2.1 The base flow

In the coordinate system $({\it\xi},{\it\eta},z)$ , the inviscid part of the base flow is given by Cowley & Hall (Reference Cowley and Hall1990)

(2.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{R}={\it\epsilon},\quad R=1,\quad \bar{U}=U=\frac{1}{(1+{\it\epsilon}^{2}\mathscr{T}^{2})^{1/2}},\quad \bar{V}=-\frac{\mathscr{T}}{(1+{\it\epsilon}^{2}\mathscr{T}^{2})^{1/2}},\quad V={\it\epsilon}\bar{V}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{W}=W=0,\quad \bar{P}=\frac{{\it\epsilon}}{{\it\gamma}M^{2}}-\frac{{\it\epsilon}({\it\gamma}-1)(1-{\it\epsilon}^{2})\mathscr{T}^{2}}{2{\it\gamma}(1+{\it\epsilon}^{2}\mathscr{T}^{2})},\quad P=\frac{1}{{\it\gamma}M^{2}},\quad \mathscr{T}=\tan {\it\sigma}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.7c ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{E}=\frac{\bar{P}}{({\it\gamma}-1)\bar{R}}+\frac{1}{2}(\bar{U}^{2}+\bar{V}^{2}),\quad E=\frac{P}{({\it\gamma}-1)R}+\frac{1}{2}(U^{2}+V^{2}), & \displaystyle\end{eqnarray}$$

The viscous boundary layer is described by the transverse variable

(2.8) $$\begin{eqnarray}\displaystyle {\tilde{Y}}=Re^{1/2}y, & & \displaystyle\end{eqnarray}$$

and its streamwise velocity profile is given by the similarity solution

(2.9) $$\begin{eqnarray}\displaystyle U_{B}=f^{\prime }({\it\zeta})\quad \text{with }{\it\zeta}=\frac{1}{\sqrt{x}}\int _{0}^{{\tilde{Y}}}R_{B}\,\text{d}{\tilde{Y}}, & & \displaystyle\end{eqnarray}$$

where $R_{B}$ in ${\it\zeta}$ (the Dorodnitsyn–Howarth variable) is the mean density profile. It can be shown that

(2.10) $$\begin{eqnarray}\displaystyle U_{B}\rightarrow {\it\lambda}{\it\mu}_{w}^{-1/2}T_{w}^{-1/2}{\tilde{Y}}\quad \text{as }{\tilde{Y}}\rightarrow 0, & & \displaystyle\end{eqnarray}$$

where ${\it\lambda}={\it\chi}x^{-1/2}$ with ${\it\chi}\approx 0.332$ , while ${\it\mu}_{w}$ and $T_{w}$ denote the shear viscosity and temperature on the wall, respectively; for the Chapman law, ${\it\mu}_{w}=T_{w}$ . For simplicity, we assume that (1) the wedge walls are insulating, i.e.  $\partial T_{B}/\partial {\tilde{Y}}=0$ on ${\tilde{Y}}=0$ and (2) the Prandtl number is unity, i.e.  $Pr=1$ . Then the temperature and density of the boundary-layer flow, $T_{B}$ and $R_{B}$ , are given by

(2.11a,b ) $$\begin{eqnarray}\displaystyle T_{B}=1+\frac{{\it\gamma}-1}{2}M^{2}(1-U_{B}^{2}),\quad R_{B}=T_{B}^{-1}. & & \displaystyle\end{eqnarray}$$

2.2 Unsteady disturbances

Upstream of the shock, an unsteady perturbation with a small amplitude of order ${\it\delta}\ll 1$ is superimposed on the steady flow. In order to derive the so-called shock condition satisfied by the disturbance downstream of the shock, it is convenient to work with the coordinate system $({\it\xi},{\it\eta},z)$ , where ${\it\xi}$ , ${\it\eta}$ are parallel and normal to the shock, respectively, and $z$ is in the spanwise direction; they are normalized by $\breve{l}$ , the distance to the leading edge. The corresponding velocity field is denoted by $(u,v,w)$ . The total flow field can be written as

(2.12) $$\begin{eqnarray}\displaystyle ({\it\rho},u,v,w,p,\mathscr{E})=(\bar{R},\bar{U},\bar{V},0,\bar{P},\bar{E})+{\it\delta}(\tilde{r}_{0},\tilde{u} _{0},\tilde{v}_{0},\tilde{w}_{0},\tilde{p}_{0},\tilde{e}_{0}), & & \displaystyle\end{eqnarray}$$

where $\mathscr{E}=p/[({\it\gamma}-1){\it\rho}]+(u^{2}+v^{2}+w^{2})/2$ .

As the background flow is uniform, the perturbation can be decomposed as

(2.13a ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{e}_{0}=\frac{1}{({\it\gamma}-1)\bar{R}}\tilde{p}_{0}-\frac{\bar{P}}{({\it\gamma}-1)\bar{R}^{2}}\tilde{r}_{0}+\bar{U}\tilde{u} _{0}+\bar{V}\tilde{v}_{0}, & \displaystyle\end{eqnarray}$$

(2.13b ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{u} _{0}=\tilde{u} _{0a}^{+}+\tilde{u} _{0a}^{-}+\tilde{u} _{0v},\quad \tilde{v}_{0}=\tilde{v}_{0a}^{+}+\tilde{v}_{0a}^{-}+\tilde{v}_{0v},\quad \tilde{w}_{0}=\tilde{w}_{0a}^{+}+\tilde{w}_{0a}^{-}+\tilde{w}_{0v}, & \displaystyle\end{eqnarray}$$

(2.13c ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{p}_{0}=\tilde{p}_{0a}^{+}+\tilde{p}_{0a}^{-},\quad \tilde{r}_{0}=\tilde{r}_{0a}^{+}+\tilde{r}_{0a}^{-}+\tilde{r}_{0e}, & \displaystyle\end{eqnarray}$$

$a$

$v$

$e$

$+/-$

Each of three types of perturbations is governed by the linearized N–S equations about the uniform mean field. Acoustic waves are irrotational, and their pressure, density and velocity are found as

(2.14) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \tilde{p}_{0a}^{\pm }=p_{I}^{\pm }\text{e}^{\text{i}{\it\nu}_{0}^{\pm }{\it\eta}}\tilde{E}_{0}+\text{c.c.},\quad \tilde{r}_{0a}^{\pm }=M_{u}^{2}\frac{\cos ^{2}{\it\sigma}}{\cos ^{2}{\it\phi}}p_{I}^{\pm }\text{e}^{\text{i}{\it\nu}_{0}^{\pm }{\it\eta}}\tilde{E}_{0}+\text{c.c.},\\ \displaystyle (\tilde{u} _{0a}^{\pm },\tilde{v}_{0a}^{\pm },\tilde{w}_{0a}^{\pm })=({\it\alpha},{\it\nu}_{0}^{\pm },{\it\beta})\frac{p_{I}^{\pm }}{({\it\omega}-{\it\alpha}\bar{U}-{\it\nu}_{0}^{\pm }\bar{V})\bar{R}}\text{e}^{\text{i}{\it\nu}_{0}^{\pm }{\it\eta}}\tilde{E}_{0}+\text{c.c.},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where we have introduced

(2.15) $$\begin{eqnarray}\displaystyle \tilde{E}_{0}=\text{e}^{\text{i}({\it\alpha}{\it\xi}+{\it\beta}z-{\it\omega}t)}, & & \displaystyle\end{eqnarray}$$

and the frequency and wavenumbers satisfy the acoustic dispersion relation

(2.16) $$\begin{eqnarray}\displaystyle {\it\omega}-{\it\alpha}\bar{U}-{\it\nu}_{0}^{\pm }\bar{V}=\pm \frac{\cos {\it\phi}}{M_{u}\cos {\it\sigma}}\sqrt{{\it\alpha}^{2}+({\it\nu}_{0}^{\pm })^{2}+{\it\beta}^{2}}. & & \displaystyle\end{eqnarray}$$

A vorticity wave has, on the other hand, no pressure fluctuation, and its velocity components take the form

(2.17) $$\begin{eqnarray}\displaystyle (\tilde{u} _{0v},\tilde{v}_{0v},\tilde{w}_{0v})=(u_{I},v_{I},w_{I})\text{e}^{\text{i}{\it\nu}_{I}{\it\eta}}\tilde{E}_{0}+\text{c.c.}, & & \displaystyle\end{eqnarray}$$

whereas an entropy mode carries merely a density (or temperature) fluctuation,

(2.18) $$\begin{eqnarray}\displaystyle \tilde{r}_{0e}=r_{I}\text{e}^{\text{i}{\it\nu}_{I}{\it\eta}}\tilde{E}_{0}+\text{c.c.} & & \displaystyle\end{eqnarray}$$

For both vorticity and entropy modes, the wavenumber in the ${\it\eta}$ -direction, ${\it\nu}_{I}$ , satisfies the dispersion relation

(2.19) $$\begin{eqnarray}\displaystyle {\it\omega}-{\it\alpha}\bar{U}-{\it\nu}_{I}\bar{V}=0. & & \displaystyle\end{eqnarray}$$

In general, whenever any one of acoustic, vorticity and entropy waves strikes a shock, all three types of fluctuations may arise altogether downstream. They will be referred to as transmitted waves if they are of the same type as the upstream ones, or generated waves if different. The generated/transmitted waves have the same frequency and the same wavelengths parallel to the shock as the disturbance upstream. For a shock in free space, the disturbances downstream are all transmitted from, and/or generated by, the upstream perturbation. However, for a shock adjacent to a wall, along which a boundary layer develops, there may also exist the free-stream signature of boundary-layer instability modes. This is the case for the present problem. We shall assume that the perturbation has a short wavelength, i.e. ${\it\alpha},{\it\beta}\gg 1$ . The perturbation downstream the shock is allowed to modulate slowly over a length scale much longer than $2{\rm\pi}/{\it\alpha}$ but much shorter than $O(1)$ . The modulation is described by the variable $x_{1}=x/\tilde{{\it\epsilon}}$ , where $2{\rm\pi}/{\it\alpha}\leqslant \tilde{{\it\epsilon}}^{2}\ll 1$ . With the above considerations, we write the downstream flow field as

(2.20) $$\begin{eqnarray}\displaystyle ({\it\rho},u,v,w,p,\mathscr{E}) & = & \displaystyle (R,U,V,0,P,E)+{\it\delta}\tilde{{\it\epsilon}}^{-1} [(\tilde{r}_{1},\tilde{u} _{1},\tilde{v}_{1},\tilde{w}_{1},\tilde{p}_{1},\tilde{e}_{1})\nonumber\\ \displaystyle & & \displaystyle +\,\tilde{{\it\epsilon}}(\tilde{r}_{2},\tilde{u} _{2},\tilde{v}_{2},\tilde{w}_{2},\tilde{p}_{2},\tilde{e}_{2})],\end{eqnarray}$$

where the $O({\it\delta}/\tilde{{\it\epsilon}})$ term represents the signature of the T–S wave that may be generated by the $O({\it\delta})$ disturbance in the oncoming flow. The larger magnitude is due to the resonant nature of the forcing. This term is present in the expansion near the neutral position of the T–S wave, but would be absent upstream of the neutral position. The disturbance in (2.20) again consists of acoustic, vortical and entropy parts, namely

(2.21a ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{e}_{j}=\frac{1}{({\it\gamma}-1)R}\tilde{p}_{j}-\frac{P}{({\it\gamma}-1)R^{2}}\tilde{r}_{j}+U\tilde{u} _{j}+V\tilde{v}_{j}, & \displaystyle\end{eqnarray}$$

(2.21b ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{u} _{j}=\tilde{u} _{ja}+\tilde{u} _{jv},\quad \tilde{v}_{j}=\tilde{v}_{ja}+\tilde{v}_{jv},\quad \tilde{w}_{j}=\tilde{w}_{ja}+\tilde{w}_{jv}, & \displaystyle\end{eqnarray}$$

(2.21c ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{p}_{j}=\tilde{p}_{ja},\quad \quad \tilde{r}_{j}=\tilde{r}_{ja}+\tilde{r}_{je},\quad (j=1,2). & \displaystyle\end{eqnarray}$$

The leading-order (i.e. $O({\it\delta}\tilde{{\it\epsilon}}^{-1})$ ) acoustic part admits the solution for the pressure

(2.22) $$\begin{eqnarray}\displaystyle \tilde{p}_{1a}=({\check{a}}_{I}\text{e}^{\text{i}{\it\nu}_{1}{\it\eta}}+{\check{a}}_{R}\text{e}^{\text{i}{\it\nu}_{2}{\it\eta}})\tilde{E}_{0}, & & \displaystyle\end{eqnarray}$$

where ${\it\nu}_{1}$ and ${\it\nu}_{2}$ are the wavenumbers of the incoming and outgoing slow acoustic waves, and they satisfy the dispersion relation

(2.23) $$\begin{eqnarray}\displaystyle {\it\omega}-{\it\alpha}U-{\it\nu}_{1,2}V=-\frac{1}{M}\sqrt{{\it\alpha}^{2}+{\it\nu}_{1,2}^{2}+{\it\beta}^{2}}, & & \displaystyle\end{eqnarray}$$

and the amplitudes of these two waves, ${\check{a}}_{I}$ and ${\check{a}}_{R}$ , are functions of $x_{1}$ . Note that a minus sign on the right-hand side of (2.23) is taken because a T–S instability mode has a small phase speed, and behaves outside the boundary layer like a slow acoustic wave. The corresponding expressions for the velocities and the solution for the induced vortical part of the disturbance are given in appendix A. The result will be used in the main text specifically for ${\it\alpha}=O(Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{-3/4})$ , ${\it\beta}=O(Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{1/4})$ and $\tilde{{\it\epsilon}}=Re^{-3/16}{\it\mu}_{w}^{3/16}T_{w}^{9/16}M^{3/8}$ .

