3.1 The governing equations

The dimensionless compressible N–S equations that govern the flow field ($u,v,w,p,\unicode[STIX]{x1D70C}$, $T$) (see (2.2)) are written in the conservative form,

(3.1)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{U}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}\boldsymbol{E}}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}\boldsymbol{F}}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}\boldsymbol{G}}{\unicode[STIX]{x2202}z}+\boldsymbol{B}=\frac{\unicode[STIX]{x2202}\boldsymbol{E}_{v}}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}\boldsymbol{F}_{v}}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}\boldsymbol{G}_{v}}{\unicode[STIX]{x2202}z},\end{eqnarray}$$

where $\boldsymbol{U}$ is the conservative flux, $\boldsymbol{E}$, $\boldsymbol{F}$ and $\boldsymbol{G}$ are the nonlinear terms, $\boldsymbol{E}_{v}$, $\boldsymbol{F}_{v}$ and $\boldsymbol{G}_{v}$ are the viscosity terms and $\boldsymbol{B}$ is a body force, introduced to render the base flow $\boldsymbol{U}_{B}$ (Blasius solution) to satisfy (3.1); the expression for each term can be found in Qin & Dong (Reference Qin and Dong2016).

Before the system (3.1) can be discretized and solved, it is necessary to calculate the unperturbed base flow, specify the free-stream disturbance, and to set up boundary conditions as well as the inflow and outflow conditions; these will be done in §§ 3.2–3.5. The simulation of receptivity is to be carried out using the Blasius solution as the base flow. Since the length scale of the roughness-induced distortion is $O(R^{-3/8}L^{\ast })$, much shorter than the scale on which the Blasius flow develops, we neglect the non-parallel effect of the Blasius solution in order to focus on the roughness-induced receptivity.

The inflow condition is imposed as a superposition of the Blasius solution and a small-amplitude acoustic signature. The boundary conditions at the perturbed wall surface are to be approximated by the equivalent ones at the flat plate via the first-order Taylor expansion. The total flow field can be decomposed into a steady base flow and perturbation, namely,

(3.2)$$\begin{eqnarray}\displaystyle & & \displaystyle [u,v,w,\unicode[STIX]{x1D70C},T,p](x,y,z,t)\nonumber\\ \displaystyle & & \displaystyle \quad =[U_{B},0,0,R_{B},T_{B},P_{B}](y)+\frac{\unicode[STIX]{x1D716}_{u}}{2}[\widetilde{u}(y),\widetilde{v}(y),\widetilde{w}(y),\widetilde{\unicode[STIX]{x1D70C}}(y),\widetilde{T}(y),\widetilde{p}(y)]\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{s}x+\unicode[STIX]{x1D6FD}_{s}z-\unicode[STIX]{x1D714}t)}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,[U_{h},V_{h},W_{h},R_{h},T_{h},P_{h}](x,y,z)+[\widehat{u},\widehat{v},\widehat{w},\widehat{\unicode[STIX]{x1D70C}},\widehat{T},\widehat{p}](x,y,z,t)+\text{c.c.},\end{eqnarray}$$

where $[U_{B},0,0,R_{B},T_{B},P_{B}]$ denotes the Blasius solution, $[\widetilde{u},\widetilde{v},\widetilde{w},\widetilde{\unicode[STIX]{x1D70C}},\widetilde{T},\widetilde{p}]$ stands for the response of the boundary layer to the free-stream sound over a smooth flat plate, with $\unicode[STIX]{x1D716}_{u}$ being the amplitude of the incident acoustic wave, $\unicode[STIX]{x1D6FC}_{s}$ and $\unicode[STIX]{x1D6FD}_{s}$ its streamwise and spanwise wavenumbers, respectively; $[U_{h},V_{h},W_{h},R_{h},T_{h},P_{h}]$ represents the localized mean-flow distortion induced by the surface roughness. The interaction of the acoustic signature with the mean-flow distortion produces the scattered field $[\widehat{u},\widehat{v},\widehat{w},\widehat{\unicode[STIX]{x1D70C}},\widehat{T},\widehat{p}]$, from which an instability mode emerges.

3.2 Blasius solution

In this subsection, we consider $[U_{B},V_{B},0,R_{B},T_{B},P_{B}]$ in (3.2). The base flow over a smooth flat plate is governed by the classical compressible boundary-layer equations.

We introduce the Dorodnitsyn–Howarth transformation,

(3.3)$$\begin{eqnarray}\unicode[STIX]{x1D709}=(1+x),\quad \unicode[STIX]{x1D702}=\sqrt{\frac{R}{2(1+x)}}\int _{0}^{y}R_{B}\,\text{d}y.\end{eqnarray}$$

In terms of $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D702}$, the solution can be expressed in the self-similar form,

(3.4)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle U_{B}(\unicode[STIX]{x1D702})=f^{\prime }(\unicode[STIX]{x1D702}),\quad V_{B}=-\sqrt{\frac{1}{2\unicode[STIX]{x1D709}R}}\left[f(\unicode[STIX]{x1D702})g(\unicode[STIX]{x1D702})-f^{\prime }(\unicode[STIX]{x1D702})\int _{0}^{\unicode[STIX]{x1D702}}g(\unicode[STIX]{x1D702})\,\text{d}\unicode[STIX]{x1D702}\right],\\ \displaystyle T_{B}(\unicode[STIX]{x1D702})=g(\unicode[STIX]{x1D702}),\quad P_{B}(\unicode[STIX]{x1D702})=\frac{1}{\unicode[STIX]{x1D6FE}M^{2}},\quad R_{B}(\unicode[STIX]{x1D702})=\frac{1}{g(\unicode[STIX]{x1D702})},\end{array}\right\}\end{eqnarray}$$

where $f$ and $g$ satisfy the coupled equations,

(3.5a,b)$$\begin{eqnarray}(Cf^{\prime \prime })^{\prime }+ff^{\prime \prime }=0,\quad (Cg^{\prime })^{\prime }+P_{r}fg^{\prime }+(\unicode[STIX]{x1D6FE}-1)M^{2}P_{r}C{f^{\prime \prime }}^{2}=0;\end{eqnarray}$$

here, $C(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D707}_{B}/T_{B}=T_{B}^{1/2}(1+110.4/T_{\infty })/(T_{B}+110.4/T_{\infty })$ when Sutherland’s law is used, the Prandtl number $Pr$ is a constant and equals 0.72 and $\unicode[STIX]{x1D6FE}=1.4$ is the ratio of the specific heats. Equation (3.5a,b) is subject to the boundary conditions,

(3.6)$$\begin{eqnarray}f^{\prime }(0)=0,\quad g(0)=T_{w};\quad f^{\prime },\quad g\rightarrow 1\quad \text{as}~\unicode[STIX]{x1D702}\rightarrow \infty .\end{eqnarray}$$

In this work, we take the wall temperature $T_{w}$ to be

(3.7)$$\begin{eqnarray}T_{w}=1+\sqrt{Pr}(\unicode[STIX]{x1D6FE}-1)M^{2}/2,\end{eqnarray}$$

which is the temperature of an adiabatic wall if $P_{r}=1$, but only approximately so if $P_{r}\neq 1$, e.g. $P_{r}=0.72$, the value that we shall use. The displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }$ is expressed as

