2.1 Base flow

We begin with our base flow $(\boldsymbol{U},P)=(U,Re^{-1/2}V,W,P)$ , which is a solution of the boundary-layer equations

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle UU_{x}+VU_{Y}+WU_{z}=-P_{x}+U_{YY}, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle P_{Y}=0, & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle UW_{x}+VW_{Y}+WW_{z}=-P_{z}+W_{YY}, & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle U_{x}+V_{Y}+W_{z}=0, & \displaystyle\end{eqnarray}$$

subject to the boundary conditions

(2.6a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{U}=\mathbf{0}\quad \text{at }Y=0,\quad (U,W)\rightarrow (U_{s},W_{s})\quad \text{as }Y\rightarrow \infty . & \displaystyle\end{eqnarray}$$

Here $Y=Re^{1/2}y$ is the $O(1)$ scaled vertical variable for the boundary layer, while $U_{s}$ and $W_{s}$ are the chordwise and spanwise slip velocities imposed by the inviscid flow external to the boundary layer. Since the pressure $P$ does not vary across the boundary layer, we can express it in terms of the slip velocities as

(2.7a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{x}=-U_{s}U_{s,x}-W_{s}U_{s,z},\quad P_{z}=-U_{s}W_{s,x}-W_{s}W_{s,z}. & \displaystyle\end{eqnarray}$$

Near the surface of the wing, $Y=0$ , each velocity component can be expressed as a Maclaurin series:

(2.8a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle U\sim \mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}_{n}^{x}Y^{n},\quad V\sim \mathop{\sum }_{n=1}^{\infty }-\frac{1}{n+1}(\unicode[STIX]{x1D706}_{n,x}^{x}+\unicode[STIX]{x1D706}_{n,z}^{z})Y^{n+1},\quad W\sim \mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}_{n}^{z}Y^{n}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{n}^{x}$ and $\unicode[STIX]{x1D706}_{n}^{z}$ are the Maclaurin coefficients for the series, given by

(2.9a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{n}^{x}(x,z):=\left.\frac{1}{n!}\frac{\unicode[STIX]{x2202}^{n}U}{\unicode[STIX]{x2202}Y^{n}}\right|_{Y=0},\quad \unicode[STIX]{x1D706}_{n}^{z}(x,z):=\left.\frac{1}{n!}\frac{\unicode[STIX]{x2202}^{n}W}{\unicode[STIX]{x2202}Y^{n}}\right|_{Y=0}. & & \displaystyle\end{eqnarray}$$

Substituting (2.8) into (2.2) and (2.4) and collecting terms at each power of $Y$ gives the following relations:

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle 2\unicode[STIX]{x1D706}_{2}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}=-U_{s}\,{\displaystyle\mathop{ U}\limits_{{\sim}}}_{s,x}-W_{s}{\displaystyle\mathop{ U}\limits_{{\sim}}}_{s,z}, & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle 6\unicode[STIX]{x1D706}_{3}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}=0, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle 24\unicode[STIX]{x1D706}_{4}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}=2\unicode[STIX]{x1D706}_{1}^{x}\unicode[STIX]{x1D706}_{1,x}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}+2\unicode[STIX]{x1D706}_{1}^{z}\unicode[STIX]{x1D706}_{1,z}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}-\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}(\unicode[STIX]{x1D706}_{1,x}^{x}+\unicode[STIX]{x1D706}_{1,z}^{z}). & \displaystyle\end{eqnarray}$$

In the ensuing analysis, there are often only minor differences between the corresponding chordwise and spanwise equations. We have thus introduced the simplifying notation ${\displaystyle\mathop{q}\limits_{{\sim}}}$ in (2.10)–(2.12), which we will make use of throughout this paper. In order to convert from the chordwise equation to its spanwise counterpart, one makes the substitutions

(2.13a-c ) $$\begin{eqnarray}\displaystyle \,{\displaystyle\mathop{x}\limits_{{\sim}}}\mapsto z,\quad \,{\displaystyle\mathop{U}\limits_{{\sim}}}\mapsto W,\quad \,{\displaystyle\mathop{u}\limits_{{\sim}}}\mapsto w. & & \displaystyle\end{eqnarray}$$

The right-hand side of (2.11) is zero because there is no imposed blowing or suction on the surface of the wing. The effects of blowing and suction on lower-branch modes have been considered in Choudhari (Reference Choudhari1994a ), where it was shown that even moderately weak suction could change the asymptotic scaling of the lower branch. Similarly, the non-parallel regime considered here is not present for the rotating-disk flow, due to the effects of the Coriolis force (Hall Reference Hall1986).

2.2 Instability modes

To the base flow $(\boldsymbol{U},P)$ , a perturbation $(\boldsymbol{u}^{\prime },p^{\prime })$ of size $\unicode[STIX]{x1D716}$ is introduced. Since we are considering the early development of the stationary crossflow instability, this perturbation is small, i.e. $\unicode[STIX]{x1D716}\ll 1$ . The perturbed flow can thus be expressed as

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle (\boldsymbol{u},p)=(\boldsymbol{U},P)+\unicode[STIX]{x1D716}(\boldsymbol{u}^{\prime }(x,Y,z),p^{\prime }(x,Y,z))+O(\unicode[STIX]{x1D716}^{2}). & \displaystyle\end{eqnarray}$$

The perturbation is governed by the linearised N–S equations,

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle (\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{\prime }+(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{U}=-\unicode[STIX]{x1D735}p^{\prime }+Re^{-1}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}^{\prime }, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }=0, & \displaystyle\end{eqnarray}$$

subject to the no-slip and impermeability conditions,

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}^{\prime }=\mathbf{0}\quad \text{on }Y=0, & \displaystyle\end{eqnarray}$$

and the attenuation condition in the far field,

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle (\boldsymbol{u}^{\prime },p^{\prime })\rightarrow (\mathbf{0},0)\quad \text{as }Y\rightarrow \infty . & \displaystyle\end{eqnarray}$$

We seek solutions to this linear system in terms of its eigenmodes. Since the length scale over which the perturbation varies across the wing is much shorter than that for the base flow, the eigenmodes of the system take the WKB form

(2.19) $$\begin{eqnarray}\displaystyle & \displaystyle (\boldsymbol{u}^{\prime },p^{\prime })=(\tilde{\boldsymbol{u}},\tilde{p})(Y;x,z)\exp \left[\text{i}\left(\int _{0}^{x}\unicode[STIX]{x1D6FC}\,\text{d}\hat{x}+\unicode[STIX]{x1D6FD}z\right)\right]+\text{c.c.}, & \displaystyle\end{eqnarray}$$

where the chordwise and spanwise wavenumbers, denoted by $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ , are large. In the regimes of interest

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\sim Re^{\unicode[STIX]{x1D6FE}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}$ is positive and takes different values depending on the regime considered. The value of $\unicode[STIX]{x1D6FE}$ is to be determined such that non-parallelism has a leading-order influence on the growth rate of the perturbation. Because the span of the wing is much larger than its chord length, the spanwise wavenumber $\unicode[STIX]{x1D6FD}=Re^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FD}_{0}$ is taken to be real and given, while the chordwise wavenumber $\unicode[STIX]{x1D6FC}$ is treated as a complex eigenvalue.

Hall (Reference Hall1986) showed for the rotating-disk flow that the wavefronts of lower-branch modes must be aligned with the local wall shear at leading order. By considering the limiting form of the dispersion relation for the triple-deck ( $\unicode[STIX]{x1D6FE}=3/8$ ) crossflow regime, Choudhari (Reference Choudhari1995) demonstrated that this also holds for swept-wing flows. The leading-order wavenumbers $Re^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FC}_{0}$ and $Re^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FD}_{0}$ are thus related through

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D706}_{1}^{x}\unicode[STIX]{x1D6FC}_{0}+\unicode[STIX]{x1D706}_{1}^{z}\unicode[STIX]{x1D6FD}_{0}=0, & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D706}_{1}^{x}$ and $\unicode[STIX]{x1D706}_{1}^{z}$ defined by (2.8). As a result of this, the rescaled leading-order chordwise wavenumber $\unicode[STIX]{x1D6FC}_{0}$ is real. In order to obtain the growth rate, it is necessary to take the analysis to a higher order and expand $\unicode[STIX]{x1D6FC}$ as

(2.22) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}=Re^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FC}_{0}+Re^{\unicode[STIX]{x1D6FE}_{np}}\unicode[STIX]{x1D6FC}_{1}+\cdots \,. & \displaystyle\end{eqnarray}$$

We thus wish to determine the correction $Re^{\unicode[STIX]{x1D6FE}_{np}}\unicode[STIX]{x1D6FC}_{1}$ to the chordwise wavenumber, which includes the leading-order contribution of non-parallelism to the growth rate.

The instability in this regime is, as expected, governed by the base-flow velocity projected in the direction of the leading-order wavenumber vector $(\unicode[STIX]{x1D6FC}_{0},\unicode[STIX]{x1D6FD}_{0})$ . This projection and its Maclaurin coefficients are defined as

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle U_{eff}:=U+\frac{\unicode[STIX]{x1D6FD}_{0}}{\unicode[STIX]{x1D6FC}_{0}}W, & \displaystyle\end{eqnarray}$$

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D706}_{n}^{0}:=\unicode[STIX]{x1D706}_{n}^{x}+\frac{\unicode[STIX]{x1D6FD}_{0}}{\unicode[STIX]{x1D6FC}_{0}}\unicode[STIX]{x1D706}_{n}^{z}. & \displaystyle\end{eqnarray}$$

We can thus express the wall-shear alignment condition (2.21) as

(2.25) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D706}_{1}^{0}=0, & \displaystyle\end{eqnarray}$$

and (2.11), which holds in the absence of wall suction, as $\unicode[STIX]{x1D706}_{3}^{0}=0$ . Close to the surface of the wing, the effective base flow thus behaves as

(2.26) $$\begin{eqnarray}\displaystyle & \displaystyle U_{eff}\sim \unicode[STIX]{x1D706}_{2}^{0}Y^{2}+\unicode[STIX]{x1D706}_{4}^{0}Y^{4}+\cdots \quad \text{as }Y\rightarrow 0. & \displaystyle\end{eqnarray}$$

Since $\unicode[STIX]{x1D706}_{1}^{0}=0$ , the flow in the projected direction resembles a boundary-layer flow at the point of marginal separation (Goldstein et al. Reference Goldstein, Leib and Cowley1987). This results in a marginally separated triple-deck structure for the perturbation: an ‘upper deck’ lies between the main boundary layer and the inviscid flow in order to account for the attenuation condition (2.18), while a thin, viscous ‘lower deck’ lies near the surface of the wing in order to satisfy the no-slip condition (2.17).

From (2.12) and (2.21), it follows that for any base flow with no imposed suction,

(2.27) $$\begin{eqnarray}\displaystyle & \displaystyle 12\unicode[STIX]{x1D706}_{4}^{0}=\unicode[STIX]{x1D706}_{1}^{x}\unicode[STIX]{x1D706}_{1,x}^{0}+\unicode[STIX]{x1D706}_{1}^{z}\unicode[STIX]{x1D706}_{1,z}^{0}, & \displaystyle\end{eqnarray}$$

where

(2.28) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{n,{\displaystyle\mathop{x}\limits_{{\sim}}}}^{0}:=\unicode[STIX]{x1D706}_{n,{\displaystyle\mathop{x}\limits_{{\sim}}}}^{x}+\frac{\unicode[STIX]{x1D6FD}_{0}}{\unicode[STIX]{x1D6FC}_{0}}\unicode[STIX]{x1D706}_{n,{\displaystyle\mathop{x}\limits_{{\sim}}}}^{z}. & & \displaystyle\end{eqnarray}$$

2.3 Wavenumber scalings

We now present a scale analysis to fix $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FE}_{np}$ . Choudhari (Reference Choudhari1994a ) derived these by balancing the pressure gradient with the contribution to the inertia from the $O(Y^{4})$ term in $U_{eff}$ (see (2.26)). We will instead focus on the inclusion of non-parallelism in the governing equation for the perturbation.

As with the neighbouring instability regimes, a triple-deck structure arises (Choudhari Reference Choudhari1994a ). The lower-deck thickness is set by the balance of the inertial and viscous terms, $\unicode[STIX]{x1D6FC}U_{eff}u^{\prime }\sim u_{YY}^{\prime }$ . Since $\unicode[STIX]{x1D6FC}\sim Re^{\unicode[STIX]{x1D6FE}}$ , $U_{eff}\sim \unicode[STIX]{x1D706}_{2}^{0}Y^{2}$ by (2.26), and $u_{YY}^{\prime }\sim u^{\prime }\,Y^{-2}$ , it follows for $\unicode[STIX]{x1D706}_{2}^{0}=O(1)$ that

(2.29) $$\begin{eqnarray}\displaystyle & \displaystyle Y\sim Re^{-\unicode[STIX]{x1D6FE}/4}. & \displaystyle\end{eqnarray}$$

As the base-flow Maclaurin series proceeds in powers of $Y$ , we similarly expand $u^{\prime }$ as an asymptotic series in powers of $Re^{-\unicode[STIX]{x1D6FE}/4}$ , i.e. $u^{\prime }=\bar{u}_{0}+Re^{-\unicode[STIX]{x1D6FE}/4}\bar{u}_{1}+\cdots \,$ , etc.

Due to the interactive nature of the triple-deck structure, the behaviour of the perturbation is governed by the lower-deck equation for $v^{\prime }$ that contains the self-induced pressure perturbation $p^{\prime }$ . The scaling for the chordwise and spanwise wavenumbers is thus set by the requirement that in the lower deck the chordwise pressure gradient balances with the non-parallel variations of the flow, $\unicode[STIX]{x1D6FC}p^{\prime }\sim u^{\prime }U_{x}$ . Since $U_{x}\sim Y\sim Re^{-\unicode[STIX]{x1D6FE}/4}$ in the lower deck, we have $p^{\prime }\sim Re^{-5\unicode[STIX]{x1D6FE}/4}u^{\prime }$ . In triple-deck structures, the pressure perturbation does not vary across the lower or main deck at leading order. It must also be the same size as the vertical velocity perturbation in the main deck, $p^{\prime }\sim v^{\prime }$ , so that in the upper deck the pressure gradient can balance with both the vertical and horizontal momenta. Since the continuity equation (2.16) gives us that $v^{\prime }\sim yu_{x}^{\prime }$ , the pressure perturbation thus scales as $p^{\prime }\sim v^{\prime }\sim Re^{\unicode[STIX]{x1D6FE}-1/2}u^{\prime }$ in the main and lower decks. Equating these two pressure scalings, we find that

(2.30) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}={\displaystyle \frac{2}{9}}. & \displaystyle\end{eqnarray}$$

The scaling for the correction to the chordwise wavenumber is set by requiring that the corresponding inertial term also balances with the pressure term, $Re^{\unicode[STIX]{x1D6FE}_{np}}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D706}_{1}^{x}Yu^{\prime }\sim \unicode[STIX]{x1D6FC}p^{\prime }$ . Now, in the main deck we find that $v^{\prime }\propto U_{eff}$ at leading order. Because of the marginally separated nature of the effective base-flow profile, the matching condition from the main deck thus imposes that in the lower deck $v^{\prime }\sim Re^{\unicode[STIX]{x1D6FE}-1/2}Y^{2}u^{\prime }\sim Re^{(\unicode[STIX]{x1D6FE}-1)/2}u^{\prime }$ , which is $Re^{-\unicode[STIX]{x1D6FE}/4}$ smaller than the scaling suggested by the continuity equation. This means that $v^{\prime }$ does not appear at leading order in the lower-deck continuity equation, and so the inertial term above must involve the sub-leading-order term $\bar{u}_{1}$ in order to make a non-zero contribution to the combined momentum equation. The balance between inertia and non-parallelism thus becomes $Re^{\unicode[STIX]{x1D6FE}_{np}}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D706}_{1}^{x}Y(Re^{-\unicode[STIX]{x1D6FE}/4}\bar{u}_{1})\sim Re^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FC}_{0}p^{\prime }$ , and so

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}_{np}={\displaystyle \frac{\unicode[STIX]{x1D6FE}}{4}}={\displaystyle \frac{1}{18}}. & \displaystyle\end{eqnarray}$$

For these scalings the inertial term $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{4}^{x}Y^{4}(Re^{-\unicode[STIX]{x1D6FE}/4}\bar{u}_{1})$ also balances with the pressure term, in agreement with Choudhari (Reference Choudhari1994a ). Our alternative arguments emphasise the importance of non-parallelism within this regime, some preliminary results about which were presented in Butler & Wu (Reference Butler and Wu2015).

3.1 Main deck

We first consider the main deck, where $Y=O(1)$ . Without loss of generality, we can take $u^{\prime }$ and $w^{\prime }$ to be $O(1)$ quantities. Balancing the terms within the continuity equation, (2.16), tells us that

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}Re^{2/9}(\unicode[STIX]{x1D6FC}_{0}+\cdots \,)u^{\prime }\sim Re^{1/2}v^{\prime }\sim \text{i}Re^{2/9}\unicode[STIX]{x1D6FD}_{0}w^{\prime }, & \displaystyle\end{eqnarray}$$

and so

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle v^{\prime }\sim Re^{-5/18}. & \displaystyle\end{eqnarray}$$

Thus the velocity and pressure of our eigenmode (2.19) are scaled as

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle (\tilde{\boldsymbol{u}},\tilde{p})=(\tilde{u} ,\unicode[STIX]{x1D700}^{5}\tilde{v},\tilde{w},\unicode[STIX]{x1D700}^{5}\tilde{p}), & \displaystyle\end{eqnarray}$$

where the small parameter $\unicode[STIX]{x1D700}$ is defined as

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}:=Re^{-1/18}, & \displaystyle\end{eqnarray}$$

and we expand each of the flow components as asymptotic series: $(\tilde{\boldsymbol{u}},\tilde{p})=\sum _{j}\unicode[STIX]{x1D700}^{j}(\tilde{\boldsymbol{u}}_{j},\tilde{p})$ . We proceed by substituting this form of the perturbation into the linearised N–S equations (2.15)–(2.16) and boundary conditions (2.15)–(2.18), and solving the resulting equations at each order of $\unicode[STIX]{x1D700}$ .