The $O({\it\delta})$ disturbance downstream the shock consists of two parts: (i) that transmitted from, or generated by, the upstream perturbation and (ii) that arising from the modulation of the leading-order (i.e. $O({\it\delta}/\tilde{{\it\epsilon}})$ ) fluctuations associated with the T–S mode. The solution for the acoustic part, which is governed by the inhomogeneous wave equation, is of the form

(2.24) $$\begin{eqnarray}\displaystyle \tilde{p}_{2a}=(\hat{a}_{I}\text{e}^{\text{i}{\it\nu}_{1}{\it\eta}}+\hat{a}_{R}\text{e}^{\text{i}{\it\nu}_{2}{\it\eta}}+\hat{p}_{2a,p})\tilde{E}_{0}, & & \displaystyle\end{eqnarray}$$

where $\hat{p}_{2a,p}$ denotes the particular solution and $\hat{a}_{I}$ and $\hat{a}_{R}$ are functions of $x_{1}$ .

For an oblique shock in free space, the relations satisfied by the disturbance on the shock were given by McKenzie & Westphal (Reference McKenzie and Westphal1968). In the presence of a wall, the analysis must allow for the acoustic wave reflected by the wall as well as for the signature of the boundary-layer perturbations. These were considered by Cowley & Hall (Reference Cowley and Hall1990) but the upstream disturbance was assumed absent. Here, we consider the case where the upstream disturbance may be comprised of any of three types of modes. Furthermore, the disturbance downstream may be spatially modulated. The detailed analysis of their interaction with a general oblique shock is performed by Qin & Wu (Reference Qin and Wu2016) for any $\tilde{{\it\epsilon}}$ satisfying $2{\rm\pi}/{\it\alpha}\leqslant \tilde{{\it\epsilon}}^{2}\ll 1$ , and the generalized shock conditions are derived. In appendix A, these conditions are specialized to (2.7a–c ), leading to (A 7) and (A 21), which relate ${\check{a}}_{I}$ and ${\check{a}}_{R}$ in (2.22), and $\hat{a}_{I}$ and $\hat{a}_{R}$ in (2.24), respectively. The relations (A 7) and (A 21) simplify significantly for disturbances on the triple-deck scales.

In the coordinate system $(x,y,z)$ , the leading-order pressure $\tilde{p}$ is governed by

(2.25) $$\begin{eqnarray}\displaystyle M^{2}\left(\frac{\partial }{\partial t}+\frac{\partial }{\partial x}\right)^{2}\tilde{p}-{\rm\nabla}^{2}\tilde{p}=0. & & \displaystyle\end{eqnarray}$$

The solution takes the form

(2.26) $$\begin{eqnarray}\displaystyle \tilde{p}={\it\delta}[a_{I}\text{e}^{\text{i}({\it\alpha}_{a}x+{\it\gamma}_{a}y+{\it\beta}_{a}z-{\it\omega}_{a}t)}+a_{R}\text{e}^{\text{i}({\it\alpha}_{a}x-{\it\gamma}_{a}y+{\it\beta}_{a}z-{\it\omega}_{a}t)}]. & & \displaystyle\end{eqnarray}$$

Substitution of (2.26) into (2.25) leads to the dispersion relation for slow acoustic waves,

(2.27) $$\begin{eqnarray}\displaystyle {\it\omega}_{a}/{\it\alpha}_{a}=1-\frac{1}{M}\sqrt{1+({\it\beta}_{a}/{\it\alpha}_{a})^{2}+({\it\gamma}_{a}/{\it\alpha}_{a})^{2}}. & & \displaystyle\end{eqnarray}$$

Note that the two coordinate systems, $(x,y,z)$ and $({\it\xi},{\it\eta},z)$ , are related to each other by

(2.28a,b ) $$\begin{eqnarray}\displaystyle x={\it\xi}\cos {\it\phi}-{\it\eta}\sin {\it\phi},\quad y={\it\xi}\sin {\it\phi}+{\it\eta}\cos {\it\phi}. & & \displaystyle\end{eqnarray}$$

The wavenumbers in the two systems are related by

(2.29a,b ) $$\begin{eqnarray}\displaystyle {\it\alpha}_{a}={\it\alpha}\cos {\it\phi}-{\it\nu}_{1,2}\sin {\it\phi},\quad {\it\gamma}_{a}={\it\alpha}\sin {\it\phi}+{\it\nu}_{1,2}\cos {\it\phi}, & & \displaystyle\end{eqnarray}$$

with which the dispersion relations (2.23) and (2.27) are equivalent as expected. When ${\it\phi}\ll 1$ , which is the case for our problem, the expressions (2.22) and (2.26) are identical on the shock ( ${\it\eta}=0$ ) provided that

(2.30a,b ) $$\begin{eqnarray}\displaystyle {\check{a}}_{I}=a_{I}\text{e}^{\text{i}{\it\gamma}_{a}y_{s}},\quad {\check{a}}_{R}=a_{R}\text{e}^{-\text{i}{\it\gamma}_{a}y_{s}}, & & \displaystyle\end{eqnarray}$$

where $y_{s}={\it\xi}\sin {\it\phi}=x\tan {\it\phi}$ is the local shock position.

2.3 Asymptotic scalings

In order to develop an asymptotic theory for receptivity, we assume the Reynolds and Mach numbers to be large: $Re\gg 1$ and $M\gg 1$ . More specifically, for the base flow we adopt the same scaling restrictions as those in Cowley & Hall (Reference Cowley and Hall1990),

(2.31a-e ) $$\begin{eqnarray}\displaystyle {\it\sigma}\ll 1,\quad M\gg 1,\quad {\it\sigma}M\gg 1,\quad {\it\epsilon}\approx 1/({\it\sigma}^{2}M^{2}),\quad {\it\gamma}-1\sim \bar{{\it\gamma}}/({\it\sigma}^{2}M^{2}),\qquad & & \displaystyle\end{eqnarray}$$

to ensure that the shock is sufficiently close to the wall to influence the boundary-layer instability. Use of (2.31) to the first equation in (2.3a,b ) shows that

(2.32) $$\begin{eqnarray}\displaystyle M_{u}^{2}=\frac{2}{\sin ^{2}{\it\sigma}[({\it\gamma}+1){\it\epsilon}-{\it\gamma}+1]}\sim \frac{2}{2-\bar{{\it\gamma}}}M^{2}, & & \displaystyle\end{eqnarray}$$

where $\bar{{\it\gamma}}$ is an $O(1)$ constant and $0<\bar{{\it\gamma}}<2$ .

In the limit of $Re\gg 1$ and $M\gg 1$ , the characteristic frequency of lower-branch T–S waves, ${\it\omega}_{TS}$ , and their wavenumbers in the $x$ - and $z$ -directions, ${\it\alpha}_{TS}$ and ${\it\beta}_{TS}$ , scale as (Smith Reference Smith1989; Cowley & Hall Reference Cowley and Hall1990)

(2.33a,b ) $$\begin{eqnarray}\displaystyle {\it\omega}_{TS}=Re^{1/4}{\it\mu}_{w}^{-1/4}T_{w}^{-3/4}M^{-1/2}\hat{{\it\omega}}_{TS},\quad ({\it\alpha}_{TS},{\it\beta}_{TS})=Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{-3/4}(\hat{{\it\alpha}}_{TS},M\hat{{\it\beta}}_{TS}). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The disturbance in the oncoming free stream is in general of broadband nature and so are those emerging downstream of the shock, among which each acoustic wave is characterized by its frequency ${\it\omega}_{a}$ , and streamwise and spanwise wavenumbers, ${\it\alpha}_{a}$ and ${\it\beta}_{a}$ (with ${\it\gamma}_{a}$ determined by the dispersion relation (2.27)). Our interest is in Fourier components which generate T–S waves. This requires ${\it\omega}_{a}$ , ${\it\alpha}_{a}$ and ${\it\beta}_{a}$ of slow acoustic waves to be on the same triple-deck scales,

(2.34a,b ) $$\begin{eqnarray}\displaystyle {\it\omega}_{a}=Re^{1/4}{\it\mu}_{w}^{-1/4}T_{w}^{-3/4}M^{-1/2}\hat{{\it\omega}}_{a},\quad ({\it\alpha}_{a},{\it\beta}_{a})=Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{-3/4}(\hat{{\it\alpha}}_{a},M\hat{{\it\beta}}_{a}). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Substitution of the scalings (2.34) into (2.27) leads to

(2.35) $$\begin{eqnarray}\displaystyle 1-\frac{1}{M}\sqrt{1+(\hat{{\it\beta}}_{a}/\hat{{\it\alpha}}_{a})^{2}M^{2}+({\it\gamma}_{a}/{\it\alpha}_{a})^{2}}=Re^{-1/8}{\it\mu}_{w}^{1/8}T_{w}^{3/8}M^{1/4}(\hat{{\it\omega}}_{a}/\hat{{\it\alpha}}_{a}). & & \displaystyle\end{eqnarray}$$

In order for the left-hand side of the equation to be positive and asymptotically small, we require that $\hat{{\it\alpha}}_{a}>\hat{{\it\beta}}_{a}$ and ${\it\gamma}_{a}=Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{1/4}\hat{{\it\gamma}}_{a}$ with $\hat{{\it\gamma}}_{a}$ being given, to leading order, by

(2.36) $$\begin{eqnarray}\displaystyle \hat{{\it\gamma}}_{a}=\sqrt{\hat{{\it\alpha}}_{a}^{2}-\hat{{\it\beta}}_{a}^{2}}. & & \displaystyle\end{eqnarray}$$

This relation specifies the propagation direction of the slow acoustic waves which have wavelengths and phase speeds comparable with those of T–S waves. Consistent with the scalings (2.34), we introduce the time, the streamwise and spanwise variables

(2.37a,b ) $$\begin{eqnarray}\displaystyle \bar{t}=Re^{1/4}{\it\mu}_{w}^{-1/4}T_{w}^{-3/4}M^{-1/2}t,\quad (\bar{x},\bar{z})=Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{-3/4}(x,Mz). & & \displaystyle\end{eqnarray}$$

In order for a slow acoustic wave on the scales (2.34) to arise downstream, upstream (fast or slow) acoustic, vortical and entropy modes must have appropriate scales. For all these modes, the wavenumber parallel to the shock

(2.38) $$\begin{eqnarray}\displaystyle {\it\alpha}\approx {\it\alpha}_{a}=Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{-3/4}\hat{{\it\alpha}}_{a}, & & \displaystyle\end{eqnarray}$$

but that normal to the shock is different. For a fast/slow upstream acoustic mode, it follows from (2.16) and (2.32) that to leading-order accuracy,

(2.39) $$\begin{eqnarray}\displaystyle {\it\nu}_{0}^{\pm }\approx Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{-3/4}\left.\left(\hat{{\it\alpha}}_{a}\pm \sqrt{\frac{2-\bar{{\it\gamma}}}{2}}\hat{{\it\beta}}_{a}\right)\!\right/\!{\it\sigma}. & & \displaystyle\end{eqnarray}$$

For upstream vorticity and entropy modes, we find from (2.19) that

(2.40) $$\begin{eqnarray}\displaystyle {\it\nu}_{I}\approx Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{-3/4}\hat{{\it\alpha}}_{a}/{\it\sigma}. & & \displaystyle\end{eqnarray}$$

At a general streamwise location $x$ where $({\it\omega}_{a},{\it\alpha}_{a},{\it\beta}_{a})\neq ({\it\omega}_{TS},{\it\alpha}_{TS},{\it\beta}_{TS})$ , an acoustic wave generates a finite response within the boundary layer. However, for each frequency ${\it\omega}_{a}$ , there exists a location $x_{0}$ at which a T–S wave with the same frequency and spanwise wavenumber is neutrally stable if the streamwise wavenumber of the slow acoustic wave also coincides with that of the neutral T–S mode, that is,

(2.41a-c ) $$\begin{eqnarray}\displaystyle \hat{{\it\omega}}_{a}=\hat{{\it\omega}}_{TS},\quad \hat{{\it\alpha}}_{a}=\hat{{\it\alpha}}_{TS},\quad \hat{{\it\beta}}_{a}=\hat{{\it\beta}}_{TS} & & \displaystyle\end{eqnarray}$$

at $x_{0}$ , the response becomes unbounded, signalling that a resonance takes place in the vicinity of $x_{0}$ . A scaling argument similar to that of Wu (Reference Wu2001b ) shows that the forced response evolves into a T–S mode under the influence of non-parallelism, and this resonance region corresponds to $x-x_{0}=O(\bar{{\it\epsilon}})$ with

