2. The interaction equations

Consider a shear flow $(u^*(y^*), 0, 0)$ with respect to Cartesian coordinates $(x^*,y^*,z^*)$, here, a $*$ denotes a dimensional variable. We suppose in the first instance that the depth of fluid is large compared with the wavelength $b$ of forcing from the boundary. We define the friction velocity $u_s=\sqrt {\nu {u^*}' (0) }$ where $\nu$ is the kinematic viscosity and use $\hat {b}={b}/{2{\rm \pi} }$ as length scale, here the factor $2{\rm \pi}$ has been introduced for convenience to simplify the later analysis. If we take a typical velocity to be $\hat {b}{u^*}' (0)$, scale time on ${1}/{{u^*}'(0)}$, and define the friction Reynolds number $R_f={u_s {b}}/{\nu }$ then the dimensionless equations of motion take the form

(2.1)\begin{gather} \frac{\textrm{D} {\boldsymbol{\hat{u}}}}{\textrm{D} t} ={-} \boldsymbol{\nabla}{\hat{p}}+\frac{1}{R}\nabla^{2}{\boldsymbol{\hat{u}}}, \end{gather}

(2.2)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\hat{u}} =0, \end{gather}

where $p$ is the pressure and the Reynolds number $R={R_f^2}/{4{\rm \pi} ^2}$ is assumed large. Because we assume the forcing wavelength is small compared with the length scale of the unperturbed flow in the wall-normal direction, and $R\gg 1$, the basic state corresponds to $\hat {u}=y$ for $0< y< \infty$. Thus in the small wavelength limit the flow stimulated by the forcing depends only on the wall shear of the unperturbed flow, and it follows that (2.1) and (2.2) apply to an arbitrary unidirectional shear flow.

If the wall forcing wavelength is comparable to the depth $d$ of the unperturbed basic flow $u^*=\bar {U}\bar {u}({y^*}/{d})$, then (2.1) and (2.2) still apply with $R={\bar {U}d}/{\nu }$ and the unperturbed flow then corresponds to ${\boldsymbol {\hat {u}}}=(\bar {u},0,0)$. We will refer to the two cases as the unbounded and bounded problems respectively but for definiteness we will take $\bar {u}=y$ so that the bounded flow is Couette flow and $d$ is the channel depth. The analysis is readily modified when $\bar {u}\ne y$ and we shall indicate later the required changes. The upshot then is that we are interested in finding solutions of (2.1) and (2.2) when the unperturbed flow is a uniform shear flow extending to $y=1$ or $y=\infty$.

Suppose that the wall is defined by

(2.3)\begin{equation} y=2\epsilon \cos\left(\alpha x+\beta z -\frac{\omega t}{R^{1/3}}\right), \end{equation}

so that a wave propagates at an angle $\varTheta =\tan ^{-1}({\alpha }/{\beta })$ to the direction of the basic flow at a speed close to that of fluid in the viscous sublayer of depth $R^{-{({1}/{3})}}$ where the flow adjusts to the perturbed wall position. For future reference we note that if $h$ is the dimensional amplitude of the wavy wall then for the unbounded problem $\epsilon ={{\rm \pi} h}/{b}$ and for the bounded problem, here Couette flow, $\epsilon ={h}/{2d}$. Other forms of the wave can be considered and HO considered ’egg-box’ roughness in pipe flow where the undulation is streamwise periodic and modulated in the azimuthal direction. Likewise, if the wall shape contains several harmonics, the theory can be trivially modified in the way indicated in HO. We further assume that $\epsilon$, the scaled undulation amplitude, is small compared with the sublayer thickness. For the transpiration problem we suppose the wall remains flat and fluid is injected into the flow such that

(2.4)\begin{equation} \hat{v}=2\delta \cos\left(\alpha x +\beta z -\frac{\omega t}{R^{1/3}}\right),\quad y=0, \end{equation}

and $\delta$ is also assumed small compared with the sublayer thickness. We will refer to the cases with $\beta =0, \beta \ne 0$ as the aligned and oblique configurations, respectively.