At the orders that are relevant to our analysis, the ${\displaystyle\mathop{x}\limits_{{\sim}}}$ -momentum equation gives

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\text{i}\unicode[STIX]{x1D6FC}_{0}U_{eff}\,\tilde{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{\,j}+\tilde{v}_{j}\,{\displaystyle\mathop{ U}\limits_{{\sim}}}_{Y}, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\text{i}\unicode[STIX]{x1D6FC}_{0}U_{eff}\,\tilde{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{3}+\text{i}\unicode[STIX]{x1D6FC}_{1}U\,\tilde{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}+\tilde{v}_{3}\,{\displaystyle\mathop{ U}\limits_{{\sim}}}_{Y}, & \displaystyle\end{eqnarray}$$

where $j\in \{0,1,2\}$ . As usual, the $y$ -momentum equation reduces to $\tilde{p}_{k,Y}=0$ , and so the pressure perturbation is constant across the main deck, at the orders we are considering. Finally, the continuity equation (2.16) gives

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\text{i}\unicode[STIX]{x1D6FC}_{0}\tilde{u} _{j}+\tilde{v}_{j,Y}+\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{w}_{j}, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\text{i}\unicode[STIX]{x1D6FC}_{0}\tilde{u} _{3}+\text{i}\unicode[STIX]{x1D6FC}_{1}\tilde{u} _{0}+\tilde{v}_{3,Y}+\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{w}_{3}. & \displaystyle\end{eqnarray}$$

We solve these at each order by using the continuity equation to combine the $x$ - and $z$ -momentum equations, leaving an equation for $\tilde{v}_{k}$ . We find that

(3.9a-c ) $$\begin{eqnarray}\displaystyle \tilde{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{\,j}=A_{j}(x,z)\,{\displaystyle\mathop{ U}\limits_{{\sim}}}_{Y},\quad \tilde{v}_{j}=-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{j}(x,z)U_{eff},\quad \tilde{p}_{j}=P_{j}(x,z), & & \displaystyle\end{eqnarray}$$

for $j=0,1,2$ , while

(3.10a-c ) $$\begin{eqnarray}\displaystyle \tilde{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{3}=A_{3}(x,z)\,{\displaystyle\mathop{ U}\limits_{{\sim}}}_{Y},\quad \tilde{v}_{3}=-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{3}(x,z)U_{eff}-\text{i}\unicode[STIX]{x1D6FC}_{1}A_{0}(x,z)U,\quad \tilde{p}_{3}=P_{3}(x,z), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where the as-yet-unknown displacement functions $A_{k}(x,z)$ and pressure functions $P_{k}(x,z)$ will be related through the upper deck.

Non-parallelism appears in the main-deck solution at the next order in the expansions. However, we can show that $\tilde{v}_{4}\sim Y$ as $Y\rightarrow 0$ , and so the $O(\unicode[STIX]{x1D700}^{4})$ solution does not contribute to the matching conditions needed to determine the growth rate.

To facilitate matching, we note that these main-deck solutions have the following behaviours for small and large $Y$ . As $Y\rightarrow 0$ ,

(3.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \tilde{{\displaystyle\mathop{u}\limits_{{\sim}}}}\sim A_{0}(\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}+2\unicode[STIX]{x1D706}_{2}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}Y+\cdots \,)+\unicode[STIX]{x1D700}A_{1}(\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}+\cdots \,)+\unicode[STIX]{x1D700}^{2}A_{2}(\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}+\cdots \,)\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D700}^{3}A_{3}(\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}+\cdots \,)+\cdots \,,\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D700}^{5}\tilde{v}\sim \unicode[STIX]{x1D700}^{5}(-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{0}(\unicode[STIX]{x1D706}_{2}^{0}Y^{2}+\cdots \,))+\unicode[STIX]{x1D700}^{6}(-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{1}(\unicode[STIX]{x1D706}_{2}^{0}Y^{2}+\cdots \,))+\unicode[STIX]{x1D700}^{7}(-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{2}(\unicode[STIX]{x1D706}_{2}^{0}Y^{2}+\cdots \,))\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,\unicode[STIX]{x1D700}^{8}(-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{3}(\unicode[STIX]{x1D706}_{2}^{0}Y^{2}+\cdots \,)-\text{i}\unicode[STIX]{x1D6FC}_{1}A_{0}(\unicode[STIX]{x1D706}_{1}^{x}Y+\cdots \,))+\unicode[STIX]{x1D700}^{9}(O(Y))+\cdots \,,\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D700}^{5}\tilde{p}=\unicode[STIX]{x1D700}^{5}P_{0}+\unicode[STIX]{x1D700}^{6}P_{1}+\unicode[STIX]{x1D700}^{7}P_{2}+\unicode[STIX]{x1D700}^{8}P_{3}+\cdots \,.\end{eqnarray}$$

On the other hand, as $Y\rightarrow \infty$ ,

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \phantom{AA}\tilde{{\displaystyle\mathop{u}\limits_{{\sim}}}}\rightarrow 0, & \displaystyle\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}^{5}\tilde{v}\sim \unicode[STIX]{x1D700}^{5}(-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{0}U_{eff_{s}})+\unicode[STIX]{x1D700}^{6}(-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{1}U_{eff_{s}})+\unicode[STIX]{x1D700}^{7}(-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{2}U_{eff_{s}}) & \displaystyle \nonumber\\ \displaystyle & \quad \displaystyle +\,\unicode[STIX]{x1D700}^{8}(-\text{i}\unicode[STIX]{x1D6FC}_{0}A_{3}U_{eff_{s}}-\text{i}\unicode[STIX]{x1D6FC}_{1}A_{0}U_{s})+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}^{5}\tilde{p}=\unicode[STIX]{x1D700}^{5}P_{0}+\unicode[STIX]{x1D700}^{6}P_{1}+\unicode[STIX]{x1D700}^{7}P_{2}+\unicode[STIX]{x1D700}^{8}P_{3}+\cdots \,, & \displaystyle\end{eqnarray}$$

where $U_{eff_{s}}:=U_{s}+(\unicode[STIX]{x1D6FD}_{0}/\unicode[STIX]{x1D6FC}_{0})W_{s}$ is the effective slip velocity. Since $v^{\prime },p^{\prime }{\nrightarrow}0$ as $Y\rightarrow \infty$ , the main-deck solution does not satisfy the attenuation condition (2.18). To rectify this, we must introduce an upper deck. Similarly since ${\displaystyle\mathop{u}\limits_{{\sim}}}^{\prime }{\nrightarrow}0$ as $Y\rightarrow 0$ , i.e. the main-deck solution does not satisfy the no-slip condition (2.17), we must introduce a viscous lower deck.

3.2 Upper deck

In the upper deck, the vertical inertia and pressure gradient terms must balance in order to satisfy the attenuation condition (2.18). It follows from the far-field behaviours of the main-deck solution, (3.15) and (3.16), that $v^{\prime }$ and $p^{\prime }$ are $O(\unicode[STIX]{x1D700}^{5})$ . The balance $\text{i}\unicode[STIX]{x1D6FC}U_{eff}v^{\prime }\sim p_{y}^{\prime }$ then indicates that $y\sim \unicode[STIX]{x1D6FC}^{-1}$ , and so in the upper deck $y\sim Re^{-2/9}$ and $Y\sim Re^{5/18}$ . We thus introduce the scaled vertical variable

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{y}}:=\unicode[STIX]{x1D700}^{-4}y. & \displaystyle\end{eqnarray}$$

From the continuity equation (2.16) we find that $u^{\prime }=O(\unicode[STIX]{x1D700}^{5})$ , and thus the velocity and pressure of the eigenmode (2.19) are scaled as

(3.18) $$\begin{eqnarray}\displaystyle (\tilde{\boldsymbol{u}},\tilde{p})({\hat{y}};x,z)=\unicode[STIX]{x1D700}^{5}(\hat{u} ,\hat{v},{\hat{w}},\hat{p}). & & \displaystyle\end{eqnarray}$$

As in the main deck, we expand each of the flow components in terms of $\unicode[STIX]{x1D700}$ . Substituting the expansions into the governing equations (2.15)–(2.16) and solving the resulting equations, we find that the upper-deck solution for $j\in \{0,1,2\}$ is

(3.19a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle (\hat{u} _{j},\hat{v}_{j},{\hat{w}}_{j})=-\left(1,{\displaystyle \frac{\text{i}\bar{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x1D6FC}_{0}}},{\displaystyle \frac{\unicode[STIX]{x1D6FD}_{0}}{\unicode[STIX]{x1D6FC}_{0}}}\right){\displaystyle \frac{P_{j}}{U_{eff_{s}}}}\text{e}^{-\bar{\unicode[STIX]{x1D6FC}}{\hat{y}}},\quad \hat{p}_{j}=P_{j}\text{e}^{-\bar{\unicode[STIX]{x1D6FC}}{\hat{y}}}, & \displaystyle\end{eqnarray}$$

while at the next order

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{u} _{3}={\displaystyle \frac{1}{U_{eff_{s}}}}\left({\displaystyle \frac{\unicode[STIX]{x1D6FC}_{1}}{\unicode[STIX]{x1D6FC}_{0}}}\left({\displaystyle \frac{U_{s}}{U_{eff_{s}}}}-1+2{\displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}^{2}}{\bar{\unicode[STIX]{x1D6FC}}}}{\hat{y}}\right)P_{0}-P_{3}\right)\text{e}^{-\bar{\unicode[STIX]{x1D6FC}}{\hat{y}}}, & \displaystyle\end{eqnarray}$$

(3.21) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{v}_{3}=\left({\displaystyle \frac{-\text{i}}{U_{eff_{s}}}}\right)\left({\displaystyle \frac{\unicode[STIX]{x1D6FC}_{1}}{\bar{\unicode[STIX]{x1D6FC}}}}\left(2\left(1-\bar{\unicode[STIX]{x1D6FC}}{\hat{y}}\right)-{\displaystyle \frac{\bar{\unicode[STIX]{x1D6FC}}^{2}U_{s}}{\unicode[STIX]{x1D6FC}_{0}^{2}U_{eff_{s}}}}\right)P_{0}+{\displaystyle \frac{\bar{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x1D6FC}_{0}}}P_{3}\right)\text{e}^{-\bar{\unicode[STIX]{x1D6FC}}{\hat{y}}}, & \displaystyle\end{eqnarray}$$

(3.22) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{w}}_{3}={\displaystyle \frac{\unicode[STIX]{x1D6FD}_{0}}{\unicode[STIX]{x1D6FC}_{0}U_{eff_{s}}}}\left({\displaystyle \frac{\unicode[STIX]{x1D6FC}_{1}}{\unicode[STIX]{x1D6FC}_{0}}}\left({\displaystyle \frac{U_{s}}{U_{eff_{s}}}}+2{\displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}^{2}}{\bar{\unicode[STIX]{x1D6FC}}}}{\hat{y}}\right)P_{0}-P_{3}\right)\text{e}^{-\bar{\unicode[STIX]{x1D6FC}}{\hat{y}}}, & \displaystyle\end{eqnarray}$$

(3.23) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{p}_{3}=\left(P_{3}-2{\displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D6FC}_{1}}{\bar{\unicode[STIX]{x1D6FC}}}}P_{0}{\hat{y}}\right)\text{e}^{-\bar{\unicode[STIX]{x1D6FC}}{\hat{y}}}, & \displaystyle\end{eqnarray}$$

where

(3.24) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{\unicode[STIX]{x1D6FC}}:=\sqrt{\unicode[STIX]{x1D6FC}_{0}^{2}+\unicode[STIX]{x1D6FD}_{0}^{2}}. & \displaystyle\end{eqnarray}$$

By matching the solution for $v^{\prime }$ as ${\hat{y}}\rightarrow 0$ with (3.15), we arrive at the pressure–displacement relations,

(3.25a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle A_{j}={\displaystyle \frac{\bar{\unicode[STIX]{x1D6FC}}P_{j}}{\unicode[STIX]{x1D6FC}_{0}^{2}U_{eff_{s}}^{2}}},\quad A_{3}={\displaystyle \frac{2\unicode[STIX]{x1D6FC}_{1}P_{0}}{\bar{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FC}_{0}U_{eff_{s}}^{2}}}\left(1-{\displaystyle \frac{\bar{\unicode[STIX]{x1D6FC}}^{2}U_{s}}{\unicode[STIX]{x1D6FC}_{0}^{2}U_{eff_{s}}}}\right)+{\displaystyle \frac{\bar{\unicode[STIX]{x1D6FC}}P_{3}}{\unicode[STIX]{x1D6FC}_{0}^{2}U_{eff_{s}}^{2}}}. & \displaystyle\end{eqnarray}$$

3.3 Lower deck

In order to satisfy the no-slip condition (2.17) in the lower deck, the chordwise inertial and viscous terms must balance. This determines the lower-deck scaling (2.29), that is $y\sim Re^{-10/18}$ and $Y\sim Re^{-1/18}$ . We thus introduce the scaled vertical variable

(3.26) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{y}:=\unicode[STIX]{x1D700}^{10}y. & \displaystyle\end{eqnarray}$$

From the small- $Y$ behaviour of the main-deck solution, (3.11)–(3.13), the perturbation in the lower deck takes the form

(3.27) $$\begin{eqnarray}\displaystyle & \displaystyle (\tilde{\boldsymbol{u}},\tilde{p})(\bar{y};x,z)=(\bar{u},\unicode[STIX]{x1D700}^{7}\bar{v},\bar{w},\unicode[STIX]{x1D700}^{5}\bar{p}), & \displaystyle\end{eqnarray}$$

with each component expanded as an asymptotic series in $\unicode[STIX]{x1D700}$ . The pressure is again constant across this region, at the orders we are considering. The relevant ${\displaystyle\mathop{x}\limits_{{\sim}}}$ -momentum equations are

(3.28) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}\bar{y}^{2}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}+\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\bar{v}_{0}=\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0,\bar{y}\bar{y}}, & \displaystyle\end{eqnarray}$$

(3.29) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}\bar{y}^{2}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{1}+\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\bar{v}_{1}+2\unicode[STIX]{x1D706}_{2}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\bar{y}\bar{v}_{0}=\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{1,\bar{y}\bar{y}}, & \displaystyle\end{eqnarray}$$

(3.30) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{4}^{0}\bar{y}^{4}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}+\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}\bar{y}^{2}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{2}+\text{i}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D706}_{1}^{x}\bar{y}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}+\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\bar{v}_{2}+2\unicode[STIX]{x1D706}_{2}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\bar{y}\bar{v}_{1}=\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{2,\bar{y}\bar{y}}, & \displaystyle\end{eqnarray}$$

(3.31) $$\begin{eqnarray}\displaystyle & \text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{5}^{0}\bar{y}^{5}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}+\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{4}^{0}\bar{y}^{4}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{1}+\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}\bar{y}^{2}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{3}+\text{i}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D706}_{1}^{x}\bar{y}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{1}+\text{i}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D706}_{2}^{x}\bar{y}^{2}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0} & \displaystyle \nonumber\\ \displaystyle & \quad +\,\unicode[STIX]{x1D706}_{1}^{x}\bar{y}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0,x}-\frac{1}{2}\left(\unicode[STIX]{x1D706}_{1,x}^{x}+\unicode[STIX]{x1D706}_{1,z}^{z}\right)\bar{y}^{2}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0,\bar{y}}+\unicode[STIX]{x1D706}_{1}^{z}\bar{y}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0,z} & \displaystyle \nonumber\\ \displaystyle & +\,\bar{u}_{0}\unicode[STIX]{x1D706}_{1,x}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}~\bar{y}+\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\bar{v}_{3}+2\unicode[STIX]{x1D706}_{2}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\bar{y}\bar{v}_{2}+4\unicode[STIX]{x1D706}_{4}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\bar{y}^{3}\bar{v}_{0}+\bar{w}_{0}\unicode[STIX]{x1D706}_{1,z}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}~\bar{y}+\text{i}\,{\displaystyle\mathop{\unicode[STIX]{x1D6FC}}\limits_{{\sim}}}_{\hspace{-0.55pt}0}\bar{p}_{0}=\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{3,\bar{y}\bar{y}}, & \displaystyle\end{eqnarray}$$

where ${\displaystyle\mathop{\unicode[STIX]{x1D6FC}}\limits_{{\sim}}}_{0}\mapsto \unicode[STIX]{x1D6FD}_{0}$ for the spanwise equation, while the continuity equation gives

(3.32) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D6FC}_{0}\bar{u}_{0}+\text{i}\unicode[STIX]{x1D6FD}_{0}\bar{w}_{0}=0, & \displaystyle\end{eqnarray}$$

(3.33) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D6FC}_{0}\bar{u}_{1}+\bar{v}_{0,\bar{y}}+\text{i}\unicode[STIX]{x1D6FD}_{0}\bar{w}_{1}=0, & \displaystyle\end{eqnarray}$$

(3.34) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D6FC}_{0}\bar{u}_{2}+\bar{v}_{1,\bar{y}}+\text{i}\unicode[STIX]{x1D6FD}_{0}\bar{w}_{2}=0, & \displaystyle\end{eqnarray}$$

