectral analysis of the linearized Jacobian as in Section 2.3 rarely agrees with experiment. Almost invariably, the transition to turbulence is observed at sub-critical $Re$, that is, below the threshold predicted by modal theory (Trefethen et al., 1993). This behavior is now well understood to be a consequence of the highly non-normal nature of the Orr-Sommerfeld-Squire (OSS) operator $\mathsfbi{S}^{\prime}$, which, in turn, arises from the off-diagonal term $\left(i\beta\mathcal{D}U-i\alpha\mathcal{D}W\right)$ driving the Squire equation; see (10) and (13). In general oblique Couette-Poiseuille flows, this forcing can evidently comprise both the streamwise and spanwise mean shear.

A non-normal operator such as $\mathsfbi{S}^{\prime}$ admits eigenfunctions that are non-orthogonal in the underlying Hilbert space. When arbitrary initial states are transformed into the basis of these eigenfunctions, they can suffer from large cross-terms in the induced norm (Schmid, 2007). An immediate consequence is that while a modal analysis might suggest asymptotic decay, energy amplification can still occur over finite time horizons. In shear flows, the transition to turbulent regimes has often been attributed to these transient phenomena, providing a potential explanation for the so-called bypass transition (Butler & Farrell, 1992). Furthermore, there is no guarantee that the long-time eigenmode is even realized, in spite of the most careful calibration, since sufficiently strong transient amplification will likely excite non-linear mechanisms in the flow and violate the linear assumption (Waleffe, 1995; Trefethen, 1997).

To explore the implications of non-normality in OCPfs, we first solve the initial-value problem in Equation (12) exactly to yield

where $\Phi\left(t,0\right)\equiv e^{-i\mathsfbi{S}t}$ is the state-transition operator, $\mathsfbi{S}=i\mathsfbi{S}^{\prime}$, and $\overline{\boldsymbol{q}}_{0}$ is the state of the system at the initial time $t=0$. Under appropriate norms in the input and output spaces, the gain can be defined as

where, due to its physical significance, we let $\left\lVert\cdot\right\rVert_{\mathrm{out}}=\left\lVert\cdot\right\rVert_{\mathrm{in}}=\left\lVert\cdot\right\rVert_{E}$ be an energy norm

over the volume $V$ defined by the Cartesian product $\left(x,y,z\right)\in\left[0,2\upi/\alpha\right]\times\left[-1,1\right]\times\left[0,2\upi/\beta\right]$. In this way, the energy of one full wavelength of a disturbance can be captured; see Butler & Farrell (1992). Here, $\left\langle\cdot\right\rangle^{\dagger}$ denotes a conjugate transpose operation, and the operator $\mathsfbi{E}$ is positive-definite and incorporates the Clenshaw-Curtis quadrature weights (Trefethen, 2000). With a Cholesky decomposition, we may write $\mathsfbi{E}=\mathsfbi{F}^{\dagger}\mathsfbi{F}$ so that

It immediately follows that

which can be computed trivially via the singular value decomposition (note, in fact, that $G=\left\lVert\Phi\left(t,0\right)\right\rVert_{E}^{2}$). The associated right and left singular functions represent, respectively, the initial condition and response pair for which the gain at time $t$ is realized.

Intuitively, no energy growth is expected if $G\leq 1$. An equivalent condition can be expressed in terms of the resolvent of $\mathsfbi{S}$. Consider an exogenous harmonic forcing profile $\boldsymbol{H}\left(x,y,z,t\right)=\boldsymbol{h}\left(x,y,z\right)e^{-i\zeta t}$ with frequency $\zeta\in\mathbb{C}$ to the linearized system, appropriately transformed into wavenumber space

The response can easily be verified to be

where the operator $\mathsfbi{R}\equiv\left(\zeta\mathsfbi{I}-\mathsfbi{S}\right)^{-1}$ is known as the resolvent. From an input-output perspective, $\mathsfbi{R}$ serves as a transfer function between the excitation and its response. The quantity $\mathcal{R}=\left\lVert\mathsfbi{R}\right\rVert_{E}$ is, therefore, of particular interest here, since for a non-normal system, it can be large even if the forcing is pseudo-resonant, that is, $\zeta\notin\Lambda\left(\mathsfbi{S}\right)$, the spectrum of $\mathsfbi{S}$ (Trefethen & Embree, 2005). Such a paradigm is especially informative for the receptivity of the flow to external disturbances (Brandt, 2014), and if $\zeta$ is restricted to real values, a physical interpretation of the resolvent is the perturbed operator that can result, for example, from external vibrations or planar imperfections (Trefethen et al., 1993). By further generalizing to the complex plane, one recovers the $\epsilon$-pseudospectra, the set of values defined as

For non-normal operators, $\Lambda_{\epsilon}$ can protrude deep into the upper-half of the complex plane, and the more pronounced this effect, the greater the potential for transient growth irrespective of the presence of linear instability. More rigorously, the Hille-Yosida Theorem states that $G\leq 1$ if and only if the $\epsilon$-pseudospectra lie sufficiently close to the lower half-plane (Reddy et al., 1993). For further details, we refer the reader to this paper, the citations within, and the text of Trefethen & Embree (2005).