(2.42) $$\begin{eqnarray}\displaystyle \bar{{\it\epsilon}}=Re^{-3/16}{\it\mu}_{w}^{3/16}T_{w}^{9/16}M^{3/8}. & & \displaystyle\end{eqnarray}$$

In order to describe the evolution of the disturbance through the resonance region, we introduce

(2.43) $$\begin{eqnarray}\displaystyle x_{1}=(x-x_{0})/\bar{{\it\epsilon}}. & & \displaystyle\end{eqnarray}$$

It follows from (2.37) and (2.43) that

(2.44a-c ) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial t}\rightarrow Re^{1/4}{\it\mu}_{w}^{-1/4}T_{w}^{-3/4}M^{-1/2}\frac{\partial }{\partial \bar{t}},\quad \frac{\partial }{\partial z}\rightarrow \bar{{\it\epsilon}}^{-2}M\frac{\partial }{\partial \bar{z}},\quad \frac{\partial }{\partial x}\rightarrow \bar{{\it\epsilon}}^{-2}\left(\frac{\partial }{\partial \bar{x}}+\bar{{\it\epsilon}}\frac{\partial }{\partial x_{1}}\right). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

We further expand ${\it\lambda}$ about $x_{0}$ ,

(2.45) $$\begin{eqnarray}\displaystyle {\it\lambda}={\it\lambda}_{0}+\bar{{\it\epsilon}}{\it\lambda}_{1}x_{1}\quad \text{with }{\it\lambda}_{1}=-{\textstyle \frac{1}{2}}{\it\chi}{x_{0}}^{-3/2}. & & \displaystyle\end{eqnarray}$$

The asymptotic structure describing the boundary-layer response and receptivity is shown in figure 2. To a sound wave with its frequency and wavenumbers being scaled as (2.34), the boundary-layer response acquires, in the wall-normal direction, the triple-deck structure that governs the inherent boundary-layer instability. A disturbance satisfying the resonance condition (2.41) evolves, in the streamwise direction, through the pre-resonance, resonance and post-resonance regimes. The response in the pre-resonance regime will be considered in the next section, and the main result is the expressions (3.21) for $a_{I}$ and $a_{R}$ , the magnitudes of the pressure waves entering, and reflected by, the boundary layer, respectively. The evolution in the resonance region will be analysed in § 4 and the main outcome is (4.43), which describes the development of the response into a T–S wave, and allows us to determine the initial amplitude of the eigenmode excited.

Figure 2.

The asymptotic structure of the boundary-layer response and receptivity.

3.1 Upper deck

In the upper deck, the transverse variable is

(3.1) $$\begin{eqnarray}\displaystyle \bar{y}=Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{1/4}y, & & \displaystyle\end{eqnarray}$$

and the expansion takes the form

(3.2) $$\begin{eqnarray}\displaystyle (u,v,w,p,{\it\rho},T)=(1,0,0,1/{\it\gamma}M^{2},1,1)+{\it\delta}(u_{a},Mv_{a},Mw_{a},p_{a},M^{2}{\it\rho}_{a},M^{2}{\it\tau}_{a})+\cdots \,. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here we assume that there is no pre-existing T–S wave. In the pre-resonance region, no T–S wave is excited yet and so the $O({\it\delta}/\bar{{\it\epsilon}})$ term is absent. The perturbation is solely induced by the $O({\it\delta})$ disturbance in the oncoming flow.

We focus on the acoustic part of the perturbation. Vortical and entropy waves are also present (see appendix A), but they are absorbed by a thin layer located at the outer edge of the boundary layer without entering the latter (Dong & Wu Reference Dong and Wu2013; see also Maslov et al. Reference Maslov, Mironov, Kudryavtsev, Poplavskaya and Tsyryulnokov2010, where such features were observed experimentally). The acoustic wave is governed by the (quasi-steady) Euler equations linearized about the uniform flow, among which the momentum equation in the $y$ -direction reads

(3.3) $$\begin{eqnarray}\displaystyle \frac{\partial v_{a}}{\partial \bar{x}}=-\frac{\partial p_{a}}{\partial \bar{y}}. & & \displaystyle\end{eqnarray}$$

The linearized Euler equations can be reduced to a single equation for $p_{a}$ ,

(3.4) $$\begin{eqnarray}\displaystyle \frac{\partial ^{2}p_{a}}{\partial \bar{y}^{2}}-\frac{\partial ^{2}p_{a}}{\partial \bar{x}^{2}}+\frac{\partial ^{2}p_{a}}{\partial \bar{z}^{2}}=0. & & \displaystyle\end{eqnarray}$$

Equations (3.4) and (3.3) can be solved in sequence to find the solution for $p_{a}$ and  $v_{a}$ ,

(3.5a,b ) $$\begin{eqnarray}\displaystyle p_{a}=(a_{I}\text{e}^{\text{i}\hat{{\it\gamma}}_{a}\bar{y}}+a_{R}\text{e}^{-\text{i}\hat{{\it\gamma}}_{a}\bar{y}})E_{a}+\text{c.c.},\quad v_{a}=-\frac{\hat{{\it\gamma}}_{a}}{\hat{{\it\alpha}}_{a}}(a_{I}\text{e}^{\text{i}\hat{{\it\gamma}}_{a}\bar{y}}-a_{R}\text{e}^{-\text{i}\hat{{\it\gamma}}_{a}\bar{y}})E_{a}+\text{c.c.}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $E_{a}=\exp \{\text{i}(\hat{{\it\alpha}}_{a}\bar{x}+\hat{{\it\beta}}_{a}\bar{z}-\hat{{\it\omega}}_{a}\bar{t})\}$ , and $a_{I}$ and $a_{R}$ depend on $x$ parametrically, and they represent respectively the amplitudes of the sound waves propagating towards and away from the wall. As $\bar{y}\rightarrow 0$ ,

(3.6a,b ) $$\begin{eqnarray}\displaystyle p_{a}\rightarrow (a_{I}+a_{R})E_{a}+\text{c.c.},\quad v_{a}\rightarrow -\frac{\hat{{\it\gamma}}_{a}}{\hat{{\it\alpha}}_{a}}(a_{I}-a_{R})E_{a}+\text{c.c.} & & \displaystyle\end{eqnarray}$$

In the pre-resonance stage, the T–S signature is absent and thus the shock condition corresponds to (A 21) with the modulation terms being set to zero, which further reduces, for the disturbance on the triple-deck scales, to (A 46). Noting that $\hat{a}_{I}=a_{I}\text{e}^{\text{i}\hat{{\it\gamma}}_{a}\bar{y}_{s}}$ and $\hat{a}_{R}=a_{R}\text{e}^{-\text{i}\hat{{\it\gamma}}_{a}\bar{y}_{s}}$ (which are similar to (2.30)), we obtain

(3.7) $$\begin{eqnarray}\displaystyle a_{I}\text{e}^{\text{i}\hat{{\it\gamma}}_{a}\bar{y}_{s}}+a_{R}\text{e}^{-\text{i}\hat{{\it\gamma}}_{a}\bar{y}_{s}}=J_{0}, & & \displaystyle\end{eqnarray}$$

where $\bar{y}_{s}=Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{1/4}y_{s}$ is the rescaled shock position, and $J_{0}$ is defined in (A 47).

3.2 Main deck

In the main deck, the expansion takes the form

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle u=U_{B}({\tilde{Y}})+{\it\delta}Re^{1/8}{\it\mu}_{w}^{3/8}T_{w}^{9/8}M^{7/4}B_{a}U_{B}^{\prime }({\tilde{Y}})E_{a}+\text{c.c.}+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle v=-{\it\delta}M\text{i}\hat{{\it\alpha}}_{a}B_{a}U_{B}({\tilde{Y}})E_{a}+\text{c.c.}+\cdots \,, & \displaystyle\end{eqnarray}$$

where $B_{a}=B_{a}(x)$ depends on $x$ parametrically. It can be shown that

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle u\rightarrow {\it\lambda}{\it\mu}_{w}^{-1/2}T_{w}^{-1/2}{\tilde{Y}}+{\it\delta}Re^{1/8}{\it\mu}_{w}^{-1/8}T_{w}^{5/8}M^{7/4}B_{a}{\it\lambda}E_{a}+\text{c.c.}\quad \text{as }{\tilde{Y}}\rightarrow 0, & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle v\rightarrow -{\it\delta}M\text{i}\hat{{\it\alpha}}_{a}B_{a}E_{a}+\text{c.c.}\quad \text{as }{\tilde{Y}}\rightarrow \infty . & \displaystyle\end{eqnarray}$$

Matching with the normal velocity at the bottom of the upper deck, as given in (3.6), yields

(3.12) $$\begin{eqnarray}\displaystyle \frac{\hat{{\it\gamma}}_{a}}{\hat{{\it\alpha}}_{a}}(a_{I}-a_{R})=\text{i}\hat{{\it\alpha}}_{a}B_{a}. & & \displaystyle\end{eqnarray}$$

3.3 Lower deck

In the lower deck, the appropriate transverse coordinate is defined as

(3.13) $$\begin{eqnarray}\displaystyle Y=Re^{5/8}{\it\mu}_{w}^{-5/8}T_{w}^{-7/8}M^{-1/4}y, & & \displaystyle\end{eqnarray}$$

and the solution can be written as

(3.14a ) $$\begin{eqnarray}\displaystyle & \displaystyle u=Re^{-1/8}{\it\mu}_{w}^{1/8}T_{w}^{3/8}M^{1/4}{\it\lambda}Y+{\it\delta}Re^{1/8}{\it\mu}_{w}^{-1/8}T_{w}^{5/8}M^{7/4}\tilde{U} _{a}E_{a}+\text{c.c.}+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.14b ) $$\begin{eqnarray}\displaystyle & \displaystyle v={\it\delta}Re^{-1/8}{\it\mu}_{w}^{1/8}T_{w}^{3/8}M^{5/4}\tilde{V}_{a}E_{a}+\text{c.c.}+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.14c ) $$\begin{eqnarray}\displaystyle & \displaystyle w={\it\delta}Re^{1/8}{\it\mu}_{w}^{-1/8}T_{w}^{5/8}M^{3/4}\tilde{W}_{a}E_{a}+\text{c.c.}+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.14d ) $$\begin{eqnarray}\displaystyle & \displaystyle p={\it\delta}\tilde{P}_{a}E_{a}+\text{c.c.}+\cdots \,. & \displaystyle\end{eqnarray}$$

(3.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\hat{{\it\alpha}}_{a}\tilde{U} _{a}+\tilde{V}_{a,Y}+\text{i}\hat{{\it\beta}}_{a}\tilde{W}_{a}=0, & \displaystyle\end{eqnarray}$$

(3.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}(\hat{{\it\alpha}}_{a}{\it\lambda}Y-\hat{{\it\omega}}_{a})\tilde{U} _{a}+{\it\lambda}\tilde{V}_{a}=\tilde{U} _{a,YY}, & \displaystyle\end{eqnarray}$$

(3.15c ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}(\hat{{\it\alpha}}_{a}{\it\lambda}Y-\hat{{\it\omega}}_{a})\tilde{W}_{a}=-\text{i}\hat{{\it\beta}}_{a}\tilde{P}_{a}+\tilde{W}_{a,YY}. & \displaystyle\end{eqnarray}$$

$\tilde{U} _{a}=\tilde{V}_{a}=\tilde{W}_{a}=0$

$Y=0$

$(\tilde{U} _{a},\tilde{W}_{a})\rightarrow ({\it\lambda}B_{a},0)$

$Y\rightarrow \infty$

(3.16a,b ) $$\begin{eqnarray}\displaystyle \tilde{V}_{a,YYY}(0)=\hat{{\it\beta}}_{a}^{2}\tilde{P}_{a};\quad \tilde{V}_{a,Y}\rightarrow -\text{i}\hat{{\it\alpha}}_{a}{\it\lambda}B_{a}\quad \text{as }Y\rightarrow \infty . & & \displaystyle\end{eqnarray}$$

Eliminating the pressure yields $[\partial ^{2}/\partial Y^{2}-\text{i}(\hat{{\it\alpha}}_{a}{\it\lambda}Y-\hat{{\it\omega}}_{a})]\tilde{V}_{a,YY}=0$ , which has the solution

(3.17) $$\begin{eqnarray}\displaystyle \tilde{V}_{a,Y}=q_{a}\int _{\tilde{{\it\eta}}_{0}}^{\tilde{{\it\eta}}}\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}, & & \displaystyle\end{eqnarray}$$

where $q_{a}$ is an unknown function of $x$ , $\text{Ai}$ denotes the Airy function and

(3.18) $$\begin{eqnarray}\displaystyle \tilde{{\it\eta}}=(\text{i}\hat{{\it\alpha}}_{a}{\it\lambda})^{1/3}Y+\tilde{{\it\eta}}_{0},\quad \tilde{{\it\eta}}_{0}=-\text{i}\hat{{\it\omega}}_{a}(\text{i}\hat{{\it\alpha}}_{a}{\it\lambda})^{-2/3}. & & \displaystyle\end{eqnarray}$$