If we take $\delta =\epsilon =0$ we recover the ‘square-vortex’ VWI problem introduced by HS. The key point of the square-vortex VWI is that the wave driving the roll part of the roll-streak flow has its critical layer at the wall. Therefore, the roll is sustained near the wall rather than away from the wall as in Hall & Sherwin (Reference Hall and Sherwin2010). The self-sustaining structures described in the latter paper can co-exist with the square-vortex VWI state but interact only weakly with the wall undulations or transpiration. The appropriate scalings are given in HS so we give only the essential details here. For $y=O(1)$ we assume the uniform shear flow is deformed into the roll-streak flow

(2.5a,b)\begin{equation} {\boldsymbol{\hat{u}}}=\left(u(y,z), \frac{v(y,z)}{R}, \frac{w(y,z)}{R}\right)+\cdots,\quad \hat{p}=\frac{p(y,z)}{R^2}+ \cdots, \end{equation}

where the flow is assumed to be periodic in $z$ with wavelength ${2 {\rm \pi} }/{k}$ and the terms represented by $\ldots$ represent higher order corrections to the roll-streak flow and a small wave of size yet to be determined. Note we have made the usual assumption that the roll flow is smaller by a factor of $R$ than the streak. For the unbounded problem we require that ${\boldsymbol {\hat {u}}}\rightarrow (y,0,0), y\rightarrow \infty$ so that the uniform shear flow is re-established. For the bounded problem ${\boldsymbol {\hat {u}}}$ must satisfy the no-slip condition at the upper wall. An important point to note is that the roll-streak flow has spanwise wavenumber $k$ whereas the wall forcing has spanwise wavenumber $\beta$, the theory we give does not require these to be the same. If the wave is sufficiently small then the roll-streak flow satisfies

(2.6)\begin{gather} \frac{\partial v}{\partial y}+\frac{\partial w}{\partial z} =0, \end{gather}

(2.7)\begin{gather}\left[\frac {\partial^2}{\partial y^2}+\frac {\partial^2}{\partial z^2}\right]u=v \frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}, \end{gather}

(2.8)\begin{gather}\left[\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right]v= \frac{\partial p}{\partial y}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}, \end{gather}

(2.9)\begin{gather}\left[\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right]w=\frac{\partial p}{\partial z}+v\frac{\partial w}{\partial y}+w\frac{\partial w} {\partial z}, \end{gather}

so the roll decouples from the streak but the latter is driven by the roll. Next, we suppose that a small wave ${\sim }\exp ({\textrm {i}[\alpha x - \omega R^{-({1}/{3})}t]})$ is imposed on the above flow and at this stage we assume there is no wall forcing. Note that the frequency has been scaled such that the wave moves with the speed of the streak in a viscous sublayer of depth $O(R^{-({1}/{3})})$ at the wall. If the wave in the viscous layer has an $x$ velocity component of size $\varDelta$ then from the continuity and streamwise momentum equations we deduce that the $y,z$ velocity components and pressure there will be of sizes $\varDelta R^{-({1}/{3})}, \varDelta,\varDelta R^{-({1}/{3})}$, respectively. The spanwise velocity component of the wave is driven solely by the wave pressure and so decays like $\zeta ^{-1}$ for large values of the scaled wall layer variable $\zeta =R^{{1}/{3}}y$. Now, consider the spanwise momentum equation and average in $x$ the contribution to $u({\partial w}/{\partial x})+v({\partial w}/{\partial y})+w({\partial w}/{\partial z})$ from the leading-order wave terms. That interaction will drive a spanwise mean flow $w_m(\zeta, z)$ in the wall layer and for large $\zeta$ the forcing decays like $\zeta ^{-2}$ so that

(2.10)\begin{equation} \frac{1}{R^{1/3}} \frac{ \partial^2 w_m }{ \partial \zeta^2 } \sim \frac{\varDelta^2}{\zeta^2},\quad \zeta \gg 1. \end{equation}

Thus for large $\zeta$ we obtain $w_m \sim \varDelta ^2 {R^{1/3}} \log \zeta \sim \varDelta ^2 {R^{1/3}}\log R$ in an overlap region between the sublayer and outer roll-streak flow (2.5a,b). Noting that the spanwise component of the roll in (2.5a,b) is of size $R^{-1}$ for $y=O(1)$ we obtain closure with the interaction in the wall layer driving the roll for $y=O(1)$ if $\varDelta {{\sqrt {\log R}}{R^{{2/3}}}}$ is $O(1)$. Note here that the size differs from that of a VWI of the type considered by Hall & Sherwin (Reference Hall and Sherwin2010) because of the different dependences of $w_m$ on $\zeta$ exiting the sublayer.