An investigation of the perturbation energy budget can reveal the mechanism of instability in OCPfs. Throughout this section, the Einstein convention is implied via repeated indices. We define the perturbation energy density $\mathcal{E}$ as

By multiplying Equation (4) throughout by $\boldsymbol{u}^{\dagger}$ and integrating over $V$, evolution equations for the total energy are recovered

where we have assumed spatial periodicity of the disturbance field in $x$ and $z$. Under the normal mode ansatz, Equation (17), the above expression reduces to

where we have defined

Two contributions to the disturbance kinetic energy can be identified, $\mathcal{P}$, the production against the background shear(s), and $\varepsilon$, the viscous dissipation. The former can be further separated into terms representing the transfer of energy from the base streamwise and spanwise flows, respectively, to the perturbation field through the action of the associated Reynolds stresses, $\tau_{u}$ and $\tau_{w}$. These have been denoted by $\mathcal{P}_{u}$ and $\mathcal{P}_{w}$. In general, (positive) production destabilizes, whereas dissipation stabilizes the disturbance field.

We begin by investigating the dynamics of the eigenspectra in OCPfs. For a sample wavenumber combination, Figure 3 illustrates the loci of the first $\approx 50$ least stable modes as the non-dimensional wall speed $\xi$ is varied at $\theta=\upi/6$. The results have been presented in terms of $c=\omega/\alpha$. A familiar $Y$-shaped distribution can be observed, with three distinct branches reminiscent of the spectrum for plane-Poiseuille flow (pPf). As the wall speed increases, this structure collectively translates further into the right half-plane, and the most unstable mode monotonically stabilizes. In doing so, the shape of the $S$-branch, comprising the so-called mean modes related to the mean velocity, remains relatively undistorted. On the other hand, a sharper change occurs in the $A$-branch – the wall modes – which separate into two distinct subsets associated, respectively, with each wall. In a somewhat similar manner, starting from its bottom half, the $P$-branch of center modes also begins to split into two noticeable sub-branches. Together, these observations are indicative of the increased Couette contribution to the base flow, since the spectra for various flavors of Couette flow are usually scattered symmetrically within two $A$-branches, e.g. Duck et al. (1994); Schmid & Henningson (2001); Liu & Liu (2012); Zou et al. (2023).

In OCPfs, the distribution of this Couette component between the base velocities is directly controlled by the shear angle $\theta$. However, its impact at the level of the Orr-Sommerfeld (OS) equation is rather subtle. Since $W$ is linear, $\mathcal{D}^{2}W=0$, and the OS operator, simplified from Equation (14), becomes

where $U$ and $W$ retain their definitions from Equation (6), instantiated with some wall speed $\xi$. We immediately observe, despite the three-dimensionality of the flow, that the spanwise velocity appears only in a single term, $\mathcal{O}_{1}$, in Equation (34). In particular, for a spanwise-independent mode, $\beta=0$, the effects of obliqueness in the base flow are, in a sense, “shut off”, since the corresponding OS operator

reduces precisely to that for ACPf under the umbrella of Squire’s Theorem (excluding, of course, the factor of $\cos\theta$ in $U$, which can essentially be lumped into the wall speed). To extend this analogy to more general disturbances, a modification must first be made. Consider the generic three-dimensional (that is, prior to an application of Squire’s result) OS operator for ACPf

where

Comparing the two operators in (34) and (36) allows us to identify crucial similarities in structure. Specifically, for a constant wave triplet $\left(\alpha,\beta,Re\right)$, while $\mathcal{O}_{3}\equiv\mathcal{A}_{3}$ is immediate, $\mathcal{O}_{2}\equiv\mathcal{A}_{2}$ follows from the fact that $\mathcal{D}^{2}U=-2=\mathcal{D}^{2}U_{\mathrm{ACPf}}$. Therefore, $\mathcal{L}_{OS}$ and $\mathcal{L}_{OS}^{\mathrm{ACPf}}$ differ exclusively in their terms $\mathcal{O}_{1}$ and $\mathcal{A}_{1}$, respectively. However, since $k^{2}$ has also been fixed by our choice of wavenumbers, $\mathcal{O}_{1}\equiv\mathcal{A}_{1}$ can be made possible by requiring

where $\gamma=\beta/\alpha$. Therefore, for an arbitrary OCPf, the Orr-Sommerfeld problem at any wavenumber pair can be exactly mapped to one for ACPf via the “effective” wall speed $\xi_{\mathrm{eff}}$

