2.1. Governing equations and base profiles

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:

(2.1)\begin{equation} \rho\left[\frac{\partial\tilde{\boldsymbol{u}}}{\partial\tilde{t}} +(\tilde{\boldsymbol{u}}\boldsymbol{\cdot}\tilde{\boldsymbol{\nabla}})\tilde{\boldsymbol{u}}\right] ={-}\tilde{\boldsymbol{\nabla}}\tilde{p} + \mu\tilde{\nabla}^2\tilde{\boldsymbol{u}}, \end{equation}

where $\tilde {\boldsymbol {u}} = (\tilde {u}\ \tilde {v} \ \tilde {w})^\intercal$ is the Eulerian velocity field, $\tilde {p}$ 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 $\tilde {y} = \pm h$, confine an incompressible fluid subject to a fixed streamwise pressure gradient $\mathcal {G} < 0$. A cross-flow is established by additionally translating the top wall with a constant velocity $U_w$ at an angle $\theta$ with respect to the positive $\tilde {x}$ axis. The steady laminar profile $\tilde {\boldsymbol {U}}$ satisfies

(2.2a–c)\begin{equation} {-}\tilde{\boldsymbol{\nabla}} \tilde{P} + \mu\tilde{\nabla}^2\tilde{\boldsymbol{U}} = 0, \quad \tilde{\boldsymbol{U}} = \begin{pmatrix} \tilde{U}(\tilde{y}) & 0 & \tilde{W}(\tilde{y}) \end{pmatrix}^\intercal, \quad\tilde{\boldsymbol{\nabla}}\tilde{P} = \begin{pmatrix} \mathcal{G} & 0 & 0 \end{pmatrix}^\intercal, \end{equation}

subject to the boundary conditions

(2.3a–c)\begin{align} \tilde{U}(\tilde{y}=h) = U_w\cos\theta,\quad \tilde{W}(\tilde{y}=h) = U_w\sin\theta, \quad\tilde{U}(\tilde{y}={-}h) = \tilde{W}(\tilde{y}={-}h) = 0. \end{align}

In particular, we can find

(2.4a,b)\begin{align} \tilde{U}(\tilde{y}) ={-}\frac{h^2\mathcal{G}}{2\mu}\left(1 - \frac{\tilde{y}^2}{h^2}\right) + \frac{U_w}{2}\left(1+\frac{\tilde{y}}{h}\right)\cos\theta,\quad \tilde{W}(\tilde{y}) = \frac{U_w}{2}\left(1+\frac{\tilde{y}}{h}\right)\sin\theta. \end{align}

The resulting system is, therefore, a viscous-induced three-dimensional boundary layer, for which the flow angle, defined as

(2.5)\begin{equation} \phi(\tilde{y}) = \tan^{{-}1}\left(\frac{\tilde{W}(\tilde{y})}{\tilde{U} (\tilde{y})}\right), \end{equation}

varies 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, $\tilde {U}$ and $\tilde {W}$ 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 cross-flow vanishes and the system reduces to the well-known aligned Couette–Poiseuille flow (ACPf), in which the pressure gradient and wall motion coincide exactly.

Figure 1.

A sketch of the three-dimensional flow geometry for oblique Couette–Poiseuille flows; here, $\mathrm {d}p/\mathrm {d}{\kern0.8pt}x = \mathcal {G} < 0$ is the constant streamwise pressure gradient. The wall at $\tilde {y} = h$ translates with velocity $U_w$ at an angle $\theta \neq 0$ to the streamwise direction, inducing a three-dimensional shear flow.

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 $\tilde {U}$ is $O(U_p)$. Therefore, to facilitate comparison with the previous literature, we choose to scale with $U_p$. More specifically, the following non-dimensionalization scheme is adopted:

(2.6a–d)\begin{equation} \boldsymbol{x} = \frac{\tilde{\boldsymbol{x}}}{h},\quad \boldsymbol{u} = \frac{\tilde{\boldsymbol{u}}}{U_p},\quad t = \frac{\tilde{t}}{h/U_p},\quad p = \frac{\tilde{p}}{\rho U_p^2}, \end{equation}

which yields the dimensionless form of the momentum equations

(2.7)$$\begin{gather} \frac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u} ={-}\boldsymbol{\nabla} p + \frac{1}{Re}\nabla^2\boldsymbol{u}, \end{gather}$$

(2.8)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u} = 0. \end{gather}$$

Here, (2.8) represents the incompressibility constraint, and $Re = \rho U_p h/\mu = U_ph/\nu$ is a Reynolds number, with $\nu$ being the kinematic viscosity. The base velocity profiles become

(2.9a,b)\begin{equation} U(y) = 1-y^2 + \frac{\xi}{2}(1+y)\cos\theta, \quad W(y) = \frac{\xi}{2}(1+y)\sin\theta, \end{equation}

where by defining $Re_w = U_wh/\nu$, we can interpret $\xi =U_w / U_p = Re_w/Re$ 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 ($\theta = n{\rm \pi} /2$ for odd $n$). Therefore, it is reasonable to limit attention to pairs $(\xi, \theta )\in [0, 1]\times [0, 2{\rm \pi} ]$, 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.

Figure 2.

From left to right, plots of the streamwise and spanwise velocities as well as the flow direction $\phi$ (normalized by ${\rm \pi}$) against the wall-normal coordinate $y$: (a–c) $\theta = {\rm \pi}/4$ and $\xi \in \{0.2, 0.4, 0.6, 0.8, 1\}$; (d–f) $\xi =0.5$ and $\theta \in \{{\rm \pi} /8, {\rm \pi}/4, 3{\rm \pi} /8, {\rm \pi}/2\}$. Formally, $\phi$ is singular near the lower wall, where $U$ and $W$ both vanish due to the no-slip condition. However, from l'Hopital's rule, the limit can be computed as $\phi (y\to -1) = \tan ^{-1}(\xi \sin \theta /(4+\xi \cos \theta ))$, evidently the angle between the wall shear stresses.