(3.8)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{\ast }=\int _{0}^{\infty }(1-R_{B}U_{B})\,\text{d}y^{\ast }=\sqrt{2}L^{\ast }R^{-1/2}\int _{0}^{\infty }[g(\unicode[STIX]{x1D702})-f^{\prime }(\unicode[STIX]{x1D702})]\,\text{d}\unicode[STIX]{x1D702}.\end{eqnarray}$$

3.3 Inflow boundary condition: response to free-stream acoustic disturbances and continuous spectrum of the Orr–Sommerfeld equation

In this subsection, we consider the acoustic signature in the boundary layer, represented by the $O(\unicode[STIX]{x1D716}_{u})$ term in (3.2). With the parallel-flow assumption, the non-dimensional base flow can be viewed as a function of $y$ only, with the dependence on $x$ becoming parametric. The disturbance satisfies the linearized continuity, momentum, energy and state equations. If we introduce $\unicode[STIX]{x1D753}$ as

(3.9)$$\begin{eqnarray}\unicode[STIX]{x1D753}(y)=\left(\widetilde{u},\frac{\text{d}\widetilde{u}}{\text{d}y},\widetilde{v},\widetilde{p},\widetilde{T},\frac{\text{d}\widetilde{T}}{\text{d}y},\widetilde{w},\frac{\text{d}\widetilde{w}}{\text{d}y}\right)^{T},\end{eqnarray}$$

then $\unicode[STIX]{x1D753}$ is governed by a system of first-order equations,

(3.10)$$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D753}}{\text{d}y}=\unicode[STIX]{x1D63F}(y)\unicode[STIX]{x1D753},\end{eqnarray}$$

where $\unicode[STIX]{x1D63F}$ is an $8\times 8$ matrix, whose expression can be found in Malik (Reference Malik1990).

At a finite Reynolds number, the incident sound wave and the signature induced in the boundary layer correspond to the acoustic branch of the continuous modes of the system (3.10). With allowance for the presence of continuous modes, the far-field asymptotic behaviour of $\unicode[STIX]{x1D753}$ can be written as

(3.11)$$\begin{eqnarray}\unicode[STIX]{x1D753}=\mathop{\sum }_{i=1}^{8}A_{i}\unicode[STIX]{x1D743}_{i}\text{e}^{\unicode[STIX]{x1D706}_{i}y}\quad \text{as}~y\rightarrow \infty ,\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{i}$ and $\unicode[STIX]{x1D743}_{i}$ are the eigenvalues and eigenvectors of the matrix $\unicode[STIX]{x1D63F}_{\infty }\equiv \lim _{y\rightarrow \infty }\unicode[STIX]{x1D63F}$, respectively, and $A_{i}$ represents the coefficient of each component. The eigenvalues are given in Balakumar & Malik (Reference Balakumar and Malik1992). The continuous mode shape is oscillatory in the free stream, and so some of the eigenvalues $\unicode[STIX]{x1D706}_{i}$ must be (almost) purely imaginary with $R\gg 1$. The dispersion relationship can be written as $\unicode[STIX]{x1D6FC}_{s}=\unicode[STIX]{x1D6FC}_{s}(\unicode[STIX]{x1D714},\unicode[STIX]{x1D6FE}_{s},\unicode[STIX]{x1D6FD}_{s})$, and for a spatially developing wave $\unicode[STIX]{x1D6FE}_{s}$, $\unicode[STIX]{x1D6FD}_{s}$ and $\unicode[STIX]{x1D714}$ are real, and $\unicode[STIX]{x1D6FC}_{s}$ is complex, whose imaginary part is $O(R^{-1})$. The imaginary part of $\unicode[STIX]{x1D6FC}_{s}$ is positive because continuous modes are decaying perturbations.

For an acoustic wave, we take $\unicode[STIX]{x1D706}_{5}=-\unicode[STIX]{x1D706}_{6}=\text{i}\unicode[STIX]{x1D6FE}_{s}$. In the limit of large Reynolds number, $R\gg 1$, the dispersion relationship to leading-order accuracy is

(3.12)$$\begin{eqnarray}-(\unicode[STIX]{x1D6FC}_{s}-\unicode[STIX]{x1D714})^{2}M^{2}+\unicode[STIX]{x1D6FC}_{s}^{2}+\unicode[STIX]{x1D6FE}_{s}^{2}+\unicode[STIX]{x1D6FD}_{s}^{2}=0,\end{eqnarray}$$

from which it follows

(3.13)$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{s}=\frac{M^{2}\unicode[STIX]{x1D714}\pm \sqrt{M^{2}\unicode[STIX]{x1D714}^{2}+(M^{2}-1)(\unicode[STIX]{x1D6FE}_{s}^{2}+\unicode[STIX]{x1D6FD}_{s}^{2})}}{M^{2}-1}.\end{eqnarray}$$

The incident angle is defined as

(3.14)$$\begin{eqnarray}\unicode[STIX]{x1D703}=\cos ^{-1}\frac{\unicode[STIX]{x1D6FC}_{s}}{\sqrt{\unicode[STIX]{x1D6FC}_{s}^{2}+\unicode[STIX]{x1D6FD}_{s}^{2}+\unicode[STIX]{x1D6FE}_{s}^{2}}}.\end{eqnarray}$$

Without loss of generality, we take $\unicode[STIX]{x1D714}>0$. If the ‘+’ sign in (3.13) is selected, the phase speed is

(3.15)$$\begin{eqnarray}c_{s}=\frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6FC}_{s}}=1-\frac{1}{M\cos \unicode[STIX]{x1D703}},\quad \unicode[STIX]{x1D703}\in \left[0,\cos ^{-1}\frac{1}{M}\right),\end{eqnarray}$$

for which $c_{s}>0$ and $\unicode[STIX]{x1D6FC}_{s}>0$. Since $c_{s}$ is smaller than the speed of the oncoming stream, this disturbance is referred to as a slow acoustic wave. On the other hand, if the ‘-’ sign in (3.13) is selected, the phase speed $c_{s}=1+1/(M\cos \unicode[STIX]{x1D703})$ can be either greater than 1 or negative, depending on the incident angle: the disturbance corresponds to a fast acoustic wave for $\unicode[STIX]{x1D703}\in [0,\unicode[STIX]{x03C0}/2)$, but to a slow acoustic wave for $\unicode[STIX]{x1D703}\in (\unicode[STIX]{x03C0}+\cos ^{-1}(1/M),3\unicode[STIX]{x03C0}/2)$. Therefore, in addition to (3.15), there also exist slow acoustic waves having phase speeds

(3.16)$$\begin{eqnarray}c_{s}=\frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6FC}_{s}}=1+\frac{1}{M\cos \unicode[STIX]{x1D703}},\quad \unicode[STIX]{x1D703}\in \left(\unicode[STIX]{x03C0}+\cos ^{-1}\frac{1}{M},\frac{3\unicode[STIX]{x03C0}}{2}\right),\end{eqnarray}$$

for which $c_{s}<0$ and $\unicode[STIX]{x1D6FC}_{s}<0$. Note that for both the cases mentioned above, the group velocity satisfies $\text{d}\unicode[STIX]{x1D714}/\text{d}\unicode[STIX]{x1D6FC}_{s}=1\mp \cos \unicode[STIX]{x1D703}/M>0$.