The above discussion explains how the wave size is fixed so as to close a VWI interaction involving a square vortex and a travelling wave of wavelength comparable to the vortex spanwise wavelength and moving downstream with speed $O(R^{-({1}/{3})})$. Thus, if the wall is now wavy as in (2.3), the streamwise and normal velocity components of the wave will be driven by the wall if we take

(2.11)\begin{equation} \epsilon =\frac{\varGamma}{R^{2/3}\sqrt{\log R}}, \end{equation}

whilst wall transpiration as given by (2.4) will drive the normal velocity component of the wave if

(2.12)\begin{equation} \delta =\frac{\varLambda}{R\sqrt{\log R}}. \end{equation}

Here, $\varGamma, \varLambda$ are $O(1)$ prescribed constants. A formal solution for the wave in the wall layer is found by expanding the velocity field in the form

(2.13)\begin{gather} \hat{u}=\frac{\zeta \lambda}{R^{1/3}}+\cdots +\varDelta (U(\zeta,z)E+C.C.)+\cdots, \end{gather}

(2.14)\begin{gather}\hat{v}=\frac{v_m(\zeta,z)}{R^{4/3}}+\cdots +\frac{\varDelta}{R^{1/3}} (V(\zeta,z)E+C.C.)+\cdots, \end{gather}

(2.15)\begin{gather}\hat{w}=\frac{w_m(\zeta,z)}{R }+\cdots +\varDelta (W(\zeta,z)E+C.C.)+\cdots, \end{gather}

(2.16)\begin{gather}\hat{p}=\frac{p_m(\zeta,z)}{R^2 }+{+}\cdots +\frac{\varDelta}{R^{1/3}}(P(z)E+C.C.)+\cdots, \end{gather}

where $\lambda (z)=u_y(0,z)$ is the wall shear of the streak, $E=\exp ({\textrm {i}(\alpha x -R^{-({1}/{3})}\omega t)})$, C.C. denotes complex conjugate and we have anticipated that the wave pressure is independent of the scaled wall layer variable $\zeta$. Substituting the above expansions into the equations of motion we find the $x,z$ momentum equations and continuity give

(2.17)\begin{gather} \textrm{i} \alpha U +V_\zeta+W_z=0, \end{gather}

(2.18)\begin{gather}-\textrm{i}\omega U +\textrm{i} \alpha \lambda \zeta U +\lambda V +\zeta W \lambda'={-}\textrm{i}\alpha P +U_{\zeta \zeta}, \end{gather}

(2.19)\begin{gather}-\textrm{i}\omega W +\textrm{i} \alpha \lambda \zeta W ={-}P_z +W_{\zeta \zeta}. \end{gather}

The normal momentum equation confirms the wave pressure is constant in the wall layer. The equations are to be solved subject to

(2.20)\begin{equation} U, V_\zeta, W \rightarrow 0,\quad \zeta \rightarrow \infty. \end{equation}

The forcing at the wall need not be specified in detail, so at this stage we simply assume it leads to inhomogeneous conditions on the wave at $\zeta =0$, thus we require

(2.21a–d)\begin{equation} U=U_0(z),\quad V=V_0(z),\quad W=0,\quad \zeta=0, \end{equation}

where for the wavy wall case

(2.22a,b)\begin{equation} U_0={-}\lambda \varGamma \,\textrm{e}^{\textrm{i}\beta z},\quad V_0={-}\textrm{i} \omega \varGamma \,\textrm{e}^{\textrm{i}\beta z}, \end{equation}

and for the transpiration case

(2.23a,b)\begin{equation} U_0=0,\quad V_0=\varLambda \,\textrm{e}^{\textrm{i}\beta z}. \end{equation}

The system (2.17)–(2.21a–d) is solved in the usual way by first adding (2.17) multiplied by $\textrm {i}\alpha$ to ${\partial }/{\partial z}$ of (2.19) and then eliminating $U,W$ using continuity. If that equation is then evaluated at the wall we obtain an equation linking the pressure to $V$ at the wall. If the equation is differentiated with respect to $\zeta$ we obtain Airy's equation for $V_{\zeta \zeta }$. The required solution of the system is