2.2. The linearized system

This section follows standard monographs on hydrodynamic stability, and we refer the reader to the works of Schmid & Henningson (Reference Schmid and Henningson2001) or Drazin & Reid (Reference Drazin and Reid2004), for example. In operator format, the Navier–Stokes equations can be rewritten as

(2.10)\begin{equation} \frac{\partial \boldsymbol{u}^*}{\partial t} = \mathcal{N}(\boldsymbol{u}^*), \end{equation}

where $\mathcal {N}$ is a nonlinear function of the state vector $\boldsymbol {u}^* = (\boldsymbol {u} \ p)^\intercal$. 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 ${\boldsymbol {u}^\prime }^* = (\boldsymbol {u}^\prime \ p^\prime )^\intercal$. In particular, we have

(2.11a–c)\begin{equation} \boldsymbol{U}^* = \begin{pmatrix} \boldsymbol{U} & P \end{pmatrix}^\intercal, \quad \boldsymbol{U} = \begin{pmatrix} U(y) & 0 & W(y) \end{pmatrix}^\intercal, \quad \boldsymbol{\nabla} P = \begin{pmatrix} -2/Re & 0 & 0 \end{pmatrix}^\intercal. \end{equation}

By Taylor expanding $\mathcal {N}$ around $\boldsymbol {U}^*$ and neglecting terms that are $O(\lVert {\boldsymbol {u}^\prime }^* \rVert ^2)$, 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 $\boldsymbol {q} = (v^\prime \ \eta ^\prime )$:

(2.12)$$\begin{gather} \left[\left(\frac{\partial}{\partial t} + U\frac{\partial}{\partial x} + W\frac{\partial}{\partial z}\right)\nabla^2 - \frac{\mathrm{d}^2 U}{\mathrm{d} y^2}\frac{\partial}{\partial x}- \frac{\mathrm{d}^2 W}{\mathrm{d} y^2}\frac{\partial}{\partial z}- \frac{1}{Re}\nabla^4\right]v' = 0, \end{gather}$$

(2.13)$$\begin{gather}\left[\frac{\partial}{\partial t} + U\frac{\partial}{\partial x} + W\frac{\partial}{\partial z} - \frac{1} {Re}\nabla^2\right]\eta'-\frac{\mathrm{d}W}{\mathrm{d} y}\frac{\partial v'}{\partial x}+\frac{\mathrm{d}U}{\mathrm{d} y}\frac{\partial v'}{\partial z}= 0, \end{gather}$$

where $\nabla ^2$ is the usual Laplacian in a Cartesian coordinate system and $\nabla ^4\langle \cdot \rangle \equiv \nabla ^2(\nabla ^2\langle \cdot \rangle )$ is the bi-harmonic operator. Here on, 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, (2.13), 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,

(2.14)\begin{equation} \bar{\boldsymbol{q}}(y,t; \alpha, \beta) = \iint_{-\infty}^\infty\boldsymbol{q}(x,y,z,t)\, {\rm e}^{-{\rm i}(\alpha x+\beta z)}\,\mathrm{d}{\kern0.8pt}x\,\mathrm{d}z, \end{equation}

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 $\bar {\boldsymbol {q}} = (\bar {v} \ \bar {\eta })$ is a block vector of Fourier coefficients. The transformed equations can be compactly written as

(2.15)\begin{equation} {{\boldsymbol{\mathsf{L}}}} \bar{\boldsymbol{q}} ={-}\frac{\partial}{\partial t}{{\boldsymbol{\mathsf{M}}}} \bar{\boldsymbol{q}}, \end{equation}

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

(2.16a,b)\begin{equation} {{\boldsymbol{\mathsf{L}}}} = \begin{pmatrix} \mathcal{L}_{OS} & 0 \\ {\rm i} \beta \mathcal{D}U - {\rm i} \alpha\mathcal{D}W & \mathcal{L}_{SQ} \end{pmatrix},\quad {{\boldsymbol{\mathsf{M}}}} = \begin{pmatrix} \mathcal{D}^2 - k^2 & 0 \\ 0 & 1 \end{pmatrix}. \end{equation}

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

(2.17)$$\begin{gather} \mathcal{L}_{OS} = ({\rm i} \alpha U+{\rm i} \beta W)(\mathcal{D}^2-k^2) - {\rm i} \alpha \mathcal{D}^2 U-{\rm i} \beta \mathcal{D}^2 W-\frac{1}{Re}(\mathcal{D}^2-k^2)^2, \end{gather}$$

(2.18)$$\begin{gather}\mathcal{L}_{SQ} = {\rm i} \alpha U +{\rm i} \beta W -\frac{1}{ Re}(\mathcal{D}^2-k^2). \end{gather}$$

Equation (2.15) forms an initial-value problem for the Fourier-transformed state vector $\bar {\boldsymbol {q}}$ 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

(2.19)\begin{equation} \begin{pmatrix} \bar{u} \\ \bar{v} \\ \bar{w} \end{pmatrix} = \frac{1}{k^2}\begin{pmatrix} {\rm i} \alpha\mathcal{D} & -{\rm i} \beta\\ k^2 & 0\\ {\rm i} \beta\mathcal{D} & {\rm i} \alpha \end{pmatrix}\begin{pmatrix} \bar{v} \\ \bar{\eta} \end{pmatrix}. \end{equation}

Details of the numerical discretization of (2.15) can be found in Appendix A.