Let matrix $\unicode[STIX]{x1D641}=[\unicode[STIX]{x1D743}_{1},\unicode[STIX]{x1D743}_{2},\unicode[STIX]{x1D743}_{3},\unicode[STIX]{x1D743}_{4},\unicode[STIX]{x1D743}_{5},\unicode[STIX]{x1D743}_{6},\unicode[STIX]{x1D743}_{7},\unicode[STIX]{x1D743}_{8}]$, or in terms their components $\unicode[STIX]{x1D60D}_{ji}=\unicode[STIX]{x1D709}_{ji}$, and let $\unicode[STIX]{x1D641}^{-1}$ be its inverse. Then $A_{i}$ can be expressed as

(3.17)$$\begin{eqnarray}A_{i}=\left(\mathop{\sum }_{j=1}^{8}\unicode[STIX]{x1D60D}_{ij}^{-1}\unicode[STIX]{x1D719}_{j}\right)\text{e}^{-\unicode[STIX]{x1D706}_{i}y}\quad \text{as}~y\rightarrow \infty \quad (i=1,2\ldots 8).\end{eqnarray}$$

The real parts of $\unicode[STIX]{x1D706}_{2}$, $\unicode[STIX]{x1D706}_{4}$ and $\unicode[STIX]{x1D706}_{8}$ are positive, and so in order to render the solution bounded, $A_{2},A_{4},A_{8}$ must be set to zero. For $\unicode[STIX]{x1D6FC}_{s}>0$ ($c_{s}>0$), the incident acoustic wave corresponds to $\unicode[STIX]{x1D706}_{5}=\text{i}\unicode[STIX]{x1D6FE}_{s}$ as it has a negative vertical group velocity, whereas the computed $A_{6}\unicode[STIX]{x1D709}_{16}$ represents the amplitude of the acoustic wave reflected by the boundary layer. In this work, $[\widetilde{u}(y),\widetilde{v}(y),\widetilde{w}(y),\widetilde{\unicode[STIX]{x1D70C}}(y),\widetilde{T}(y),\widetilde{p}(y)]$ are normalized such that the streamwise velocity amplitude of the incident acoustic wave is unity, i.e. $A_{5}\unicode[STIX]{x1D709}_{15}=1$. Thus, the full boundary conditions for continuous modes are

(3.18)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \widetilde{u}(0)=\widetilde{v}(0)=\widetilde{w}(0)=\widetilde{T}(0)=0\quad \text{at}~y=0,\\ \displaystyle \mathop{\sum }_{j=1}^{8}F_{2j}^{-1}\unicode[STIX]{x1D719}_{j}=0,\quad \!\mathop{\sum }_{j=1}^{8}F_{4j}^{-1}\unicode[STIX]{x1D719}_{j}=0,\quad \!\mathop{\sum }_{j=1}^{8}F_{8j}^{-1}\unicode[STIX]{x1D719}_{j}=0,\quad \!\mathop{\sum }_{j=1}^{8}F_{5j}^{-1}\unicode[STIX]{x1D719}_{j}\text{e}^{-\unicode[STIX]{x1D706}_{5}y}=\frac{1}{\unicode[STIX]{x1D709}_{15}}\quad \text{as}~y\rightarrow \infty .\end{array}\right\}\end{eqnarray}$$

The reflection coefficient is defined as the ratio of the amplitude of the reflected acoustic wave to that of the incident, i.e.

(3.19)$$\begin{eqnarray}\mathscr{R}=A_{6}\unicode[STIX]{x1D709}_{16}.\end{eqnarray}$$

In the case of $\unicode[STIX]{x1D6FC}_{s}<0$ ($c_{s}<0$), the roles of $A_{5}$ and $A_{6}$ are exchanged. Thus, the streamwise velocity in the free stream is expressed as

(3.20)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \widetilde{u}(y)=\text{e}^{\text{i}\unicode[STIX]{x1D6FE}_{s}y}+\mathscr{R}\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}_{s}y}\quad \text{for}~\unicode[STIX]{x1D6FC}_{s}>0,\\ \displaystyle \widetilde{u}(y)=\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}_{s}y}+\mathscr{R}\text{e}^{\text{i}\unicode[STIX]{x1D6FE}_{s}y}\quad \text{for}~\unicode[STIX]{x1D6FC}_{s}<0,\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}_{s}>0$.

3.4 Wall boundary conditions

The no-slip and isothermal boundary conditions are imposed at the roughness surface, and so the total velocity and temperature field at the surface are given as

(3.21)$$\begin{eqnarray}[u,v,w,T](x,h(x,z),z,t)=[0,0,0,T_{w}],\end{eqnarray}$$

where $h(x,z)=h^{\ast }(x,z)/L^{\ast }$ is the normalized shape function of the roughness. Calculations will be performed for a roughness array which is spanwise periodic and streamwise localized with a Gaussian shape,

(3.22)$$\begin{eqnarray}h(x,z)=h_{0}\text{e}^{-x^{2}/d^{2}}\cos (\unicode[STIX]{x1D6FD}_{w}z),\end{eqnarray}$$

where $d$ and $h_{0}$ are dimensionless measures of the roughness width and height respectively, and $\unicode[STIX]{x1D6FD}_{w}$ the spanwise wavenumber. Applying the first-order Taylor expansion to (3.21) for $h\ll 1$, we obtain the approximate boundary conditions,

(3.23)$$\begin{eqnarray}[u,v,w,T]+h\left[\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y},\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}y},\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}y},\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}y}\right]=[0,0,0,T_{w}]\quad \text{at}~y=0,\end{eqnarray}$$

which is of the mixed type. If the first two terms in (3.2) are used in (3.23), there follows, to leading-order accuracy, the Dirichlet boundary conditions,

(3.24)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle u+h\frac{\unicode[STIX]{x2202}U_{B}}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x1D716}_{u}}{2}h\frac{\unicode[STIX]{x2202}\widetilde{u}}{\unicode[STIX]{x2202}y}\text{e}^{\text{i}[\unicode[STIX]{x1D6FC}_{s}(x-x_{0})+\unicode[STIX]{x1D6FD}_{s}z-\unicode[STIX]{x1D714}t]}+\text{c.c.}=0,\\ \displaystyle v+\frac{\unicode[STIX]{x1D716}_{u}}{2}h\frac{\unicode[STIX]{x2202}\widetilde{v}}{\unicode[STIX]{x2202}y}\text{e}^{\text{i}[\unicode[STIX]{x1D6FC}_{s}(x-x_{0})+\unicode[STIX]{x1D6FD}_{s}z-\unicode[STIX]{x1D714}t]}+\text{c.c.}=0,\\ \displaystyle w+\frac{\unicode[STIX]{x1D716}_{u}}{2}h\frac{\unicode[STIX]{x2202}\widetilde{w}}{\unicode[STIX]{x2202}y}\text{e}^{\text{i}[\unicode[STIX]{x1D6FC}_{s}(x-x_{0})+\unicode[STIX]{x1D6FD}_{s}z-\unicode[STIX]{x1D714}t]}+\text{c.c.}=0,\\ \displaystyle T+h\frac{\unicode[STIX]{x2202}T_{B}}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x1D716}_{u}}{2}h\frac{\unicode[STIX]{x2202}\widetilde{T}}{\unicode[STIX]{x2202}y}\text{e}^{\text{i}[\unicode[STIX]{x1D6FC}_{s}(x-x_{0})+\unicode[STIX]{x1D6FD}_{s}z-\unicode[STIX]{x1D714}t]}+\text{c.c.}=T_{w},\end{array}\right\}\end{eqnarray}$$

where $x_{0}$ represents the streamwise coordinate of the inlet. Note that the flow field represents the mean flow induced by the wall roughness when $\unicode[STIX]{x1D716}_{u}=0$. Use of either boundary conditions (3.23) or (3.24) allows both the mean-flow distortion and the acoustic signature to be computed in the DNS.