(2.24)\begin{gather} V=V_0+\hat{A} \int_{\xi_0}^\xi \,\textrm{d}\theta \int_\infty^\theta Ai(\eta)\,\textrm{d}\eta -\frac{\lambda_z P_z}{2 \lambda \sigma} [L''(\xi)-L''(\xi_0)-2\xi_0L(\xi )], \end{gather}

(2.25)\begin{gather}W=\sigma^{-({2}/{3})} P_z{\mathcal{L}}(\xi), \end{gather}

where $Ai$ is the Airy function,

(2.26a–e)\begin{equation} \left.\begin{gathered} \sigma=\textrm{i} \alpha \lambda,\quad \xi=\sigma^{1/3}\zeta+\xi_0,\quad \xi_0={-}\frac{\textrm{i}^{1/3}\omega}{[\alpha\lambda]^{2/3}},\\ \hat{A}\kappa =\frac{\textrm{i} \alpha U_0}{\sigma^{1/3}}- \frac{\lambda'P_z\xi_0 \kappa}{2\lambda \sigma Ai(\xi_0)},\quad \kappa=\int_{\xi_0}^\infty Ai(\eta)\,\textrm{d}\eta, \end{gathered}\right\} \end{equation}

and ${\mathcal {L}}$ satisfies

(2.27a–c)\begin{equation} {\mathcal{L}}''-\xi{\mathcal{L}}=1,\quad {\mathcal{L}}=0,\quad \xi=\xi_0,\infty. \end{equation}

The pressure is then given by

(2.28)\begin{equation} \frac{\textrm{d}^2 P}{\textrm{d} z^2} -{\mathcal{F}}\frac{\lambda'}{\lambda} \frac{\textrm{d} P}{\textrm{d} z}-\alpha^2 P={-} \alpha \omega U_0- \frac{\textrm{i} \alpha \sigma^{2/3}Ai'(\xi_0) }{\kappa } U_0-\sigma V_0, \end{equation}

where the function ${\mathcal {F}}$ is defined by

(2.29)\begin{equation} {\mathcal{F}}=\frac{1}{2}\left(3+\frac{\xi_0 Ai'(\xi_0)}{Ai(\xi_0)}\left[1+\frac{\kappa \xi_0}{Ai'(\xi_0)}\right]\right). \end{equation}

In the absence of forcing from the wall the right-hand side of (2.28) is zero and the equation reduces to that given in HS. It remains for us to consider the spanwise momentum equation averaged over one streamwise wavelength. As mentioned earlier the forcing of the latter equation varies like $\xi ^{-2}$ far from the wall, so that a spanwise mean flow varying like $\log \xi$ is induced there. Substituting the far field form of the wave into the $z$ momentum equation we find that, exactly as in HS, the spanwise mean then drives the spanwise roll velocity component in the $y=O(1)$ region through the boundary condition

(2.30)\begin{equation} w={-}\frac{1}{3}(\frac{1}{\lambda^2}\frac{\partial}{\partial z} [\vert P\vert^2+\alpha^{{-}2}\vert P_z \vert^2]),\quad y=0. \end{equation}

The term on the right-hand side above is, apart from the factor $\frac {1}{3}$, arising from the slightly different scaling used here, the same as in HS so the forcing does not modify the matching condition. It is useful to now write down the fully nonlinear interactive problem associated with wall forcing

(2.31)\begin{gather} \frac{\partial v}{\partial y}+\frac{\partial w}{\partial z} =0, \end{gather}

(2.32)\begin{gather}\left[\frac {\partial^2}{\partial y^2}+\frac {\partial^2}{\partial z^2}\right]u=v \frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}, \end{gather}

(2.33)\begin{gather}\left[\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right]v= \frac{\partial p}{\partial y}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}, \end{gather}

(2.34)\begin{gather}\left[\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right]w= \frac{\partial p}{\partial z}+v\frac{\partial w}{\partial y}+w\frac{\partial w} {\partial z}, \end{gather}

subject to

(2.35)\begin{gather} {\boldsymbol{u}} =(0,0, -\frac{1}{3}(\frac{1}{\lambda^2} \frac{\partial}{\partial z}[ \vert P\vert^2+\alpha^{{-}2}\vert P_z \vert^2]), \quad y=0, \end{gather}

(2.36)\begin{gather}{\boldsymbol{u}} \rightarrow (y,0,0),\quad y \rightarrow \infty \quad \textrm{or} \quad {\boldsymbol{u}}=(1,0,0),\quad y=1, \end{gather}