2.3. Modal analysis

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

(2.20)\begin{equation} \hat{\boldsymbol{q}}(y; \alpha, \beta, \omega) = \int_{-\infty}^\infty \bar{\boldsymbol{q}}(y, t;\alpha, \beta)\, {\rm e}^{{\rm i} \omega t}\,\mathrm{d}t, \end{equation}

where $\omega = \omega _r + {\rm i} \omega _i \in \mathbb {C}$ is the complex wave frequency. Equation (2.15) then reduces to a generalized eigenvalue problem described by the linear operator pencil $({{\boldsymbol{\mathsf{L}}}},{{\boldsymbol{\mathsf{M}}}})$:

(2.21)\begin{equation} {{\boldsymbol{\mathsf{L}}}}\hat{\boldsymbol{q}} = {\rm i} \omega{{\boldsymbol{\mathsf{M}}}}\hat{\boldsymbol{q}}, \end{equation}

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

(2.22)\begin{equation} \omega_i(\alpha, \beta, Re, \xi, \theta) = 0. \end{equation}

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 e.g. Mack Reference Mack1984; Schmid & Henningson Reference Schmid and Henningson2001), 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 $(\xi, \theta )$, since a full stability portrait requires a sweep through the $(\alpha, \beta, Re)$-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 § 3, we demonstrate that from the perspective of modal stability, OCPfs 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 OS operator, (2.17). Furthermore, since neither component of the base velocity is inflectional, OCPfs do not admit an inviscid cross-flow-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:

(2.23)\begin{equation} \mathcal{D}^2U + \gamma\mathcal{D}^2W = 0, \end{equation}

where $\gamma = \beta /\alpha$. Although (2.23) will, for general flows, vary in wavenumber space, the linearity of $W$ implies that $\mathcal {D}^2W = 0$ for OCPfs. Thus, since $\mathcal {D}^2U = -2$, the instability must be viscous in nature.

2.4. Non-modal analysis

For most shear flows, a spectral analysis of the linearized Jacobian as in § 2.3 rarely agrees with experiment. In fact, quite often, the transition to turbulence is observed at subcritical $Re$, that is, below the threshold predicted by modal theory (Trefethen et al. Reference Trefethen, Trefethen, Reddy and Driscoll1993). This behaviour is now well understood to be a consequence of the highly non-normal nature of the OSS operator ${{\boldsymbol{\mathsf{S}}}}^\prime$, which, in part, arises from the off-diagonal term $({\rm i} \beta \mathcal {D} U - {\rm i} \alpha \mathcal {D}W)$ driving the Squire equation; see (2.13) and (2.16a,b). In general OCPfs, this forcing can evidently comprise both the streamwise and spanwise mean shear.

A non-normal operator such as ${{\boldsymbol{\mathsf{S}}}}^\prime$ 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 Reference Schmid2007). 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 Reference Butler and Farrell1992). 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 nonlinear mechanisms in the flow and violate the linear assumption (Waleffe Reference Waleffe1995b; Trefethen Reference Trefethen1997)

To explore the implications of non-normality in OCPfs, we first solve the initial-value problem in (2.15) exactly to yield

(2.24)\begin{equation} \bar{\boldsymbol{q}}(t) = \varPhi(t, 0)\bar{\boldsymbol{q}}_0, \end{equation}

where $\varPhi (t,0)\equiv {\rm e}^{{\rm i} {{\boldsymbol{\mathsf{S}}}} t}$ is the state-transition operator, ${{\boldsymbol{\mathsf{S}}}} = i {{\boldsymbol{\mathsf{S}}}}^\prime$, and $\bar {\boldsymbol {q}}_0$ 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

(2.25)\begin{equation} G(\alpha, \beta, Re, \xi, \theta, t) = \sup_{\bar{\boldsymbol{q}}_0\neq 0}\frac{\lVert \bar{\boldsymbol{q}}\rVert_{{out}}^2}{\lVert \bar{\boldsymbol{q}}_0\rVert_{in}^2}, \end{equation}

where, due to its physical significance, we let $\lVert \cdot \rVert _{out}=\lVert \cdot \rVert _{in}=\lVert \cdot \rVert _E$ be an energy norm,

(2.26)\begin{equation} \lVert \bar{\boldsymbol{q}}\rVert_E^2 = \int_{{-}1}^1\bar{v}^{{{\dagger}}}\bar{v} + \frac{1}{k^2}\left(\bar{\eta}^{{{\dagger}}}\bar{\eta} + \frac{\partial\bar{v}^{{{\dagger}}}}{\partial y}\frac{\partial\bar{v}}{\partial y}\right)\,\mathrm{d} y\simeq \bar{\boldsymbol{q}}^{\dagger}{{\boldsymbol{\mathsf{E}}}}\bar{\boldsymbol{q}}, \end{equation}

over the volume $V$ defined by the Cartesian product $(x,y,z)\in [0,2{\rm \pi} /\alpha ]\times [-1,1]\times [0,2{\rm \pi} /\beta ]$. In this way, the energy of one full wavelength of a disturbance can be captured (see Butler & Farrell Reference Butler and Farrell1992). Here, $\langle \cdot \rangle ^{\dagger}$ denotes a conjugate transpose operation, and the operator ${{\boldsymbol{\mathsf{E}}}}$ is positive-definite and incorporates the Clenshaw–Curtis quadrature weights (Trefethen Reference Trefethen2000). With a Cholesky decomposition, we may write ${{\boldsymbol{\mathsf{E}}}} = {{\boldsymbol{\mathsf{F}}}}^{\dagger} {{\boldsymbol{\mathsf{F}}}}$ so that