3.5 Outflow and upper boundary conditions

A buffer region is imposed as the outflow condition downstream so that the waves will not be reflected back into the computational domain. The Navier–Stokes characteristic boundary condition (NSCBC; see Poinsot & Lele Reference Poinsot and Lele1992) is employed at the upper boundary, i.e. for the outgoing wave, its values at the boundary are extrapolated using the information at interior points, while the boundary values of the incoming wave are specified as the incident free-stream acoustic disturbance.

3.6 Discretization

With the boundary conditions now being specified on the flat surface, we discretize (3.1) on a chosen cubic domain, $x_{l}^{\ast }/\unicode[STIX]{x1D6FF}^{\ast }\times y_{l}^{\ast }/\unicode[STIX]{x1D6FF}^{\ast }\times z_{l}^{\ast }/\unicode[STIX]{x1D6FF}^{\ast }$, and denote each grid by $(x_{i},y_{j},z_{k})$ with $i\in [0,In]$, $j\in [0,Jn]$ and $k\in [0,Kn]$. Uniform grids are employed in both the streamwise and spanwise directions, while non-uniform grids clustered in the near-wall region are used in the wall-normal direction.

The nonlinear terms are discretized by using the fifth-order finite-difference scheme with flux splitting according to positive and negative eigenvalues of the Jacobian matrix, while the viscous terms are discretized by the sixth-order central difference scheme. The second-order Runge–Kutta method is used for time evolution. The schemes and the method to generate the wall-normal grids are described in Qin & Dong (Reference Qin and Dong2016).

3.7 Receptivity coefficient

In this section, we consider the scattered field $[\widehat{u},\widehat{v},\widehat{w},\widehat{\unicode[STIX]{x1D70C}},\widehat{T},\widehat{p}]$ in (3.2). The DNS give the total flow field, from which the pre-calculated base flow, the roughness-induced local mean-flow distortion and the acoustic signature are subtracted out to obtain the scattered signature. A main aim of studying receptivity is to determine the amplitude of the instability mode excited in terms of external disturbances, presently, the free-stream acoustic wave and the surface roughness. Since the disturbance is complex near the roughness, the modal amplitude can only be defined by considering the asymptotic behaviour of the scattered field sufficiently downstream, where the mode emerges and amplifies exponentially, e.g. the streamwise velocity $\widehat{u}(x,y,z,t)$ behaves as

(3.25)$$\begin{eqnarray}|\widehat{u}|\equiv \max _{y,z,t}(|\widehat{u}(x,y,z,t)|)\sim \text{e}^{-\unicode[STIX]{x1D6FC}_{Ti}x};\end{eqnarray}$$

where here the modal growth rate $-\unicode[STIX]{x1D6FC}_{Ti}$ is independent of $x$ since we have neglected the non-parallelism of the base flow over a flat wall. We introduce the concept of ‘equivalent amplitude’, namely, the amplitude of the mode at the roughness site ($x=0$) whose downstream evolution is the same as that of the disturbance excited (Wu Reference Wu2001a). In this sense, the equivalent amplitude of the streamwise velocity can be defined as

(3.26)$$\begin{eqnarray}A_{u0}\equiv \lim _{x\rightarrow \infty }\max _{y,z,t}|\widehat{u}(x,y,z)|\text{e}^{\unicode[STIX]{x1D6FC}_{Ti}x}.\end{eqnarray}$$

This equivalent amplitude is what is customarily referred to as the ‘initial amplitude’. The receptivity coefficient is defined as the ratio of the streamwise-velocity amplitude of the excited instability mode to that of the incident acoustic wave,

(3.27)$$\begin{eqnarray}{\mathcal{K}}=A_{u0}/\unicode[STIX]{x1D716}_{u}.\end{eqnarray}$$

4.1 Linearized solution for the mean flow induced by roughness

In this section, we consider $(\widetilde{u}_{r},\widetilde{v}_{r},\widetilde{w}_{r},\widetilde{a}_{r},\widetilde{p}_{r})$ in (4.15). After inserting (4.15) into (4.8), we obtain at $O(H)$ the governing equations (Smith et al. Reference Smith, Sykes and Brighton1977)

(4.16)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{u}_{r}}{\unicode[STIX]{x2202}X}+\frac{\unicode[STIX]{x2202}\widetilde{v}_{r}}{\unicode[STIX]{x2202}Y}+\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{w}\widetilde{w}_{r}=0,\\ \displaystyle Y\frac{\unicode[STIX]{x2202}\widetilde{u}_{r}}{\unicode[STIX]{x2202}X}+\widetilde{v}_{r}=-\frac{\unicode[STIX]{x2202}\widetilde{p}_{r}}{\unicode[STIX]{x2202}X}+\frac{\unicode[STIX]{x2202}^{2}\widetilde{u}_{r}}{\unicode[STIX]{x2202}Y^{2}},\\ \displaystyle Y\frac{\unicode[STIX]{x2202}\widetilde{w}_{r}}{\unicode[STIX]{x2202}X}=-\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{w}\widetilde{p}_{r}+\frac{\unicode[STIX]{x2202}^{2}\widetilde{w}_{r}}{\unicode[STIX]{x2202}Y^{2}}.\end{array}\right\}\end{eqnarray}$$

Taylor expansion of the no-slip conditions in (4.9) on the roughness $Y=HF$ for $H\ll 1$ yields the boundary conditions,

(4.17)$$\begin{eqnarray}\widetilde{u}_{r}=-F(X),\quad \widetilde{v}_{r}=\widetilde{w}_{r}=0\quad \text{at}~Y=0.\end{eqnarray}$$

Matching with the main-deck solution requires that

(4.18)$$\begin{eqnarray}\widetilde{u}_{r}\rightarrow \widetilde{a}_{r}(X),\quad \widetilde{w}_{r}\rightarrow 0\quad \text{as}~Y\rightarrow \infty .\end{eqnarray}$$

Define the Fourier transforms of a function $f(x)$ with respect to $X$ as

(4.19)$$\begin{eqnarray}\widehat{f}(k)=\frac{1}{\sqrt{2\unicode[STIX]{x03C0}}}\int _{-\infty }^{\infty }\widetilde{f}(X)\text{e}^{-\text{i}kX}\,\text{d}X,\end{eqnarray}$$

where a tilde and a hat represent the physical and spectral quantities, respectively. Performing Fourier transform on (4.16)–(4.18), we obtain