(2.37)\begin{gather}\frac{\textrm{d}^2 P}{\textrm{d} z^2} -{\mathcal{F}}\frac{\lambda'}{\lambda}\frac{\textrm{d} P}{\textrm{d} z} -\alpha^2 P={-} \alpha \omega U_0-\frac{\textrm{i} \alpha \sigma^{2/3}Ai'(\xi_0) }{\kappa } U_0-\sigma V_0. \end{gather}

Note that in (2.36) the two conditions refer to the unbounded and bounded problems respectively and that the coupling of the pressure with the streak $u$ is through the wall shear $\lambda (z)$. The above system corresponds to a wave-like disturbance with $x,y$ velocity components proportional to $U_0E+C.C., V_0E+C.C.$ imposed at $y=0$. For the wavy wall and transpiration cases the pressure equation respectively simplifies to

(2.38)\begin{gather} \frac{\textrm{d}^2 P}{\textrm{d} z^2} -{\mathcal{F}}\frac{\lambda'}{\lambda}\frac{\textrm{d} P}{\textrm{d} z}-\alpha^2 P= \frac{\varGamma \sigma^{5/3}Ai'(\xi_0) }{\kappa }\textrm{e}^{\textrm{i}\beta z}, \end{gather}

(2.39)\begin{gather}\frac{\textrm{d}^2 P}{\textrm{d} z^2} -{\mathcal{F}}\frac{\lambda'}{\lambda}\frac{\textrm{d} P}{\textrm{d} z}-\alpha^2 P={-}\sigma \varLambda \,\textrm{e}^{\textrm{i}\beta z}. \end{gather}

Other forms of the wall forcing can be considered, for example for an ‘egg-box’ wavy wall as in HO, we replace $\textrm {e}^{\textrm {i} \beta z}$ in (2.38) by $\cos \beta z.$ The basic state where no streamwise vortex is present corresponds to $P=A_0 \,\textrm {e}^{\textrm {i} \beta z}$, where $A_0$ is a constant, and the right-hand side of (2.35) then vanishes. But the system (2.31)–(2.37) also describes fully nonlinear solutions supporting a streamwise vortex and, for given complex values of $U_0, V_0,$ it constitutes a nonlinear eigenvalue problem for say the drag at the wall as a function of $\alpha, \beta, k,$ and $\omega$. We stress that, for the unbounded problem, the interactive system (2.31)–(2.37) is appropriate to any shear flow, the details of any particular unbounded flow only entering the problem through the friction velocity.

3. Linear instability using the interaction equations

Now we use the interaction equations (2.31)–(2.37) to derive the linear instability mechanism first described in H1. The present approach is simpler, applies to different kinds of wall forcing and is readily extended into the weakly nonlinear regime. We analyse the linear instability problem for wall undulations and then quote the results for the transpiration problem. Suppose that the strongly nonlinear state described above is close to the unperturbed linear shear flow. In that case for both the bounded and unbounded problems we write

(3.1)\begin{equation} {\boldsymbol{u}}=(y,0,0)+A(u_1(y) \cos kz, v_1(y) \cos kz, w_1(y) \sin kz)+\cdots, \end{equation}

where the vortex amplitude $A\ll 1$. The wall shear $\lambda (z)$ felt by the wave in the wall layer takes the form

(3.2)\begin{equation} \lambda=1+A u_1'(0) \cos kz +\cdots. \end{equation}

If we substitute from above into the roll-streak equations, equate terms $O(A)$, and then eliminate $w_1$ and the corresponding disturbance pressure $p_1$, we obtain

(3.3a,b)\begin{equation} \left[\frac{\textrm{d}^2}{{\textrm{d}y}^2}-k^2\right]u_1=v_1,\quad \left[\frac{\textrm{d}^2}{{\textrm{d}y}^2}-k^2\right]^2v_1=0. \end{equation}

For the unbounded problem these equations must be solved subject to ${\boldsymbol {u_1}}\rightarrow 0,\ y \rightarrow \infty$ whilst for the bounded case ${\boldsymbol {u_1}}=0,\ y=1$. At the wall $y=0$ we require $u_1=v_1=0$ but $w_1$ is forced by the linearised form of (2.35). Hence, we solve the above equations for $u_1, v_1$ subject to the appropriate upper boundary condition and $u_1(0)=v_1(0)=0.$ We normalise the solution with $w_1(0)=-1$, thus for the unbounded problem we obtain