(2.27)\begin{equation} \lVert \bar{\boldsymbol{q}}\rVert_E^2 \simeq \bar{\boldsymbol{q}}^{\dagger}{{\boldsymbol{\mathsf{F}}}}^{\dagger}{{\boldsymbol{\mathsf{F}}}}\bar{\boldsymbol{q}} = \lVert {{\boldsymbol{\mathsf{F}}}} \bar{\boldsymbol{q}}\rVert_2^2. \end{equation}

It immediately follows that

(2.28)\begin{equation} G = \sup_{\bar{\boldsymbol{q}}_0\neq 0}\frac{\lVert{{\boldsymbol{\mathsf{F}}}}\varPhi(t,0) \bar{\boldsymbol{q}}_0\rVert_2^2}{\lVert{{\boldsymbol{\mathsf{F}}}} \bar{\boldsymbol{q}}_0\rVert_2^2} = \sup_{\bar{\boldsymbol{q}}_0\neq 0}\frac{\lVert{{\boldsymbol{\mathsf{F}}}}\varPhi(t,0) {{\boldsymbol{\mathsf{F}}}}^{{-}1}{{\boldsymbol{\mathsf{F}}}} \bar{\boldsymbol{q}}_0\rVert_2^2}{\lVert{{\boldsymbol{\mathsf{F}}}} \bar{\boldsymbol{q}}_0\rVert_2^2} =\lVert{{\boldsymbol{\mathsf{F}}}}\varPhi(t,0){{\boldsymbol{\mathsf{F}}}}^{{-}1}\rVert_2^2, \end{equation}

which can be computed trivially via the singular value decomposition (note, in fact, that $G = \lVert \varPhi (t,0)\rVert _E^2$). 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 ${{\boldsymbol{\mathsf{S}}}}$. Consider an exogenous harmonic forcing profile $\boldsymbol {H}(x,y,z,t) = \boldsymbol {h}(x,y,z)\, {\rm e}^{-{\rm i}\zeta t}$ with frequency $\zeta \in \mathbb {C}$ to the linearized system, appropriately transformed into wavenumber space:

(2.29)\begin{equation} \bar{\boldsymbol{h}}(y; \alpha, \beta) = \iint_{-\infty}^\infty\boldsymbol{h}(x,y,z)\, {\rm e}^{-{\rm i}(\alpha x+\beta z)}\,\mathrm{d}{\kern0.8pt}x\,\mathrm{d}z. \end{equation}

The long-time response, assuming asymptotic stability, can easily be verified to be

(2.30)\begin{equation} \bar{\boldsymbol{q}} = {\rm i}{\rm e}^{-{\rm i}\zeta t}(\zeta {{\boldsymbol{\mathsf{I}}}} - {{\boldsymbol{\mathsf{S}}}})^{{-}1}\bar{\boldsymbol{h}}, \end{equation}

where the operator ${{\boldsymbol{\mathsf{R}}}} \equiv (\zeta {{\boldsymbol{\mathsf{I}}}}-{{\boldsymbol{\mathsf{S}}}})^{-1}$ is known as the resolvent. From an input–output perspective, ${{\boldsymbol{\mathsf{R}}}}$ serves as a transfer function between the excitation and its response. The quantity $\mathcal {R} = \lVert {{\boldsymbol{\mathsf{R}}}}\rVert _E$ is, therefore, of particular interest here, since for a non-normal system, it can be large even if the forcing is pseudoresonant, that is, $\zeta \notin \varLambda ({{\boldsymbol{\mathsf{S}}}})$, the spectrum of ${{\boldsymbol{\mathsf{S}}}}$ (Trefethen & Embree Reference Trefethen and Embree2005). Such a paradigm is especially informative for the receptivity of the flow to external disturbances (Brandt Reference Brandt2014), 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. Reference Trefethen, Trefethen, Reddy and Driscoll1993). By further generalizing to the complex plane, one recovers the $\epsilon$-pseudospectra, the set of values defined as

(2.31)\begin{equation} \varLambda_\epsilon({{\boldsymbol{\mathsf{S}}}}) = \{\zeta\in\mathbb{C}\colon\mathcal{R}\geq \epsilon^{{-}1}\}. \end{equation}

For non-normal operators, $\varLambda _\epsilon$ 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. Reference Reddy, Schmid and Henningson1993). For further details, we refer the reader to that paper, the citations within and the text of Trefethen & Embree (Reference Trefethen and Embree2005).

2.5. Energy budget analysis

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

(2.32)\begin{equation} \mathcal{E} = \tfrac{1}{2}\boldsymbol{u}^{\dagger}\boldsymbol{u} = \tfrac{1}{2}u_i^{\dagger} u_i = \tfrac{1}{2}(|u|^2 + |v|^2 + |w|^2). \end{equation}

By multiplying equation (2.7) throughout by $\boldsymbol {u}^{\dagger}$ and integrating over $V$, evolution equations for the total energy are recovered:

(2.33)\begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\int_V\mathcal{E}\,\mathrm{d}V =\int_V \frac{\mathrm{d}\mathcal{E}}{\mathrm{d}t}\,\mathrm{d}V={-}\int_V \frac{1}{2}(u_i^{\dagger} u_j+u_i u_j^{\dagger})\frac{\partial U_i}{\partial x_j}\,\mathrm{d}V -\frac{1}{Re}\int_V\frac{\partial u_i^{\dagger}}{\partial x_j}\frac{\partial u_i}{\partial x_j}\,\mathrm{d}V, \end{equation}

where we have assumed spatial periodicity of the disturbance field in $x$ and $z$. Under the normal mode ansatz, (2.20), the above expression reduces to