(4.20a)$$\begin{eqnarray}\displaystyle & \displaystyle \text{i}k\widehat{u}_{r}+\widehat{v}_{r}^{\prime }+\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{w}\widehat{w}_{r}=0, & \displaystyle\end{eqnarray}$$

(4.20b)$$\begin{eqnarray}\displaystyle & \displaystyle \text{i}kY\widehat{u}_{r}+\widehat{v}_{r}=-\text{i}k\widehat{p}_{r}+\widehat{u}_{r}^{\prime \prime }, & \displaystyle\end{eqnarray}$$

(4.20c)$$\begin{eqnarray}\displaystyle & \displaystyle \text{i}kY\widehat{w}_{r}=-\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{w}\widehat{p}_{r}+\widehat{w}_{r}^{\prime \prime }, & \displaystyle\end{eqnarray}$$

(4.20d)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{u}_{r}=-\widehat{F},\quad \widehat{v}_{r}=\widehat{w}_{r}=0\quad \text{at}~Y=0, & \displaystyle\end{eqnarray}$$

(4.20e)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{u}_{r}\rightarrow \widehat{a}_{r}\quad \widehat{w}_{r}\rightarrow 0\quad \text{as}~Y\rightarrow \infty , & \displaystyle\end{eqnarray}$$

$(\,)^{\prime }\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}Y$

Consideration of the solution in the upper deck yields the pressure-displacement relation,

(4.21)$$\begin{eqnarray}\widehat{p}_{r}=\frac{k^{2}}{[\widehat{\unicode[STIX]{x1D6FD}}_{w}^{2}/(M^{2}-1)-k^{2}]^{1/2}}\widehat{a}_{r},\end{eqnarray}$$

where the branch of $[\widehat{\unicode[STIX]{x1D6FD}}_{w}^{2}/(M^{2}-1)-k^{2}]^{1/2}$ is chosen such that the pressure attenuates exponentially, or propagates outwardly. Here, for brevity, we omit the derivation, which can be found in Smith et al. (Reference Smith, Sykes and Brighton1977).

By using the two homogeneous solutions of the Airy equation, $\text{Ai}(\unicode[STIX]{x1D702})$ and $\text{Bi}(\unicode[STIX]{x1D702})$, the solution $\widehat{w}_{r}$ to the inhomogeneous equation (4.20c) that remains bounded as $Y\rightarrow \infty$ can be derived as

(4.22)$$\begin{eqnarray}\widehat{w}_{r}=-\text{i}\unicode[STIX]{x03C0}\widehat{\unicode[STIX]{x1D6FD}}_{w}\widehat{p}_{r}(\text{i}k)^{-2/3}[\text{Gi}(\unicode[STIX]{x1D702})-\text{Gi}(0)\text{Ai}(\unicode[STIX]{x1D702})/\text{Ai}(0)],\end{eqnarray}$$

with $\text{Gi}$ denoting the combinations of Airy functions defined by Abramowitz & Stegun (Reference Abramowitz and Stegun1964, p. 448). After solving the boundary-value problem (4.20) supplemented by (4.21), the solution for $\widehat{p}_{r}$, $\widehat{u}_{r}$ and $\widehat{v}_{r}$ is found as (Smith Reference Smith1973; Smith et al. Reference Smith, Sykes and Brighton1977)

(4.23)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \widehat{p}_{r}(k)=-\frac{(\text{i}k)^{5/3}}{k^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{w}^{2}}\text{Ai}^{\prime }(0)\widehat{F}/\mathscr{D},\\ \displaystyle \widehat{u}_{r}=\widehat{F}\left[\int _{0}^{\unicode[STIX]{x1D702}}\text{Ai}(\unicode[STIX]{x1D702})\,\text{d}\unicode[STIX]{x1D702}/\mathscr{D}-1\right]-\widehat{\unicode[STIX]{x1D6FD}}_{w}\widehat{w}_{r}/k,\\ \displaystyle \widehat{v}_{r}=\widehat{u}_{r}^{\prime \prime }-\text{i}k\widehat{p}_{r}-\text{i}kY\widehat{u}_{r},\end{array}\right\}\end{eqnarray}$$

where

(4.24)$$\begin{eqnarray}\mathscr{D}(k)=\int _{0}^{\infty }\text{Ai}(\unicode[STIX]{x1D702})\,\text{d}\unicode[STIX]{x1D702}-\frac{\text{Ai}^{\prime }(0)[\widehat{\unicode[STIX]{x1D6FD}}_{w}^{2}/(M^{2}-1)-k^{2}]^{1/2}}{(\text{i}k)^{1/3}(k^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{w}^{2})}.\end{eqnarray}$$

4.2 Acoustic signature in the boundary layer: impingement and reflection

In order to calculate the instability mode to be excited, it is necessary first to find the solution for the response of the boundary layer to the acoustic wave, which is represented by $(\widehat{u}_{s},\widehat{v}_{s},\widehat{w}_{s},\widehat{a}_{s},\widehat{p}_{s})$ in (4.15).

Substitution of (4.15) into (4.8) shows, at $O(\tilde{\unicode[STIX]{x1D716}})$, that the induced disturbance in the lower deck is governed by the linearized boundary-layer equations,

(4.25a)$$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{u}_{s}+\widehat{v}_{s}^{\prime }+\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{w}_{s}=0, & \displaystyle\end{eqnarray}$$

(4.25b)$$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\unicode[STIX]{x1D6FA}\widehat{u}_{s}+\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}Y\widehat{u}_{s}+\widehat{v}_{s}=-\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{p}_{s}+\widehat{u}_{s}^{\prime \prime }, & \displaystyle\end{eqnarray}$$

(4.25c)$$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\unicode[STIX]{x1D6FA}\widehat{w}_{s}+\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}Y\widehat{w}_{s}=-\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{p}_{s}+\widehat{w}_{s}^{\prime \prime }, & \displaystyle\end{eqnarray}$$

(4.26a)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{u}_{s}=\widehat{v}_{s}=\widehat{w}_{s}=0\quad \text{at}~Y=0; & \displaystyle\end{eqnarray}$$

(4.26b)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{u}_{s}\rightarrow \widehat{a}_{s},\quad \widehat{w}_{s}\rightarrow 0\quad \text{as}~Y\rightarrow \infty . & \displaystyle\end{eqnarray}$$

Elimination of $\widehat{p}_{s}$ and $\widehat{v}_{s}$ among (4.25a)–(4.25c) gives

(4.27)$$\begin{eqnarray}(\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{u}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{w}_{s})^{\prime \prime \prime }+\text{i}(\unicode[STIX]{x1D6FA}-\widehat{\unicode[STIX]{x1D6FC}}_{s}Y)(\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{u}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{w}_{s})^{\prime }=0.\end{eqnarray}$$

The solution to this equation satisfying the boundary condition (4.26a) is

(4.28)$$\begin{eqnarray}\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{u}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{w}_{s}=C_{2}\int _{\unicode[STIX]{x1D701}_{0}}^{\unicode[STIX]{x1D701}}\text{Ai}(\unicode[STIX]{x1D701})\,\text{d}\unicode[STIX]{x1D701},\end{eqnarray}$$

where $C_{2}$ is an arbitrary constant, and

(4.29)$$\begin{eqnarray}\unicode[STIX]{x1D701}=(\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s})^{1/3}Y+\unicode[STIX]{x1D701}_{0},\quad \unicode[STIX]{x1D701}_{0}=-\text{i}\unicode[STIX]{x1D6FA}(\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s})^{-2/3}.\end{eqnarray}$$