Application of the boundary conditions (3.16) gives

(3.19a,b ) $$\begin{eqnarray}\displaystyle (\text{i}\hat{{\it\alpha}}_{a}{\it\lambda})^{2/3}\text{Ai}^{\prime }(\tilde{{\it\eta}}_{0})q_{a}=\hat{{\it\beta}}_{a}^{2}\tilde{P}_{a},\quad q_{a}\int _{\tilde{{\it\eta}}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}=-\text{i}\hat{{\it\alpha}}_{a}{\it\lambda}B_{a}. & & \displaystyle\end{eqnarray}$$

That the pressure does not change across the main deck implies that

(3.20) $$\begin{eqnarray}\displaystyle a_{I}+a_{R}=\tilde{P}_{a}. & & \displaystyle\end{eqnarray}$$

Equations (3.7), (3.12) and (3.19)–(3.20) are solved to give

(3.21) $$\begin{eqnarray}\displaystyle a_{I,R}(\hat{{\it\omega}}_{a},\hat{{\it\alpha}}_{a},\hat{{\it\beta}}_{a};x)=-\frac{1}{2}J_{0}\left.\left[{\it\lambda}^{5/3}\hat{{\it\gamma}}_{a}\,\text{Ai}^{\prime }(\tilde{{\it\eta}}_{0})\pm \text{i}(\text{i}\hat{{\it\alpha}}_{a})^{1/3}\hat{{\it\beta}}_{a}^{2}\int _{\tilde{{\it\eta}}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}\right]\right/\!{\it\Delta}, & & \displaystyle\end{eqnarray}$$

where the $\pm$ signs refer to the subscripts $I$ and $R$ , respectively, and

(3.22) $$\begin{eqnarray}\displaystyle {\it\Delta}(\hat{{\it\omega}}_{a},\hat{{\it\alpha}}_{a},\hat{{\it\beta}}_{a};x)=(\text{i}\hat{{\it\alpha}}_{a})^{1/3}\hat{{\it\beta}}_{a}^{2}\sin (\hat{{\it\gamma}}_{a}\bar{y}_{s})\int _{\tilde{{\it\eta}}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}-{\it\lambda}^{5/3}\hat{{\it\gamma}}_{a}\,\text{Ai}^{\prime }(\tilde{{\it\eta}}_{0})\cos (\hat{{\it\gamma}}_{a}\bar{y}_{s})\qquad & & \displaystyle\end{eqnarray}$$

is the dispersion function of the T–S wave (Cowley & Hall Reference Cowley and Hall1990) as it will transpire. We introduce a reflection coefficient

(3.23) $$\begin{eqnarray}\displaystyle \mathscr{R}\equiv a_{R}/a_{I}=\frac{2\text{i}\sin (\hat{{\it\gamma}}_{a}\bar{y}_{s}){\it\lambda}^{5/3}\hat{{\it\gamma}}_{a}\,\text{Ai}^{\prime }(\tilde{{\it\eta}}_{0})}{[\text{i}\sin (\hat{{\it\gamma}}_{a}\bar{y}_{s})-\cos (\hat{{\it\gamma}}_{a}\bar{y}_{s})]{\it\lambda}^{5/3}\hat{{\it\gamma}}_{a}\,\text{Ai}^{\prime }(\tilde{{\it\eta}}_{0})-{\it\Delta}}-1. & & \displaystyle\end{eqnarray}$$

We now examine the behaviour of $a_{I}$ and $a_{R}$ when the resonant condition (2.41) is satisfied at $x_{0}$ , where ${\it\Delta}(\hat{{\it\omega}}_{a},\hat{{\it\alpha}}_{a},\hat{{\it\beta}}_{a};x_{0})=0$ . A Taylor expansion of the expression (3.22) about $x_{0}$ shows that

(3.24) $$\begin{eqnarray}\displaystyle a_{I,R}\rightarrow \left[\frac{x_{0}J_{0}[\tan (\hat{{\it\gamma}}_{TS}\bar{y}_{s})\pm \text{i}]}{b\cos (\hat{{\it\gamma}}_{TS}\bar{y}_{s})}\int _{{\it\eta}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}\right](x-x_{0})^{-1}\quad \text{as }x\rightarrow x_{0}, & & \displaystyle\end{eqnarray}$$

where

(3.25) $$\begin{eqnarray}\displaystyle b & = & \displaystyle \frac{2}{3}\tan (\hat{{\it\gamma}}_{TS}\bar{y}_{s}){\it\eta}_{0}\,\text{Ai}({\it\eta}_{0})-\tan (\hat{{\it\gamma}}_{TS}\bar{y}_{s})\int _{{\it\eta}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2\hat{{\it\gamma}}_{TS}(\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0})^{5/3}}{3\hat{{\it\alpha}}_{TS}^{2}\hat{{\it\beta}}_{TS}^{2}}[\text{Ai}^{\prime }({\it\eta}_{0})-{\it\eta}_{0}^{2}\,\text{Ai}({\it\eta}_{0})].\end{eqnarray}$$

As (3.24) indicates, the amplitudes $a_{I}$ and $a_{R}$ blow up on approaching the neutral point $x_{0}$ . In the $O(\bar{{\it\epsilon}})$ vicinity of the neutral point, the rate of change of the forced response, $a_{I}^{\prime }/a_{I}$ and $a_{R}^{\prime }/a_{R}$ , becomes comparable with the growth rate of the T–S mode, implying that the streamwise variable no longer plays the role of a parameter but acts a variable (Wu Reference Wu1999) and consequently the preceding analysis is no longer valid. Non-parallelism strongly affects the evolution of the disturbance in this resonance region, which will be considered in the next section.

4.1 Upper deck

In the upper deck, the expansion takes the form

(4.1) $$\begin{eqnarray}\displaystyle (u,v,w,p,{\it\rho},T) & = & \displaystyle (1,0,0,1/{\it\gamma}M^{2},1,1)\nonumber\\ \displaystyle & & \displaystyle +\,({\it\delta}/\bar{{\it\epsilon}})(u_{TS},Mv_{TS},Mw_{TS},p_{TS},M^{2}{\it\rho}_{TS},M^{2}{\it\tau}_{TS})\tilde{E}\nonumber\\ \displaystyle & & \displaystyle +\,({\it\delta}/\bar{{\it\epsilon}})M({\it\sigma}u_{v},v_{v},{\it\sigma}Mw_{v},0,{\it\sigma}^{-1}{\it\rho}_{e},{\it\sigma}^{-1}{\it\tau}_{e})E_{v}\nonumber\\ \displaystyle & & \displaystyle +\,{\it\delta}(u_{2},Mv_{2},Mw_{2},p_{2},M^{2}{\it\rho}_{2},M^{2}{\it\tau}_{2})+\text{c.c.}+\cdots \,,\end{eqnarray}$$

where the $O({\it\delta}/\bar{{\it\epsilon}})$ terms with the subscripts ‘ $TS$ ’ stand for the signature of the T–S mode that is generated through the resonance, and its carrier wave

(4.2) $$\begin{eqnarray}\displaystyle \tilde{E}=\exp \{\text{i}(\hat{{\it\alpha}}_{TS}\bar{x}+\hat{{\it\beta}}_{TS}\bar{z}-\hat{{\it\omega}}_{TS}\bar{t})\}; & & \displaystyle\end{eqnarray}$$

the terms of the same order of magnitude but with the subscript ‘ $v$ ’ denote the vorticity–entropy waves, which arise along with the T–S mode in order to satisfy the shock condition, and their carrier is

(4.3) $$\begin{eqnarray}\displaystyle E_{v}=\exp \{\text{i}(Re^{-1/8}{\it\mu}_{w}^{1/8}T_{w}^{3/8}M^{1/4}\hat{{\it\alpha}}_{v}\bar{x}+\hat{{\it\beta}}_{TS}\bar{z}-\hat{{\it\omega}}_{TS}\bar{t})\}, & & \displaystyle\end{eqnarray}$$

where $\hat{{\it\alpha}}_{v}$ is the scaled streamwise wavenumber.

Substitution of (4.1) into (2.5) yields, at leading order, the linearized Euler equations. The pressure $p_{TS}$ and the normal velocity $v_{TS}$ satisfy

(4.4a,b ) $$\begin{eqnarray}\displaystyle \frac{\partial ^{2}p_{TS}}{\partial \bar{y}^{2}}+(\hat{{\it\alpha}}_{TS}^{2}-\hat{{\it\beta}}_{TS}^{2})p_{TS}=0,\quad \text{i}\hat{{\it\alpha}}_{TS}v_{TS}=-p_{TS,\bar{y}}. & & \displaystyle\end{eqnarray}$$

The solution for $p_{TS}$ is found as

(4.5) $$\begin{eqnarray}\displaystyle p_{TS}=A(x_{1})[\bar{a}_{I}\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}+\bar{a}_{R}\text{e}^{-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}], & & \displaystyle\end{eqnarray}$$

where $\hat{{\it\gamma}}_{TS}=(\hat{{\it\alpha}}_{TS}^{2}-\hat{{\it\beta}}_{TS}^{2})^{1/2}$ , $\bar{a}_{I}$ and $\bar{a}_{R}$ are constants to be determined, and $A(x_{1})$ is the amplitude of the T–S wave. The form of the solution indicates that shock-induced T–S modes exhibit the character of sound in the upper deck. The shock condition (A 18) implies that $p_{TS}=0$ at $\bar{y}=\bar{y}_{s}$ , and thus (Cowley & Hall Reference Cowley and Hall1990)

(4.6) $$\begin{eqnarray}\displaystyle \bar{a}_{I}\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}}+\bar{a}_{R}\text{e}^{-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}}=0, & & \displaystyle\end{eqnarray}$$

use of which in (4.5) gives

(4.7) $$\begin{eqnarray}\displaystyle p_{TS}=A(x_{1})\bar{a}_{I}(\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}-\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}). & & \displaystyle\end{eqnarray}$$

It follows from the transverse momentum equation in (4.4) that

(4.8) $$\begin{eqnarray}\displaystyle v_{TS}=-\frac{\hat{{\it\gamma}}_{TS}}{\hat{{\it\alpha}}_{TS}}\bar{a}_{I}A(x_{1})(\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}+\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}). & & \displaystyle\end{eqnarray}$$

Now turn to the vorticity–entropy waves, which accompany the T–S mode. Comparing (4.1) with (2.20)–(2.21) and noting (A 4) and (A 19), we find that

(4.9) $$\begin{eqnarray}\displaystyle (u_{v},v_{v},w_{v},{\it\rho}_{e},{\it\tau}_{e})=2\bar{a}_{I}A(x_{1})(\hat{{\it\alpha}}_{TS},-\hat{{\it\beta}}_{TS}^{2}/\hat{{\it\alpha}}_{TS},\hat{{\it\beta}}_{TS},\hat{{\it\alpha}}_{TS},-\hat{{\it\alpha}}_{TS})\text{e}^{-\text{i}\hat{{\it\gamma}}_{v}\bar{y}}\!\!\left/\!\sqrt{\hat{{\it\alpha}}_{TS}^{2}-\hat{{\it\beta}}_{TS}^{2}}\right..\quad & & \displaystyle\end{eqnarray}$$

By rewriting (A 4) in terms of $\bar{x}$ and $\bar{y}$ , and using the dispersion relation (A 5), the vertical wavenumber, $\hat{{\it\gamma}}_{v}\equiv -Re^{-3/8}{\it\mu}_{w}^{3/8}T_{w}^{9/8}M^{-1/4}[{\it\alpha}\sin {\it\phi}+{\it\nu}_{s}\cos {\it\phi}]$ , is found to be

(4.10) $$\begin{eqnarray}\displaystyle \hat{{\it\gamma}}_{v}\approx [\hat{{\it\alpha}}-Re^{-1/8}{\it\mu}_{w}^{1/8}T_{w}^{3/8}M^{1/4}\hat{{\it\omega}}_{TS}-\bar{{\it\epsilon}}\text{i}A^{\prime }(x_{1})/A]/(MV)\approx -({\it\sigma}M)\hat{{\it\alpha}}_{TS}, & & \displaystyle\end{eqnarray}$$

while the streamwise wavenumber

(4.11) $$\begin{eqnarray}\displaystyle \hat{{\it\alpha}}_{v}=\hat{{\it\omega}}_{TS}+Re^{-1/16}{\it\mu}_{w}^{1/16}T_{w}^{3/16}M^{1/8}\text{i}A^{\prime }/A, & & \displaystyle\end{eqnarray}$$

consistent with the advective equations satisfied by vorticity–entropy waves in the system $(x,y,z)$ . It is worth noting that the streamwise and spanwise velocity components of the vorticity wave are much greater than those of the T–S wave. As its phase speed is approximately unity, this part of the disturbance is trapped in a thin layer at the outer reach of the boundary layer and hence has little influence on the latter (Dong & Wu Reference Dong and Wu2013).