(3.35) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D6FC}_{0}\bar{u}_{3}+\text{i}\unicode[STIX]{x1D6FC}_{1}\bar{u}_{0}+\bar{v}_{2,\bar{y}}+\text{i}\unicode[STIX]{x1D6FD}_{0}\bar{w}_{3}=0. & \displaystyle\end{eqnarray}$$

As before, we combine the corresponding $x$ - and $z$ -momentum equations through the continuity equation to obtain an equation for $\bar{v}_{k}$ . Since $\unicode[STIX]{x1D706}_{1}^{0}=0$ , the leading-order combination cancels and we must go to the next order to determine $\bar{v}_{0}$ . We find that

(3.36) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}\bar{y}^{4}\left[{\displaystyle \frac{\bar{v}_{0}}{\bar{y}^{2}}}\right]_{\bar{y}}=\bar{v}_{0,\bar{y}\bar{y}\bar{y}}. & \displaystyle\end{eqnarray}$$

This is satisfied by the continuation of the main-deck solution, $\bar{v}_{0}=-\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}A_{0}\bar{y}^{2}$ , which can then be substituted into (3.28) to find $\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}$ . On introducing the variable

(3.37) $$\begin{eqnarray}\displaystyle & \displaystyle s:=(4\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0})^{1/4}\bar{y}, & \displaystyle\end{eqnarray}$$

the leading-order ${\displaystyle\mathop{x}\limits_{{\sim}}}$ -momentum equation becomes

(3.38) $$\begin{eqnarray}\displaystyle {\mathcal{L}}[\,{\displaystyle\mathop{{\mathcal{U}}}\limits_{{\sim}}}]:=\left[\frac{\text{d}^{2}}{\text{d}s^{2}}-{\displaystyle \frac{1}{4}}s^{2}\right]\,{\displaystyle\mathop{{\mathcal{U}}}\limits_{{\sim}}}=0, & & \displaystyle\end{eqnarray}$$

where

(3.39) $$\begin{eqnarray}\displaystyle & \,{\displaystyle\mathop{{\mathcal{U}}}\limits_{{\sim}}}\displaystyle :=\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}-\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{0}. & \displaystyle\end{eqnarray}$$

For the standard triple-deck structure, the perturbation in the lower deck is usually governed by an Airy equation. However, because of our wall-shear alignment condition that $\unicode[STIX]{x1D706}_{1}^{0}=0$ we instead have a parabolic cylinder equation; the solution satisfying the no-slip condition (2.17) and the matching condition (3.11) is (Hall Reference Hall1986; Choudhari Reference Choudhari1994a )

(3.40) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}=\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{0}\left(1-{\displaystyle \frac{{\mathcal{D}}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}}\right). & \displaystyle\end{eqnarray}$$

The parabolic cylinder function ${\mathcal{D}}_{-1/2}(s)$ decays exponentially as $s\rightarrow \infty$ along the real line. For ${\mathcal{D}}_{-1/2}(s)$ to also decay for the complex variable $s$ as $\bar{y}\rightarrow \infty$ , we find that the appropriate branch for our complex root must be taken such that

(3.41) $$\begin{eqnarray}\displaystyle & \displaystyle s=(4\unicode[STIX]{x1D6FC}_{0}|\unicode[STIX]{x1D706}_{2}^{0}|)^{1/4}\text{e}^{\pm \text{i}(\unicode[STIX]{x03C0}/8)}\bar{y}, & \displaystyle\end{eqnarray}$$

where $\pm =\text{sign}\{\unicode[STIX]{x1D706}_{2}^{0}\}$ . Here we have assumed that $\unicode[STIX]{x1D6FC}_{0}>0$ without losing generality.

The leading-order solution for the lower deck is then found to be

(3.42a-c ) $$\begin{eqnarray}\displaystyle \bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}=\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{0}\left(1-{\displaystyle \frac{{\mathcal{D}}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}}\right),\quad \bar{v}_{0}=-\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}A_{0}\bar{y}^{2},\quad \bar{p}_{0}=P_{0}={\displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}^{2}U_{eff_{s}}^{2}}{\bar{\unicode[STIX]{x1D6FC}}}}A_{0}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

At the following order, we find that $\bar{v}_{1}$ also satisfies (3.36) and the full solution is

(3.43a-c ) $$\begin{eqnarray}\displaystyle \bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{1}=\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{1}\left(1-{\displaystyle \frac{{\mathcal{D}}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}}\right)+2\unicode[STIX]{x1D706}_{2}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{0}\bar{y},\quad \bar{v}_{1}=-\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}A_{1}\bar{y}^{2},\quad \bar{p}_{1}=P_{1}={\displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}^{2}U_{eff_{s}}^{2}}{\bar{\unicode[STIX]{x1D6FC}}}}A_{1}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The equations at the next order allow us to derive the dispersion relation that determines the chordwise wavenumber $\unicode[STIX]{x1D6FC}_{1}$ . Combining the chordwise and spanwise forms of (3.31) via the continuity equation, we arrive at

(3.44) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{1}{\text{i}\unicode[STIX]{x1D6FC}_{0}}}\bar{v}_{2,\bar{y}\bar{y}\bar{y}}-\unicode[STIX]{x1D706}_{2}^{0}\bar{y}^{2}\bar{v}_{2,\bar{y}}+2\unicode[STIX]{x1D706}_{2}^{0}\bar{y}\bar{v}_{2}=\frac{\unicode[STIX]{x1D6FC}_{1}}{\unicode[STIX]{x1D6FC}_{0}}\unicode[STIX]{x1D706}_{1}^{x}\{\bar{y}\bar{v}_{0,\bar{y}}-\bar{v}_{0}\}+\unicode[STIX]{x1D706}_{4}^{0}\bar{y}^{3}\left\{\bar{y}\bar{v}_{0,\bar{y}}-4\bar{v}_{0}\right\}+\left\{\frac{\bar{\unicode[STIX]{x1D6FC}}^{2}}{\text{i}\unicode[STIX]{x1D6FC}_{0}}\bar{p}_{0}\right\} & \displaystyle \nonumber\\ \displaystyle & \displaystyle -\bar{y}\left\{\unicode[STIX]{x1D706}_{1}^{x}\left(\bar{u}_{0,x}+\frac{\unicode[STIX]{x1D6FD}_{0}}{\unicode[STIX]{x1D6FC}_{0}}\bar{w}_{0,x}\right)+\unicode[STIX]{x1D706}_{1}^{z}\left(\bar{u}_{0,z}+\frac{\unicode[STIX]{x1D6FD}_{0}}{\unicode[STIX]{x1D6FC}_{0}}\bar{w}_{0,z}\right)+\left(\bar{u}_{0}\unicode[STIX]{x1D706}_{1,x}^{0}+\bar{w}_{0}\unicode[STIX]{x1D706}_{1,z}^{0}\right)\right\}. & \displaystyle\end{eqnarray}$$

On the right-hand side of (3.44), the first set of terms enclosed in braces represents the effect of the correction to the chordwise wavenumber, the second set that of the fourth-order derivative of $U_{eff}$ at $Y=0$ and the third the effect of the self-induced pressure. Each of these is present in Choudhari (Reference Choudhari1994a ). The fourth set of terms here represents the non-parallel-flow effects, which consist of two parts. The first two groupings of terms are the contribution from the streamwise and spanwise variations of the eigenfunction, while the third is that from the streamwise and spanwise variations of the base flow itself.

Since the flow is wall-shear aligned $(\unicode[STIX]{x1D706}_{1}^{0}=0)$ , it can be shown that

(3.45) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{u}_{0,{\displaystyle\mathop{x}\limits_{{\sim}}}}+\frac{\unicode[STIX]{x1D6FD}_{0}}{\unicode[STIX]{x1D6FC}_{0}}\bar{w}_{0,{\displaystyle\mathop{x}\limits_{{\sim}}}}={\displaystyle \frac{\unicode[STIX]{x1D706}_{1,{\displaystyle\mathop{x}\limits_{{\sim}}}}^{0}}{\unicode[STIX]{x1D706}_{1}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}}}\,\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}. & \displaystyle\end{eqnarray}$$

It follows that the non-parallel contributions from the perturbation and the base flow are thus the same, and so are of equal importance in the development of the eigenmode; these were both overlooked in Choudhari (Reference Choudhari1994a ).

To take account of the matching condition (3.12) for $v^{\prime }$ , we define

(3.46) $$\begin{eqnarray}\displaystyle \bar{v}_{2}=: & \displaystyle -\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}A_{2}\bar{y}^{2}-\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{4}^{0}A_{0}\bar{y}^{4}+{\mathcal{V}}, & \displaystyle\end{eqnarray}$$

so that ${\mathcal{V}}\sim -\text{i}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D706}_{1}^{x}A_{0}\bar{y}\text{ as }\bar{y}\rightarrow \infty$ . Using (3.45) and (3.46), we can then rewrite (3.44) in terms of ${\mathcal{V}}$ and $s=(4\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0})^{1/4}\bar{y}$ to obtain

(3.47) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6EC}_{s}[{\mathcal{V}}]:=\left[\frac{\text{d}^{3}}{\text{d}s^{3}}-\frac{1}{4}s^{2}\frac{\text{d}}{\text{d}s}+\frac{1}{2}s\right]{\mathcal{V}}=f^{{\mathcal{V}}}(s), & \displaystyle\end{eqnarray}$$

where

(3.48) $$\begin{eqnarray}\displaystyle & \displaystyle f^{{\mathcal{V}}}(s):={\displaystyle \frac{6\unicode[STIX]{x1D706}_{4}^{0}}{\unicode[STIX]{x1D706}_{2}^{0}}}A_{0}s-{\displaystyle \frac{\text{i}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D706}_{1}^{x}}{4}}A_{0}\left(4\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}\right)^{-1/4}s^{2}-{\displaystyle \frac{\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x1D706}_{1}^{x}}}{\displaystyle \frac{\bar{u}_{0}}{2\unicode[STIX]{x1D706}_{2}^{0}}}s+\bar{\unicode[STIX]{x1D6FC}}^{2}\left(4\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}\right)^{-3/4}\bar{p}_{0}. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here we have introduced

(3.49) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}:=\unicode[STIX]{x1D706}_{1}^{x}\unicode[STIX]{x1D706}_{1,x}^{0}+\unicode[STIX]{x1D706}_{1}^{z}\unicode[STIX]{x1D706}_{1,z}^{0}. & \displaystyle\end{eqnarray}$$

Our choice of notation should again be made clear:

(3.50) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D706}_{n,{\displaystyle\mathop{x}\limits_{{\sim}}}}^{0}:=\unicode[STIX]{x1D706}_{n,{\displaystyle\mathop{x}\limits_{{\sim}}}}^{x}+\frac{\unicode[STIX]{x1D6FD}_{0}}{\unicode[STIX]{x1D6FC}_{0}}\unicode[STIX]{x1D706}_{n,{\displaystyle\mathop{x}\limits_{{\sim}}}}^{z}\not \equiv \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}{\displaystyle\mathop{x}\limits_{{\sim}}}}(\unicode[STIX]{x1D706}_{n}^{0}). & \displaystyle\end{eqnarray}$$

This is the projection of the derivatives, rather than the derivative of the projection.

3.4 Dispersion relation

Rather than attempting to solve (3.47) directly, we can instead construct a solvability condition that will determine the dispersion relation for $\unicode[STIX]{x1D6FC}_{1}$ . As usual, this is derived through the use of the adjoint operator, $\unicode[STIX]{x1D6EC}_{s}^{\dagger }$ , which can be found by multiplying $\unicode[STIX]{x1D6EC}_{s}[{\mathcal{V}}]$ by a test function $\unicode[STIX]{x1D710}(s)$ and integrating by parts:

(3.51) $$\begin{eqnarray}\displaystyle \displaystyle \int _{0}^{\infty }\!\!\unicode[STIX]{x1D6EC}_{s}[{\mathcal{V}}]\unicode[STIX]{x1D710}\,\text{d}s=\int _{0}^{\infty }\!{\mathcal{V}}\underbrace{\left(-\unicode[STIX]{x1D710}_{sss}+{\displaystyle \frac{1}{4}}s^{2}\unicode[STIX]{x1D710}_{s}+s\unicode[STIX]{x1D710}\right)}_{\unicode[STIX]{x1D6EC}_{s}^{\dagger }[\unicode[STIX]{x1D710}]}\,\text{d}s+\left[{\mathcal{V}}_{ss}\unicode[STIX]{x1D710}-{\mathcal{V}}_{s}\unicode[STIX]{x1D710}_{s}+{\mathcal{V}}\unicode[STIX]{x1D710}_{ss}-\frac{s^{2}}{4}{\mathcal{V}}\unicode[STIX]{x1D710}\right]_{0}^{\infty }. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Conveniently, both terms on the right-hand side are zero for $\unicode[STIX]{x1D710}(s)=s{\mathcal{D}}_{-1/2}(s)$ . The dispersion relation thus comes from satisfying

(3.52) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\infty }f^{{\mathcal{V}}}s{\mathcal{D}}_{-1/2}(s)\,\text{d}s=0. & \displaystyle\end{eqnarray}$$

In order to simplify the result, we need the following identities:

(3.53) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle \int _{0}^{\infty }s^{2}{\mathcal{D}}_{-1/2}(s)\,\text{d}s=-4{{\mathcal{D}}^{\prime }}_{-1/2}(0),\quad \displaystyle \unicode[STIX]{x1D6FF}_{1}:=\frac{{{\mathcal{D}}^{\prime }}_{-1/2}(0)}{{\mathcal{D}}_{-1/2}(0)}\approx -0.478,\\[12.0pt] \displaystyle \int _{0}^{\infty }s^{3}{\mathcal{D}}_{-1/2}(s)\,\text{d}s=4{\mathcal{D}}_{-1/2}(0),\quad \displaystyle \unicode[STIX]{x1D6FF}_{2}:=\int _{0}^{\infty }{\displaystyle \frac{s{\mathcal{D}}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}}\,\text{d}s\approx 1.198,\\[12.0pt] \displaystyle \int _{0}^{\infty }\!\bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}s^{2}{\mathcal{D}}_{-1/2}(s)\,\text{d}s=4\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{0}\!\int _{0}^{\infty }\!{\displaystyle \frac{\left[{{\mathcal{D}}^{\prime }}_{-1/2}(s)\right]^{2}}{{\mathcal{D}}_{-1/2}(0)}}\,\text{d}s,\quad \displaystyle \unicode[STIX]{x1D6FF}_{3}:=\!\int _{0}^{\infty }\!\!\left(\!{\displaystyle \frac{{{\mathcal{D}}^{\prime }}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}}\!\right)^{\!2}\!\text{d}s\approx 0.357.\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The numerical values of $\unicode[STIX]{x1D6FF}_{1}$ and $\unicode[STIX]{x1D6FF}_{2}$ agree with those previously found (Choudhari Reference Choudhari1994a ).

Applying these to (3.52), using the leading-order solutions in (3.42), and taking the appropriate root of $s^{1/4}$ as given in (3.41), we arrive after considerable manipulation at the dispersion relation:

(3.54) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{1}=\left\{\pm \unicode[STIX]{x1D6FC}_{0}^{5/2}C_{P}\mp \unicode[STIX]{x1D6FC}_{0}^{1/4}(C_{4}+C_{np})\tan \frac{\unicode[STIX]{x03C0}}{8}\right\}+\text{i}\{\unicode[STIX]{x1D6FC}_{0}^{5/2}C_{P}+\unicode[STIX]{x1D6FC}_{0}^{1/4}(C_{4}+C_{np})\}, & \displaystyle\end{eqnarray}$$

where the coefficients are

(3.55) $$\begin{eqnarray}\displaystyle & \displaystyle C_{P}:=-\left(1+\left({\displaystyle \frac{\unicode[STIX]{x1D706}_{1}^{x}}{\unicode[STIX]{x1D706}_{1}^{z}}}\right)^{2}\right)^{1/2}{\displaystyle \frac{U_{eff_{s}}\unicode[STIX]{x1D6FF}_{2}}{2^{3/2}|\unicode[STIX]{x1D706}_{2}^{0}|^{1/2}\unicode[STIX]{x1D706}_{1}^{x}}}, & \displaystyle\end{eqnarray}$$

(3.56) $$\begin{eqnarray}\displaystyle & \displaystyle C_{4}:=\pm 2^{3/2}\cos \left(\frac{\unicode[STIX]{x03C0}}{8}\right){\displaystyle \frac{12\unicode[STIX]{x1D706}_{4}^{0}\unicode[STIX]{x1D6FF}_{1}}{|\unicode[STIX]{x1D706}_{2}^{0}|^{3/4}\unicode[STIX]{x1D706}_{1}^{x}}}, & \displaystyle\end{eqnarray}$$

(3.57) $$\begin{eqnarray}\displaystyle & \displaystyle C_{np}:=\pm 2^{3/2}\cos \left(\frac{\unicode[STIX]{x03C0}}{8}\right){\displaystyle \frac{\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D6FF}_{3}}{|\unicode[STIX]{x1D706}_{2}^{0}|^{3/4}\unicode[STIX]{x1D706}_{1}^{x}}}, & \displaystyle\end{eqnarray}$$

with $\pm =\text{sign}\{\unicode[STIX]{x1D706}_{2}^{0}\}$ .