Setting $Y=0$ in (4.25b) and (4.25c), leads to $(\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{u}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{w}_{s})^{\prime \prime }(0)=\text{i}(\widehat{\unicode[STIX]{x1D6FC}}_{s}^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{s}^{2})\widehat{p}_{s}$, use of which in (4.28) gives

(4.30)$$\begin{eqnarray}C_{2}(\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s})^{2/3}\text{Ai}^{\prime }(\unicode[STIX]{x1D701}_{0})=\text{i}(\widehat{\unicode[STIX]{x1D6FC}}_{s}^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{s}^{2})\widehat{p}_{s}.\end{eqnarray}$$

It follows from the matching condition (4.26b) that

(4.31)$$\begin{eqnarray}C_{2}\int _{\unicode[STIX]{x1D701}_{0}}^{\infty }\text{Ai}(\unicode[STIX]{x1D701})\,\text{d}\unicode[STIX]{x1D701}=\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{a}_{s}.\end{eqnarray}$$

The pressure in the upper deck is expressed as $\widetilde{\unicode[STIX]{x1D716}}\widehat{P}_{s}\text{e}^{\text{i}(\widehat{\unicode[STIX]{x1D6FC}}_{s}X+\widehat{\unicode[STIX]{x1D6FD}}_{s}Z-\unicode[STIX]{x1D6FA}\widehat{t})}+\text{c.c.}$, with $\widehat{P}_{s}(\overline{Y})$ being governed by the boundary-value problem,

(4.32a)$$\begin{eqnarray}\displaystyle & \displaystyle (M^{2}-1)\frac{\unicode[STIX]{x2202}^{2}\widehat{P}_{s}}{\unicode[STIX]{x2202}\overline{Y}^{2}}+[M^{2}(\widehat{\unicode[STIX]{x1D6FC}}_{s}-\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FA})^{2}-\widehat{\unicode[STIX]{x1D6FC}}_{s}^{2}-\widehat{\unicode[STIX]{x1D6FD}}_{s}^{2}]\widehat{P}_{s}=0, & \displaystyle\end{eqnarray}$$

(4.32b)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\widehat{P}_{s}}{\unicode[STIX]{x2202}\overline{Y}}=-(\widehat{\unicode[STIX]{x1D6FC}}_{s}-\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FA})^{2}\widehat{a}_{s}\quad \text{at}~\overline{Y}=0, & \displaystyle\end{eqnarray}$$

$\widehat{\unicode[STIX]{x1D6FE}}_{s}=[[M^{2}(\widehat{\unicode[STIX]{x1D6FC}}_{s}-\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FA})^{2}-\widehat{\unicode[STIX]{x1D6FC}}_{s}^{2}-\widehat{\unicode[STIX]{x1D6FD}}_{s}^{2}]/(M^{2}-1)]^{1/2}$

$$\begin{eqnarray}\unicode[STIX]{x1D716}_{0}=\unicode[STIX]{x1D706}^{1/4}(M^{2}-1)^{-1/8}C^{1/8}T_{w}^{1/2}\unicode[STIX]{x1D716}.\end{eqnarray}$$

Here, in order to characterize the acoustic wave fully, the unsteady terms are retained, despite the fact that they are of higher order. Corresponding to and consistent with (3.20), the pressure of the incident and reflected sound waves may be expressed as

(4.33)$$\begin{eqnarray}\widehat{P}_{s}=\left\{\begin{array}{@{}ll@{}}P_{I}(\text{e}^{\text{i}\widehat{\unicode[STIX]{x1D6FE}}_{s}\overline{Y}}+\mathscr{R}\text{e}^{-\text{i}\widehat{\unicode[STIX]{x1D6FE}}_{s}\overline{Y}}) & \text{for}~\widehat{\unicode[STIX]{x1D6FC}}_{s}>0,\\ P_{I}(\text{e}^{-\text{i}\widehat{\unicode[STIX]{x1D6FE}}_{s}\overline{Y}}+\mathscr{R}\text{e}^{\text{i}\widehat{\unicode[STIX]{x1D6FE}}_{s}\overline{Y}}) & \text{for}~\widehat{\unicode[STIX]{x1D6FC}}_{s}<0,\end{array}\right.\end{eqnarray}$$

where $P_{I}$ represents the rescaled amplitude of the incident wave, and the constant $\mathscr{R}$ is recognized as the reflection coefficient, which is to be found as part of the solution. It follows from (4.14) and (4.32b) that

(4.34)$$\begin{eqnarray}\widehat{p}_{s}=(1+\mathscr{R})P_{I},\quad \displaystyle (\widehat{\unicode[STIX]{x1D6FC}}_{s}-\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FA})^{2}\widehat{a}_{s}=\left\{\begin{array}{@{}ll@{}}-\text{i}\widehat{\unicode[STIX]{x1D6FE}}_{s}(1-\mathscr{R})P_{I} & \text{for}~\widehat{\unicode[STIX]{x1D6FC}}_{s}>0,\\ \text{i}\widehat{\unicode[STIX]{x1D6FE}}_{s}(1-\mathscr{R})P_{I} & \text{for}~\widehat{\unicode[STIX]{x1D6FC}}_{s}<0.\end{array}\right.\end{eqnarray}$$

Elimination of $P_{I}$ yields

(4.35)$$\begin{eqnarray}\widehat{p}_{s}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\text{i}(\widehat{\unicode[STIX]{x1D6FC}}_{s}-\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FA})^{2}(1+\mathscr{R})}{\widehat{\unicode[STIX]{x1D6FE}}_{s}(1-\mathscr{R})}\widehat{a}_{s} & \text{for}~\widehat{\unicode[STIX]{x1D6FC}}_{s}>0,\\ \displaystyle \frac{-\text{i}(\widehat{\unicode[STIX]{x1D6FC}}_{s}-\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FA})^{2}(1+\mathscr{R})}{\widehat{\unicode[STIX]{x1D6FE}}_{s}(1-\mathscr{R})}\widehat{a}_{s} & \text{for}~\widehat{\unicode[STIX]{x1D6FC}}_{s}<0.\end{array}\right.\end{eqnarray}$$

The streamwise velocity in the upper deck is expressed as $\widetilde{\unicode[STIX]{x1D716}}\widehat{U}_{s}\text{e}^{\text{i}(\widehat{\unicode[STIX]{x1D6FC}}_{s}X+\widehat{\unicode[STIX]{x1D6FD}}_{s}Z-\unicode[STIX]{x1D6FA}\widehat{t})}+\text{c.c.}$, and so the streamwise momentum equation determines the velocity–pressure relation,

(4.36)$$\begin{eqnarray}\widehat{U}_{s}=-M\cos \unicode[STIX]{x1D703}\widehat{P}_{s}\quad \text{for}~\widehat{\unicode[STIX]{x1D6FC}}_{s}>0;\quad \widehat{U}_{s}=M\cos \unicode[STIX]{x1D703}\widehat{P}_{s}\quad \text{for}~\widehat{\unicode[STIX]{x1D6FC}}_{s}<0.\end{eqnarray}$$