(3.4a–c)\begin{equation} u_1={-}\left[\frac{k^2 y^2+ky}{4k^2}\right]\textrm{e}^{{-}ky},\quad v_1=ky\,\textrm{e}^{{-}ky},\quad w_1(y)=(ky-1)\,\textrm{e}^{{-}ky}. \end{equation}

These functions are shown in figure 1(a), they are similar to Taylor–Görtler instabilities and we note for future reference that

(3.5)\begin{equation} \frac{w_1(0)}{4ku_1'(0)}=1. \end{equation}

For the bounded problem corresponding to Couette flow we need $u_1, v_1,w_1$ to vanish at $y=1$, we find that

(3.6)\begin{equation} \frac{w_1(0)}{4ku_1'(0)}={\mathcal{H}}(k)=\left[\frac{\sinh ^2(2k)-4k^2 }{\sinh^2(2k)-8k^3 \coth(2k)}\right], \end{equation}

and we note that ${\mathcal {H}}(\infty )=1$ so that (3.6) reduces to (3.5) for $k\gg 1.$ The linearised form of (2.35) provides a second relationship between $w_1(0),u_1'(0)$ and the consistency of that equation with either (3.5) or (3.6) determines the eigenrelation for neutrally stable modes. Note that (3.4a–c) and (3.5) are independent of the wall forcing and so apply to both the wavy wall and transpiration problems. For the wavy wall we perturb the wave pressure equation by writing

(3.7)\begin{equation} P=\frac{\varGamma [\textrm{i} \alpha]^{5/3}Ai_1' }{\kappa_1 }\textrm{e}^{\textrm{i}\beta z} (Q_0+A[Q_{c} \cos kz +\textrm{i}Q_{s}\sin kz ]u_1'(0)+\cdots), \end{equation}

where $\xi _1=-({\textrm {i}^{1/3}\omega }/{\alpha ^{2/3}}),\ Ai_1=Ai(\xi _1),$ and for later use we define $Ai_1'=Ai'(\xi _1), \kappa _1=\kappa ( \xi _1).$ We substitute the above expansion into (2.38) with $\lambda$ given by (3.2) and, equating terms $O(A)$, we obtain

(3.8)\begin{equation} Q_0={-}\frac{1}{\alpha^2+\beta^2}. \end{equation}

Next, at $O(A)$ we find that

(3.9)\begin{gather} {}[\gamma^2-4 \beta^2k^2]Q_{c}=\frac{2\beta^2k^2{\mathcal{F}}_1}{\alpha^2+\beta^2}-\frac{\gamma}{3}[5-S_1], \end{gather}

(3.10)\begin{gather}{}[\gamma^2-4 \beta^2k^2]Q_{s}=\frac{2\beta k[5-S_1]}{3}-\frac{\gamma \beta k {\mathcal{F}}_1}{\alpha^2+\beta^2 }, \end{gather}

where $\gamma =\alpha ^2+\beta ^2+k^2$, ${\mathcal {F}}_1$ is ${\mathcal {F}}$ defined by (2.29) evaluated at $\xi _0=\xi _1$ and $S_1=2\xi _1Ai_1[{\xi _1}/{Ai'_1}+{1}/{\kappa _1}].$ It remains to write down the condition on $w$ found by expanding the right-hand side of (2.35) in powers of $A$. Substituting for $P$ from above into that condition we obtain

(3.11)\begin{align} {w (0)} &={-}\frac{Au_1'(0)\varGamma^2k\alpha^{{4}/{3}}\vert Ai_1\vert^2}{3\vert \kappa_1\vert^2 } \left[\left(Q_c +\frac{ \beta k {Q_s}}{\alpha^2+\beta^2}\right)\sin kz \right.\nonumber\\ &\left.\quad -\textrm{i}\left(Q_s+\frac{ \beta k Q_c }{\alpha^2+\beta^2}\right)\cos kz\right]+C.C.+O(A^2). \end{align}

This equation must of course be consistent with (3.1) and (3.5) or (3.6) depending on whether we are considering the bounded or unbounded problems. There is an inconsistency unless the coefficient of $\cos kz$ in (3.11) is zero. In fact, we see from (3.9) and (3.10) that the coefficient does indeed vanish if either $\beta$ or $\omega$ are zero. So let us first assume that either $\beta$ or $\omega$ vanish and write down the equations which determine the neutral curves in the $k-\varGamma$ plane. We will comment at the end of this section on how to deal with the case when neither of $\beta, \omega$ vanishes.