With the corollary

we conclude that the stability of any oblique Couette-Poiseuille flow can be prescribed entirely by comparison with the appropriate ACPf configuration(s). A stronger result, and one perhaps in the same spirit as Squire’s Theorem, is as follows: if $\mathfrak{O}$ denotes the set of all possible OS operators for OCPf and $\mathfrak{A}$ the equivalent set for ACPf, then $\mathfrak{O}\subseteq\mathfrak{A}$.

The influence of the shear angle on modal behavior can now be made precise. We start by noting that $\xi_{\mathrm{eff}}$ is $2\upi$-periodic and

so that it varies strongly even throughout wavenumber space. Mathematically, at a fixed triplet $\left(\alpha,\beta,Re\right)$, its action in $\theta$ seems to be to accentuate or mask the strength of the wall speed. As an example, Figure 4 shows how changes in $\xi_{\mathrm{eff}}$ with $\theta$ affect the real and imaginary components of the most unstable eigenvalue for an arbitrarily chosen wavenumber pair. It is evident that the periodicity of $\xi_{\mathrm{eff}}$ directly translates to that of the spectrum, which itself becomes, at a minimum, $2\upi$-periodic. Furthermore, we found (not shown here; refer to Figure 3) that variations in $\xi_{\mathrm{eff}}$ modified the distribution of the eigenmodes in the complex plane much in the same fashion as variations in $\xi$ for fixed $\theta$, e.g., increasing $\xi_{\mathrm{eff}}$ increased $c_{r}$, and vice versa. When taken together, these observations, combined with Equation (40) and the interpretation of $\xi_{\mathrm{eff}}$, suggest that the manifold of marginal stability for OCPfs is contained wholly within that for ACPf. Accounting for a non-trivial directionality in the flow affects perhaps only the subset of the latter that is ultimately accessed.

In this section, we present the findings of a comprehensive investigation into the modal stability of oblique Couette-Poiseuille flows. We introduce the critical Reynolds number, denoted $Re_{c}$, which represents the minimum Reynolds number below which the flow remains linearly stable. Beyond this value, at least one disturbance, characterized by the critical wavenumbers $\left(\alpha_{c},\beta_{c}\right)$, becomes unstable. When analyzing two-dimensional flows, Squire’s Theorem (Squire, 1933) allows us to focus solely on disturbances that are independent of the spanwise direction, that is, $\beta_{c}=0$. However, for general three-dimensional profiles, an accurate assessment of stability necessitates the consideration of modes with non-zero $\beta$. Consequently, in the case of an oblique Couette-Poiseuille flow $\left(\xi,\theta\right)$, a thorough exploration of the entire three-dimensional $\left(\alpha,\beta,Re\right)$-space is required.

To reduce the degree of computation, we now consider some important simplifications. First, we note that in the stability literature for ACPf, the analysis for $\xi<0$ is typically neglected, since the modal growth rates are symmetric around $\xi=0$ (although the corresponding real parts might not be). Potter (1966) rationalized this by adopting the coordinate transformation $y\to-y$. Since $\xi_{\mathrm{eff}}\left(\theta+\upi\right)=-\xi_{\mathrm{eff}}\left(\theta\right)$, a similar argument allows us to restrict our attention to $\theta\in\left[0,\upi\right]$. However, a second reduction is also possible and can be achieved by noting that, at a constant $\xi$, if $\left(\alpha_{c},\beta_{c},Re_{c}\right)$ is the critical tuple for $\theta=\theta^{\prime}$, then $\left(\alpha_{c},-\beta_{c},Re_{c}\right)$ is necessarily the critical tuple for $\theta=\upi-\theta^{\prime}$. This result is immediate from the definition of $\xi_{\mathrm{eff}}$ in Equation (39), since

where we have assumed $\alpha>0$. Thus, it suffices to explore the range $\theta\in\left[0,\upi/2\right]$. For the Fourier wavenumbers, we focused on small to intermediate values, in particular, $\left(\alpha,\beta\right)\in\left[-3,3\right]\times\left[-3,3\right]$. This is generally the subspace of the wavenumber plane within which linear instability is first encountered in most canonical shear flows, and particularly for ACPfs (Potter, 1966). In total, $O\left(10^{10}\right)$ gridpoints were investigated, and our numerical procedure, including our method for traversing such an unwieldy parameter space, is outlined in Appendix A. In what follows, all results are presented for values of $\theta$ in degrees rather than in radians.