From (4.30), (4.31) and (4.35), we find that the reflection coefficient

(4.37)$$\begin{eqnarray}\mathscr{R}=1-2\int _{\unicode[STIX]{x1D701}_{0}}^{\infty }\text{Ai}(\unicode[STIX]{x1D701})\,\text{d}\unicode[STIX]{x1D701}/\triangle _{s}(\widehat{\unicode[STIX]{x1D6FC}}_{s},\widehat{\unicode[STIX]{x1D6FD}}_{s},\unicode[STIX]{x1D6FA},M),\end{eqnarray}$$

where we have put

(4.38)$$\begin{eqnarray}\triangle _{s}(\widehat{\unicode[STIX]{x1D6FC}}_{s},\widehat{\unicode[STIX]{x1D6FD}}_{s},\unicode[STIX]{x1D6FA},M)=\int _{\unicode[STIX]{x1D701}_{0}}^{\infty }\text{Ai}(\unicode[STIX]{x1D701})\,\text{d}\unicode[STIX]{x1D701}-\frac{|\widehat{\unicode[STIX]{x1D6FC}}_{s}|\widehat{\unicode[STIX]{x1D6FE}}_{s}(\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s})^{2/3}}{(\widehat{\unicode[STIX]{x1D6FC}}_{s}-\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FA})^{2}(\widehat{\unicode[STIX]{x1D6FC}}_{s}^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{s}^{2})}\text{Ai}^{\prime }(\unicode[STIX]{x1D701}_{0}).\end{eqnarray}$$

Similar to the solution (4.22) for $\widehat{w}_{r}$, the solution $\widehat{w}_{s}$ to the inhomogeneous equation (4.25c) is

(4.39)$$\begin{eqnarray}\widehat{w}_{s}=-\text{i}\unicode[STIX]{x03C0}\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{p}_{s}(\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s})^{-2/3}[\text{Gi}(\unicode[STIX]{x1D701})-\text{Gi}(\unicode[STIX]{x1D701}_{0})\text{Ai}(\unicode[STIX]{x1D701})/\text{Ai}(\unicode[STIX]{x1D701}_{0})].\end{eqnarray}$$

The solution for $\widehat{p}_{s},\widehat{u}_{s}$ and $\widehat{v}_{s}$ is found as

(4.40a)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{p}_{s}=\frac{-2|\widehat{\unicode[STIX]{x1D6FC}}_{s}|\widehat{\unicode[STIX]{x1D6FE}}_{s}P_{I}}{(\widehat{\unicode[STIX]{x1D6FC}}_{s}-\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FA})^{2}\triangle _{s}}\frac{(\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s})^{2/3}\text{Ai}^{\prime }(\unicode[STIX]{x1D701}_{0})}{(\widehat{\unicode[STIX]{x1D6FC}}_{s}^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{s}^{2})}, & \displaystyle\end{eqnarray}$$

(4.40b)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{u}_{s}=\left.\left[\frac{-2\text{i}|\widehat{\unicode[STIX]{x1D6FC}}_{s}|\widehat{\unicode[STIX]{x1D6FE}}_{s}P_{I}}{(\widehat{\unicode[STIX]{x1D6FC}}_{s}-\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FA})^{2}\triangle _{s}}\int _{\unicode[STIX]{x1D701}_{0}}^{\unicode[STIX]{x1D701}}\text{Ai}(\unicode[STIX]{x1D701})\,\text{d}\unicode[STIX]{x1D701}-\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{w}_{s}\right]\right/\widehat{\unicode[STIX]{x1D6FC}}_{s}, & \displaystyle\end{eqnarray}$$

(4.40c)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{v}_{s}=[(\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{u}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{w}_{s})^{\prime \prime }+\text{i}(\unicode[STIX]{x1D6FA}-\widehat{\unicode[STIX]{x1D6FC}}_{s}Y)(\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{u}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{w}_{s})-\text{i}(\widehat{\unicode[STIX]{x1D6FC}}_{s}^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{s}^{2})\widehat{p}_{s}]/\widehat{\unicode[STIX]{x1D6FC}}_{s}. & \displaystyle\end{eqnarray}$$

4.3 The scattered wave field

We now consider the scattered field $(\widetilde{u}_{i},\widetilde{v}_{i},\widetilde{w}_{i},\widetilde{a}_{i},\widetilde{p}_{i})$ in (4.15). It satisfies the inhomogeneous linearized boundary-layer equations

(4.41a)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\widetilde{u}_{i}}{\unicode[STIX]{x2202}X}+\frac{\unicode[STIX]{x2202}\widetilde{v}_{i}}{\unicode[STIX]{x2202}Y}+\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widetilde{w}_{i}=0, & \displaystyle\end{eqnarray}$$

(4.41b)$$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\unicode[STIX]{x1D6FA}\widetilde{u}_{i}+Y\frac{\unicode[STIX]{x2202}\widetilde{u}_{i}}{\unicode[STIX]{x2202}X}+\widetilde{v}_{i}=-\frac{\unicode[STIX]{x2202}\widetilde{p}_{i}}{\unicode[STIX]{x2202}X}+\frac{\unicode[STIX]{x2202}^{2}\widetilde{u}_{i}}{\unicode[STIX]{x2202}Y^{2}}-N_{1}\text{e}^{\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}X}, & \displaystyle\end{eqnarray}$$

(4.41c)$$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\unicode[STIX]{x1D6FA}\widetilde{w}_{i}+Y\frac{\unicode[STIX]{x2202}\widetilde{w}_{i}}{\unicode[STIX]{x2202}X}=-\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widetilde{p}_{i}+\frac{\unicode[STIX]{x2202}^{2}\widetilde{w}_{i}}{\unicode[STIX]{x2202}Y^{2}}-N_{2}\text{e}^{\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}X}, & \displaystyle\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle & \displaystyle N_{1}=\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}\widetilde{u}_{r}\widehat{u}_{s}+\widehat{u}_{s}\frac{\unicode[STIX]{x2202}\widetilde{u}_{r}}{\unicode[STIX]{x2202}X}+\widetilde{v}_{r}\frac{\unicode[STIX]{x2202}\widehat{u}_{s}}{\unicode[STIX]{x2202}Y}+\widehat{v}_{s}\frac{\unicode[STIX]{x2202}\widetilde{u}_{r}}{\unicode[STIX]{x2202}Y}+\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{s}\widetilde{w}_{r}\widehat{u}_{s}+\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{w}\widehat{w}_{s}\widetilde{u}_{r}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle N_{2}=\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}\widetilde{u}_{r}\widehat{w}_{s}+\widehat{u}_{s}\frac{\unicode[STIX]{x2202}\widetilde{w}_{r}}{\unicode[STIX]{x2202}X}+\widetilde{v}_{r}\frac{\unicode[STIX]{x2202}\widehat{w}_{s}}{\unicode[STIX]{x2202}Y}+\widehat{v}_{s}\frac{\unicode[STIX]{x2202}\widetilde{w}_{r}}{\unicode[STIX]{x2202}Y}+\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widetilde{w}_{r}\widehat{w}_{s}. & \displaystyle \nonumber\end{eqnarray}$$

The boundary condition approximated by its first-order Taylor expansion about $Y=0$ and the matching condition are