(2.34)\begin{equation} 2\omega_i\int_{{-}1}^1\frac{1}{2}\hat{u}_i^{\dagger}\hat{u}_i\,\mathrm{d} y = \int_{{-}1}^1\mathcal{P}\,\mathrm{d} y-\int_{{-}1}^1\varepsilon\,\mathrm{d} y, \end{equation}

where we have defined

(2.35)$$\begin{gather} \mathcal{P} =\underbrace{-\overbrace{\frac{1}{2}(\hat{u}^{\dagger}\hat{v} + \hat{u}\hat{v}^{\dagger})}^{\tau_u}\frac{\partial U}{\partial y}}_{\mathcal{P}_{u}}\underbrace{-\overbrace{\frac{1}{2}(\hat{w}^{\dagger}\hat{v} + \hat{w}\hat{v}^{\dagger})}^{\tau_w}\frac{\partial W}{\partial y}}_{\mathcal{P}_{w}}, \end{gather}$$

(2.36)$$\begin{gather}\varepsilon =\frac{1}{Re}[(\mathcal{D}\hat{u}_i)^{\dagger}\mathcal{D}\hat{u}_i + k^2 \hat{u}_i^{\dagger}\hat{u}_i]. \end{gather}$$

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.

3.1. Characteristics of the eigenspectra

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 $\theta = {\rm \pi}/6$. 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 centre 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 flavours of Couette flow are usually scattered symmetrically within two $A$-branches (e.g. Duck, Erlebacher & Hussaini Reference Duck, Erlebacher and Hussaini1994; Schmid & Henningson Reference Schmid and Henningson2001; Zou et al. Reference Zou, Bi, Zhong, Yuan and Tang2023).

Figure 3.

The locus of the eigenspectrum for $(\alpha,\beta ) = (1, 0.5)$ at $Re = 5700$ and $\theta = {\rm \pi}/6$ for (a–f) $\xi \in \{0, 0.2, 0.4, 0.6, 0.8, 1\}$. The $A$, $P$ and $S$ branches have been appropriately labelled. On each plot, a grey dashed line denotes the stability boundary, $c_i = 0$.

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 OS equation is rather subtle. Since $W$ is linear, $\mathcal {D}^2W = 0$, and the OS operator, simplified from (2.17), becomes

(3.1)\begin{equation} \mathcal{L}_{OS} = \overbrace{({\rm i} \alpha U+{\rm i} \beta W)(\mathcal{D}^2-k^2)}^{\mathcal{O}_1} - \overbrace{{\rm i} \alpha \mathcal{D}^2 U}^{\mathcal{O}_2}-\overbrace{\frac{1}{Re}(\mathcal{D}^2 - k^2)^2}^{\mathcal{O}_3}, \end{equation}

where $U$ and $W$ retain their definitions from (2.9a,b), 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 (3.1). 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

(3.2)\begin{equation} \mathcal{L}_{OS} = {\rm i} \alpha U(\mathcal{D}^2-\alpha^2) - {\rm i} \alpha \mathcal{D}^2 U-\frac{1}{Re}(\mathcal{D}^2 - \alpha^2)^2 \end{equation}

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

(3.3)\begin{equation} \mathcal{L}_{OS}^{ACPf} = \overbrace{{\rm i} \alpha U_{{ACPf}}(\mathcal{D}^2-k^2)}^{\mathcal{A}_1} - \overbrace{{\rm i} \alpha \mathcal{D}^2 U_{{ACPf}}}^{\mathcal{A}_2}-\overbrace{\frac{1}{Re}(\mathcal{D}^2 - k^2)^2}^{\mathcal{A}_3}, \end{equation}

where

(3.4)\begin{equation} U_{ACPf} = 1-y^2 + \frac{\xi_{ACPf}}{2}(1+y). \end{equation}

Comparing the two operators in (3.1) and (3.3) allows us to identify crucial similarities in structure. Specifically, for a constant wave triplet $(\alpha,\beta, Re)$, while $\mathcal {O}_3\equiv \mathcal {A}_3$ is immediate, $\mathcal {O}_2\equiv \mathcal {A}_2$ follows from the fact that $\mathcal {D}^2U=-2=\mathcal {D}^2 U_{{ACPf}}$. Therefore, $\mathcal {L}_{OS}$ and $\mathcal {L}_{OS}^{ACPf}$ 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

(3.5)\begin{equation} {\rm i} \alpha U_{ACPf} = {\rm i} \alpha U + {\rm i} \beta W\implies \xi_{ACPf} = \xi(\cos\theta + \gamma\sin\theta), \end{equation}

where $\gamma = \beta /\alpha$. Therefore, for an arbitrary OCPf, the OS problem at any wavenumber pair can be exactly mapped to one for ACPf via the ‘effective’ wall speed $\xi _{{eff}}$:

(3.6)\begin{equation} \xi_{{eff}} \equiv \xi(\cos\theta + \gamma\sin\theta). \end{equation}

With the corollary

(3.7)\begin{equation} \omega(\alpha,\beta,Re,\xi,\theta\neq0) = \omega(\alpha,\beta,Re,\xi_{{eff}},\theta=0), \end{equation}

we conclude that the stability of any OCPf 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 behaviour can now be made precise. We start by noting that $\xi _{eff}$ is $2{\rm \pi}$-periodic and

(3.8)\begin{equation} {-}\xi k/\alpha \leq \xi_{eff} \leq \xi k/\alpha, \end{equation}

so that it varies strongly even throughout wavenumber space. Mathematically, at a fixed triplet $(\alpha, \beta, Re)$, 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 $\xi _{eff}$ 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 $\xi _{eff}$ directly translates to that of the spectrum, which itself becomes, at a minimum, $2{\rm \pi}$-periodic. Furthermore, we found (not shown here; refer to figure 3) that variations in $\xi _{eff}$ modified the distribution of the eigenmodes in the complex plane in much the same fashion as variations in $\xi$ for fixed $\theta$, e.g. increasing $\xi _{eff}$ increased $c_r$, and vice versa. When taken together, these observations, combined with (3.7) and the interpretation of $\xi _{eff}$, 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.