In general, the introduction of skewness in Couette-Poiseuille flows was found to be destabilizing, at least relative to ACPf. However, two qualitative regimes could still be identified in $\theta$. The first, denoted $\Theta_{1}$, comprises $0^{\circ}<\theta\lessapprox 20^{\circ}$ and is arguably the most interesting of the two, as it exhibits drastic changes in stability throughout its extent. Since oblique Couette-Poiseuille flows reduce to the standard aligned case as $\theta\to 0$, it is natural to expect the stability characteristics of ACPf to continue at least to modest $\theta$. Figure 5 supports this intuition. For all $\theta\in\Theta_{1}$, a short range of stabilization is followed by an inflection point in the $Re_{c}$-curves between $0.2\lessapprox\xi\lessapprox 0.4$ and then further growth, a trend that is precisely reminiscent of ACPf (Potter, 1966).

Perhaps the most striking feature is the fact that linear instability seems to persist throughout the entire range of wall speeds considered here. This behavior was observed even for “small” angles, such as $\theta\in\left\{1^{\circ},1.5^{\circ}\right\}$ (and even down to $\theta\in\left\{0.5^{\circ},0.75^{\circ}\right\}$, not shown here), where the wall motion is approximately parallel to the pressure gradient. This is in stark contrast to ACPf, which achieves unconditional linear stability, $Re_{c}\to\infty$, against infinitesimal disturbances beyond the so-called cutoff wall speed $\xi_{A}\approx 0.7$ (Potter, 1966; Cowley & Smith, 1985). A crude explanation for this is that the inclusion of a spanwise velocity makes $\beta$ a relevant stability parameter, providing, in light of the effective wall speed $\xi_{\mathrm{eff}}$ and the analysis outlined at the end of Section 3.1, an additional buffer for an OCPf to return to a region of parameter space that is linearly unstable for ACPfs. In fact, in Section 3.4, we argue, using a modified version of the long-wavelength approximation of Cowley & Smith (1985), that all OCPfs for which $\theta\neq 0$ are always linearly unstable: there is no cutoff wall speed for OCPfs. The absence of such a threshold for OCPfs seems to manifest itself in terms of the appearance of a limiting regime in the critical parameters. Here, the existence of linear instability seems to become entirely independent of $\xi$, as evidenced, in part, by the flattening of the $Re_{c}$-curves in Figure 5.

We note that before achieving the respective asymptotes in their $Re_{c}$-curves, the destabilization experienced by OCPfs in $\Theta_{1}$ at any wall speed is not necessarily monotonic with $\theta$. Beginning with inset $(a)$ in Figure 5, we see that increasing the shear angle is conclusively destabilizing up to $\xi\approx 0.2$. However, between approximately $0.2<\xi\lessapprox 0.275$, some of the more modest angles, say $\theta\gtrapprox 10^{\circ}$, tend to stabilize, while the values of $Re_{c}$ for even larger angles, $\theta\gtrapprox 18.5^{\circ}$, remain below those of ACPf; see Figure 5 $(b)$. Around $\xi\approx 0.3$ lies the crossing point $\mathcal{I}$, which initiates a region of monotonic stabilization for $\theta\gtrapprox 10^{\circ}$, a pattern that persists until around $\xi\approx 0.375$. Here, as seen in Figure 5 $(c)$, all $Re_{c}$-curves experience a turning point, which occurs at increasing wall speeds with $\theta$, and begin to ascend toward their eventual plateaus. Beyond this location, the stabilization becomes monotonic throughout $\theta>0$, as can be verified in Figure 5 $(d)$.