The result (3.54) amounts to an explicit expression for the scaled growth rate. The coefficients $C_{P}$ , $C_{4}$ and $C_{np}$ represent three contributing mechanisms. The term $C_{P}$ , expressed in terms of the projected slip velocity $U_{eff_{s}}$ via the pressure–displacement relations (3.25), represents the effect of the self-induced pressure through viscous–inviscid interaction. For $\unicode[STIX]{x1D706}_{1}^{x}>0$ this self-induced pressure term is destabilising, as in the other stationary crossflow regimes (Choudhari Reference Choudhari1994a , Reference Choudhari1995) and as for TS waves in two-dimensional boundary layers. The $C_{4}$ term is the contribution due to the fourth-order vertical derivatives of the base flow at the surface of the wing, and is a local quantity. Finally, the effects of non-parallelism are incorporated into the $C_{np}$ term. These consist of equal contributions from the streamwise variations of the base flow and the eigenfunction, which are both related to the spatial development of the wall shear through $\unicode[STIX]{x1D6E4}$ .

The coefficients $C_{4}$ and $C_{np}$ are in fact closely related. It follows from (2.27) and (3.49) that for a flow with no imposed suction on the wing surface, we have $12\unicode[STIX]{x1D706}_{4}^{0}=\unicode[STIX]{x1D6E4}$ , and hence

(3.58) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}_{1}C_{np}=\unicode[STIX]{x1D6FF}_{3}C_{4}. & \displaystyle\end{eqnarray}$$

Despite this relation, these represent distinct physical mechanisms. An O–S approach, such as that carried out by Choudhari (Reference Choudhari1994a ), would include the effects due to $C_{P}$ and $C_{4}$ , but would fail to incorporate the non-parallel-flow effects due to $C_{np}$ .

For a given spanwise wavenumber, the critical value of $\unicode[STIX]{x1D6FC}_{0}$ for which the growth rate of the mode, $-\text{Im}\{\unicode[STIX]{x1D6FC}_{1}\}$ , is zero is given by

(3.59) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{nt}=\left[-{\displaystyle \frac{C_{4}+C_{np}}{C_{P}}}\right]^{4/9}\,\left(\text{if }\,{\displaystyle \frac{C_{4}+C_{np}}{C_{P}}}<0\right). & \displaystyle\end{eqnarray}$$

Since $C_{4}$ and $C_{np}$ are so closely related through (3.58), the condition for the existence of a neutral wavenumber reduces simply to

(3.60) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D706}_{2}^{0}\unicode[STIX]{x1D706}_{4}^{0}<0. & \displaystyle\end{eqnarray}$$

The presence of a neutral point in this regime has important implications for the receptivity of the flow here. The generation of stationary vortices through the base flow interacting with surface roughness elements will be most effective around this point of neutral stability, as the nascent mode will develop over the entire region of growth without an initial period of decay.

For $\unicode[STIX]{x1D6FD}_{0}\gg 1$ , the self-induced pressure term $C_{P}$ dominates (3.54), and the dispersion relation matches appropriately with that for the wall-shear-aligned regime (Choudhari Reference Choudhari1995). For $\unicode[STIX]{x1D6FD}_{0}\ll 1$ , the self-induced pressure drops out of the dispersion relation and the instability takes on a classical, non-interactive structure. This regime is also governed by non-parallelism but lies upstream of the neutral point, where the perturbation decays.

3.5 Instability regimes and chordwise variation of the growth rate

Rather than focusing on a specific chordwise location on the wing and varying the spanwise wavenumber of the perturbation, we can instead consider a mode with a fixed physical spanwise wavelength $\unicode[STIX]{x1D706}^{\ast }$ at varying distances from the leading edge. The locations of the distinguished instability regimes are then determined by the scaling

(3.61) $$\begin{eqnarray}\displaystyle & \displaystyle \left({\displaystyle \frac{2\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D706}^{\ast }}}\right)x^{\ast }=\unicode[STIX]{x1D6FD}^{\ast }x^{\ast }=\unicode[STIX]{x1D6FD}\sim Re^{\unicode[STIX]{x1D6FE}}=\left({\displaystyle \frac{V_{\infty }^{\ast }Q_{s}(x)x^{\ast }}{\unicode[STIX]{x1D708}}}\right)^{\unicode[STIX]{x1D6FE}}, & \displaystyle\end{eqnarray}$$

which can be rearranged as

(3.62) $$\begin{eqnarray}\displaystyle & \displaystyle x={\displaystyle \frac{x^{\ast }}{L^{\ast }}}\sim \left({\displaystyle \frac{\unicode[STIX]{x1D706}}{2\unicode[STIX]{x03C0}}}\right)^{1/(1-\unicode[STIX]{x1D6FE})}\left(Re_{L}Q_{s}(x)\right)^{\unicode[STIX]{x1D6FE}/(1-\unicode[STIX]{x1D6FE})}. & \displaystyle\end{eqnarray}$$

Here we introduce $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}^{\ast }/L^{\ast }$ and the global Reynolds number

(3.63) $$\begin{eqnarray}\displaystyle & \displaystyle Re_{L}=V_{\infty }^{\ast }L^{\ast }/\unicode[STIX]{x1D708}^{\ast }, & \displaystyle\end{eqnarray}$$

which is based on the free-stream velocity $V_{\infty }^{\ast }$ and the chord length of the wing $L^{\ast }$ . The relevant regime locations from (3.62) for our present investigation are those for the non-parallel regime we have introduced here ( $\unicode[STIX]{x1D6FE}=2/9$ ), and the neighbouring wall-shear-aligned regime ( $\unicode[STIX]{x1D6FE}=2/7$ ). Since (3.62) is an asymptotic relation it does not dictate precise locations for the regimes, but how the instability characteristics change in sequence as the distance is increased.

The growth rate of the eigenmode is given by the imaginary part of $\unicode[STIX]{x1D6FC}$ , $-\text{Im}\{\unicode[STIX]{x1D6FC}\}$ , and it follows from (2.22) and the dispersion relation (3.54) that

(3.64) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{Im}\{\unicode[STIX]{x1D6FC}\}=-\text{Im}\{Re^{2/9}\unicode[STIX]{x1D6FC}_{0}+Re^{1/18}\unicode[STIX]{x1D6FC}_{1}+\cdots \,\}=-Re^{1/18}(\unicode[STIX]{x1D6FC}_{0}^{5/2}C_{P}+\unicode[STIX]{x1D6FC}_{0}^{1/4}(C_{4}+C_{np})). & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

This expression is valid in the vicinity of the location described by (3.62) for $\unicode[STIX]{x1D6FE}=2/9$ .

Figure 3.

Comparison of the growth rates with and without $C_{np}$ , for $\unicode[STIX]{x1D706}=6\times 10^{-3}$ and $Re_{L}=3\times 10^{6}$ on a Joukowskii aerofoil with $b/a=0.025$ .

The growth rate of the instability, (3.64), involves the base-flow properties through the Maclaurin coefficients  which must be determined numerically from (2.10)–(2.12). In order to demonstrate the effects of the non-parallel term $C_{np}$ , we focus on the particular case of a Joukowskii aerofoil, for which the inviscid flow can be obtained using the Joukowskii conformal mapping. The numerical scheme used to calculate the base flow around a spanwise-infinite, swept Joukowskii aerofoil is based on a 2-D code described in Cebeci & Cousteix (Reference Cebeci and Cousteix2005), which has then been extended to accommodate a spanwise-infinite 3-D flow. A typical growth-rate plot for this base flow is shown in figure 3, where the prediction with the effect of the non-parallel term $C_{np}$ neglected is also displayed. Here we have taken the chord length of the aerofoil to be $L^{\ast }=2~\text{m}$ , the global Reynolds number $Re_{L}=3\times 10^{6}$ , the spanwise slip velocity $W_{s}=1$ , the sweep angle of the wing $\unicode[STIX]{x1D719}=45^{\circ }$ and the fixed physical spanwise wavelength $\unicode[STIX]{x1D706}^{\ast }=12~\text{mm}$ . The non-dimensional spanwise wavelength is then $\unicode[STIX]{x1D706}=6\times 10^{-3}$ , and the rescaled local spanwise wavenumber is given by $\unicode[STIX]{x1D6FD}_{0}=-2\unicode[STIX]{x03C0}Re^{-2/9}x^{\ast }/\unicode[STIX]{x1D706}^{\ast }$ .

For a blunt-nosed aerofoil, which has the slip velocity and arclength related by $U_{s}\sim {\mathcal{S}}$ near the front stagnation point, $\unicode[STIX]{x1D706}_{2}^{0}\unicode[STIX]{x1D706}_{4}^{0}$ is in general negative between the front stagnation point and the zero of $\unicode[STIX]{x1D706}_{4}^{0}$ and then again after the zero of $\unicode[STIX]{x1D706}_{2}^{0}$ , and so a neutral point exists in each of these regions. The flow upstream of the neutral point is stable, and so the present non-parallel regime represents the genesis of stationary crossflow vortices. After the first neutral point that occurs close to the leading edge, but before the zero of $\unicode[STIX]{x1D706}_{2}^{0}$ , non-parallelism has a stabilising effect, as can be seen in figure 3. After this zero, non-parallelism has a destabilising effect on the perturbation. Despite this, the growth rate is much larger just after the first neutral point than the second. Figure 3 also demonstrates that the position of the second neutral point is strongly affected by non-parallelism. The effect of non-parallelism is highlighted in figure 4, where the accumulated growth, or $N$ factor, is plotted with and without $C_{np}$ for a mode generated at each neutral point. By including non-parallelism, the $N$ factor can differ by as much as $2$ over the regions considered.

A discontinuity occurs in the dispersion relation (3.54) when $\unicode[STIX]{x1D706}_{2}^{0}=0$ , separating the growth-rate curve into two branches (branch I and branch II). Since this is the local growth rate for the non-parallel regime at each point on the wing, the ultimate behaviour of a particular mode as it is traced downstream will depend on whether or not it encounters this point while governed by the non-parallel regime.

Branch-II modes entering the non-parallel regime downstream of the location where $\unicode[STIX]{x1D706}_{2}^{0}=0$ , for example by being initially triggered after this point, will follow the second branch depicted in figure 3, become unstable at the second neutral point and subsequently grow and follow the sequence of linear regimes described by Choudhari (Reference Choudhari1994a , Reference Choudhari1995), until nonlinear effects take over.

Branch-I modes, which will typically include all of those generated near the leading edge, will behave similarly if they are able to grow sufficiently to pass into the wall-shear-aligned regime before $\unicode[STIX]{x1D706}_{2}^{0}=0$ . However, if they encounter this point while still governed by the non-parallel regime then the scaling for the lower deck must be adjusted and a new dispersion relation determined in the vicinity of this point. A new type of instability operates in this region, as we will show in the next section.

Figure 4.

The effect of non-parallelism on the $N$ factor $N(x)=\int _{x_{n}}^{x}(-\unicode[STIX]{x1D6FC}_{i})\,\text{d}s$ for the two neutral points depicted in figure 3. The location of the second neutral point is altered significantly by non-parallelism.

4.1 New lower-deck equations

The derivation of the governing lower-deck equations in the vicinity of the effective pressure minimum follows a process similar to that in § 3.3, and so is consigned to the appendix A. From the leading-order continuity equation, we find that

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D706}_{1}^{x}\bar{w}_{0}=\unicode[STIX]{x1D706}_{1}^{z}\bar{u}_{0}, & \displaystyle\end{eqnarray}$$

and so we need only consider $\bar{u}_{0}$ and $\bar{v}_{0}$ . To account for the near-wall behaviours of the main-deck solution, (3.11) and (3.12), which remain valid subject to scaling adjustment, we introduce

(4.13a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{u}_{0}=:\unicode[STIX]{x1D706}_{1}^{x}A_{0}+{\mathcal{U}},\quad \bar{v}_{0}=:-f(\bar{y})A_{0}+{\mathcal{V}}, & \displaystyle\end{eqnarray}$$

where

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle f(\bar{y}):=\text{i}\unicode[STIX]{x1D6FC}_{0}\left(\dot{\unicode[STIX]{x1D706}}_{2}^{0}\bar{x}\bar{y}^{2}+\unicode[STIX]{x1D706}_{4}^{0}\bar{y}^{4}\right)+\text{i}\tilde{\unicode[STIX]{x1D6FC}}_{1}\unicode[STIX]{x1D706}_{1}^{x}\bar{y}. & \displaystyle\end{eqnarray}$$

After considerable algebraic work (appendix A), we eventually find that the velocity components ${\mathcal{U}}$ and ${\mathcal{V}}$ are governed by

(4.15) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{L}}[{\mathcal{U}}]:=\left[\frac{\text{d}^{2}}{\text{d}\bar{y}^{2}}-f\right]{\mathcal{U}}=\unicode[STIX]{x1D706}_{1}^{x}{\mathcal{V}}, & \displaystyle\end{eqnarray}$$

(4.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6EC}[\unicode[STIX]{x1D706}_{1}^{x}{\mathcal{V}}]:=\left[\frac{\text{d}^{3}}{\text{d}\bar{y}^{3}}-f\frac{\text{d}}{\text{d}\bar{y}}+f^{\prime }\right]\unicode[STIX]{x1D706}_{1}^{x}{\mathcal{V}}=-f^{\prime \prime \prime }{\mathcal{U}}. & \displaystyle\end{eqnarray}$$

Here $f^{\prime }:=\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}\bar{y}$ , with $\bar{x}$ fixed.

There are three notable differences between these governing equations and the corresponding pair (3.38) and (3.47) for the $\unicode[STIX]{x1D706}_{2}^{0}=O(1)$ non-parallel regime: these equations are coupled; $f$ is quartic rather than quadratic; and $f$ contains the chordwise wavenumber $\tilde{\unicode[STIX]{x1D6FC}}_{1}$ , making the problem implicit in the sense that the quantity we wish to determine $(\tilde{\unicode[STIX]{x1D6FC}}_{1})$ is part of the differential operator, as in classical stability theory, rather than the forcing term. We may then expect this system to be less amenable to analytical treatment, and so a primarily numerical calculation is to be taken.

By applying the differential operator $\unicode[STIX]{x1D6EC}$ to (4.15), we can reduce this system to

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{\text{d}^{4}}{\text{d}\bar{y}^{4}}-2f\frac{\text{d}^{2}}{\text{d}\bar{y}^{2}}-2f^{\prime }\frac{\text{d}}{\text{d}\bar{y}}+(f^{2}-3f^{\prime \prime })\right]\unicode[STIX]{x1D6FA}=0, & \displaystyle\end{eqnarray}$$

where we have introduced

(4.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}:=\frac{\text{d}{\mathcal{U}}}{\text{d}\bar{y}}. & \displaystyle\end{eqnarray}$$

The governing equation (4.17) requires four boundary conditions. From the no-slip and impermeability conditions, (2.17), we have that

(4.19a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{U}}=-\unicode[STIX]{x1D706}_{1}^{x}A_{0},\quad {\mathcal{V}}=0,\quad {\mathcal{V}}_{\bar{y}}=-\text{i}\tilde{\unicode[STIX]{x1D6FC}}_{1}\unicode[STIX]{x1D706}_{1}^{x}A_{0}\quad \text{at }\bar{y}=0, & \displaystyle\end{eqnarray}$$

which, through (4.15) and (4.16), give us

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}^{\prime }(0)=\unicode[STIX]{x1D6FA}^{\prime \prime }(0)=0. & \displaystyle\end{eqnarray}$$

The solution of (4.17) has exponential behaviour as $\bar{y}\rightarrow \infty$ . Our remaining boundary conditions come from the requirement that $\unicode[STIX]{x1D6FA}$ decays in this limit, so that it is possible to match with the main-deck solution:

(4.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA},\,\unicode[STIX]{x1D6FA}^{\prime }\rightarrow 0\quad \text{as}~\bar{y}\rightarrow \infty . & & \displaystyle\end{eqnarray}$$

Due to the linear and homogeneous nature of (4.17) and its boundary conditions (4.20) and (4.21), we can also impose the normalisation condition

(4.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}(0)=1. & & \displaystyle\end{eqnarray}$$

4.2 Eigenvalue problem

Rather than attempting to seek an analytical solution to (4.17), we rewrite it as an eigenvalue problem for $\tilde{\unicode[STIX]{x1D6FC}}_{1}$ and solve this numerically. By splitting $f$ into

(4.23) $$\begin{eqnarray}\displaystyle & \displaystyle f(\bar{y}):=k(\bar{y})+\tilde{\unicode[STIX]{x1D6FC}}_{1}\unicode[STIX]{x1D70E}(\bar{y})=(\text{i}\unicode[STIX]{x1D6FC}_{0}(\dot{\unicode[STIX]{x1D706}}_{2}^{0}\bar{x}\bar{y}^{2}+\unicode[STIX]{x1D706}_{4}^{0}\bar{y}^{4}))+\tilde{\unicode[STIX]{x1D6FC}}_{1}(\text{i}\unicode[STIX]{x1D706}_{1}^{x}\bar{y}), & \displaystyle\end{eqnarray}$$

we can redistribute (4.17) into

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\underbrace{\left\{\frac{\text{d}^{4}}{\text{d}\bar{y}^{4}}-2k\frac{\text{d}^{2}}{\text{d}\bar{y}^{2}}-2k^{\prime }\frac{\text{d}}{\text{d}\bar{y}}+k^{2}\right\}}_{{\mathcal{A}}}+\tilde{\unicode[STIX]{x1D6FC}}_{1}\underbrace{\left\{-2\unicode[STIX]{x1D70E}\frac{\text{d}^{2}}{\text{d}\bar{y}^{2}}-2\unicode[STIX]{x1D70E}^{\prime }\frac{\text{d}}{\text{d}\bar{y}}+2k\unicode[STIX]{x1D70E}\right\}}_{{\mathcal{B}}}+\tilde{\unicode[STIX]{x1D6FC}}_{1}^{2}\underbrace{\left\{\unicode[STIX]{x1D70E}^{2}\right\}}_{{\mathcal{C}}}\right]\unicode[STIX]{x1D6FA}=0, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $^{\prime }$ denotes the derivative with respect to $\bar{y}$ .