Figure 4.

At $Re=10\ 000$ and $\xi =0.2$, the variation with $\xi _{eff}$ of the least stable eigenmode for $(\alpha, \beta ) = (1,0.25)$. The only dashed grey line marks the boundary $c_i=0$. Both components change in tandem with $\xi _{eff}$, and when juxtaposed with the information in figure 3, lend weight to $\xi _{eff}$ serving as an effective wall speed. Note that $c_i$ is, in fact, ${\rm \pi}$-periodic, the underlying mechanism being precisely that which allows for symmetric growth rates around $\xi = 0$ for ACPf (see § 3.2).

3.2. Exploring criticality in OCPfs

In this section, we present the findings of a comprehensive investigation into the modal stability of OCPfs. We introduce the critical Reynolds number, denoted $Re_c$, which represents the minimum Reynolds number below which the flow remains linearly stable. At this value, at least one disturbance, characterized by the critical wavenumbers $(\alpha _c,\beta _c)$, must achieve neutral stability. When analysing two-dimensional flows, Squire's theorem (Squire Reference Squire1933) 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 OCPf $(\xi,\theta )$, a thorough exploration of the entire three-dimensional $(\alpha, \beta, Re)$-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 (Reference Potter1966) rationalized this by adopting the coordinate transformation $y\to -y$. Since $\xi _{eff}(\theta + {\rm \pi}) = -\xi _{eff}(\theta )$, a similar argument allows us to restrict our attention to $\theta \in [0, {\rm \pi}]$. However, a second reduction is also possible and can be achieved by noting that, at a constant $\xi$, if $(\alpha _c, \beta _c, Re_c)$ is the critical tuple for $\theta =\theta ^\prime$, then $(\alpha _c, -\beta _c, Re_c)$ is necessarily the critical tuple for $\theta = {\rm \pi}- \theta ^\prime$. This result is immediate from the definition of $\xi _{eff}$ in (3.6), since

(3.9)\begin{equation} \xi_{{eff}}(\alpha, \beta, \xi, \theta^\prime) ={-}\xi_{{eff}}(\alpha, -\beta, \xi, {\rm \pi}-\theta^\prime), \end{equation}

where we have assumed $\alpha > 0$. Thus, it suffices to explore the range $\theta \in [0, {\rm \pi}/2]$. For the Fourier wavenumbers, we focused on small to intermediate values, in particular, $(\alpha, \beta ) \in [-3, 3]\times [-3, 3]$. 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 Reference Potter1966). In total, $O(10^{10})$ different parameter combinations were investigated, and our numerical procedure, including our method for traversing such an unwieldy 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 $\varTheta _1$, 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 OCPfs 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 $\theta \in \varTheta _1$, a short range of stabilization in the $Re_c$ curves is followed by an inflection point between $0.2\lessapprox \xi \lessapprox 0.4$ and then further growth, a trend that is precisely reminiscent of ACPf (Potter Reference Potter1966).

Figure 5.

The critical Reynolds number $Re_c$ against $\xi$ for $\varTheta _1\equiv (0, 20^\circ ]$. Throughout this figure, a dashed line indicates the equivalent plot for ACPf. The insets magnify regions of particular interest that have been discussed in the text. A circle in inset (b) denotes the crossing point $\mathcal {I}$. In this range of shear angles, a typical $Re_c$ curve mimics that for ACPf when $\xi \lessapprox \xi _A$, but appears to have been ‘dragged’ down from infinity when $\xi > \xi _A$, yielding a finite $Re_c$ even beyond this threshold wall speed.

Perhaps the most striking feature is the fact that linear instability seems to persist throughout the entire range of wall speeds considered here. This behaviour was observed even for ‘small’ angles, such as $\theta \in \{1^\circ, 1.5^\circ \}$ (and even down to $\theta \in \{0.5^\circ, 0.75^\circ \}$, 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 Reference Potter1966; Cowley & Smith Reference Cowley and Smith1985). 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 $\xi _{{eff}}$ and the analysis outlined at the end of § 3.1, an additional buffer for an OCPf to return to a region of linear instability for ACPfs. The absence of an equivalent cutoff wall speed for OCPfs seems to manifest itself in terms of the appearance of a limiting regime in the critical parameters. Here, 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 $\varTheta _1$ 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 5b). 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 towards 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 $\theta \in \varTheta _1$. Consistent with the trends observed for ACPf, the critical streamwise wavenumber $\alpha _c$ generally displays an initial monotonic decline towards $\alpha _c = 0$, 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 behaviour 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 $\varTheta _1$ also experience an inflection point. The latter is a key stability feature in ACPf, and, in this range, Potter (Reference Potter1966) 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 $\varTheta _1$ 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 $\varTheta _2$, the second stability regime defined by $20^\circ < \theta \leq 90^\circ$.

Figure 6.

Curves of the critical wavenumbers (a) $\alpha _c$ and (b) $\beta _c$ versus $\xi$ for $\varTheta _1\equiv (0, 20^\circ ]$. As usual, a dashed line represents the equivalent plots for ACPf (note that for the latter, Squire's theorem implies $\beta _c = 0$ for all linearly unstable wall speeds). Asymptotic behaviour similar to the curves for $Re_c$ is observed for all $\theta$. Furthermore, at wall speeds beyond the cutoff value $\xi _A$ for ACPf, the $\alpha _c$ curves appear to once again have been pulled away from $\alpha _c = 0$ as $\theta$ increases.