Figure 6 presents the spatial wavenumbers at criticality for $\theta\in\Theta_{1}$. Consistent with the trends observed for ACPf, the critical streamwise wavenumber $\alpha_{c}$ generally displays an initial monotonic decline toward $\alpha_{c}=0$, the latter limit corresponding to the complete loss of linear instability in ACPf beyond $\xi=\xi_{A}$. However, even for the smallest shear angles treated here, we see that the critical streamwise wavenumber for OCPfs is only close to, but never exactly, zero. Furthermore, in conjunction with the critical Reynolds number, the $\alpha_{c}$-curves also level out at sufficiently high wall speeds, reinforcing the presence of a limiting regime in modal stability. Meanwhile, the critical spanwise wavenumber $\beta_{c}$ is generally non-zero, a reminder of the three-dimensional nature of the base flow and the consequent inapplicability of Squire’s Theorem. An especially interesting behavior is observed in Figure 6 $(b)$ for a short range around $\xi\approx 0.3$, where $\beta_{c}\approx 0$. Here, as shown in Figure 5, the $Re_{c}$-curves for OCPfs in $\Theta_{1}$ also experience an inflection point. The latter is a key stability feature in ACPf, and, in this range, Potter (1966) had noted that $\xi\approx c_{r,c}$, the real part of the $x$-phase speed at criticality, hinting at some sort of link between this equality and the accompanying destabilization. We were able to verify this relation for $\Theta_{1}$ as well, suggesting that its secondary effect here is a preference for spanwise-independent modes (note that this is implicit for ACPf). Finally, just before this inflectional region, for $\theta=20^{\circ}$, we can resolve a rather dramatic trough in the $\alpha_{c}$-curve, which seems to temporarily terminate at the corresponding asymptotic ($\xi$-independent value) before recovering to its original trajectory. We interpret this as the first indication of an imminent departure from the modal characteristics of ACPf, which naturally leads to a discussion of $\Theta_{2}$, the second stability regime defined by $20^{\circ}<\theta\leq 90^{\circ}$.

In particular, Figure 7 highlights for $\Theta_{2}$ a noticeable shift in the stability characteristics of OCPfs. The $Re_{c}$-curves lose their inflectional nature as in $\Theta_{1}$, and while increasing the shear angle still induces destabilization, the decrease in $Re_{c}$ at any $\xi$ is uniform in $\theta$. Furthermore, all critical parameters reach their asymptotic values at smaller wall speeds, doing so by following relatively smoother trajectories (compare, for example, with the $\beta_{c}$-curves in Figure 6). From the perspective of the critical Reynolds number, the most unstable OCPf configurations occur as $\theta\to 90^{\circ}\in\Theta_{2}$, when the wall motion is perfectly orthogonal to the direction of the pressure gradient. In this limit, the streamwise and spanwise velocities reduce to

which are, respectively, pPf and Couette profiles. At this angle, we found that the critical parameters remained invariant for all the wall speeds studied here, approaching, in fact, the equivalent tuple for pPf. Specifically, we had

effectively indicating, for this $\theta$, a superposition of the stability of the individual velocity components (note that the Couette flow is always linearly stable, see Romanov (1973)). To rationalize this, we recall that the influence of the spanwise crossflow on the OS operator is partially modulated by the wavenumbers, particularly through the effective wall speed $\xi_{\mathrm{eff}}$. Since the corresponding eigenvalue problem can always be mapped to an equivalent one for ACPf, one would wish to somehow negate the Couette contribution, which is known to be stabilizing, in order to “maximize” instability. This is achieved most optimally in disturbances with $\beta=0$, immediately reducing the OS operator to that for pPf under Squire’s Theorem and yielding the critical tuple in Equation (44).

For full contour plots of the critical parameters in the $\left(\xi,\theta\right)$-plane as well as a discussion of the critical phase speeds and growth rates, we refer the reader to Appendix B.

In the previous section, it was observed that when the wall speed is sufficiently high, the stability of oblique Couette-Poiseuille flows becomes independent of $\xi$. The values of $\xi_{f}$, which represents the approximate wall speed that initiates this limiting regime, are shown in Figure 8. It can be seen that $\xi_{f}$ generally decreases with $\theta$, following a roughly linear relationship within the range $\Theta_{2}$, where OCPfs demonstrate the strongest deviations from the stability characteristics of ACPf. In this section, by adopting a simple juxtaposition with known results on the linear stability of pPf and ACPf, we aim to derive analytical formulae for the asymptotic values of the critical parameters. A small part of the following argument was briefly mentioned earlier when discussing criticality for $\theta=90^{\circ}$, but will now be elaborated upon.

For purely streamwise velocity profiles, a classic argument due to Squire (Squire, 1933) is that transversal ($\beta=0$) modes must become unstable at a lower $Re$ than both longitudinal ($\alpha=0$) and oblique ($\alpha,\beta\neq 0$) modes. The proof proceeds by defining a two-dimensional ($\beta=0$) Orr-Sommerfeld problem and arguing that, at criticality, the corresponding Reynolds number $Re_{c,2D}$ cannot be larger than that of the three-dimensional ($\beta\neq 0$) case $Re_{c,3D}$. Now, consider ACPf, $\theta=0$, and recall that in the limit $\xi\to 0$, pPf can be recovered. In particular, an application of Squire’s Theorem to both these flows yields