For the $O({\it\delta})$ disturbance in the expansion (4.1), it suffices to consider $p_{2}$ and $v_{2}$ only, which are governed by the equations

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial ^{2}p_{2}}{\partial \bar{y}^{2}}-\frac{\partial ^{2}p_{2}}{\partial \bar{x}^{2}}+\frac{\partial ^{2}p_{2}}{\partial \bar{z}^{2}}=2\text{i}\hat{{\it\alpha}}_{TS}\bar{a}_{I}A^{\prime }(x_{1})(\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}-\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}})\tilde{E}, & \displaystyle\end{eqnarray}$$

(4.13) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial v_{2}}{\partial \bar{x}}=-\frac{\partial p_{2}}{\partial \bar{y}}-\frac{\partial v_{TS}}{\partial x_{1}}\tilde{E}. & \displaystyle\end{eqnarray}$$

The solution for $p_{2}$ and $v_{2}$ takes the form $(p_{2},v_{2})=(\hat{P}_{2},\hat{V}_{2})\tilde{E}+\text{c.c.}$ , where $\hat{P}_{2}$ and $\hat{V}_{2}$ are found from (4.12) and (4.13) as

(4.14) $$\begin{eqnarray}\displaystyle \hat{P}_{2} & = & \displaystyle H_{1}(x_{1})\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}+H_{2}(x_{1})\text{e}^{-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}+\frac{\hat{{\it\alpha}}_{TS}}{\hat{{\it\gamma}}_{TS}}\bar{a}_{I}A^{\prime }(x_{1})(\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}+\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}})(\bar{y}-\bar{y}_{s}),\qquad\end{eqnarray}$$

(4.15) $$\begin{eqnarray}\displaystyle \hat{V}_{2} & = & \displaystyle -\frac{\hat{{\it\gamma}}_{TS}}{\hat{{\it\alpha}}_{TS}}[H_{1}\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}-H_{2}\text{e}^{-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}]+\frac{\text{i}\hat{{\it\beta}}_{TS}^{2}}{\hat{{\it\gamma}}_{TS}\hat{{\it\alpha}}_{TS}^{2}}\bar{a}_{I}A^{\prime }(x_{1})(\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}+\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}})\nonumber\\ \displaystyle & & \displaystyle -\,\bar{a}_{I}A^{\prime }(x_{1})(\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}-\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}})(\bar{y}-\bar{y}_{s}).\end{eqnarray}$$

Clearly, as $\bar{y}\rightarrow 0$ ,

(4.16) $$\begin{eqnarray}\displaystyle & \displaystyle p_{2}\rightarrow \left\{H_{1}+H_{2}-\frac{\hat{{\it\alpha}}_{TS}}{\hat{{\it\gamma}}_{TS}}\bar{a}_{I}A^{\prime }(1+\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}})\bar{y}_{s}\right\}\tilde{E}, & \displaystyle\end{eqnarray}$$

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle v_{2}\rightarrow -\left\{\frac{\hat{{\it\gamma}}_{TS}}{\hat{{\it\alpha}}_{TS}}(H_{1}-H_{2})-\frac{\text{i}\hat{{\it\beta}}_{TS}^{2}}{\hat{{\it\gamma}}_{TS}\hat{{\it\alpha}}_{TS}^{2}}\bar{a}_{I}A^{\prime }(1+\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}})-\bar{a}_{I}A^{\prime }(1-\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}})\bar{y}_{s}\right\}\tilde{E}.\qquad & \displaystyle\end{eqnarray}$$

In addition, the shock condition (A 46) gives

(4.18) $$\begin{eqnarray}\displaystyle H_{1}(x_{1})\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}}+H_{2}(x_{1})\text{e}^{-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}}=J_{0}, & & \displaystyle\end{eqnarray}$$

where $J_{0}$ is defined in (A 47) for fast/slow acoustic, vortical and entropy modes.

4.2 Main deck

Since no further interaction takes place in the main deck, the perturbation on the triple-deck scales arises merely as the response to the upper deck and satisfies the standard main-deck equations. The solution for the velocities is found as

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle u=U_{B}({\tilde{Y}})+{\it\delta}Re^{5/16}{\it\mu}_{w}^{3/16}T_{w}^{9/16}M^{11/8}B(x_{1})U_{B}^{\prime }({\tilde{Y}})\tilde{E}+\text{c.c.}, & \displaystyle\end{eqnarray}$$

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle v=-{\it\delta}Re^{-3/16}{\it\mu}_{w}^{3/16}T_{w}^{9/16}M^{11/8}\frac{\partial }{\partial x}\{B(x_{1})U_{B}({\tilde{Y}})\tilde{E}+\text{c.c.}\}. & \displaystyle\end{eqnarray}$$

In order to match with the upper-deck solution, the displacement function $B$ expands as $B=A(x_{1})B_{1}+\bar{{\it\epsilon}}B_{2}(x_{1})$ with $B_{1}$ being an undetermined constant and $B_{2}$ a function of $x_{1}$ . It can be shown that

(4.21) $$\begin{eqnarray}\displaystyle u & \rightarrow & \displaystyle {\it\lambda}{\it\mu}_{w}^{-1/2}T_{w}^{-1/2}{\tilde{Y}}+{\it\delta}Re^{5/16}{\it\mu}_{w}^{-5/16}T_{w}^{1/16}M^{11/8}\nonumber\\ \displaystyle & & \displaystyle \times \,[{\it\lambda}_{0}AB_{1}+\bar{{\it\epsilon}}({\it\lambda}_{0}B_{2}+{\it\lambda}_{1}x_{1}AB_{1})]\tilde{E}+\text{c.c.}\quad \text{as }{\tilde{Y}}\rightarrow 0,\end{eqnarray}$$

(4.22) $$\begin{eqnarray}\displaystyle v & \rightarrow & \displaystyle -{\it\delta}Re^{3/16}{\it\mu}_{w}^{-3/16}T_{w}^{-9/16}M^{5/8}\nonumber\\ \displaystyle & & \displaystyle \times \,[\text{i}\hat{{\it\alpha}}_{TS}AB_{1}+\bar{{\it\epsilon}}(\text{i}\hat{{\it\alpha}}_{TS}B_{2}+A^{\prime }B_{1})]\tilde{E}+\text{c.c.}\quad \text{as }{\tilde{Y}}\rightarrow \infty .\end{eqnarray}$$

Matching between the upper-deck and main-deck solutions for $v$ implies that

(4.23) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\hat{{\it\gamma}}_{TS}}{\hat{{\it\alpha}}_{TS}}\bar{a}_{I}(1+\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}})=\text{i}\hat{{\it\alpha}}_{TS}B_{1}, & \displaystyle\end{eqnarray}$$

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\hat{{\it\gamma}}_{TS}}{\hat{{\it\alpha}}_{TS}}(H_{1}-H_{2})-\frac{\text{i}\hat{{\it\beta}}_{TS}^{2}}{\hat{{\it\gamma}}_{TS}\hat{{\it\alpha}}_{TS}^{2}}\bar{a}_{I}A^{\prime }(1+\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}})-\bar{a}_{I}A^{\prime }(1-\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}})\bar{y}_{s}=\text{i}\hat{{\it\alpha}}_{TS}B_{2}+A^{\prime }B_{1}. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

4.3 Lower deck

In the lower deck, the solution can be written as

(4.25a ) $$\begin{eqnarray}\displaystyle u & = & \displaystyle Re^{-1/8}{\it\mu}_{w}^{1/8}T_{w}^{3/8}M^{1/4}({\it\lambda}_{0}+\bar{{\it\epsilon}}{\it\lambda}_{1}x_{1})Y\nonumber\\ \displaystyle & & \displaystyle +\,{\it\delta}Re^{1/8}{\it\mu}_{w}^{-1/8}T_{w}^{5/8}M^{7/4}/\bar{{\it\epsilon}}[A(x_{1})\tilde{U} _{1}+\bar{{\it\epsilon}}\tilde{U} _{2}]\tilde{E}+\text{c.c.}+\cdots \,,\end{eqnarray}$$

(4.25b ) $$\begin{eqnarray}\displaystyle v & = & \displaystyle {\it\delta}Re^{-1/8}{\it\mu}_{w}^{1/8}T_{w}^{3/8}M^{5/4}/\bar{{\it\epsilon}}[A(x_{1})\tilde{V}_{1}+\bar{{\it\epsilon}}\tilde{V}_{2}]\tilde{E}+\text{c.c.}+\cdots \,,\end{eqnarray}$$

(4.25c ) $$\begin{eqnarray}\displaystyle w & = & \displaystyle {\it\delta}Re^{1/8}{\it\mu}_{w}^{-1/8}T_{w}^{5/8}M^{3/4}/\bar{{\it\epsilon}}[A(x_{1})\tilde{W}_{1}+\bar{{\it\epsilon}}\tilde{W}_{2}]\tilde{E}+\text{c.c.}+\cdots \,,\end{eqnarray}$$

(4.25d ) $$\begin{eqnarray}\displaystyle p & = & \displaystyle {\it\delta}/\bar{{\it\epsilon}}[A(x_{1})\tilde{P}_{1}+\bar{{\it\epsilon}}\tilde{P}_{2}]\tilde{E}+\text{c.c.}+\cdots \,.\end{eqnarray}$$

(4.26a ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\hat{{\it\alpha}}_{TS}\tilde{U} _{1}+\tilde{V}_{1,Y}+\text{i}\hat{{\it\beta}}_{TS}\tilde{W}_{1}=0, & \displaystyle\end{eqnarray}$$

(4.26b ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}(\hat{{\it\alpha}}_{TS}{\it\lambda}_{0}Y-\hat{{\it\omega}}_{TS})\tilde{U} _{1}+{\it\lambda}_{0}\tilde{V}_{1}=\tilde{U} _{1,YY}, & \displaystyle\end{eqnarray}$$

(4.26c ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}(\hat{{\it\alpha}}_{TS}{\it\lambda}_{0}Y-\hat{{\it\omega}}_{TS})\tilde{W}_{1}=-\text{i}\hat{{\it\beta}}_{TS}\tilde{P}_{1}+\tilde{W}_{1,YY}. & \displaystyle\end{eqnarray}$$

$\tilde{U} _{1}=\tilde{V}_{1}=\tilde{W}_{1}=0$

$Y=0$

$(\tilde{U} _{1},\tilde{W}_{1})\rightarrow ({\it\lambda}_{0}B_{1},0)$

$Y\rightarrow \infty$

(4.27a,b ) $$\begin{eqnarray}\displaystyle \tilde{V}_{1,YYY}(0)=\hat{{\it\beta}}_{TS}^{2}\tilde{P}_{1};\quad \tilde{V}_{1,Y}\rightarrow -\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0}B_{1}\quad \text{as }Y\rightarrow \infty . & & \displaystyle\end{eqnarray}$$

As in the previous section, it is found that

(4.28) $$\begin{eqnarray}\displaystyle \tilde{V}_{1,Y}=\int _{{\it\eta}_{0}}^{{\it\eta}}\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}, & & \displaystyle\end{eqnarray}$$

where $\text{Ai}$ denotes the Airy function and

(4.29) $$\begin{eqnarray}\displaystyle {\it\eta}=(\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0})^{1/3}Y+{\it\eta}_{0},\quad {\it\eta}_{0}=-\text{i}\hat{{\it\omega}}_{TS}(\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0})^{-2/3}. & & \displaystyle\end{eqnarray}$$

Application of the boundary conditions (4.27) gives

(4.30a,b ) $$\begin{eqnarray}\displaystyle (\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0})^{2/3}\,\text{Ai}^{\prime }({\it\eta}_{0})=\hat{{\it\beta}}_{TS}^{2}\tilde{P}_{1},\quad \int _{{\it\eta}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}=-\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0}B_{1}. & & \displaystyle\end{eqnarray}$$

The fact that the pressure does not vary across the main and lower decks leads to

(4.31) $$\begin{eqnarray}\displaystyle \bar{a}_{I}(1-\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}})=\tilde{P}_{1}. & & \displaystyle\end{eqnarray}$$

From (4.23), (4.30) and (4.31) follows the dispersion relation of T–S modes (Cowley & Hall Reference Cowley and Hall1990)

(4.32) $$\begin{eqnarray}\displaystyle {\it\Delta}(\hat{{\it\omega}}_{TS},\hat{{\it\alpha}}_{TS},\hat{{\it\beta}}_{TS};x_{0})=0, & & \displaystyle\end{eqnarray}$$

where the dispersion function ${\it\Delta}$ is defined by (3.22). The neutral wavenumbers and frequency can be determined by

(4.33a-d ) $$\begin{eqnarray}\displaystyle \hat{{\it\alpha}}_{TS}={{\it\lambda}_{0}}^{5/4}{\it\alpha}_{N},\quad \hat{{\it\beta}}_{TS}={{\it\lambda}_{0}}^{5/4}{\it\beta}_{N},\quad \hat{{\it\omega}}_{TS}={{\it\lambda}_{0}}^{3/2}{\it\omega}_{N},\quad \bar{y}_{s}={{\it\lambda}_{0}}^{-5/4}y_{N}, & & \displaystyle\end{eqnarray}$$

with ${\it\alpha}_{N}$ , ${\it\beta}_{N}$ and ${\it\omega}_{N}$ satisfying

(4.34a,b ) $$\begin{eqnarray}\displaystyle {\it\alpha}_{N}^{1/3}{\it\beta}_{N}^{2}\tan [({\it\alpha}_{N}^{2}-{\it\beta}_{N}^{2})^{1/2}y_{N}]=d_{1}({\it\alpha}_{N}^{2}-{\it\beta}_{N}^{2})^{1/2},\quad {\it\omega}_{N}=d_{2}{\it\alpha}_{N}^{2/3}, & & \displaystyle\end{eqnarray}$$

where $d_{1}\approx 1.001$ and $d_{2}\approx 2.299$ .

Figure 3.