In particular, figure 7 highlights for $\varTheta _2$ a noticeable shift in the stability characteristics of OCPfs. The $Re_c$ curves lose their inflectional nature as in $\varTheta _1$, 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 $\theta \to 90^\circ \in \varTheta _2$, when the wall motion is perfectly orthogonal to the direction of the pressure gradient. In this limit, the streamwise and spanwise velocities reduce to

(3.10a,b)\begin{equation} U = 1-y^2, \quad W = \frac{\xi}{2}(1+y), \end{equation}

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

(3.11)\begin{equation} (\alpha_c, \beta_c, Re_c)_{\theta=90^\circ} = (1.02, 0, 5773.22), \end{equation}

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 Reference Romanov1973). To rationalize this, we recall that the influence of the spanwise cross-flow on the OS operator is partially modulated by the wavenumbers, particularly through the effective wall speed $\xi _{{eff}}$. 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 (3.11).

Figure 7.

The critical Reynolds numbers and Fourier wavenumbers plotted against $\xi$ for some choices of $\theta \in \varTheta _2\equiv (20^\circ, 90^\circ ]$. The black arrow depicts the direction of increasing $\theta$ in increments of $10^\circ$ from $\theta = 30^\circ$ to $\theta = 90^\circ$ (perfect orthogonality). At the latter angle, the critical triplet is constant in $\xi$ and equal to that obtained from an analysis of the two-dimensional OS equation for Poiseuille flow.

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

3.3. The limiting regime of modal stability

In the previous section, it was observed that when the wall speed is sufficiently high, the stability of OCPfs 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 $\varTheta _2$, 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.

Figure 8.

The variation with $\theta$ of $\xi _f$, the wall speed at which the critical parameters asymptote. A dashed line indicates the linear law $\xi _f\sim \theta$, which holds well in $\varTheta _2$.

For purely streamwise velocity profiles, a classic argument due to Squire (Reference Squire1933) 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 ($\beta\,{=}\,0$) OS 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

(3.12a,b)\begin{equation} Re_{c, 3D}^{pPf} > Re_{c, 2D}^{pPf},\quad Re_{c, 3D}^{ACPf} > Re_{c, 2D}^{ACPf}. \end{equation}

However, from the work of Potter (Reference Potter1966), we know that $Re_{c, 2D}^{ACPf} \geq Re_{c, 2D}^{pPf}$, which immediately allows us to conclude that

(3.13)\begin{equation} Re_{c, 3D}^{ACPf} > Re_{c, 2D}^{pPf}, \end{equation}

as well. For OCPfs, due to the mean three-dimensionality, Squire's theorem is no longer valid. However, (3.7) 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

(3.14)\begin{equation} Re_c (\theta\neq 0, \xi\neq 0; \alpha,\beta) = Re_c (\theta = 0, \xi = \xi_{eff}; \alpha,\beta)\geq Re_{c, 2D}^{pPf}. \end{equation}

Here, the merits of the effective wall speed $\xi _{eff}$ 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 (3.14) approaching equality, the OS operator should degenerate precisely into the two-dimensional analogue for pPf. This can happen if and only if

(3.15a–c)\begin{equation} \xi_{eff} = \xi(\cos\theta+\gamma \sin\theta) = 0, \quad \alpha^2+ \beta^2 = \alpha_{c,{pPf}}^2, \quad \alpha Re = \alpha_{c,{pPf}}Re_{c,{pPf}}, \end{equation}

where $\gamma = \beta /\alpha$ and $(\alpha _{c,{pPf}}, Re_{c,{pPf}})\approx (1.02, 5773.22)$ (see Orszag Reference Orszag1971). Assuming $\xi \neq 0$, the first of these three equations yields

(3.16)\begin{equation} \gamma ={-}\cot\theta, \end{equation}

a result that can be combined with the remaining constraints in (3.15a–c) to obtain the following closed solutions:

(3.17a–c)\begin{equation} \alpha = \alpha_{c,{pPf}} |\sin \theta|, \quad \beta ={-}\alpha_{c,{pPf}} \,\mathrm{sgn}(\sin \theta)\cos \theta ,\quad Re = Re_{c,{pPf}}|\csc\theta|, \end{equation}

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 $\langle \cdot \rangle _f$, obtained from our numerical results overlaid with the analytical solution in (3.17a–c). An almost exact match is observed up to the resolution error of the domain sweep.

Figure 9.

The asymptotic values of ($a$,$b$) the critical streamwise and spanwise wavenumbers and (c) the critical Reynolds number versus $\theta$. The solid line denotes the theoretical estimate provided in (3.17a–c).

Some crucial remarks can now be made. As the shear angle approaches zero, (3.17a–c) 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 behaviour, so (3.17a–c) 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 $\theta \to {\rm \pi}/2$, an asymptotic spanwise wavenumber $\beta _f=0$ is predicted, which validates our findings in § 3.2. Finally, an additional interesting consequence arises when considering the wavenumber vector $\boldsymbol {k} = (\alpha \ \beta )^\intercal$. 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 (3.16), we can then conclude that the asymptotic eigenmode propagates at an angle $\psi =\theta - {\rm \pi}/2$ to the pressure gradient, that is, exactly perpendicular to the wall motion.

3.4. Eigenmodes and linear energetics

Figure 10 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. For $\theta =10^\circ \in \varTheta _1$, 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 Tollmien–Schlichting 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 $\theta = 45^\circ \in \varTheta _2$ 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 (3.16) and is oblique. Consistent with the argument presented in § 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.

Figure 10.

Iso-surfaces of the (a,c,e,g) streamwise $u$ and (b,d, f,h) spanwise $w$ velocity fluctuations for the most unstable eigenmode at $\xi =0.35$ for different $\theta$. For each case, the blue and red contours represent 25 % of the (signed) minimum and maximum values of the perturbations, respectively.