By discretising our solution domain and derivatives, which is to be described in the next section, we can convert (4.24) into the generalised linear eigenvalue problem

(4.25) $$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}cc@{}}-\unicode[STIX]{x1D63C} & \unicode[STIX]{x1D7EC}\\ \unicode[STIX]{x1D7EC} & \unicode[STIX]{x1D644}\end{array}\right]\unicode[STIX]{x1D733}=\tilde{\unicode[STIX]{x1D6FC}}_{1}\left[\begin{array}{@{}cc@{}}\unicode[STIX]{x1D63D} & \unicode[STIX]{x1D63E}\\ \unicode[STIX]{x1D644} & \unicode[STIX]{x1D7EC}\end{array}\right]\unicode[STIX]{x1D733}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D733}=(\unicode[STIX]{x1D734},\tilde{\unicode[STIX]{x1D6FC}}_{1}\unicode[STIX]{x1D734})^{\text{T}}$ . It now remains to simply determine the matrices $\unicode[STIX]{x1D63C}$ , $\unicode[STIX]{x1D63D}$ and $\unicode[STIX]{x1D63E}$ , which are the discretised forms of the operators ${\mathcal{A}}$ , ${\mathcal{B}}$ and ${\mathcal{C}}$ respectively, as well as the corresponding forms of the boundary conditions (4.20) and (4.21) for our chosen numerical scheme, and then solve (4.25) to find $\tilde{\unicode[STIX]{x1D6FC}}_{1}$ .

4.3 Numerical method and solutions

We begin by mapping our domain $[0,\infty ]$ to $[-1,1]$ via $\unicode[STIX]{x1D70F}=1-2\text{e}^{-\bar{y}}$ in order to utilise Chebyshev polynomials, following the method discussed by Trefethen (Reference Trefethen2000). Our boundary conditions at the surface of the wing, (4.20), map to

(4.26) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}^{\prime }(-1)=\unicode[STIX]{x1D6FA}^{\prime \prime }(-1)=0, & \displaystyle\end{eqnarray}$$

while the boundary conditions at $\bar{y}=\infty$ , (4.21), are transferred to

(4.27) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}(1)=\unicode[STIX]{x1D6FA}^{\prime }(1)=0. & \displaystyle\end{eqnarray}$$

With these imposed we solve the linear eigenvalue problem (4.25) and finally apply the normalisation condition

(4.28) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}(-1)=1 & \displaystyle\end{eqnarray}$$

to the resulting eigenfunctions.

As $|\bar{x}|\rightarrow \pm \infty$ , the chordwise wavenumber $\tilde{\unicode[STIX]{x1D6FC}}_{1}$ can match with the inner limit of our previous dispersion relation:

(4.29) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D6FC}}_{1}\sim \unicode[STIX]{x1D6FC}_{0}^{1/4}2^{3/2}\cos \left(\frac{\unicode[STIX]{x03C0}}{8}\right){\displaystyle \frac{12\unicode[STIX]{x1D706}_{4}^{0}(\unicode[STIX]{x1D6FF}_{1}+\unicode[STIX]{x1D6FF}_{3})}{|\dot{\unicode[STIX]{x1D706}}_{2}^{0}\bar{x}|^{3/4}\unicode[STIX]{x1D706}_{1}^{x}}}\left(-\tan {\displaystyle \frac{\unicode[STIX]{x03C0}}{8}}\pm \text{i}\right)\quad \text{for }\bar{x}\gtrless 0,\;\bar{x}\gg 1. & \displaystyle\end{eqnarray}$$

To find the eigenvalue $\tilde{\unicode[STIX]{x1D6FC}}_{1}$ for $\bar{x}=O(1)$ , we use the MATLAB routine $\mathtt{eigs}$ to track the value of $\tilde{\unicode[STIX]{x1D6FC}}_{1}$ as we march up- or downstream, starting with (4.29) as an initial condition.

For the numerical values of the local flow parameters involved, we have used the Joukowskii aerofoil flow seen previously in figure 3. These values are

(4.30a-e ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{0}=1.780,\quad & \unicode[STIX]{x1D706}_{1}^{x}=1.592,\quad \dot{\unicode[STIX]{x1D706}}_{2}^{0}=1.980,\quad \unicode[STIX]{x1D706}_{4}^{0}=-0.147,\quad \unicode[STIX]{x1D6F6}=-0.121. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

For this base flow, we have found two branches of solutions in the effective pressure minimum (EPM) region: a branch of ‘downstream’ modes that matches with (4.29) as $\bar{x}\rightarrow \infty$ , and an ‘upstream’ branch that matches (4.29) as $\bar{x}\rightarrow -\infty$ . These are displayed in figure 5.

Figure 5.

The (a) downstream and (b) upstream effective pressure minimum (EPM) modes. Solid lines represent the modes within the EPM region, while dashed lines show the asymptotic behaviour from the non-parallel regime solution.

The downstream EPM mode, shown in figure 5(a), is damped ( $-\text{Im}\{\tilde{\unicode[STIX]{x1D6FC}}_{1}\}<0$ ), with a wavenumber much larger than that of the non-parallel regime mode upstream of the effective pressure minimum. The behaviour of the eigenfunction for this mode at different $\bar{x}$ locations is displayed in figure 6. Upstream, the damped mode is confined in the wall region but spreads out near the effective pressure minimum. Downstream, the mode becomes trapped again in the wall region and develops into the branch-II mode considered in § 3, eventually becoming unstable at the second neutral point and following the sequence of regimes described by Choudhari (Reference Choudhari1994a , Reference Choudhari1995).

The upstream EPM mode has a more interesting behaviour. It is the continuation of the unstable branch-I mode into the EPM region, and remains unstable throughout (figure 5 b). An important fact to note is that from $\bar{x}\geqslant 0$ , a critical layer forms at the maximum of the effective velocity and is thus of the kind identified by Cowley, Hocking & Tutty (Reference Cowley, Hocking and Tutty1985). The behaviour of the chordwise velocity profile of the mode is shown in figure 7: downstream of the effective pressure minimum, the perturbation velocity profile alternates sign rapidly with one or more prominent peaks, and is concentrated in the critical layer, which lifts up from the surface of the wing. At $\bar{x}\approx 10$ our numerical results become spurious as this critical layer reaches the upper end of the numerical domain, where the grid for the physical $\bar{y}$ domain becomes too coarse to properly resolve the critical layer due to the exponential mapping used.

Figure 6.

Chordwise velocity profiles of the downstream EPM mode. Velocity profiles 1–3 are for $\bar{x}<0$ , while 4–6 are for $\bar{x}>0$ .

Figure 7.

Chordwise velocity profiles of the upstream EPM mode at different streamwise positions.

4.4 Development into Cowley, Hocking and Tutty modes

4.4.1 Lower-deck analysis

The results shown in figure 7 indicate the emergence of a critical layer downstream of $\bar{x}=0$ . In the lower deck, where $\bar{y}=Re^{29/54}y=O(1)$ , the perturbation is governed by (4.15) and (4.16), in which the polynomial $f(\bar{y})$ , defined by (4.14), consists of the leading-order terms from the lower-deck expansion of the effective velocity

(4.31) $$\begin{eqnarray}\displaystyle & \displaystyle \mathbb{U}=Re^{-2/9}[\unicode[STIX]{x1D6FC}U+\unicode[STIX]{x1D6FD}W]. & \displaystyle\end{eqnarray}$$

We now show that the upstream EPM mode develops into the type introduced by Cowley, Hocking & Tutty (Reference Cowley, Hocking and Tutty1985). This instability, referred to as a CHT instability, can occur if, at some transverse location $\bar{y}_{c}$ , the effective velocity is equal to the local wave speed (i.e. zero for a stationary mode) and has zero shear. This corresponds to

(4.32) $$\begin{eqnarray}\displaystyle 0=f_{0}(\bar{y}_{c})=\text{i}\unicode[STIX]{x1D6FC}_{0}\left(\dot{\unicode[STIX]{x1D706}}_{2}^{0}\bar{x}\bar{y}_{c}^{2}+\unicode[STIX]{x1D706}_{4}^{0}\bar{y}_{c}^{4}\right)+\text{i}\tilde{\unicode[STIX]{x1D6FC}}_{10}\unicode[STIX]{x1D706}_{1}^{x}\bar{y}_{c}, & & \displaystyle\end{eqnarray}$$

(4.33) $$\begin{eqnarray}\displaystyle 0=f_{0}^{\prime }(\bar{y}_{c})=\text{i}\unicode[STIX]{x1D6FC}_{0}\left(2\dot{\unicode[STIX]{x1D706}}_{2}^{0}\bar{x}\bar{y}_{c}+4\unicode[STIX]{x1D706}_{4}^{0}\bar{y}_{c}^{3}\right)+\text{i}\tilde{\unicode[STIX]{x1D6FC}}_{10}\unicode[STIX]{x1D706}_{1}^{x}. & & \displaystyle\end{eqnarray}$$

Here we have expanded $\tilde{\unicode[STIX]{x1D6FC}}_{1}$ and $f(\bar{y})$ as

(4.34a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D6FC}}_{1}=\tilde{\unicode[STIX]{x1D6FC}}_{10}+\tilde{\unicode[STIX]{x1D6FC}}_{11},\quad f(\bar{y})=f_{0}(\bar{y})+\text{i}\tilde{\unicode[STIX]{x1D6FC}}_{11}\bar{y}, & \displaystyle\end{eqnarray}$$

with $\tilde{\unicode[STIX]{x1D6FC}}_{11}\ll \tilde{\unicode[STIX]{x1D6FC}}_{10}$ for $\bar{x}\gg 1$ . For a critical layer away from the surface of the wing, (4.32) and (4.33) yield

(4.35) $$\begin{eqnarray}\displaystyle \bar{y}_{c}=\sqrt{{\displaystyle \frac{\dot{\unicode[STIX]{x1D706}}_{2}^{0}}{3|\unicode[STIX]{x1D706}_{4}^{0}|}}}\bar{x}^{1/2}=:\tilde{q}\bar{x}^{1/2}, & & \displaystyle\end{eqnarray}$$

(4.36) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D6FC}}_{10}=-{\displaystyle \frac{2\unicode[STIX]{x1D6FC}_{0}}{\unicode[STIX]{x1D706}_{1}^{x}}}\sqrt{{\displaystyle \frac{(\dot{\unicode[STIX]{x1D706}}_{2}^{0})^{3}}{27|\unicode[STIX]{x1D706}_{4}^{0}|}}}\bar{x}^{3/2}. & & \displaystyle\end{eqnarray}$$

Since $\tilde{\unicode[STIX]{x1D6FC}}_{10}$ is real, it is $\tilde{\unicode[STIX]{x1D6FC}}_{11}$ that describes the development of the instability.

Such a critical layer can only occur for $\bar{x}>0$ , as $\dot{\unicode[STIX]{x1D706}}_{2}^{0}>0$ and $\unicode[STIX]{x1D706}_{4}^{0}<0$ . Within this layer, where $\bar{y}-\bar{y}_{c}=\hat{\unicode[STIX]{x1D6FF}}\ll 1$ , we can expand our effective velocity profile as

(4.37) $$\begin{eqnarray}\displaystyle \displaystyle f(\bar{y}) & {\sim} & \displaystyle f_{0}(\bar{y}_{c})+f_{0}^{\prime }(\bar{y}_{c})\hat{\unicode[STIX]{x1D6FF}}+{\textstyle \frac{1}{2}}f_{0}^{\prime \prime }(\bar{y}_{c})\hat{\unicode[STIX]{x1D6FF}}^{2}+O(\hat{\unicode[STIX]{x1D6FF}}^{3})+\text{i}\tilde{\unicode[STIX]{x1D6FC}}_{11}\bar{y}\nonumber\\ \displaystyle & {\sim} & \displaystyle {\textstyle \frac{1}{2}}(-2\text{i}\unicode[STIX]{x1D6FC}_{0}\dot{\unicode[STIX]{x1D706}}_{2}^{0}\bar{x})\hat{\unicode[STIX]{x1D6FF}}^{2}+\text{i}\tilde{\unicode[STIX]{x1D6FC}}_{11}\bar{y}+\cdots \,.\end{eqnarray}$$

Within the critical layer, viscosity and inertia must balance in order to prevent the solution from becoming singular. From (4.15) and (4.35), we thus have

(4.38) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}{\mathcal{U}}}{\text{d}\bar{y}^{2}}\sim f(\bar{y}){\mathcal{U}}, & \displaystyle\end{eqnarray}$$

which imposes the behaviour

(4.39a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D6FF}}\sim \bar{x}^{-1/4},\quad \tilde{\unicode[STIX]{x1D6FC}}_{11}\sim \bar{x}^{0}. & \displaystyle\end{eqnarray}$$

In order to determine $\tilde{\unicode[STIX]{x1D6FC}}_{11}$ , we can expand (4.16) in terms of $\bar{x}$ and compare the leading-order terms with equation (7a) in Cowley et al. (Reference Cowley, Hocking and Tutty1985). On letting

(4.40a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D6FF}}=:\bar{x}^{-1/4}\mathtt{y},\quad f_{0}^{\prime \prime }(\bar{y}_{c})\hat{\unicode[STIX]{x1D6FF}}^{2}=-2\text{i}\unicode[STIX]{x1D6FC}_{0}\dot{\unicode[STIX]{x1D706}}_{2}^{0}\bar{x}^{1/2}\mathtt{y}^{2}=:{\textstyle \frac{1}{2}}\mathbb{U}_{c}^{\prime \prime }\bar{x}^{1/2}\mathtt{y}^{2}, & \displaystyle\end{eqnarray}$$

the leading-order terms in (4.16) can be written as

(4.41) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}{\mathcal{V}}_{\mathtt{y}\mathtt{y}\mathtt{y}}=-(-\text{i})\left({\textstyle \frac{1}{2}}\mathbb{U}_{c}^{\prime \prime }\mathtt{y}^{2}-(-\text{i}\unicode[STIX]{x1D706}_{1}^{x}\tilde{q}\tilde{\unicode[STIX]{x1D6FC}}_{11})\right){\mathcal{V}}_{\mathtt{y}}+(-\text{i})\mathbb{U}_{c}^{\prime \prime }\mathtt{y}{\mathcal{V}}, & \displaystyle\end{eqnarray}$$

where $\tilde{q}$ and $\mathbb{U}_{c}^{\prime \prime }$ are defined in (4.35) and (4.40) respectively. Comparing (4.41) with equation (7a) in Cowley et al. (Reference Cowley, Hocking and Tutty1985), we see that their parameters $K_{0}$ and $c_{1}$ are here given by $K_{0}=-\text{i}$ and $c_{1}=-\text{i}\unicode[STIX]{x1D706}_{1}^{x}\tilde{q}\tilde{\unicode[STIX]{x1D6FC}}_{11}$ . The eigenvalues corresponding to equation (10) in Cowley et al. (Reference Cowley, Hocking and Tutty1985) are thus

(4.42) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D6FC}}_{11}=\frac{(4N+7)}{\unicode[STIX]{x1D706}_{1}^{x}}\sqrt{{\displaystyle \frac{3\unicode[STIX]{x1D6FC}_{0}|\unicode[STIX]{x1D706}_{4}^{0}|}{2}}}(1+\text{i})\quad \text{for }N\in \{-2,-1,0,\ldots \}. & \displaystyle\end{eqnarray}$$

The most (and only) unstable mode, for which $-\text{Im}\{\tilde{\unicode[STIX]{x1D6FC}}_{1}\}>0$ , is the $N=-2$ mode. Our critical-layer chordwise wavenumber is thus given by

(4.43) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D6FC}}_{1}=-{\displaystyle \frac{2\unicode[STIX]{x1D6FC}_{0}}{\unicode[STIX]{x1D706}_{1}^{x}}}\sqrt{{\displaystyle \frac{(\dot{\unicode[STIX]{x1D706}}_{2}^{0})^{3}}{27|\unicode[STIX]{x1D706}_{4}^{0}|}}}\bar{x}^{3/2}-\frac{1}{\unicode[STIX]{x1D706}_{1}^{x}}\sqrt{{\displaystyle \frac{3\unicode[STIX]{x1D6FC}_{0}|\unicode[STIX]{x1D706}_{4}^{0}|}{2}}}(1+\text{i}). & \displaystyle\end{eqnarray}$$

The asymptotic solution for $\bar{x}\gg 1$ , (4.43), is compared with the numerical solution of (4.25) in figure 8. Clearly, there is very good agreement – even from as early as $\bar{x}\approx 2$ . Thus the upstream EPM mode does evolve into a CHT mode.

Figure 8.

Comparison of numerical results with the asymptotic prediction (4.43) for the Cowley, Hocking and Tutty (CHT) instability wavenumber.

4.4.2 Downstream behaviour of the critical layer

When $(x-x_{min})$ becomes $O(1)$ , two things occur: the height of the critical layer $\bar{y}_{c}$ becomes $O(Re^{1/27})$ as the critical layer moves into the main deck of the boundary layer; and $\tilde{\unicode[STIX]{x1D6FC}}_{1}$ becomes $O(Re^{1/9})$ and adjusts the chordwise wavenumber at leading order, according to (4.8). The growth rate, however, remains $O(Re^{1/9})$ .

Figure 9.

(a) Location of the critical layer and (b) wavenumber ratio for the non-parallel regime and CHT modes.