The neutral curves ${\it\alpha}_{N}={\it\alpha}_{N}({\it\beta}_{N})$ for (a) $y_{N}=0.2$ , (b) $y_{N}=1$ and (c) $y_{N}=2$ .

Figure 4.

Normalized wavenumber ${\it\alpha}_{r}$ (a) and normalized spatial growth rate $-{\it\alpha}_{i}$ (b) versus ${\it\omega}_{N}$ calculated for $y_{N}=2$ . Mode 1: solid line for ${\it\beta}_{N}=0.5$ and dashed line for ${\it\beta}_{N}=1$ . Modes 2 and 3: solid lines for ${\it\beta}=1$ and dashed lines for ${\it\beta}_{N}=2$ .

We computed neutral modes for several values of $y_{N}$ . As is shown in figure 3, the presence of a shock allows for a countable set of neutral modes. We denote the mode with the smallest ${\it\alpha}_{N}$ for a given ${\it\beta}_{N}$ as mode 1, and modes 2, 3 and etc. have increasingly larger ${\it\alpha}_{N}$ . The bandwidth of these shock-induced instability modes appears to be inversely proportional to $y_{N}$ . Their existence is restricted in the region ${\it\beta}_{N}<{\it\alpha}_{N}$ except mode-1 family, which continues to the region ${\it\beta}_{N}>{\it\alpha}_{N}$ . Modes in this region approach, in the limit $y_{N}\rightarrow \infty$ , the more familiar T–S modes whose eigenfunctions attenuate in the upper deck (Smith Reference Smith1989).

Cowley & Hall (Reference Cowley and Hall1990) calculated only temporal growth rates. Here, we perform a spatial stability analysis, which is considered to be more relevant. Figure 4 displays the wavenumbers and spatial amplification rates of modes 1, 2 and 3. For a fixed ${\it\beta}_{N}$ , as ${\it\omega}_{N}$ is increased from the neutral frequency, the wavenumber increases; the growth rate increases first, reaches a peak and then decreases, but always remains positive. This is in contrast with the temporal growth rate, which changes its sign repeatedly as the wavenumber increases (Cowley & Hall Reference Cowley and Hall1990).

Figure 5.

Contours of wavenumbers (a) and spatial amplification rates (b) of growing modes 1 for the case of $y_{N}=2$ . The dashed line in (a) marks the neutral curve, and the decaying modes below it are not considered.

Figure 6.

Contours of wavenumbers (a) and spatial amplification rates (b) of growing modes 2 for the case of $y_{N}=2$ . The dashed line in (a) marks the neutral curve, and the decaying modes below it are not considered.

In order to provide further information about the instability, we compute the growth rate $(-{\it\alpha}_{i})$ and the associated wavenumber for modes 1 and 2 in a fairly large range of ${\it\omega}_{N}$ and ${\it\beta}_{N}$ . Figures 5 and 6 show the contours of $-{\it\alpha}_{i}({\it\omega}_{N},{\it\beta}_{N})$ and ${\it\alpha}_{r}({\it\omega}_{N},{\it\beta}_{N})$ on the $({\it\omega}_{N},{\it\beta}_{N})$ plane. Overall, the second (and higher) modes have smaller growth rates than those in mode-1 family. For each ${\it\omega}_{N}$ , the most unstable mode is oblique. The shock-induced neutral modes form the lower branch of the neutral curve, the unstable region above which appears to extend to infinity suggesting that an upper branch is absent in the triple-deck scaling regime. The results imply that once a lower-branch mode is excited, it will amplify since for a mode with a fixed physical frequency and spanwise wavelength, its non-dimensional frequency and spanwise wavenumber increase as the boundary layer becomes thicker.

The second-order terms in the perturbation expansion (4.25) are governed by equations

(4.35a ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\hat{{\it\alpha}}_{TS}\tilde{U} _{2}+\tilde{V}_{2,Y}+\text{i}\hat{{\it\beta}}_{TS}\tilde{W}_{2}=-A^{\prime }(x_{1})\tilde{U} _{1}, & \displaystyle\end{eqnarray}$$

(4.35b ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}(\hat{{\it\alpha}}_{TS}{\it\lambda}_{0}Y-\hat{{\it\omega}}_{TS})\tilde{U} _{2}+{\it\lambda}_{0}\tilde{V}_{2}=\tilde{U} _{2,YY}-{\it\Lambda}(x_{1})Y\tilde{U} _{1}-{\it\lambda}_{1}x_{1}A(x_{1})\tilde{V}_{1}, & \displaystyle\end{eqnarray}$$

(4.35c ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}(\hat{{\it\alpha}}_{TS}{\it\lambda}_{0}Y-\hat{{\it\omega}}_{TS})\tilde{W}_{2}=-\text{i}\hat{{\it\beta}}_{TS}\tilde{P}_{2}+\tilde{W}_{2,YY}-{\it\Lambda}(x_{1})Y\tilde{W}_{1}, & \displaystyle\end{eqnarray}$$

${\it\Lambda}$

(4.36) $$\begin{eqnarray}\displaystyle {\it\Lambda}={\it\lambda}_{0}A^{\prime }(x_{1})+\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{1}x_{1}A(x_{1}). & & \displaystyle\end{eqnarray}$$

The no-slip condition on the wall and matching with the main-deck solution imply that $\tilde{U} _{2}=\tilde{V}_{2}=\tilde{W}_{2}=0$ at $Y=0$ and $(\tilde{U} _{2},\tilde{W}_{2})\rightarrow ({\it\lambda}_{0}B_{2}+{\it\lambda}_{1}x_{1}AB_{1},0)$ as $Y\rightarrow \infty$ , or equivalently,

(4.37a,b ) $$\begin{eqnarray}\displaystyle \tilde{V}_{2,YYY}(0)=\hat{{\it\beta}}_{TS}^{2}\tilde{P}_{2};\quad \tilde{V}_{2,Y}\rightarrow -\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0}B_{2}-{\it\Lambda}(x_{1})B_{1}\quad \text{as }Y\rightarrow \infty . & & \displaystyle\end{eqnarray}$$

Eliminating the pressure $\tilde{P}_{2}$ from (4.35) yields

(4.38) $$\begin{eqnarray}\displaystyle \left\{\frac{\partial ^{2}}{\partial Y^{2}}-\text{i}(\hat{{\it\alpha}}_{TS}{\it\lambda}_{0}Y-\hat{{\it\omega}}_{TS})\right\}\tilde{V}_{2,YY}={\it\Lambda}(x_{1})Y\tilde{V}_{1,YY}={\it\Lambda}(x_{1})({\it\eta}-{\it\eta}_{0})\,\text{Ai}({\it\eta}), & & \displaystyle\end{eqnarray}$$

which has the solution (cf. Wu Reference Wu1999)

(4.39) $$\begin{eqnarray}\displaystyle \tilde{V}_{2,Y}=\frac{1}{3}(\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0})^{-1}{\it\Lambda}(x_{1})[({\it\eta}-3{\it\eta}_{0})\,\text{Ai}({\it\eta})+2{\it\eta}_{0}\,\text{Ai}({\it\eta}_{0})]+q_{2}\int _{{\it\eta}_{0}}^{{\it\eta}}\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}, & & \displaystyle\end{eqnarray}$$

where $q_{2}$ is an unknown function of $x_{1}$ . The boundary conditions (4.37) lead to

(4.40) $$\begin{eqnarray}\displaystyle & \displaystyle {\textstyle \frac{2}{3}}(\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0})^{-1/3}{\it\Lambda}[\text{Ai}^{\prime }({\it\eta}_{0})-{\it\eta}_{0}\,\text{Ai}^{\prime \prime }({\it\eta}_{0})]+(\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0})^{2/3}q_{2}\,\text{Ai}^{\prime }({\it\eta}_{0})=\hat{{\it\beta}}_{TS}^{2}\tilde{P}_{2}, & \displaystyle\end{eqnarray}$$

(4.41) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{2}{3}(\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0})^{-1}{\it\Lambda}{\it\eta}_{0}\,\text{Ai}({\it\eta}_{0})+q_{2}\int _{{\it\eta}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}=-\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0}B_{2}-{\it\Lambda}B_{1}. & \displaystyle\end{eqnarray}$$

Since the pressure does not change across the main and lower decks, we have

(4.42) $$\begin{eqnarray}\displaystyle H_{1}+H_{2}-\frac{\hat{{\it\alpha}}_{TS}}{\hat{{\it\gamma}}_{TS}}\bar{a}_{I}A^{\prime }(1+\text{e}^{2\text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}})\bar{y}_{s}=\tilde{P}_{2}. & & \displaystyle\end{eqnarray}$$

Eliminating $B_{2}$ and $\tilde{P}_{2}$ from (4.18), (4.24), (4.40)–(4.42) and making use of (4.30)–(4.31), we finally obtain the amplitude equation for $A$ ,

(4.43) $$\begin{eqnarray}\displaystyle A^{\prime }(x_{1})={\it\varsigma}x_{1}A+N, & & \displaystyle\end{eqnarray}$$

where

(4.44a,b ) $$\begin{eqnarray}\displaystyle {\it\varsigma}=-\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{1}b/({\it\lambda}_{0}a),\quad N=-{\it\lambda}_{0}\hat{{\it\gamma}}_{TS}\sec (\hat{{\it\gamma}}_{TS}\bar{y}_{s})J_{0}/a, & & \displaystyle\end{eqnarray}$$

with constants $b$ being given by (3.25), and

(4.45) $$\begin{eqnarray}\displaystyle a & = & \displaystyle \frac{2}{3}\tan (\hat{{\it\gamma}}_{TS}\bar{y}_{s}){\it\eta}_{0}\,\text{Ai}({\it\eta}_{0})+\left[\frac{\hat{{\it\alpha}}_{TS}^{2}\bar{y}_{s}}{\hat{{\it\gamma}}_{TS}\cos ^{2}(\hat{{\it\gamma}}_{TS}\bar{y}_{s})}-\frac{\hat{{\it\beta}}_{TS}^{2}}{\hat{{\it\gamma}}_{TS}^{2}}\tan (\hat{{\it\gamma}}_{TS}\bar{y}_{s})\right]\int _{{\it\eta}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2\hat{{\it\gamma}}_{TS}(\text{i}\hat{{\it\alpha}}_{TS}{\it\lambda}_{0})^{5/3}}{3\hat{{\it\alpha}}_{TS}^{2}\hat{{\it\beta}}_{TS}^{2}}[\text{Ai}^{\prime }({\it\eta}_{0})-{\it\eta}_{0}^{2}\,\text{Ai}({\it\eta}_{0})].\end{eqnarray}$$

5.1 Matching with the pre-resonance region

On assuming that the amplitude of the T–S wave vanishes as $x_{1}\rightarrow -\infty$ , the amplitude equation (4.43) has the solution

(5.1) $$\begin{eqnarray}\displaystyle A(x_{1})=N\text{e}^{{\it\varsigma}{x_{1}}^{2}/2}\int _{-\infty }^{x_{1}}\text{e}^{-{\it\varsigma}{\it\xi}^{2}/2}\,\text{d}{\it\xi}. & & \displaystyle\end{eqnarray}$$

More precisely, it can be shown that

(5.2) $$\begin{eqnarray}\displaystyle A(x_{1})\rightarrow (N/{\it\varsigma})(-x_{1})^{-1}+O(x_{1}^{-3})\quad \text{as }x_{1}\rightarrow -\infty . & & \displaystyle\end{eqnarray}$$

The above behaviour implies matching with the upstream response. To illustrate this, we take the pressure $p$ as an example. In the upper deck, the pressure $p$ is given by

(5.3) $$\begin{eqnarray}\displaystyle p={\it\delta}Re^{3/16}{\it\mu}_{w}^{-3/16}T_{w}^{-9/16}M^{-3/8}A(x_{1})[\bar{a}_{I}\text{e}^{\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}+\bar{a}_{R}\text{e}^{-\text{i}\hat{{\it\gamma}}_{TS}\bar{y}}]\tilde{E}+\text{c.c.}, & & \displaystyle\end{eqnarray}$$

where $\bar{a}_{I}$ and $\bar{a}_{R}$ are found from (4.30)–(4.31) as

(5.4) $$\begin{eqnarray}\displaystyle \bar{a}_{I,R}=\mp \frac{\hat{{\it\alpha}}_{TS}\text{e}^{\mp \text{i}\hat{{\it\gamma}}_{TS}\bar{y}_{s}}}{2{\it\lambda}_{0}\hat{{\it\gamma}}_{TS}\cos (\hat{{\it\gamma}}_{TS}\bar{y}_{s})}\int _{{\it\eta}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}. & & \displaystyle\end{eqnarray}$$

Using (5.2), (4.44) and (2.45), we find that

(5.5) $$\begin{eqnarray}\displaystyle \bar{a}_{I,R}A(x_{1})\rightarrow \left\{\frac{x_{0}J_{0}[\tan (\hat{{\it\gamma}}_{TS}\bar{y}_{s})\pm \text{i}]}{b\cos (\hat{{\it\gamma}}_{TS}\bar{y}_{s})}\int _{{\it\eta}_{0}}^{\infty }\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}\right\}x_{1}^{-1}\quad \text{as }x_{1}\rightarrow -\infty . & & \displaystyle\end{eqnarray}$$

Comparison with (3.24) shows that

(5.6) $$\begin{eqnarray}\displaystyle Re^{3/16}{\it\mu}_{w}^{-3/16}T_{w}^{-9/16}M^{-3/8}A(x_{1})\bar{a}_{I,R}\rightarrow a_{I,R}\quad \text{as }x_{1}\rightarrow -\infty , & & \displaystyle\end{eqnarray}$$

which is precisely what is required for the matching of the pressure in the two stages.