The most interesting behaviour is observed for $\theta = 90^\circ$, when the pressure gradient and the wall velocity vectors are exactly orthogonal. First, we recall from (3.11) that despite the value of $\xi$ for this configuration, $\beta _c = 0$, and the OS operator can always be identified with that for the pPf. Thus, the wall-normal and streamwise components (see (2.19)) of the disturbance are identically invariant with the wall speed, precisely reducing to the Tollmien–Schlichting 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 cross-flow shear in the off-diagonal forcing term, (2.16a,b). 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.

Figure 11 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.

Figure 11.

At criticality, the spatial distribution of the perturbation energy budget terms for $\theta =30^\circ$ and wall speeds between $\xi =0.1$ and $\xi =0.8$ in increments of 0.1. (a) The streamwise $\mathcal {P}_u$ and (b) the spanwise $\mathcal {P}_w$ production, (c) the viscous dissipation $\varepsilon$, (d) the total production ($\mathcal {P}=\mathcal {P}_u+\mathcal {P}_w$) and the Reynolds stresses (e) $\tau _u$ and ( f) $\tau _w$.

The spatial variation of the Reynolds stresses is also shown in figure 11(e, f). Regarding $\tau _u$, we see that while it is primarily negative near the stationary wall, it is always positive near the moving wall. This is in stark contrast to ACPf and many of its variants (e.g. Sadeghi & Higgins Reference Sadeghi and Higgins1991; Nouar & Frigaard Reference Nouar and Frigaard2009; Guha & Frigaard Reference Guha and Frigaard2010), for which an increasing wall speed also generates a region of negative stress in the upper half of the channel. Since the base streamwise shear in ACPfs is (typically) also negative throughout this region, the latter phenomenon decreases the overall energy production, stabilizing the flow (see Appendix A). A related commentary can be made on the changes and eventual disappearance of the critical layers at each wall, which, for strictly streamwise flows, are the wall-normal locations where the streamwise velocity matches the real part $c_r$ of the $x$-phase speed. In ACPf, the critical layer near the moving wall is known to vanish as $\xi$ increases, which is often identified with stabilization. For OCPfs, due to the structure of the OS equation, an analogous argument can be constructed using the effective velocity profile:

(3.18)\begin{equation} U_{eff} = U + \gamma W = 1-y^2 + \frac{\xi_{eff}}{2}(1+y), \end{equation}

which essentially represents a projection of the base velocity in the direction of the wavenumber vector (Schmid & Henningson Reference Schmid and Henningson2001). Here, we are interested in the wall-normal location(s) $y_c$ such that $U_{eff}(y_c) = c_r$ (note that since we are focusing on criticality, $c = c_r$). In general, since the effective velocity is quadratic in $y$, two such points can exist, associated with each wall, and are explicitly given in closed form as follows:

(3.19)\begin{equation} y_c = \tfrac{1}{4}\left(\xi_{eff} \pm \sqrt{(4+\xi_{eff})^2 - 16c_r}\right). \end{equation}

The distance of each layer from the wall can then be expressed as $\delta _c = 1 - |y_c|$ (see e.g. Nouar & Frigaard Reference Nouar and Frigaard2009), and is shown in figure 12 for criticality at $\theta =10^\circ \in \varTheta _1$ and $\theta =60^\circ \in \varTheta _2$. For the former case, similar to ACPf (see Appendix A), both layers initially approach the channel boundaries. Eventually, past the point of inflection in the associated $Re_c$ curve, the critical layer near the moving wall vanishes (i.e. $\delta _c\to 0$), whereas the one near the lower wall begins to move further towards the centreline, stabilizing the flow. Mathematically, this can be attributed to changes in the asymmetry of $U_{eff}$, which, in turn, are influenced by variations in the effective wall speed $\xi _{eff}$. However, a particularly intriguing behaviour is observed when $\xi \geq \xi _f$. In this case, $\delta _c$ for the moving wall experiences a sudden increase from zero to a roughly constant value approximately equal to that of the lower wall. This can be explained by the constraints on the asymptotic wavenumber pair, (3.15a–c), which enforce $\xi _{eff}\approx 0$ in the asymptotic regime and reduce $U_{eff}$ to a symmetric parabolic profile. As a result, the absolute values of the roots in (3.19) coalesce and remain $|y_c| \leq 1$, which appears to be consistent with the plateauing observed in the critical parameters (figures 5 and 6). In a similar vein, for $\theta =60^\circ$, because the asymptotic regime is accessed earlier (that is, at smaller $\xi$), the effective velocity profile is almost always perfectly symmetric, so that the distance $\delta _c$ remains roughly identical and non-zero for both walls throughout most of the $\xi$ range explored here. On a separate note, for $\theta =60^\circ$, figure 12(b) highlights an initial increase in $\delta _c$ near the stationary wall, which seems to support the monotonicity of the corresponding $Re_c$ curve in this range (figure 7a). Furthermore, considering the behaviour of the critical layers when the wall speed approaches the point of inflection in the $Re_c$ curves for ACPf and OCPfs in $\varTheta _1$, it is likely that the absence of such a feature for $\varTheta _2$ is related to the fact that both critical layers remain intact at these angles.

Figure 12.

For representative $\theta$, the development of $\delta _c$ for each critical layer. A solid versus dashed line is used to distinguish the lower, stationary, wall from the upper, moving, wall. In each case, an inset illustrates the effective mean velocity profiles $U_{eff}$ for values of the wall speed between $\xi = 0.2$ and $\xi =1$ in increments of 0.1. (a) For $\theta =10^\circ$, an arrow depicts the direction of increasing $\xi$ (note that $\xi \to \xi _f$ implies $\xi _{eff} \to 0$). (b) For $\theta =60^\circ$, while not immediately apparent, the effective velocity profiles for the wall speeds chosen here coincide almost exactly.