However, from the work of Potter (1966), we know that $Re_{c,2D}^{\mathrm{ACPf}}\geq Re_{c,2D}^{\mathrm{pPf}}$, which immediately allows us to conclude that

as well. For oblique Couette-Poiseuille flows, due to the mean three-dimensionality, Squire’s Theorem is no longer valid. However, Equation (40) implies that there exists a one-to-one mapping of the OS eigenproblem for an arbitrary OCPf, initialized with any combination of the wavenumbers, to an equivalent (in general) three-dimensional one for ACPf. Therefore, we have that

Here, the merits of the effective wall speed $\xi_{\mathrm{eff}}$ are once again apparent, as it can be made as large or as small as possible due to its dependence on $\theta$ and, more importantly, on the wavenumbers themselves. Therefore, to optimize the instability for an OCPf, which is equivalent to “$\geq$” in Equation (47) approaching equality, the OS operator should degenerate precisely into the two-dimensional analog for pPf. This can happen if and only if

where $\gamma=\beta/\alpha$ and $\left(\alpha_{c,\mathrm{pPf}},Re_{c,\mathrm{pPf}}\right)\approx\left(1.02,5773.22\right)$; see Orszag (1971). Assuming $\xi\neq 0$, the first of these three equations yields

a result that can be combined with the remaining constraints in Equation (48) to obtain the following closed solutions

where $\mathrm{sgn}$ represents the signum function and we have selected the positive solution for $\alpha$. Figure 9 presents the asymptotic values of the critical parameters, denoted by the subscript $\left\langle\cdot\right\rangle_{f}$, obtained from our numerical results overlaid with the analytical solution in Equation (50). An almost exact match is observed up to the resolution error of the domain sweep. Some crucial remarks can now be made. As the shear angle approaches zero, Equation (50) claims that the asymptotic streamwise wavenumber vanishes, while the asymptotic spanwise wavenumber experiences a discontinuity. Although this may seem erroneous at first glance, we note that the critical parameters in ACPf continuously vary with wall speed, showing no limiting behavior, so Equation (50) has no meaning in this limit anyway. On the other hand, Figure 6 $(a)$ seems to suggest that a flattening of the critical parameters for ACPf, if it occurs, should do so for $\alpha\to 0$. Meanwhile, we note that for $\theta\to\upi/2$, an asymptotic spanwise wavenumber $\beta_{f}=0$ is predicted, which validates our findings in Section 3.2.

In Figure 10, we present $\gamma_{c}$, the ratio $\gamma$ at criticality for various symmetrically chosen shear angles around $\theta=\upi/2$, which should be interpreted in light of Equation (49). We first see that $\gamma_{c}\to 0$ as $\xi\to 0$; this is expected, of course, since pPf is recovered in this limit, for which the most unstable disturbance has a vanishing spanwise wavenumber. Furthermore, we observe that $\gamma_{c}\left(\upi-\theta\right)=-\gamma_{c}\left(\theta\right)$, which supports the initial restriction of the state space described at the beginning of Section 3.2. Finally, for $\xi\geq\xi_{f}$, the $\gamma_{c}$-curves in Figure 10 plateau precisely at $\gamma_{c}=-\cot\theta$, as predicted by Equation (49). An interesting consequence arises when considering the wavenumber vector $\boldsymbol{k}=\begin{pmatrix}\alpha&\beta\end{pmatrix}^{\intercal}$. In wave theory, the wavenumber vector encodes the direction of wave motion, and a wave with wavenumber vector $\boldsymbol{k}$ propagates at an angle $\psi$ to the positive streamwise direction, where $\tan\psi=\beta/\alpha=\gamma$. From Equation (49), we can then conclude that the asymptotic eigenmode propagates at an angle $\psi=\theta-\upi/2$ to the pressure gradient, that is, exactly perpendicular to the wall motion.

A notable feature in the stability analysis of aligned Couette-Poiseuille flow is the unconditional stabilization achieved beyond $\xi=\xi_{A}$, the so-called cutoff velocity. Potter (1966) estimated this wall speed to be $\xi_{A}\approx 0.7$, which was later validated by Cowley & Smith (1985) through a weakly nonlinear analysis using the scaling $\alpha\sim Re^{-1}$ of the lower and upper branches of the neutral curves in the high-$Re$ limit. The authors employed a long-wavelength distinguished limit to the Orr-Sommerfeld equations, allowing $\left(\alpha,Re\right)\to\left(0,\infty\right)$ and fixing the product $\lambda^{-1}=\alpha Re$. Numerically, a parameter sweep can then be performed in the $\left(\xi,\lambda\right)$-plane to determine the maximum $\xi$ for which the asymptotic Orr-Sommerfeld operator displays neutral stability; this is the cutoff value.