5.2 Coupling coefficients

We now turn to the downstream limit, $x_{1}\rightarrow +\infty$ . In this limit,

(5.7) $$\begin{eqnarray}\displaystyle A(x_{1})\rightarrow A_{\infty }\text{e}^{{\it\varsigma}{x_{1}}^{2}/2},\quad \text{with }A_{\infty }=N\int _{-\infty }^{\infty }\text{e}^{-{\it\varsigma}{\it\xi}^{2}/2}\,\text{d}{\it\xi}=N\sqrt{2{\rm\pi}/{\it\varsigma}}. & & \displaystyle\end{eqnarray}$$

In the post-resonance stage corresponding to $x_{1}=O(\bar{{\it\epsilon}}^{-1})$ , the amplitude of the excited T–S mode, $A\text{e}^{\text{i}\hat{{\it\alpha}}_{TS}\bar{x}}$ , is replaced by the usual WKBJ form

(5.8) $$\begin{eqnarray}\displaystyle A_{\infty }\exp \left\{\text{i}Re^{3/8}{\it\mu}_{w}^{-3/8}T_{w}^{-9/8}M^{-3/4}\int _{x_{0}}^{x}\hat{{\it\alpha}}_{TS}(x)\,\text{d}x\right\}, & & \displaystyle\end{eqnarray}$$

where the complex wavenumber $\hat{{\it\alpha}}_{TS}(x)$ is determined by the local parallel stability theory. Clearly, the constant $A_{\infty }$ appears as the (scaled) initial amplitude.

In order to be precise, we use the maximum value of the streamwise velocity of the T–S wave as an indicator of its magnitude. First, inserting $\tilde{P}_{1}$ found from (4.30) into the spanwise momentum equation in (4.26), we may express the spanwise velocity as $\tilde{W}_{1}=(\text{i}\,\text{Ai}^{\prime }({\it\eta}_{0})/\hat{{\it\beta}}_{TS})L({\it\eta})$ , where $L({\it\eta})$ is the solution to the boundary-value problem:

(5.9) $$\begin{eqnarray}\displaystyle L^{\prime \prime }-{\it\eta}L=1,\quad L({\it\eta}_{0})=0;\quad L\rightarrow 0\quad \text{as }{\it\eta}\rightarrow \infty . & & \displaystyle\end{eqnarray}$$

From the continuity equation, the streamwise velocity is obtained and its maximum magnitude at the neutral point is given by (cf. Wu Reference Wu1999)

(5.10) $$\begin{eqnarray}\displaystyle u_{TS}={\it\delta}Re^{5/16}{\it\mu}_{w}^{-5/16}T_{w}^{1/16}M^{11/8}\hat{{\it\alpha}}_{TS}^{-1}A_{\infty }U_{m}, & & \displaystyle\end{eqnarray}$$

where

(5.11) $$\begin{eqnarray}\displaystyle U_{m}=\max _{{\it\eta}}\left|\int _{{\it\eta}_{0}}^{{\it\eta}}\,\text{Ai}({\it\eta})\,\text{d}{\it\eta}-\text{Ai}^{\prime }({\it\eta}_{0})L({\it\eta})\right|. & & \displaystyle\end{eqnarray}$$

Following the practice in experiments, we normalize the dimensional frequency of the disturbance $\breve{{\it\omega}}$ as

(5.12) $$\begin{eqnarray}\displaystyle f=\breve{{\it\omega}}\breve{{\it\nu}}_{s}/\breve{U}_{s}^{2}, & & \displaystyle\end{eqnarray}$$

For the rescalings (2.33), we find by using (4.33)–(4.34) that the neutral position, non-dimensionalized by $\breve{{\it\mu}}_{s}/\breve{{\it\rho}}_{s}\breve{U}_{s}$ , is given by

(5.13) $$\begin{eqnarray}\displaystyle \tilde{x}_{0}\equiv \frac{\breve{{\it\rho}}_{s}\breve{U}_{s}\breve{x}_{0}}{\breve{{\it\mu}}_{s}}={\it\mu}_{w}^{-1/3}T_{w}^{-1}M^{-2/3}{\it\lambda}_{0}^{2}(f/{\it\omega}_{N})^{-4/3}. & & \displaystyle\end{eqnarray}$$

The streamwise length scale $\breve{l}$ can be taken as $\breve{x}_{0}$ so that $x_{0}=1$ , ${\it\lambda}_{0}={\it\chi}$ and the Reynolds number $Re$ is actually $\tilde{x}_{0}$ .

We define the coupling coefficient for the fast/slow acoustic waves respectively as

(5.14) $$\begin{eqnarray}\displaystyle C_{A}^{\pm }=u_{TS}/{\it\delta}p_{I}^{\pm }, & & \displaystyle\end{eqnarray}$$

and it follows from (5.7), (4.44) and (5.12)–(5.13) that

(5.15) $$\begin{eqnarray}\displaystyle C_{A}^{\pm }=\frac{2}{2-\bar{{\it\gamma}}}{\it\sigma}^{2}{\it\mu}_{w}^{-5/12}T_{w}^{-1/4}M^{19/6}f^{-5/12}{\it\kappa}, & & \displaystyle\end{eqnarray}$$

where

(5.16) $$\begin{eqnarray}\displaystyle {\it\kappa}({\it\beta}_{N},y_{N})=\sqrt{2{\rm\pi}}{\it\chi}^{13/8}{\it\omega}_{N}^{5/12}\sqrt{1-({\it\beta}_{N}/{\it\alpha}_{N})^{2}}\sec \left(\sqrt{{\it\alpha}_{N}^{2}-{\it\beta}_{N}^{2}}y_{N}\right)U_{m}/|a\sqrt{{\it\varsigma}}|.\qquad & & \displaystyle\end{eqnarray}$$

Similarly, we define the coupling coefficients for the vorticity and entropy waves respectively as

(5.17a,b ) $$\begin{eqnarray}\displaystyle C_{V}\equiv u_{TS}/{\it\delta}v_{I},\quad C_{E}\equiv u_{TS}/{\it\delta}r_{I}. & & \displaystyle\end{eqnarray}$$

It is found that $C_{V}$ and $C_{E}$ scale with $f$ and $M$ as

(5.18a,b ) $$\begin{eqnarray}\displaystyle C_{V}=2{\it\sigma}^{-1}{\it\mu}_{w}^{-5/12}T_{w}^{-1/4}M^{-5/6}f^{-5/12}{\it\kappa},\quad C_{E}={\it\sigma}^{2}{\it\mu}_{w}^{-5/12}T_{w}^{-1/4}M^{7/6}f^{-5/12}{\it\kappa}.\qquad & & \displaystyle\end{eqnarray}$$

6.1 Boundary-layer response in the pre-resonance region

The boundary-layer responds to a broadband of free-stream disturbances, including to vortical and entropy disturbances since they generate acoustic waves downstream of the shock. We first consider the characteristics of the response at a fixed streamwise location $x$ . The dependence on $x$ is through ${\it\lambda}$ with $x$ acting as a parameter, and it can be scaled out by setting

(6.1a-d ) $$\begin{eqnarray}\displaystyle \hat{{\it\alpha}}_{a}={\it\lambda}^{5/4}{\it\alpha}_{A},\quad \hat{{\it\beta}}_{a}={\it\lambda}^{5/4}{\it\beta}_{A},\quad \hat{{\it\omega}}_{a}={\it\lambda}^{3/2}{\it\omega}_{A},\quad \bar{y}_{s}={\it\lambda}^{-5/4}y_{A}. & & \displaystyle\end{eqnarray}$$

Figure 7.

Contours of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ in the ${\it\alpha}_{A}{-}{\it\omega}_{A}$ plane for $y_{A}=1$ and ${\it\beta}_{A}=0.5$ . The dashed lines in (a) represent the loci of (6.4) for $n=1,2$ .

As an illustration, calculations are performed for $y_{A}=1$ and ${\it\beta}_{A}=0.5$ with ${\it\alpha}_{A}$ and ${\it\omega}_{A}$ in the range $1\leqslant {\it\alpha}_{A}\leqslant 6$ and $0<{\it\omega}_{A}\leqslant 10$ . Figure 7 shows the contours of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ ; contours of $|a_{R}/J_{0}|$ look similar to those of $|a_{I}/J_{0}|$ and hence are not shown. The response is finite for all $({\it\alpha}_{A},{\it\omega}_{A})$ except when $({\it\alpha}_{A},{\it\omega}_{A})=(1.46,2.95)$ and $(4.65,6.40)$ , which correspond to neutral modes 1 and 2 respectively. A striking feature is that there is an extremely strong response to a broadband of high-frequency free-stream disturbances within a narrow band of ${\it\alpha}_{A}$ , represented by the two narrow and almost vertical strips in the figure. In order to explain this, an analysis of (3.21) is performed in the high-frequency limit $\hat{{\it\omega}}_{a}\gg 1$ . For a general ${\it\alpha}_{A}$ , both the numerator and the denominator ${\it\Delta}$ are of $O(1)$ . However, the latter reduces to $O({\it\omega}_{A}^{-5/2})$ if ${\it\alpha}_{A}$ is sufficiently close to

(6.2) $$\begin{eqnarray}\displaystyle {\it\alpha}_{c}=\left[\left[\frac{(2n-1){\rm\pi}}{2y_{A}}\right]^{2}+{\it\beta}_{A}^{2}\right]^{1/2}\quad (n=1,2,\ldots ), & & \displaystyle\end{eqnarray}$$

while the numerator remains of $O(1)$ so that $a_{I,R}/J_{0}$ raises to $O({\it\omega}_{A}^{5/2})$ . More precisely, the analysis shows that

(6.3) $$\begin{eqnarray}\displaystyle a_{I,R}/J_{0}\sim (-1)^{n}\frac{(2n-1){\rm\pi}\text{e}^{-{\rm\pi}\text{i}/4}}{4{\it\alpha}_{c}^{2}{\it\beta}_{A}^{2}y_{A}}{\it\omega}_{A}^{5/2}, & & \displaystyle\end{eqnarray}$$

when

(6.4a,b ) $$\begin{eqnarray}\displaystyle {\it\alpha}_{A}={\it\alpha}_{c}-({\it\beta}_{A}^{2}/y_{A}){\it\omega}_{A}^{-1}-\frac{2{\it\beta}_{A}^{6}}{(2n-1)^{2}{\rm\pi}^{2}{\it\alpha}_{c}}{\it\omega}_{A}^{-2}\quad \text{and}\quad {\it\omega}_{A}\gg 1. & & \displaystyle\end{eqnarray}$$

The loci of (6.4) are marked by the dashed lines in figure 7(a). Figure 8 compares the numerical result with the asymptotic approximation (6.3). A good agreement is observed. The strong response is therefore due to the fact that the dispersion function ${\it\Delta}$ almost vanishes for ${\it\alpha}_{A}$ and ${\it\omega}_{A}$ specified by (6.4), and we will refer to this as quasi- or near resonance (which differs from the exact resonance corresponding to ${\it\Delta}=0$ ). The quasi-resonance is of broadband nature with respect to the frequency but highly selective with respect to the wavenumber. For each ${\it\beta}_{A}$ , there exists a series of strips centred at the curves as specified by (6.4). In the presence of a broadband of free-stream disturbances, the boundary layer acts as a high-pass amplifier. Figure 7(b) indicates that for relatively low frequencies $|\mathscr{R}|<1$ , that is, the boundary layer absorbs some of the incident acoustic wave energy. In contrast, for high-frequency waves, $|\mathscr{R}|>1$ , implying that the wave reflected by the viscous boundary layer is stronger than the incident wave, a phenomenon referred to as over-reflection.

Figure 8.

Peak values of $|a_{I}/J_{0}|$ at quasi-resonance. Solid lines: numerical solution calculated from (3.21); dashed lines: asymptotic approximation (6.3) for ${\it\omega}_{A}\gg 1$ .

Figure 9.

Variation of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ with ${\it\omega}_{A}$ for ${\it\alpha}_{A}$ being the wavenumber of mode 1.