Figure 1.

(a) Shows the linear velocity components of the streamwise vortex away from the wall. (b) Shows the variation of ${\varGamma ^2}/{\varGamma _0^2}$ with the frequency in the limit $k\rightarrow 0$ for the wavy wall in the unbounded case. Note that the most dangerous mode corresponds to $k\rightarrow 0$.

3.1. The neutral curves for wall undulations

Consider the case of undulations propagating in the direction of the unperturbed flow so that $\beta =0.$ From (3.6) and (3.11) we find that the neutral curve for a given $\omega$ is given by

(3.12)\begin{equation} \varGamma^2=\left[\frac{18(\alpha^2+k^2)\vert \kappa_1 \vert ^2 } {\vert Ai_1' \vert^2 \alpha^{{4}/{3}} \left\{5-2\left(\xi_1 Ai_1\left[\frac{\xi_1}{Ai_1'}+\dfrac{1}{\kappa_1}\right]\right)_r\right\}}\right] {\mathcal{H}}_1(k), \end{equation}

where a subscript $r$ denotes the real part and ${\mathcal {H}}_1={\mathcal {H}}(k)$ for the bounded problem and ${\mathcal {H}}_1={\mathcal {H}}(\infty )=1$ for the unbounded case with ${\mathcal {H}}$ given by (3.6). The function ${\mathcal {H}}_1$ is positive for all $k$ and so physically acceptable solutions of (3.12) having $\varGamma ^2>0$ exist only when the quantity inside the curly brackets is positive.

Next suppose the undulations are not propagating and the wave crests are no longer perpendicular to the unperturbed flow. In this case we set $\omega =0$ and the consistency of (3.6) and (3.11) then yields

(3.13)\begin{equation} \varGamma^2=\frac {4(\gamma^2-4\beta^2k^2)(\alpha^2+\beta^2)^2}{Ai'^2(0)\alpha^{4/3} [\gamma(10\{\alpha^2+\beta^2\}^2+9\beta^2k^2 )-38\beta^2k^2\{\alpha^2+\beta^2\}]}{\mathcal{H}}_1 (k). \end{equation}

We postpone a discussion of the above eigenrelations to the following section.

3.2. The neutral curves for wall transpiration

The analysis leading to the two above equations can be repeated for the case of wall transpiration, the only difference is that we now use the pressure equation (2.39) rather than (2.38). Once again we consider the situation when either $\beta$ or $\omega$ vanish. If the wave propagates in the same direction as the unperturbed flow then we take $\beta =0$ and the neutral curve is defined by

(3.14)\begin{equation} \varLambda^2=6(\alpha^2+k^2) {\mathcal{H}}_1(k), \end{equation}

with ${\mathcal {H}}_1$ defined as given earlier. Thus we see that the neutral configuration is independent of $\omega$ and so is unchanged from the stationary case. In the case when the wall transpiration wave is inclined at an angle to the unperturbed flow direction, but not propagating, the neutral curve is defined by

(3.15)\begin{equation} \varLambda^2=\frac {12(\gamma^2-4\beta^2k^2)(\alpha^2+\beta^2)^2}{[\gamma(2\{\alpha^2+\beta^2\}^2+3\beta^2k^2 )-10\beta^2k^2\{\alpha^2+\beta^2\}]}{\mathcal{H}}_1(k). \end{equation}

We postpone a discussion of the implications of the above results for transpiration until § 5. In the case when neither of $\beta, \omega$ vanish (3.1) must be modified to allow the vortex to propagate in the $z$ direction with some speed $c$, and the assumed forms for the vortex must include both sine and cosine terms. The matching condition on $w$ given by the forcing in the wall layer is then consistent with the assumed form of $w$ and, for given $\alpha, \beta, k,$ and $\omega$, the amplitudes $\varGamma$ or $\varLambda$ together with $c$ are determined from the consistency condition. We do not pursue that case further here but note that in that situation the streamwise vortex drifts in the spanwise direction.