In Section 3.2, we saw that, unlike ACPf, OCPfs remained linearly unstable beyond $\xi=\xi_{A}$, even for small but non-zero $\theta$, instead accessing a $\xi$-independent asymptotic regime described by the convergence of the critical parameters. This suggests the absence of an equivalent cutoff threshold when $\theta\neq 0$. To verify this claim, we augment the long-wavelength approach of Cowley & Smith (1985) to account for the three-dimensionality of OCPfs. Our modification is simple and involves treating the additional factor of $\gamma\equiv\beta/\alpha$ in $\xi_{\mathrm{eff}}$ through an appropriate limit for the spanwise wavenumber. In particular, we propose $\beta\to 0$ and allow $\gamma$ to remain finite, supporting this choice as follows.

First, to achieve complete linear stability in aligned Couette-Poiseuille flow, a vanishing spanwise wavenumber is implicit from Squire’s Theorem. To establish this, note that for a purely streamwise base flow, Squire’s argument proceeds by reducing the three-dimensional Orr-Sommerfeld operator to a two-dimensional one (for which $\beta=0$) by considering, among others, the following relation between the wavenumbers of the two eigenproblems

For a temporal stability problem as in this study, the wavenumbers are always assumed to be strictly real. Therefore, if $\alpha_{2D}\to 0$ at the cutoff wall speed, the positive semi-definiteness of the right-hand side of Equation (51) implies that each of $\alpha_{3D}$ and $\beta_{3D}$ must vanish identically. Despite this, we recall that Squire’s result has no immediate equivalent for mean three-dimensional base flows. However, our stability results and those of other canonical shear flows that are known to be unstable suggest that marginal stability, if it exists, usually does so only for small to intermediate – or even vanishing – values of $\gamma$.

For the modified version of the long-wavelength analysis, the Orr-Sommerfeld eigenvalue problem becomes

Note that as $\theta\to 0$ or $\gamma\to 0$, we recover the standard long-wavelength equations for ACPf as in Cowley & Smith (1985). For OCPfs, the state-space of the asymptotic problem is $\left\{\xi,\theta,\lambda,\gamma\right\}$, and we focus particularly on the quantity $\xi_{0}$, defined as follows

At a fixed $\theta$, $\xi_{0}$ represents the largest wall speed capable of sustaining neutral stability for wavenumber ratios $\gamma$. Numerically, therefore, Equation (53) is equivalent to determining the cutoff wall speeds in two-dimensional $\gamma$-slices of the three-dimensional stability manifold embedded in $\left(\xi,\lambda,\gamma\right)$-space. Figure 11 summarizes the results of this approach. A benchmark run was performed by constructing the long-wavelength neutral curve in the $\left(\lambda,\xi;\gamma=0\right)$-plane for ACPf, as shown in Figure 11 $(a)$ – the cutoff wall speed was calculated as $\xi_{A}\approx 0.70370$ and compares well with the ground truth. Figure 11 $(b)$, on the other hand, plots $\xi_{0}$ for some representative values of $\theta$. We see that, irrespective of the direction of wall motion, there exist peaks in $\xi_{0}$ that appear to extend to infinity. In fact, this divergence occurs precisely when $\gamma=-\cot\theta$, that is, when $\gamma$ obeys the relation provided in Equation (49). This result should, of course, not be surprising given the nature of the asymptotic regime. As discussed in Section 3.2, OCPfs experience a modal instability throughout $\xi\in\left[0,1\right]$, whose dependence on wall speed ultimately drops completely when $\xi\geq\xi_{f}$. Since the existence of a cutoff velocity in ACPf is necessarily related to the strength of wall motion and not to any inherent skewness of the flow, we conclude that in this range, likely even as $\xi\to\infty$, instability must always persist given the right wavenumber combination. In other words, for OCPfs, if a cutoff threshold for unconditional stability exists, it cannot do so at finite $\xi$. Note that no formal conclusions can be drawn about the critical Reynolds number under this paradigm; however, while it may very well be extremely large, it must remain finite.

Interestingly, for all $\theta$ considered here, Figure 11 $(b)$ highlights the presence of a long-wavelength instability at the asymptotic wavenumber ratio $\gamma=-\cot\theta$ even for $\xi\leq\xi_{f}$. At the same time, in this range, Sections 3.2 and 3.3 (see, for example, Figure 10) demonstrate that the associated critical wavenumbers do not necessarily follow the asymptotic laws derived in Equations (49) and (50). Therefore, for these wall speeds, we can infer that although disturbances with wavenumber ratios $\gamma=-\cot\theta$ can potentially suffer exponential modal growth, they are not the fastest to do so (in the sense of a minimal Reynolds number required to achieve a positive growth rate). However, as $\xi$ increases beyond $\xi_{f}$, while other wavenumber ratios stabilize (again, in the sense of a minimal Reynolds number), the effect, if any, on the asymptotic value is likely negligible – we discuss this further in Appendix C.