We are primarily interested in cases where an exact resonance with a T–S wave can take place, and thus in subsequent calculations for each ${\it\beta}_{A}$ the rescaled wavenumber ${\it\alpha}_{A}$ is taken to be that determined by the first relation in (4.34) for modes 1 and 2. The computed $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ as functions of ${\it\omega}_{A}$ are shown in figure 9 for mode 1; $|a_{R}/J_{0}|$ is not shown as it resembles $|a_{I}/J_{0}|$ . The response is finite for ${\it\omega}_{A}\neq {\it\omega}_{N}$ , but becomes infinite as the frequency approaches the local neutral frequency ${\it\omega}_{N}$ of mode 1, indicating an exact resonance between the sound wave and the neutral T–S mode. Figure 9(a) shows that $|a_{I}/J_{0}|$ is larger for smaller spanwise wavenumbers. Figure 9(b) indicates $|\mathscr{R}|<1$ for low frequencies and $|\mathscr{R}|>1$ for high frequencies as observed earlier. Interestingly, (3.23) indicates that

(6.5) $$\begin{eqnarray}\displaystyle \mathscr{R}\rightarrow \frac{\tan (\hat{{\it\gamma}}_{a}\bar{y}_{s})-\text{i}}{\tan (\hat{{\it\gamma}}_{a}\bar{y}_{s})+\text{i}}\quad \text{as }{\it\omega}_{A}\rightarrow {\it\omega}_{N}, & & \displaystyle\end{eqnarray}$$

so that $|\mathscr{R}|\rightarrow 1$ . The parameter region of over-reflection, ${\it\omega}_{A}>{\it\omega}_{N}$ , corresponds to the unstable regime above the neutral curve. The present consideration of reflection thus provides a physical interpretation of the shock-induced instability. A similar instability arises in the supersonic twin boundary layers developing along two parallel plates (Wu Reference Wu2014).

Figure 10.

Variation of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ with ${\it\omega}_{A}$ for ${\it\alpha}_{A}$ being the wavenumber of mode 2.

Figure 10(a,b) depicts the variations of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ with the frequency ${\it\omega}_{A}$ for different spanwise wavenumbers ${\it\beta}_{A}$ with ${\it\alpha}_{A}$ being the wavenumber of mode 2. The amplitudes $|a_{I}/J_{0}|$ and $|a_{R}/J_{0}|$ (not shown) first decrease and then increase as the spanwise wavenumber ${\it\beta}_{A}$ varies from $0.5$ to $5.0$ in both the low- and high-frequency regions. This may be associated with the non-monotonicity of the neutral curve of mode 2 (see figure 3). The reflection coefficient $|\mathscr{R}|$ exhibits a similar feature to that in figure 9(b): $|\mathscr{R}|<1$ for low-frequency waves and $|\mathscr{R}|>1$ for high-frequency waves with $|\mathscr{R}|=1$ on the neutral curve, signalling that the onset of the instability corresponds to over-reflection.

Next, we consider the response at different streamwise locations for fixed physical frequencies $\hat{{\it\omega}}_{a}$ and wavenumbers $\hat{{\it\alpha}}_{a}$ and $\hat{{\it\beta}}_{a}$ . The latter two are specified as

(6.6a,b ) $$\begin{eqnarray}\displaystyle \hat{{\it\alpha}}_{a}={\it\chi}^{5/4}{\it\alpha}_{N},\quad \hat{{\it\beta}}_{a}={\it\chi}^{5/4}{\it\beta}_{N}, & & \displaystyle\end{eqnarray}$$

where ${\it\chi}=0.332$ , and ${\it\alpha}_{N}$ and ${\it\beta}_{N}$ are on the neutral curves of mode 1 or 2. The calculations were performed for $\bar{y}_{s}={\it\chi}^{-5/4}x$ with $0<x\leqslant 1.2$ . For the case of ${\it\beta}_{N}=0.5$ and ${\it\alpha}_{N}$ being on the neutral curve of mode 1, contours of $|a_{I}/J_{0}|$ are displayed in figure 11. A broadband of strong high-frequency disturbances appears at almost the same streamwise location, owing again to the quasi-resonance. A high-frequency analysis of (3.21) for fixed ${\it\alpha}_{N}$ and ${\it\beta}_{N}$ shows that the response raises to $O((\hat{{\it\omega}}_{a}/{\it\chi}^{3/2})^{5/2})$ , or more precisely,

(6.7) $$\begin{eqnarray}\displaystyle a_{I,R}/J_{0}\sim (-1)^{n}\frac{({\it\alpha}_{N}^{2}-{\it\beta}_{N}^{2})^{1/2}\text{e}^{-{\rm\pi}\text{i}/4}}{2{\it\alpha}_{N}^{2}{\it\beta}_{N}^{2}{\it\chi}^{15/4}}\hat{{\it\omega}}_{a}^{5/2}, & & \displaystyle\end{eqnarray}$$

when

(6.8a,b ) $$\begin{eqnarray}\displaystyle x=x_{c}-\frac{{\it\alpha}_{N}{\it\beta}_{N}^{2}x_{c}^{1/2}{\it\chi}^{3/2}}{{\it\alpha}_{N}^{2}-{\it\beta}_{N}^{2}}\hat{{\it\omega}}_{a}^{-1}-\frac{{\it\alpha}_{N}^{2}{\it\beta}_{N}^{4}{\it\chi}^{3}}{2({\it\alpha}_{N}^{2}-{\it\beta}_{N}^{2})^{2}}\hat{{\it\omega}}_{a}^{-2}\quad \text{and}\quad \hat{{\it\omega}}_{a}\gg 1, & & \displaystyle\end{eqnarray}$$

where

(6.9) $$\begin{eqnarray}\displaystyle x_{c}=\frac{(2n-1){\rm\pi}}{2({\it\alpha}_{N}^{2}-{\it\beta}_{N}^{2})^{1/2}}\quad (n=1,2,\ldots ). & & \displaystyle\end{eqnarray}$$

The locus of (6.8), in the vicinity of which the response is the strongest, is indicated by the dashed line in figure 11.

Figure 11.

Contours of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ in the $x{-}\hat{{\it\omega}}_{a}/{\it\chi}^{3/2}$ plane for $\bar{y}_{s}={\it\chi}^{-5/4}x$ and ${\it\beta}_{N}=0.5$ . The dashed line represents the locus of (6.8).

Exact resonance occurs (at $x=x_{0}=1$ ) when $\hat{{\it\omega}}_{a}={\it\chi}^{3/2}{\it\omega}_{N}$ . Figure 12 shows $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ as functions of $x$ when the frequency is that of mode 1. The variation pattern of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ is similar to that displayed in figures 9 and 10. The response is finite upstream of the neutral position $x_{0}=1$ , but becomes infinite as $x\rightarrow x_{0}$ . The response is stronger for disturbances with larger spanwise wavenumbers. Figure 12(b) indicates that $|\mathscr{R}|<1$ for $x<x_{0}=1$ , and $|\mathscr{R}|\rightarrow 1$ when $x$ approaches the neutral point, downstream of which $|\mathscr{R}|>1$ (not shown), i.e. over-reflection takes place in the unstable region. Figure 13(a,b) displays the variation of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ with $x$ for different spanwise wavenumbers and the frequency being that of mode 2. Again, the amplitude of the response first decreases with the spanwise wavenumber and then increases, possibly due to the non-monotonicity of the neutral curve of mode 2. The reflection coefficient $|\mathscr{R}|$ (figure 13 b) exhibits a similar feature to that of mode 1.

Figure 12.

Variation of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ with $x$ for ${\it\alpha}_{N}$ being the wavenumber of mode 1.

Figure 13.

Variation of $|a_{I}/J_{0}|$ and $|\mathscr{R}|$ with $x$ for ${\it\alpha}_{N}$ being the wavenumber of mode 2.

The response of a supersonic boundary layer to incident slow acoustic waves was studied previously by Gaponov (Reference Gaponov1977), who used the linearized N–S equations but neglected the streamwise and spanwise diffusion as well as the effect of a shock. Over-reflection was observed, but an exact resonance with an eigenmode does not occur. Both phenomena may take place when acoustic waves interact with a supersonic vortex sheet (Ribner Reference Ribner1957), where the eigenmodes are a compressible version of Kelvin–Helmholtz instability waves.

Figure 14.

Variation of ${\it\kappa}$ with ${\it\beta}_{N}$ for (a) $y_{N}=0.2$ , (b) $y_{N}=1$ and (c) $y_{N}=2$ . ——, mode 1; – ⋅ – ⋅ –, mode 2; – – –, mode 3. The curves —▿— in (a) and (b) represent ${\it\kappa}/200$ and ${\it\kappa}/20$ respectively.

6.2 Results for coupling coefficients

In order to gain quantitative information about the efficiency of the receptivity, the dependence of ${\it\kappa}$ on the parameters is investigated. Figure 14 shows ${\it\kappa}$ for modes 1, 2 and 3, and for different shock locations. The mode-1 family stands out: the coupling coefficient is significantly larger than those for modes 2 and 3, and especially for modes in the region ${\it\beta}_{N}>{\it\alpha}_{N}$ , where ${\it\kappa}$ is approximately 100 and 10 times larger for $y_{N}=0.2$ and $1$ , respectively. It may be worth noting that when ${\it\beta}_{N}>{\it\alpha}_{N}$ , the induced acoustic perturbation downstream of the shock is not purely sinusoidal in the direction normal to the shock. Rather it consists of two components, among which one amplifies while the other attenuates exponentially. Despite this difference, the perturbation exhibits, in terms of its phase speed, the key characteristics of a slow acoustic wave. For small ${\it\beta}_{N}$ , the value of ${\it\kappa}$ for mode 2 is larger than that for mode 3, but there is a cross-over when ${\it\beta}_{N}$ exceeds a critical value. As the shock position $y_{N}$ increases, the value of ${\it\kappa}$ decreases for all three modes, implying that the coupling is weaker when the shock becomes more distant to the wall.

Figure 15.

Variation of coupling coefficients $C_{A}^{\pm }$ , $C_{V}$ and $C_{E}$ with ${\it\beta}_{N}$ for a wedge with a half-angle of 20°, $M_{u}=20$ and $Re=5.0\times 10^{6}$ , which corresponds to $y_{N}\approx 0.2$ .

Figure 14 indicates that ${\it\kappa}=O(1)$ for typical $O(1)$ values of ${\it\beta}_{N}$ and $y_{N}$ . However, the factors multiplying ${\it\kappa}$ in $C_{A}^{\pm }$ , $C_{V}$ and $C_{E}$ are large and so are these coupling coefficients. A calculation performed for a wedge with a half-angle of ${\rm\pi}/9$ (i.e. 20°), $M_{u}=20$ ( $M\approx 14$ ) and $Re=5.0\times 10^{6}$ (for which $y_{N}\approx 0.2$ ). The coupling coefficients for mode 1 with $0<{\it\beta}_{N}<2$ are shown in figure 15. The result indicates extraordinarily efficiency, which is caused by the resonant nature of the receptivity mechanism. Among three types of free-stream disturbances, acoustic waves are most efficient due to the amplification effect of shock at large- $M$ limit as (A 43) indicates. Entropy fluctuations come the second and vortical perturbations are least efficient of the three by comparison. Specifically for ${\it\beta}_{N}=1.5$ , it is found that for mode 1 ( ${\it\omega}_{N}=6.1$ ), ${\it\kappa}=2.01$ ,

(6.10a-c ) $$\begin{eqnarray}\displaystyle C_{A}^{\pm }=0.144\times 10^{6}\,{\it\kappa},\quad C_{V}=0.846\times 10^{2}\,{\it\kappa},\quad C_{E}=0.360\times 10^{3}\,{\it\kappa}, & & \displaystyle\end{eqnarray}$$

and for mode 2 ( ${\it\omega}_{N}=17.8$ ), ${\it\kappa}=0.43$ ,

(6.11a-c ) $$\begin{eqnarray}\displaystyle C_{A}^{\pm }=0.092\times 10^{6}\,{\it\kappa},\quad C_{V}=0.542\times 10^{2}\,{\it\kappa},\quad C_{E}=0.230\times 10^{3}\,{\it\kappa}. & & \displaystyle\end{eqnarray}$$

The absolute initial amplitude of the excited T–S mode depends on the level of fluctuations in the oncoming flow. The presence of a pressure fluctuation ${\it\delta}p_{I}=O(10^{-7})$ (which corresponds to a velocity fluctuation of $O(M_{u}{\it\delta}p_{I})=O(10^{-6})$ ), ${\it\delta}v_{I}=O(10^{-4})$ and ${\it\delta}r_{I}=O(10^{-4})$ would generate T–S modes with initial amplitudes as large as a few per cent. The turbulence (noise) level in wind tunnels is usually higher than these by one or two orders of magnitude. Perturbations in flight conditions are thought to be weak, e.g. at cruise altitudes of civil aircrafts, the turbulence level, in terms of velocity fluctuations, is thought to be of $O(10^{-4})$ with respect to a transonic flight speed. That translates to $O(10^{-5})$ velocity fluctuation at $M_{u}=20$ . On the other hand, density fluctuations of $O(10^{-5})$ (which is independent of the flight speed) are very likely. Vortical or density perturbations of this level are capable of generating T–S waves with initial magnitudes in the range of $0.1-1\%$ , which is quite substantial. If velocity fluctuations of $O(10^{-5})$ are primarily acoustic, the amplitude of T–S modes excited could be as large as $10\%$ , but of course in this case the validity of the present linear theory may be questioned.

Finally, it is worth noting that ${\it\kappa}\rightarrow 0$ as ${\it\beta}_{N}\rightarrow 0$ . The present mechanism thus operates primarily for three-dimensional disturbances. Two-dimensional ones are probably more likely to excite instability modes through the leading-edge mechanism of Fedorov & Khokhlov (Reference Fedorov and Khokhlov1991) and Fedorov (Reference Fedorov2003). However, if planar modes on the triple-deck scales are to be excited, new mechanisms must be sought.