In the main part of the boundary layer, the equations equivalent to (4.32) and (4.33) that determine the critical level are $\unicode[STIX]{x1D6FC}_{c}U+\unicode[STIX]{x1D6FD}W=0$ , $\unicode[STIX]{x1D6FC}_{c}U^{\prime }+\unicode[STIX]{x1D6FD}W^{\prime }=0$ . In order for a solution $(\unicode[STIX]{x1D6FC}_{c},\unicode[STIX]{x1D6FD})\neq (0,0)$ to exist, the determinant $[UW^{\prime }-WU^{\prime }]$ must be zero. This allows us to determine the location $Y_{c}\neq 0$ where this occurs, with which we can then find the chordwise wavenumber from

(4.44) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x1D6FC}_{c}}{\unicode[STIX]{x1D6FD}}}=-{\displaystyle \frac{W(Y_{c})}{U(Y_{c})}}. & \displaystyle\end{eqnarray}$$

We calculate this ratio and the location of the critical layer for the Joukowskii aerofoil discussed in § 3.5. The results are shown in figure 9. We again find that a CHT critical layer can exist within the boundary layer downstream of the effective pressure minimum, until it reaches the artificial limit of our numerical boundary layer.

However, the instability will not simply continue to rise until it eventually leaves the boundary layer. The downstream behaviour of the instability will be similar to that of the CHT mode considered by Zhao (Reference Zhao2011), as well as that originally described by Cowley et al. (Reference Cowley, Hocking and Tutty1985). Zhao demonstrated that as the boundary layer thickens the local wavenumbers increase, and the perturbation will take on an interactive structure further downstream when the self-induced pressure reappears at leading order. The perturbation ultimately develops into an inviscid Rayleigh mode with its wavelength comparable with the boundary-layer thickness.

A similar scaling argument to that used by Zhao allows us to determine when the self-induced pressure terms will reappear in the leading-order governing equations. The flow has a five-tiered structure: a lower deck near the surface of the wing in order to satisfy the no-slip condition; a main deck lying either side of a critical layer located at the maximum of the effective base flow; and an upper deck in order to satisfy the attenuation condition. It is within this upper deck that the self-induced pressure is generated, and via a scaling argument similar to that in § 2.3 we find that $p^{\prime }\sim \unicode[STIX]{x1D6FC}Re^{-1/2}$ in each of the layers.

The self-induced pressure terms appear at leading-order within the projected critical-layer equations when

(4.45) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}U_{eff}(\unicode[STIX]{x1D6FC}u^{\prime }+\unicode[STIX]{x1D6FD}w^{\prime })\sim (\unicode[STIX]{x1D6FC}^{2}+\unicode[STIX]{x1D6FD}^{2})p^{\prime }\sim (\unicode[STIX]{x1D6FC}u^{\prime }+\unicode[STIX]{x1D6FD}w^{\prime })_{YY}. & \displaystyle\end{eqnarray}$$

Since near the critical layer $U_{eff}\sim \unicode[STIX]{x1D6FF}^{2}$ , where $\unicode[STIX]{x1D6FF}=(Y-Y_{c})$ , and in the main deck $(\unicode[STIX]{x1D6FC}u^{\prime }+\unicode[STIX]{x1D6FD}w^{\prime })\propto U_{eff,Y}$ , we have that

(4.46) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}^{2}\cdot \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x1D6FC}^{2}\cdot p^{\prime }\sim \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}\cdot \unicode[STIX]{x1D6FF}^{-2}, & \displaystyle\end{eqnarray}$$

and so $\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x1D6FC}^{-1/4}$ and $p^{\prime }\sim \unicode[STIX]{x1D6FC}^{-3/4}$ . Comparing the two pressure scalings, we find that the self-induced pressure terms appear at leading order in the governing equations for the critical layer when

(4.47) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}\sim Re^{2/7}, & \displaystyle\end{eqnarray}$$

agreeing with Cowley et al. (Reference Cowley, Hocking and Tutty1985). This happens to be the scaling for the wall-shear-aligned regime, however the structure of the perturbed flow is now different due to the presence of the critical layer; the mode is not aligned with the local wall shear at leading order, as demonstrated in figure 9(b). A brief calculation reveals that the wavenumber takes on this scaling in terms of the local Reynolds number when $x^{\ast }/x_{min}^{\ast }\sim Re_{min}^{4/45}$ , where $Re_{min}$ is the local Reynolds number at the effective pressure minimum. For a typical local Reynolds number of the order of $10^{3}$ , this corresponds to $x^{\ast }\approx 2x_{min}^{\ast }$ . Further downstream, where the ratio of the boundary-layer thickness to the wavelength of the mode is larger, the role of viscosity will diminish and the perturbation will take on the form of an inviscid Rayleigh mode when $\unicode[STIX]{x1D6FC}\sim Re^{1/2}$ .

The development of the crossflow mode after the effective pressure minimum thus differs from the standard sequence of regimes laid out by Choudhari (Reference Choudhari1994a , Reference Choudhari1995), due to the presence of this critical layer. It would be worthwhile to perform an in-depth study of this new pathway for the crossflow mode, beyond the scaling arguments carried out here.

5.1 Main deck

The analysis in Fourier space for the main deck, where $Y=Re^{1/2}y=O(1)$ , proceeds just as in § 3.1. We find that for $j\in \{0,\ldots ,3\}$ ,

(5.7a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{j}^{+}=A_{j}^{+}(k;x,z){\displaystyle\mathop{ U}\limits_{{\sim}}}_{Y},\quad \tilde{v}_{j}^{+}=-\text{i}kA_{j}^{+}\tilde{U} _{eff},\quad \tilde{p}_{j}^{+}=P_{j}^{+}(k;x,z), & \displaystyle\end{eqnarray}$$

where the effective base flow is now given by $\tilde{U} _{eff}:=U+\left(\unicode[STIX]{x1D6FD}_{0}/k\right)W$ . For $Y\ll 1$ , the perturbation velocity components behave as

(5.8a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{j}^{+}\sim A_{j}^{+}(\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}+2\unicode[STIX]{x1D706}_{2}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}Y+\cdots \,),\quad \tilde{v}_{j}^{+}\sim -\text{i}kA_{j}^{+}(\tilde{\unicode[STIX]{x1D706}}_{1}^{0}Y+\tilde{\unicode[STIX]{x1D706}}_{2}^{0}Y^{2}+\cdots \,), & \displaystyle\end{eqnarray}$$

where use has been made of the Maclaurin coefficients defined in (2.9) and their projected counterparts for the effective base flow, $\tilde{\unicode[STIX]{x1D706}}_{n}^{0}:=\unicode[STIX]{x1D706}_{n}^{x}+(\unicode[STIX]{x1D6FD}_{0}/k)\unicode[STIX]{x1D706}_{n}^{z}$ . Since the perturbation now has a full wavenumber spectrum, the first projected Maclaurin coefficient $\tilde{\unicode[STIX]{x1D706}}_{1}^{0}(k)$ is in general non-zero.

Note that $\tilde{v}_{0}^{+}\rightarrow 0$ as $Y\rightarrow 0$ , but $\tilde{u} _{0}$ and $\tilde{w}_{0}$ do not satisfy the boundary condition (5.5) (a priori). We must introduce a lower deck in order to address this. We similarly need to introduce an upper deck in order to satisfy the attenuation condition (2.18). The analysis for the upper deck follows that in § 3.2, and so will not be repeated here. The balance between inertia and viscosity in the lower deck, as discussed in § 3.3, now takes the form

(5.9) $$\begin{eqnarray}\displaystyle & \displaystyle Re^{2/9}(\tilde{\unicode[STIX]{x1D706}}_{1}^{0}Y+\tilde{\unicode[STIX]{x1D706}}_{2}^{0}Y^{2}+\cdots \,)\tilde{u} ^{+}\sim \tilde{u} ^{+}/Y^{2}. & \displaystyle\end{eqnarray}$$

The thickness of the lower deck is determined by the dominant term on the left, which in turn depends on whether $\tilde{\unicode[STIX]{x1D706}}_{1}^{0}=O(1)$ or $\tilde{\unicode[STIX]{x1D706}}_{1}^{0}\ll 1$ :

(5.10a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle y\sim Re^{-31/54},\quad Y\sim Re^{-2/27}\quad \text{for }\tilde{\unicode[STIX]{x1D706}}_{1}^{0}=O(1), & \displaystyle\end{eqnarray}$$

(5.11a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle y\sim Re^{-5/9},\quad Y\sim Re^{-1/18}\quad \text{for }\tilde{\unicode[STIX]{x1D706}}_{1}^{0}\ll 1. & \displaystyle\end{eqnarray}$$

The roughness-induced response thus exhibits different behaviours depending on the chordwise wavenumber $k$ . The resulting partition of the wavenumber spectrum into two distinct regions as depicted in figure 10 – those components aligned with the wall shear and those that are misaligned – is a feature of the present receptivity problem.

Figure 10.

Schematic of the lower deck for the receptivity problem.

In order for nonlinear interactions in the lower deck to be negligible, the flow perturbation must be much smaller than the effective base flow, i.e. $u^{\prime }\ll \tilde{U} _{eff}$ , so that we can neglect terms that are quadratic in the perturbation. Here this corresponds to $h\ll \min \{Re^{-2/27},Re^{-1/9}\}=Re^{-1/9}$ . If the roughness height $h=O(Re^{-1/9})$ , we would need to include nonlinear interactions when $k$ is close to the wall-shear-aligned wavenumber $\unicode[STIX]{x1D6FC}_{0}$ (which satisfies $\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{1}^{x}+\unicode[STIX]{x1D6FD}_{0}\unicode[STIX]{x1D706}_{1}^{z}=0$ ). Finally, nonlinearity can also appear through the roughness boundary condition (5.4). The full spectral content of the roughness shape must be taken into account when $h=O(Re^{-1/6})$ ; this will be considered in detail in § 5.6.

We will not consider these sources of nonlinearity here, instead assuming $h\ll Re^{-1/9}$ . In passing, we note that in a neighbourhood of the effective pressure minimum, where $U_{eff}\sim Y^{4}$ , the linearity condition for $h$ is restricted further to $h\ll Re^{-4/27}$ .

5.2 Lower deck – wall-shear-misaligned components

We first consider the Fourier components in the response with a general wavenumber $k$ , for which $\tilde{\unicode[STIX]{x1D706}}_{1}^{0}=O(1)$ . According to (5.10), the lower-deck variable is (figure 10)

(5.12) $$\begin{eqnarray}\displaystyle & \displaystyle \check{y}=Re^{2/27}Y=O(1). & \displaystyle\end{eqnarray}$$

The small- $Y$ behaviour of the solution in the main deck, (5.8), suggests that the perturbation here takes the form

(5.13) $$\begin{eqnarray}\displaystyle & \displaystyle (\boldsymbol{u}^{\prime +},p^{\prime +})(k,\check{y};x,z)=\left({\check{u}}^{+},Re^{-19/54}\check{v}^{+},\check{w}^{+},Re^{-15/54}\check{p}^{+}\right). & \displaystyle\end{eqnarray}$$

As well as the main-deck matching conditions, this solution is subject to the roughness boundary conditions (5.5).

At leading order, the governing equations for the perturbation are

(5.14a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\text{i}k\tilde{\unicode[STIX]{x1D706}}_{1}^{0}\check{y}\,\check{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}^{+}+\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\check{v}_{0}^{+}-\check{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0,\check{y}\check{y}}^{+},\quad 0=\check{p}_{0,\check{y}}^{+},\quad 0=\text{i}k{\check{u}}_{0}^{+}+\check{v}_{0,\check{y}}^{+}+\text{i}\unicode[STIX]{x1D6FD}_{0}\check{w}_{0}^{+}. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

It is worth noting that the pressure gradient does not appear in the momentum equations. Combining the chordwise and spanwise momentum equations via the continuity equation and rewriting the resultant equation in terms of $\unicode[STIX]{x1D701}=(\text{i}k\tilde{\unicode[STIX]{x1D706}}_{1}^{0})^{1/4}\check{y}$ brings us to

(5.15) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{\text{d}^{3}}{\text{d}\unicode[STIX]{x1D701}^{3}}-\unicode[STIX]{x1D701}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D701}}+1\right]\check{v}_{0}^{+}=0. & \displaystyle\end{eqnarray}$$

The general solution of this can be expressed in terms of Airy functions. However, the solution that satisfies the appropriate boundary conditions is simply

(5.16) $$\begin{eqnarray}\displaystyle & \displaystyle \check{v}_{0}^{+}(k,\unicode[STIX]{x1D701})=(\text{i}k\tilde{\unicode[STIX]{x1D706}}_{1}^{0})^{2/3}F^{+}(k)\unicode[STIX]{x1D701}. & \displaystyle\end{eqnarray}$$

The leading-order ${\displaystyle\mathop{x}\limits_{{\sim}}}$ -momentum equation can then be written as

(5.17) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{\text{d}^{2}}{\text{d}\unicode[STIX]{x1D701}^{2}}-\unicode[STIX]{x1D701}\right](\check{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}^{+}+\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}F^{+})=0. & \displaystyle\end{eqnarray}$$

The solution that satisfies (5.5) and can be matched with its main-deck counterpart is

(5.18) $$\begin{eqnarray}\displaystyle & \displaystyle \check{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}^{+}(k,\unicode[STIX]{x1D701})=-\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}F^{+}(k), & \displaystyle\end{eqnarray}$$

and matching this with the main-deck solution gives us the relation

(5.19) $$\begin{eqnarray}\displaystyle & \displaystyle A_{0}^{+}(k)=-F^{+}(k). & \displaystyle\end{eqnarray}$$

Interestingly, the lower-deck solution is simply the continuation of the inviscid main-deck solution. This is a consequence of the pressure gradient being absent in the momentum equations.

5.3 Lower deck – almost wall-shear-aligned components

We now turn to components with $k$ such that $|k-\unicode[STIX]{x1D6FC}_{0}|=O(Re^{-1/6})$ (figure 10), where $\unicode[STIX]{x1D6FC}_{0}$ is given by (2.21). According to (5.11), the lower-deck variable is

(5.20) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{y}=Re^{1/18}Y=O(1). & \displaystyle\end{eqnarray}$$

In keeping with the wavenumber scaling derived in § 2.3, we write $k$ as

(5.21) $$\begin{eqnarray}\displaystyle k=\unicode[STIX]{x1D6FC}_{0}+Re^{-1/6}k_{1}. & & \displaystyle\end{eqnarray}$$

The matching conditions with the main-deck solution then become

(5.22) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \,{\displaystyle\mathop{u}\limits_{{\sim}}}^{\prime +}\sim \unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{0}^{+}+\unicode[STIX]{x1D700}(2\unicode[STIX]{x1D706}_{2}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{0}^{+}\bar{y}+\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{1}^{+})+\unicode[STIX]{x1D700}^{2}(2\unicode[STIX]{x1D706}_{2}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{1}^{+}\bar{y}+\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}A_{2}^{+})+O(\unicode[STIX]{x1D700}^{3}),\end{eqnarray}$$

(5.23) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle v^{\prime +}\sim \unicode[STIX]{x1D700}^{7}\left(-\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}A_{0}^{+}\bar{y}^{2}\right)+\unicode[STIX]{x1D700}^{8}\left(-\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}A_{1}^{+}\bar{y}^{2}\right)\nonumber\\ \displaystyle & & \displaystyle +\unicode[STIX]{x1D700}^{9}\left(-\text{i}k_{1}\unicode[STIX]{x1D706}_{1}^{x}A_{0}^{+}\bar{y}-\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}A_{2}^{+}\bar{y}^{2}-\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{4}^{0}A_{0}^{+}\bar{y}^{4}\right)+O(\unicode[STIX]{x1D700}^{10}),\end{eqnarray}$$

(5.24) $$\begin{eqnarray}\displaystyle p^{\prime +} & & \displaystyle \displaystyle \sim \unicode[STIX]{x1D700}^{5}P_{0}^{+}+\unicode[STIX]{x1D700}^{6}P_{1}^{+}+\unicode[STIX]{x1D700}^{7}P_{2}^{+}+O(\unicode[STIX]{x1D700}^{8}),\end{eqnarray}$$

where $\unicode[STIX]{x1D700}=Re^{-1/18}$ and $\unicode[STIX]{x1D706}_{n}^{0}=\unicode[STIX]{x1D706}_{n}^{x}+(\unicode[STIX]{x1D6FD}_{0}/\unicode[STIX]{x1D6FC}_{0})\unicode[STIX]{x1D706}_{n}^{z}$ . The lower-deck perturbation now takes the form

(5.25) $$\begin{eqnarray}\displaystyle & \displaystyle (\boldsymbol{u}^{\prime +},p^{\prime +})(k,\bar{y};x;z)=(\bar{u}^{+},\unicode[STIX]{x1D700}^{7}\bar{v}^{+},\bar{w}^{+},\unicode[STIX]{x1D700}^{5}\bar{p}^{+}), & \displaystyle\end{eqnarray}$$

and we expand each component as $\bar{u}^{+}=\bar{u}_{0}^{+}+\unicode[STIX]{x1D700}\bar{u}_{1}^{+}+\cdots \,$ , etc. The governing equations here are equivalent to those in § 3.3, namely (3.28)–(3.31) and (3.32)–(3.35), but are now subject to the boundary conditions

(5.26a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}^{+}(k,0;x,z)=-\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}F^{+}(k),\quad \bar{v}_{0}^{+}(k,0;x,z)=0. & \displaystyle\end{eqnarray}$$

At leading order, we again find that $\bar{v}_{0}^{+}=-\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}A_{0}^{+}\bar{y}^{2}$ , which is simply the continuation of the main-deck solution. The corresponding solution to the parabolic cylinder equation (3.38) is

(5.27) $$\begin{eqnarray}\displaystyle \bar{{\displaystyle\mathop{u}\limits_{{\sim}}}}_{0}^{+}(k,s)=\unicode[STIX]{x1D706}_{1}^{\phantom{x}}\hspace{0.0pt}^{\,{\displaystyle\mathop{x}\limits_{{\sim}}}}\left(A_{0}^{+}-(F^{+}+A_{0}^{+})\frac{{\mathcal{D}}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}\right). & & \displaystyle\end{eqnarray}$$

As before, the adjusted vertical velocity term ${\mathcal{V}}=\bar{v}_{2}^{+}+\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}A_{2}^{+}\bar{y}^{2}+\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{4}^{0}A_{0}^{+}\bar{y}^{4}$ is governed by

(5.28) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{\text{d}^{3}}{\text{d}s^{3}}-\frac{1}{4}s^{2}\frac{\text{d}}{\text{d}s}+{\displaystyle \frac{1}{2}}s\right]{\mathcal{V}}(k,s)=f^{+}(k_{1},s), & \displaystyle\end{eqnarray}$$

where $f^{+}(k_{1},s)$ is obtained from (3.48) by replacing $\unicode[STIX]{x1D6FC}_{1}$ by $k_{1}$ and using the spectral functions $A_{0}^{+}$ , $\bar{u}_{0}^{+}$ , and $\bar{p}_{0}^{+}$ .