Figure 12 illustrates for $\xi=0.35$ the spatial distributions of $u$ and $w$, which are, respectively, the streamwise and spanwise velocity perturbations associated with the most unstable eigenmode at criticality. In a two-dimensional shear flow such as ACPf, this instability is instigated by the Tollmien-Schlichting (TS) wave, which comprises spanwise-elongated streamwise-propagating rolls. The TS instability supports the so-called “classical” route to turbulence, where, under sufficient exponential amplification, secondary instabilities are generated and lead to non-linear breakdown and, thereafter, transition (Herbert, 1988; Schmid & Henningson, 2001; Cossu & Brandt, 2004).

For $\theta=10^{\circ}\in\Theta_{1}$, it is observed that both disturbance components propagate approximately parallel to the streamwise direction. This can be attributed, in part, to the critical spanwise wavenumber being close to zero, which, as shown in Figure 6 $(b)$, is typically the case for intermediate wall speeds within this range of angles. The streamwise fluctuations bear some resemblance to a TS wave, suggesting a possible similarity between the mechanisms of modal transition in ACPf and weakly skewed OCPfs. However, the spanwise fluctuations are non-zero and consist of weakly parallel flattened structures that are localized near the lower wall. The exact role of these structures in the transition process is not immediately clear and requires further numerical investigation, which is beyond the scope of this paper. On the other hand, for $\theta=45^{\circ}\in\Theta_{2}$ and $\theta=135^{\circ}$, $\xi=0.35\approx\xi_{f}$, indicating that the stability characteristics of both flows are close to the asymptotic regime. Therefore, the most unstable wavenumber pair satisfies Equation (49) and is oblique. Consistent with the argument presented in Section 3.3, we observe that the associated eigenmode propagates exactly perpendicular to the direction of motion of the wall. The streamwise and spanwise fluctuations are qualitatively similar, both consisting of vortices tilted slightly away from each end of the channel. The cross-sections of these vortices for $u$ are somewhat distorted, while for $w$ they have a more regular, elliptical shape. Furthermore, we found that between $\theta$ and $180^{\circ}-\theta$, the support of these structures moved from the lower to the upper wall, although this effect was not very noticeable.

The most interesting behavior is observed for $\theta=90^{\circ}$, when the pressure gradient and the wall velocity vectors are exactly orthogonal. First, we recall from Equation (44) that despite the value of $\xi$ for this configuration, $\beta_{c}=0$, and the Orr-Sommerfeld operator can always be identified with that for the plane-Poiseuille flow. Thus, the wall-normal and streamwise components (see Equation (16)) of the disturbance are identically invariant with the wall speed, precisely reducing to the TS instability found in pPf. However, unlike the latter flow, for which $\beta\to 0$ induces a vanishing normal vorticity in the (resulting) homogeneous Squire equation, the spanwise fluctuations are no longer zero due to the spanwise shear in the off-diagonal forcing term, Equation (13). They comprise bands of transverse arch-like structures that, to the best of our knowledge, have not been previously recorded in the linearized analysis of any canonical shear flow. Moreover, varying $\xi$ had little effect on the shape of these modes, appearing to change only their energy.

Figure 13 shows the linear energy budget at criticality for the angle $\theta=30^{\circ}$. As the wall speed increases from $\xi=0.1$ to $\xi=0.2$, the streamwise production $\mathcal{P}_{u}$ decreases somewhat dramatically in the lower half of the channel, consistent with the initial rise in $Re_{c}$ seen in Figure 7 $(a)$. However, for $\xi\geq\xi_{f}$, it eventually stagnates near the stationary wall, whereas a continuous, although noticeably slow, decline is observed near the moving wall. On the other hand, while $\mathcal{P}_{w}$ seems to always suffer from a region of negative production near the upper wall, it operates at least one order of magnitude lower than $\mathcal{P}_{u}$. As a result, the total production $\mathcal{P}\approx\mathcal{P}_{u}$ remains positive throughout most of the channel. Viscous dissipation decreases with increasing $\xi$ and, similar to $\mathcal{P}_{u}$ (and, therefore, $\mathcal{P}$), it converges to some extent for high wall speeds. Therefore, even from the standpoint of the linear energy budget, there is a clear indication of modal stability in OCPfs approaching asymptotic regimes, where the potential for exponential amplification at criticality seems to become entirely agnostic to the wall speed.