Using the continuity equation (3.35), the boundary conditions in (5.5) translate to

(5.29a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{V}}(0)=0,\quad {\mathcal{V}}_{\bar{y}}(0)=-\text{i}k_{1}\bar{u}_{1}(0)=\text{i}k_{1}\unicode[STIX]{x1D706}_{1}^{x}F^{+}(k). & \displaystyle\end{eqnarray}$$

In the presence of roughness, the solvability condition (3.52) becomes

(5.30) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\infty }f^{+}s{\mathcal{D}}_{-1/2}(s)\,\text{d}s=(4\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0})^{-1/4}\text{i}k_{1}\unicode[STIX]{x1D706}_{1}^{x}F^{+}{\mathcal{D}}_{-1/2}(0). & \displaystyle\end{eqnarray}$$

Evaluating (5.30), we eventually arrive at

(5.31) $$\begin{eqnarray}\displaystyle & \displaystyle (k_{1}-g)F^{+}+(k_{1}-\unicode[STIX]{x1D6FC}_{1})A_{0}^{+}=0, & \displaystyle\end{eqnarray}$$

where

(5.32) $$\begin{eqnarray}\displaystyle & \displaystyle g=\left(-\frac{2^{3/2}\unicode[STIX]{x1D6FC}_{0}^{1/4}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x1D706}_{1}^{x}|\unicode[STIX]{x1D706}_{2}^{0}|^{3/4}}(\unicode[STIX]{x1D6FF}_{1}+\unicode[STIX]{x1D6FF}_{3})\left(\sin {\displaystyle \frac{\unicode[STIX]{x03C0}}{8}}\mp \text{i}\cos {\displaystyle \frac{\unicode[STIX]{x03C0}}{8}}\right)\right), & \displaystyle\end{eqnarray}$$

(5.33) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{1}=g+{\displaystyle \frac{\bar{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FC}_{0}^{3/2}U_{eff_{s}}^{2}}{2^{3/2}\unicode[STIX]{x1D706}_{1}^{x}|\unicode[STIX]{x1D706}_{2}^{0}|^{1/2}}}\unicode[STIX]{x1D6FF}_{2}(\mp 1-\text{i}). & \displaystyle\end{eqnarray}$$

The choice of $\pm$ is given by $\text{sign}\{\unicode[STIX]{x1D706}_{2}^{0}\}$ , and the numerical values of the $\unicode[STIX]{x1D6FF}_{j}\,(j=1,2,3)$ can be found in (3.53). This receptivity relation can be expressed in a generic form as

(5.34) $$\begin{eqnarray}\displaystyle & \displaystyle A_{0}^{+}(k_{1})=-F^{+}(k)(k_{1}-g)/\unicode[STIX]{x1D6E5}(k_{1},\unicode[STIX]{x1D6FD}_{0}), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6E5}(k_{1},\unicode[STIX]{x1D6FD}_{0})=(k_{1}-\unicode[STIX]{x1D6FC}_{1})$ is the dispersion relation (3.54). The amplitude $A_{0}^{+}(k_{1})$ matches appropriately with the general- $k$ solution, (5.19), as $k_{1}\rightarrow \infty$ .

5.4 Inverse Fourier transform

In the lower deck, we have found in spectral space that for $k_{1}=Re^{1/6}(k-\unicode[STIX]{x1D6FC}_{0})=O(1)$ ,

(5.35a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{u}_{0}^{+}=\unicode[STIX]{x1D706}_{1}^{x}\left(A_{0}^{+}(k)-\left(A_{0}^{+}(k)+F^{+}(k)\right)\frac{{\mathcal{D}}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}\right),\quad A_{0}^{+}(k)=-F^{+}(k)\frac{\left(k_{1}-g\right)}{\left(k_{1}-\unicode[STIX]{x1D6FC}_{1}\right)}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

while for $k_{1}\gg 1$ ,

(5.36a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{u}_{0}^{+}=-\unicode[STIX]{x1D706}_{1}^{x}F^{+}(k),\quad A_{0}^{+}(k)=-F^{+}(k). & \displaystyle\end{eqnarray}$$

In physical space, the leading-order chordwise velocity component in the lower deck is thus given by

(5.37) $$\begin{eqnarray}\displaystyle & & \displaystyle \bar{u}_{0}(\tilde{x},\bar{y};x,z)\text{e}^{\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{z}}=\frac{1}{2\unicode[STIX]{x03C0}}\left(\int _{-\infty }^{\unicode[STIX]{x1D6FC}_{0}-\unicode[STIX]{x1D6F4}}+\int _{\unicode[STIX]{x1D6FC}_{0}+\unicode[STIX]{x1D6F4}}^{\infty }\left[-\unicode[STIX]{x1D706}_{1}^{x}F^{+}(k)\right]\text{e}^{\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{z}}\text{e}^{\text{i}k\tilde{x}}\,\text{d}k\right)\nonumber\\ \displaystyle & & \displaystyle \quad +\,{\displaystyle \frac{1}{2\unicode[STIX]{x03C0}}}\int _{\unicode[STIX]{x1D6FC}_{0}-\unicode[STIX]{x1D6F4}}^{\unicode[STIX]{x1D6FC}_{0}+\unicode[STIX]{x1D6F4}}\left[-{\displaystyle \frac{\unicode[STIX]{x1D706}_{1}^{x}F^{+}(k)}{(k_{1}-\unicode[STIX]{x1D6FC}_{1})}}\left((k_{1}-g)-(\unicode[STIX]{x1D6FC}_{1}-g){\displaystyle \frac{{\mathcal{D}}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}}\right)\right]\text{e}^{\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{z}}\text{e}^{\text{i}k\tilde{x}}\,\text{d}k,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

for some suitable choice of $\unicode[STIX]{x1D6F4}$ such that the separation of the two expressions for $\bar{u}_{0}^{+}$ is valid. We can express (5.36) in terms of the large- $k_{1}$ limit of (5.35), and so rewrite the integrand within the first two integrals in (5.37) in order to extend the third integral to the whole of the real line. Doing so, we arrive at

(5.38) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \bar{u}_{0}(\tilde{x},\bar{y};x,z)\text{e}^{\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{z}}\nonumber\\ \displaystyle & & \displaystyle \quad ={\displaystyle \frac{1}{2\unicode[STIX]{x03C0}}}\int _{-\infty }^{\infty }\left[-\unicode[STIX]{x1D706}_{1}^{x}\frac{F^{+}(k)}{(k_{1}-\unicode[STIX]{x1D6FC}_{1})}\left((k_{1}-\unicode[STIX]{x1D6FC}_{1})+\left(\unicode[STIX]{x1D6FC}_{1}-g\right)\left(1-{\displaystyle \frac{{\mathcal{D}}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}}\right)\right)\right]\text{e}^{\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{z}}\text{e}^{\text{i}k\tilde{x}}\,\text{d}k\nonumber\\ \displaystyle & & \displaystyle \quad +\,O\left(\displaystyle {\displaystyle \frac{1}{2\unicode[STIX]{x03C0}}}\left(\int _{-\infty }^{\unicode[STIX]{x1D6FC}_{0}-\unicode[STIX]{x1D6F4}}+\int _{\unicode[STIX]{x1D6FC}_{0}+\unicode[STIX]{x1D6F4}}^{\infty }{\displaystyle \frac{F^{+}(k)}{Re^{1/6}(k-\unicode[STIX]{x1D6FC}_{0})}}\text{e}^{\text{i}k\tilde{x}}\,\text{d}k\right)\right).\end{eqnarray}$$

If we impose that $F(\tilde{x})\in C^{n}$ and is bounded, then by the Riemann–Lebesgue lemma this correction term is $O(Re^{-(n+2)/6}\unicode[STIX]{x1D6F4}^{-\left(n+1\right)})$ . This is much smaller than $O(Re^{-1/6})$ , the order of magnitude of the first term, if $\unicode[STIX]{x1D6F4}\gg Re^{-1/6}$ .

The disturbance has a complicated structure for $\tilde{x}=O(1)$ , due to the contribution from the branch cut for $s^{1/4}$ , but sufficiently downstream the response of the flow is dominated by the unstable mode corresponding to the simple pole given by $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FD}_{0})=0$ (Wu Reference Wu2001b ). Using Cauchy’s residue theorem, we find that as $\tilde{x}\rightarrow \infty$ ,

(5.39) $$\begin{eqnarray}\displaystyle & \displaystyle [\bar{u}_{0}+\unicode[STIX]{x1D706}_{1}^{x}F(\tilde{x})]\text{e}^{\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{z}}\sim Re^{-1/6}\left[\unicode[STIX]{x1D706}_{1}^{x}\left(1-\frac{{\mathcal{D}}_{-1/2}(s)}{{\mathcal{D}}_{-1/2}(0)}\right)\text{e}^{\text{i}\left(\unicode[STIX]{x1D6FC}_{cf}\tilde{x}+\unicode[STIX]{x1D6FD}_{0}\tilde{z}\right)}\right][-\text{i}(\unicode[STIX]{x1D6FC}_{1}-g)F^{+}(\unicode[STIX]{x1D6FC}_{0})], & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{cf}=\unicode[STIX]{x1D6FC}_{0}+Re^{-1/6}\unicode[STIX]{x1D6FC}_{1}$ . By the linearity of the problem, the chordwise perturbation for the generated crossflow mode can be separated into its constituent parts and written in a similar fashion to Choudhari & Duck (Reference Choudhari and Duck1996) as

(5.40) $$\begin{eqnarray}\displaystyle & \displaystyle u_{cf}^{\prime }:=h\left(\bar{u}_{0}+\unicode[STIX]{x1D706}_{1}^{x}F\right)\text{e}^{\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{z}}=Re^{-1/6}\left({\displaystyle \frac{h^{\ast }}{\unicode[STIX]{x1D6FF}^{\ast }}}\right)F^{+}(\unicode[STIX]{x1D6FC}_{0})u_{E}(\unicode[STIX]{x1D6FD}_{0},\bar{y})\,{\mathcal{C}}(\unicode[STIX]{x1D6FD}_{0})\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{cf}\tilde{x}+\unicode[STIX]{x1D6FD}_{0}\tilde{z})}. & \displaystyle\end{eqnarray}$$

Here $h^{\ast }$ is the dimensional height of the surface roughness, while $\unicode[STIX]{x1D6FF}^{\ast }$ is the dimensional boundary-layer thickness at the chordwise location we are considering. The eigenfunction $u_{E}(\unicode[STIX]{x1D6FD}_{0},\bar{y})$ is given by

(5.41) $$\begin{eqnarray}\displaystyle u_{E}(\unicode[STIX]{x1D6FD}_{0},\bar{y})=\left(1-\frac{{\mathcal{D}}_{-1/2}((4\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0})^{1/4}\bar{y})}{{\mathcal{D}}_{-1/2}(0)}\right), & & \displaystyle\end{eqnarray}$$

and the coupling coefficient ${\mathcal{C}}(\unicode[STIX]{x1D6FD}_{0})$ by

(5.42) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{C}}(\unicode[STIX]{x1D6FD}_{0})={\displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}^{5/2}U_{eff_{s}}^{2}}{2^{3/2}|\unicode[STIX]{x1D706}_{1}^{z}||\unicode[STIX]{x1D706}_{2}^{0}|^{1/2}}}((\unicode[STIX]{x1D706}_{1}^{x})^{2}+(\unicode[STIX]{x1D706}_{1}^{z})^{2})^{1/2}\unicode[STIX]{x1D6FF}_{2}(-1\pm \text{i}). & \displaystyle\end{eqnarray}$$

This coupling coefficient is independent of the specific form of the surface roughness, whose effect is encapsulated in $F^{+}(\unicode[STIX]{x1D6FC}_{0})$ . Because of the form of $g$ in (5.34), $C_{P}$ is the only term from the flat-wing dispersion relation (3.54) to contribute to ${\mathcal{C}}(\unicode[STIX]{x1D6FD}_{0})$ . The receptivity to surface roughness of stationary vortices in this $\unicode[STIX]{x1D6FE}=2/9$ regime, where non-parallelism plays a leading-order role in the absence of roughness, is thus governed solely by pressure effects. This peculiarity of the receptivity problem within this non-parallel regime means that an O–S approach would give the correct value for the initial amplitude of the mode through (5.34) and (5.42), despite neglecting non-parallelism. However, the corresponding growth rate would be incorrect, resulting in a significantly different modal amplitude further downstream, as demonstrated in figures 3 and 4.

5.5 Response to surface roughness

The solution found above, (5.40), is the far-field form of the perturbation sufficiently downstream of the roughness element. An important measure of the response of the flow to roughness is $\max \{u^{\prime \ast }/h^{\ast }\}$ . This quantity can be measured downstream of the roughness element, once any unstable modes have grown to a discernible amplitude. We can extrapolate the solution upstream to $\tilde{x}=0$ to determine the ‘initial amplitude’ of the corresponding roughness-free crossflow mode that would produce the measured results at our given location. To determine the maximum amplitude at some point downstream of the roughness element, we simply need to find the maximum over $\unicode[STIX]{x1D6FD}_{0}$ of the initial amplitude at $\tilde{x}=0$ .

The solution downstream of the roughness element is simply the crossflow mode solution discussed in § 3:

(5.43) $$\begin{eqnarray}\displaystyle u_{cf}^{\prime }=\unicode[STIX]{x1D716}{\mathcal{A}}u_{E}(\unicode[STIX]{x1D6FD}_{0},\bar{y})\exp \left[\text{i}\left(\unicode[STIX]{x1D6FC}_{cf}(\unicode[STIX]{x1D6FD}_{0})\tilde{x}+\unicode[STIX]{x1D6FD}_{0}\tilde{z}\right)\right]. & & \displaystyle\end{eqnarray}$$

Extrapolating upstream to the roughness element at $\tilde{x}=0$ , we see that $\unicode[STIX]{x1D716}=hRe^{-1/6}$ and

(5.44) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{A}}(\unicode[STIX]{x1D6FD}_{0})=F^{+}(\unicode[STIX]{x1D6FC}_{0}){\mathcal{C}}(\unicode[STIX]{x1D6FD}_{0}). & \displaystyle\end{eqnarray}$$

The wavenumber dependence of ${\mathcal{A}}$ can be expressed as

(5.45) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D6FC}_{0}):=\unicode[STIX]{x1D6FC}_{0}^{5/2}F^{+}(\unicode[STIX]{x1D6FC}_{0}), & \displaystyle\end{eqnarray}$$

the maximum of which occurs when

(5.46) $$\begin{eqnarray}\displaystyle & \displaystyle {\textstyle \frac{5}{2}}F^{+}(\unicode[STIX]{x1D6FC}_{0})+\unicode[STIX]{x1D6FC}_{0}{F^{+}}^{\prime }(\unicode[STIX]{x1D6FC}_{0})=0. & \displaystyle\end{eqnarray}$$

In deriving (3.54) and (5.34) we assumed $\unicode[STIX]{x1D6FC}_{0}>0$ , which for $\unicode[STIX]{x1D706}_{1}^{x},\unicode[STIX]{x1D706}_{1}^{z}>0$ corresponds through (2.21) to choosing $\unicode[STIX]{x1D6FD}_{0}<0$ . The experimentally measurable amplitude of the mode, ${\mathcal{A}}_{s}=\unicode[STIX]{x1D716}{\mathcal{A}}u_{E}$ , also depends on the behaviour of the eigenfunction $u_{E}(\unicode[STIX]{x1D6FD}_{0},\bar{y})$ . The maximum of this is

(5.47) $$\begin{eqnarray}\displaystyle & \displaystyle \max _{\bar{y}\in [0,\infty )}|u_{E}(\unicode[STIX]{x1D6FD}_{0},\bar{y})|\approx 1.0352. & \displaystyle\end{eqnarray}$$

and occurs at $(4\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0})^{1/4}\bar{y}\approx 4.85\pm 2.01\text{i}$ for $\unicode[STIX]{x1D706}_{2}^{0}\gtrless 0$ .

We will now consider two particular roughness profiles: a Gaussian profile, and a smoothed top-hat profile. For a Gaussian roughness element given by $F(\tilde{x})=\exp [-a\tilde{x}^{2}]$ , where $a^{-1/2}$ represents its chordwise width, the maximum amplitude occurs for

(5.48a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{max}=\sqrt{5a},\quad \unicode[STIX]{x1D6FD}_{max}=-\sqrt{5a}(\unicode[STIX]{x1D706}_{1}^{x}/\unicode[STIX]{x1D706}_{1}^{z}), & \displaystyle\end{eqnarray}$$

and the corresponding complex eigenmode downstream of the roughness element is given by

(5.49) $$\begin{eqnarray}\displaystyle & \displaystyle u_{cf}^{\prime }=Re^{-1/6}h\sqrt{\unicode[STIX]{x03C0}/a}\text{e}^{-5/4}{\mathcal{C}}(\unicode[STIX]{x1D6FD}_{max})u_{E}(\unicode[STIX]{x1D6FD}_{max},\bar{y}). & \displaystyle\end{eqnarray}$$

This scales like $a^{1/4}$ , indicating that streamwise-steeper roughness arrays generate stronger vortices.

Because of the $5/2$ exponent in (5.45), the roughness element must be at least $L^{3}$ -integrable in order for the response function $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D6FC}_{0})$ to decay as $|\unicode[STIX]{x1D6FC}_{0}|\rightarrow \infty$ , and so have a finite value for $\unicode[STIX]{x1D6FC}_{max}$ . For roughness profiles that are less smooth – such as the standard top-hat profile (figure 11 a), for which (5.46) reduces to $\unicode[STIX]{x1D6FC}_{0}\cot (\unicode[STIX]{x1D6FC}_{0})=-3/2$ – there are infinitely many solutions of (5.46) and the corresponding maxima of $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D6FC}_{0})$ grow as $|\unicode[STIX]{x1D6FC}_{0}|\rightarrow \infty$ , as depicted in figure 11(b).

Figure 11.

(a) Standard top-hat roughness profile and (b) roughness response for varying values of the chordwise wavenumber $\unicode[STIX]{x1D6FC}_{0}$ .

Figure 12.

(a) The ‘tanh-smoothed’ top-hat roughness profile with $d=2,\unicode[STIX]{x1D705}=10^{-1}$ and (b) maximum response and the corresponding wavenumber against various values of $\unicode[STIX]{x1D705}$ .

A more realistic shape for the top-hat roughness element is given by the profile

(5.50) $$\begin{eqnarray}\displaystyle & \displaystyle F(\tilde{x})=\frac{1}{2}\left(1+\text{tanh}\left({\displaystyle \frac{{\displaystyle \frac{d}{2}}-|\tilde{x}|}{\unicode[STIX]{x1D705}}}\right)\right), & \displaystyle\end{eqnarray}$$

where $d$ is the diameter of the roughness element and $\unicode[STIX]{x1D705}$ the characteristic length over which the transition between $0$ and $1$ takes place. The usual $L^{1}$ -integrable top-hat profile corresponds to the limit $\unicode[STIX]{x1D705}\rightarrow 0$ . The profile for (5.50) is shown in figure 12, along with the maximum value of the response function $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D6FC}_{0})$ and the corresponding wavenumber $\unicode[STIX]{x1D6FC}_{max}$ for a range of values of $\unicode[STIX]{x1D705}$ . This shows that for a suitably smoothed profile, $\unicode[STIX]{x1D6FD}_{max}$ and the corresponding value of the coupling coefficient remain finite. The surface roughness, which is narrowly spaced in the spanwise direction, becomes sharper in the chordwise direction as $\unicode[STIX]{x1D705}\rightarrow 0$ . In this limit, the amplitude of the excited vortex increases. Since our theory and results for this regime are only valid for $\unicode[STIX]{x1D6FD}_{0}=O(1)$ , we must be careful when drawing conclusions for $\unicode[STIX]{x1D6FD}_{0}\gg 1$ . However from this behaviour – and that of profiles that are less than $L^{3}$ integrable – we can say that streamwise sharper, spanwise narrowly spaced roughness elements will produce a stronger response. This provides a possible explanation for the high sensitivity of three-dimensional boundary-layer transition to roughness elements near the leading edge whose height is only a few per cent of the local boundary-layer thickness, as seen by Radeztsky et al. (Reference Radeztsky, Reibert and Saric1999). Further calculations will be performed for a roughness shape akin to those used in DNS and experiments, and the resulting coupling coefficients presented in § 5.6.

Downstream of the roughness, the induced perturbation takes the form of the non-parallel mode considered in § 3. The influence of the surface roughness will be strongest when placed near a neutral point, but the subsequent behaviour of the generated mode will depend on which of the neutral points depicted in figure 3 the roughness is located near. A mode generated at the second neutral point, where $\unicode[STIX]{x1D706}_{2}^{0}>0$ , will grow according to the second branch of the growth curve until it passes into the $\unicode[STIX]{x1D6FD}\sim Re^{2/7}$ wall-shear-aligned regime, followed by the remaining regimes laid out by Choudhari (Reference Choudhari1994a , Reference Choudhari1995). On the other hand, a mode generated closer to the first neutral point will have a larger initial growth rate in the non-parallel regime, and depending on the location of the effective pressure minimum will either pass into the wall-shear-aligned regime and follow Choudhari’s sequence of regimes, or enter the localised region around the effective pressure minimum discussed in § 4 and take the form of the unstable EPM mode studied there. Downstream of this region it will then evolve into a CHT mode, and ultimately an inviscid Rayleigh mode.

Both routes thus lead to the inviscid regime at the end of the region of linear growth. However, the $O(Re^{3/14})$ growth rate in the wall-shear-aligned regime is significantly larger than that in the effective pressure minimum region, where it is $O(Re^{1/9})$ . While the precise behaviour will depend on the specifics of the intermediate interactive regime, the presence of the effective pressure minimum is likely to have an appreciable effect on the amplitudes of the roughness-generated modes when they reach the inviscid regime further downstream. This not only applies to the idealised Joukowskii aerofoil considered here, but also to modern wings. The pressure minimum on supercritical aerofoils can occur very close to the leading edge (see, for example, figure 7 in Harris (Reference Harris1990)). The results of § 4 may be of particular relevance to those aerofoil designs.

5.6 Super-linear response

We have so far restricted our attention to roughness elements that are small enough that only linear effects need to be considered, and that consist of only a single Fourier component. For larger roughness elements, nonlinear forcing in the boundary conditions may become important, even though the perturbation to the flow field remains linear (to leading-order accuracy). If the no-slip boundary condition (5.4) applied at $Y_{w}=hF(\tilde{x},\tilde{z})$ is extended to include $O(h^{2})$ terms, then the perturbation must satisfy

(5.51) $$\begin{eqnarray}\displaystyle & \displaystyle \mathbf{0}=\bigg(\left.h\frac{\unicode[STIX]{x2202}\boldsymbol{U}}{\unicode[STIX]{x2202}Y}\right|_{Y=0}F(\tilde{x},\tilde{z})+\left.{\displaystyle \frac{1}{2}}h^{2}\frac{\unicode[STIX]{x2202}^{2}\boldsymbol{U}}{\unicode[STIX]{x2202}Y^{2}}\right|_{Y=0}F^{2}(\tilde{x},\tilde{z})+\cdots \bigg)+h(\boldsymbol{u}^{\prime }|_{Y=0}+\cdots \,)+\cdots \,. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

We can express a general spanwise-periodic surface roughness with spanwise wavelength $\unicode[STIX]{x1D706}^{\ast }$ as

(5.52) $$\begin{eqnarray}\displaystyle & \displaystyle Y_{w}=hF(\tilde{x},\tilde{z})=h\mathop{\sum }_{n=-\infty }^{\infty }a_{n}(\tilde{x})\text{e}^{n\text{i}\unicode[STIX]{x1D6FD}_{0}\tilde{z}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}_{0}=-2\unicode[STIX]{x03C0}x^{\ast }/\unicode[STIX]{x1D706}^{\ast }$ . In general, the $a_{n}$ will not be identically zero. This roughness array will thus generate crossflow modes corresponding to each of the harmonics present in the spanwise wavenumber spectrum. The contribution to the fundamental mode, i.e. the mode with spanwise wavenumber $\unicode[STIX]{x1D6FD}_{0}$ , due to the quadratic interaction of the harmonics through $F^{2}(\tilde{x},\tilde{z})$ is then given by

(5.53) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D71A}(\tilde{x}):=\mathop{\sum }_{n=-\infty }^{\infty }a_{n}a_{1-n}. & \displaystyle\end{eqnarray}$$

Let $a_{1}^{+}(k)$ and $\unicode[STIX]{x1D71A}^{+}(k)$ represent the Fourier transforms of $a_{1}(\tilde{x})$ and $\unicode[STIX]{x1D71A}(\tilde{x})$ , respectively. For wall-shear-aligned modes, the projected form of the terms in the first pair of parentheses in (5.51) corresponding to the fundamental mode can then be written as

(5.54) $$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{1}^{0}+Re^{-1/6}k_{1}\unicode[STIX]{x1D706}_{1}^{x})a_{1}^{+}(k)+h^{2}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}\unicode[STIX]{x1D71A}^{+}(k)+O(h^{3},h^{2}Re^{-1/6}). & & \displaystyle\end{eqnarray}$$

Since $\unicode[STIX]{x1D706}_{1}^{0}=0$ in the wall-shear-aligned region of the Fourier spectrum, the leading-order term in (5.54) is $O(hRe^{-1/6})$ , rather than $O(h)$ . Quadratic terms will thus also appear at leading order when

(5.55) $$\begin{eqnarray}\displaystyle & \displaystyle h\sim Re^{-1/6}. & \displaystyle\end{eqnarray}$$

Since $h\ll Re^{-1/9}$ , nonlinear interactions in the bulk of the flow are still only higher-order effects, as discussed in § 5.1. We introduce the $O(1)$ scaled height $H$ by writing $h$ as

(5.56) $$\begin{eqnarray}\displaystyle & \displaystyle h=HRe^{-1/6}. & \displaystyle\end{eqnarray}$$

Figure 13.

The response of the flow to surface roughness of height $h^{\ast }/\unicode[STIX]{x1D6FF}^{\ast }\sim Re^{-1/6}$ , for varying values of $\unicode[STIX]{x1D6FD}_{0}$ , $\unicode[STIX]{x1D705}$ and $h$ . In (a) $h=0.15$ , while in (b) $\unicode[STIX]{x1D705}=0.017$ . The two sets of curves correspond to purely linear forcing (–  –  –) and nonlinear forcing (——).

Focusing solely on the development of the fundamental mode, the appropriate boundary conditions for the almost wall-shear-aligned components of § 5.3 corresponding to (5.29) are found through (5.4) to be

(5.57a,b ) $$\begin{eqnarray}\displaystyle {\mathcal{V}}(0)=0,\quad {\mathcal{V}}_{\bar{y}}(0)=\text{i}k_{1}\unicode[STIX]{x1D706}_{1}^{x}a_{1}^{+}+\text{i}\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}H\unicode[STIX]{x1D71A}^{+}. & & \displaystyle\end{eqnarray}$$

The analysis proceeds as before, with $F^{+}(k)$ replaced by $a_{1}(k)$ , and $g$ , defined in (5.32), by

(5.58) $$\begin{eqnarray}\displaystyle \displaystyle \tilde{g}=g-H{\displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}}{\unicode[STIX]{x1D706}_{1}^{x}}}{\displaystyle \frac{\unicode[STIX]{x1D71A}^{+}}{a_{1}^{+}}}. & & \displaystyle\end{eqnarray}$$

The resulting coupling coefficient is then

(5.59) $$\begin{eqnarray}\displaystyle \displaystyle {\mathcal{C}}(\unicode[STIX]{x1D6FD}_{0})={\displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}^{5/2}U_{eff_{s}}^{2}}{2^{3/2}|\unicode[STIX]{x1D706}_{1}^{z}||\unicode[STIX]{x1D706}_{2}^{0}|^{1/2}}}((\unicode[STIX]{x1D706}_{1}^{x})^{2}+(\unicode[STIX]{x1D706}_{1}^{z})^{2})^{1/2}\unicode[STIX]{x1D6FF}_{2}(-1\pm \text{i})-\text{i}H\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D706}_{2}^{0}{\displaystyle \frac{\unicode[STIX]{x1D71A}^{+}(\unicode[STIX]{x1D6FC}_{0})}{a_{1}^{+}(\unicode[STIX]{x1D6FC}_{0})}}. & & \displaystyle\end{eqnarray}$$

The response of the flow to the surface roughness, $u_{cf}^{\prime }$ , is thus of the form $ha_{1}^{+}(\unicode[STIX]{x1D6FC}_{0})(c_{1}+c_{2}h)$ , i.e. has a super-linear dependence on the height $h$ of the roughness.

This behaviour was investigated by Kurz & Kloker (Reference Kurz and Kloker2014) using DNS for an array of micron-sized roughness elements. As a demonstration of this effect, we compute numerically the response of the flow to their surface roughness element, which is a cylindrical version of the tanh-smoothed top-hat element (5.50) considered above:

(5.60) $$\begin{eqnarray}\displaystyle & \displaystyle F(\tilde{x},\tilde{z})=\frac{1}{2}\left(1+\tanh \left[{\displaystyle \frac{{\displaystyle \frac{d}{2}}-\sqrt{\tilde{x}^{2}+\tilde{z}^{2}}}{\unicode[STIX]{x1D705}}}\right]\right). & \displaystyle\end{eqnarray}$$

For a particular set of parameter values, we have translated case 5 from table 1 of Kurz & Kloker (Reference Kurz and Kloker2014) into our coordinate system. The roughness element array located at $2\,\%$ chord, with a local boundary-layer thickness of $\unicode[STIX]{x1D6FF}^{\ast }=1.7\times 10^{-4}~\text{m}$ and spanwise wavelength $\unicode[STIX]{x1D706}^{\ast }=23.53\unicode[STIX]{x1D6FF}^{\ast }$ , corresponds here to

(5.61a-e ) $$\begin{eqnarray}\displaystyle Re=1.18\times 10^{4},\quad \unicode[STIX]{x1D6FD}_{0}=-7.83,\quad h=0.15,\quad d=0.201,\quad \unicode[STIX]{x1D705}=0.017.\qquad & & \displaystyle\end{eqnarray}$$

For the base-flow parameters we use the values for the Joukowskii base flow, discussed in § 3.5, at $2\,\%$ chord:

(5.62a-d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{1}^{x}=2.96,\quad \unicode[STIX]{x1D706}_{1}^{z}=2.71,\quad \unicode[STIX]{x1D706}_{2}^{0}=-0.421,\quad U_{s}=1.057. & & \displaystyle\end{eqnarray}$$

Figure 13(a) shows the rescaled coupling coefficient $|a_{1}^{+}(\unicode[STIX]{x1D6FC}_{0}){\mathcal{C}}(\unicode[STIX]{x1D6FD}_{0})|$ versus $\unicode[STIX]{x1D6FD}_{0}$ for the values $\unicode[STIX]{x1D705}=0.01,0.1$ . The predictions accounting for linear forcing, represented by dashed lines, demonstrate that the coupling coefficient increases with $\unicode[STIX]{x1D6FD}_{0}$ , at least for $O(1)$ values of the wavenumber. The corresponding solid lines represent the prediction when nonlinear effects are included. The coupling coefficient is much larger for these than in the linear cases, particularly for smaller values of $\unicode[STIX]{x1D705}$ , indicating a significant super-linear behaviour. Again we see that streamwise sharper, spanwise narrowly spaced roughness elements produce a stronger response. This super-linearity can be further illustrated by plotting the coupling coefficient against $h$ for fixed values of $\unicode[STIX]{x1D6FD}_{0}$ , as shown in figure 13(b). As opposed to the constant behaviour of the purely linear prediction, the coupling coefficient varies with $h$ in the nonlinear case. Moderate $h$ reduces the coupling coefficient slightly, due to the opposing signs of the two terms in the imaginary part of (5.59), while for larger values the coupling coefficient grows linearly with $h$ .

The inclusion of the nonlinear forcing from the boundary for these roughness elements has a surprisingly strong effect on the predicted amplitude of the generated crossflow mode. This is because $U_{eff_{s}}^{2}$ , present in the linear contribution to (5.59), is approximately equal to $(1-\unicode[STIX]{x1D706}_{1}^{x}/\unicode[STIX]{x1D706}_{1}^{z})^{2}$ and thus numerically small as a result of the fairly small difference between $\unicode[STIX]{x1D706}_{1}^{x}$ and $\unicode[STIX]{x1D706}_{1}^{z}$ at the chosen location on the Joukowskii aerofoil considered here. This may not be the case at other locations or for other types of aerofoils: a larger value for $U_{eff_{s}}^{2}$ would then correspond to a larger value of the coupling coefficient, and so a larger amplitude for the resultant crossflow mode.

This super-linearity arises as a result of the wall-shear alignment of the flow, rather than non-parallelism. It is thus also a leading-order feature for crossflow modes generated in the neighbouring wall-shear-aligned ( $\unicode[STIX]{x1D6FE}=2/7$ ) regime, where non-parallelism merely contributes at higher order. However, it is of particular importance in the non-parallel regime because of the presence of the lower-branch neutral point. This super-linear behaviour could thus also be included in O–S calculations, in an ad hoc manner, by retaining $O(h^{2})$ terms in the roughness boundary conditions. We would then only expect a significant super-linear effect for modes that are closely aligned with the local wall shear. However, the exclusion of non-parallel-flow effects would lead to an incorrect value for the growth rate of the mode, and hence for the amplitude of the mode downstream of its inception.