(4.42a)$$\begin{eqnarray}\displaystyle & \displaystyle [\widetilde{u}_{i}(X,0),\widetilde{v}_{i}(X,0),\widetilde{w}_{i}(X,0)]=[-\widehat{u}_{s}^{\prime }(0),0,-\widehat{w}_{s}^{\prime }(0)]\text{e}^{\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}X}F(X)\quad \text{at}~Y=0,\qquad & \displaystyle\end{eqnarray}$$

(4.42b)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{i}\rightarrow \widetilde{a}_{i},\quad \widetilde{w}_{i}\rightarrow 0\quad \text{as}~Y\rightarrow \infty .\qquad & \displaystyle\end{eqnarray}$$

As with the mean-flow distortion, the system (4.41)–(4.42) can be solved in spectral space. Performing Fourier transform on (4.41)–(4.42), we obtain

(4.43a)$$\begin{eqnarray}\displaystyle & \displaystyle \text{i}k\widehat{u}_{i}+\widehat{v}_{i}^{\prime }+\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widehat{w}_{i}=0, & \displaystyle\end{eqnarray}$$

(4.43b)$$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\unicode[STIX]{x1D6FA}\widehat{u}_{i}+\text{i}kY\widehat{u}_{i}+\widehat{v}_{i}=-\text{i}k\widehat{p}_{i}+\widehat{u}_{i}^{\prime \prime }-\widehat{N}_{1}, & \displaystyle\end{eqnarray}$$

(4.43c)$$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\unicode[STIX]{x1D6FA}\widehat{w}_{i}+\text{i}kY\widehat{w}_{i}=-\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widehat{p}_{i}+\widehat{w}_{i}^{\prime \prime }-\widehat{N}_{2}, & \displaystyle\end{eqnarray}$$

(4.43d)$$\begin{eqnarray}\displaystyle & \displaystyle [\widehat{u}_{i},\widehat{v}_{i},\widehat{w}_{i}]=[-\widehat{u}_{s}^{\prime }(0),0,-\widehat{w}_{s}^{\prime }(0)]\widehat{F}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})\quad \text{at}~Y=0, & \displaystyle\end{eqnarray}$$

(4.43e)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{u}_{i}\rightarrow \widehat{a}_{i},\quad \widehat{w}_{i}\rightarrow 0\quad \text{as}~Y\rightarrow \infty , & \displaystyle\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{N}_{1}=\text{i}k\widehat{u}_{r}\widehat{u}_{s}+\widehat{v}_{r}\widehat{u}_{s}^{\prime }+\widehat{v}_{s}\widehat{u}_{r}^{\prime }+\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{s}\widehat{w}_{r}\widehat{u}_{s}+\text{i}\widehat{\unicode[STIX]{x1D6FD}}_{w}\widehat{w}_{s}\widehat{u}_{r}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \widehat{N}_{2}=\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{s}\widehat{u}_{r}\widehat{w}_{s}+\text{i}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})\widehat{u}_{s}\widehat{w}_{r}+\widehat{v}_{r}\widehat{w}_{s}^{\prime }+\widehat{v}_{s}\widehat{w}_{r}^{\prime }+\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widehat{w}_{r}\widehat{w}_{s}; & \displaystyle \nonumber\end{eqnarray}$$

here, the quantities having subscript $r$ are evaluated at $(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})$. Similar to that for the mean-flow distortion, the pressure-displacement relation holds (cf. (4.21)),

(4.44)$$\begin{eqnarray}\widehat{p}_{i}=\frac{k^{2}}{[(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})^{2}/(M^{2}-1)-k^{2}]^{1/2}}\widehat{a}_{i}.\end{eqnarray}$$

The homogeneous version of the boundary-value problem (4.43) with (4.44) describes the lower-branch Mack mode instability.

The boundary-value problem (4.43a)–(4.43e) supplemented by (4.44) can be solved to obtain the solution in terms of Airy functions. The procedure is rather straightforward but involves complex algebra. The reader may skip the derivation and go to the main final result (4.60). We proceed by eliminating $\widehat{u}_{i}$ and $\widehat{w}_{i}$ from (4.43a)–(4.43c) to obtain

(4.45)$$\begin{eqnarray}\widehat{v}_{i}^{\prime \prime \prime }+\text{i}(\unicode[STIX]{x1D6FA}-kY)\widehat{v}_{i}^{\prime }+\text{i}k\widehat{v}_{i}=[k^{2}+(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})^{2}]\widehat{p}_{i}-\text{i}k\widehat{N}_{1}-\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widehat{N}_{2}.\end{eqnarray}$$

Differentiation of (4.45) with respect to $Y$ leads to

(4.46)$$\begin{eqnarray}\widehat{v}_{i}^{\prime \prime \prime \prime }+\text{i}(\unicode[STIX]{x1D6FA}-kY)\widehat{v}_{i}^{\prime \prime }=\widehat{G}(Y),\end{eqnarray}$$

where

$$\begin{eqnarray}\widehat{G}(Y)=-\text{i}k\widehat{N}_{1}^{\prime }-\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widehat{N}_{2}^{\prime }.\end{eqnarray}$$

Use of (4.43a), (4.43d) gives

(4.47)$$\begin{eqnarray}\widehat{v}_{i}^{\prime }(0)=\widehat{F}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})[\text{i}k\widehat{u}_{s}^{\prime }(0)+\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widehat{w}_{s}^{\prime }(0)].\end{eqnarray}$$

Setting $Y=0$ in (4.45) and using (4.47), we have

(4.48)$$\begin{eqnarray}\widehat{v}_{i}^{\prime \prime \prime }(0)=[k^{2}+(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})^{2}]\widehat{p}_{i}+\widehat{F}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})\unicode[STIX]{x1D6FA}[k\widehat{u}_{s}^{\prime }(0)+(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widehat{w}_{s}^{\prime }(0)].\end{eqnarray}$$

The solution $\widehat{v}_{i}^{\prime \prime }$ to equation (4.46) can be expressed as (Goldstein Reference Goldstein1985),

(4.49)$$\begin{eqnarray}\widehat{v}_{i}^{\prime \prime }=C_{4}(\text{i}k)^{1/3}\text{Ai}(\unicode[STIX]{x1D709})+\widehat{S}_{v},\end{eqnarray}$$

where $C_{4}$ is a function of $k$ to be found, and $\widehat{S}_{v}$ denotes the particular solution

(4.50)$$\begin{eqnarray}\widehat{S}_{v}=-(\text{i}k)^{-1/3}\unicode[STIX]{x03C0}\left\{\text{Ai}(\unicode[STIX]{x1D709})\int _{0}^{Y}\text{Bi}(\unicode[STIX]{x1D709})\widehat{G}(Y)\,\text{d}Y-\text{Bi}(\unicode[STIX]{x1D709})\int _{\infty }^{Y}\text{Ai}(\unicode[STIX]{x1D709})\widehat{G}(Y)\,\text{d}Y\right\},\end{eqnarray}$$

with

(4.51)$$\begin{eqnarray}\unicode[STIX]{x1D709}=(\text{i}k)^{1/3}Y+\unicode[STIX]{x1D709}_{0},\quad \unicode[STIX]{x1D709}_{0}=-\text{i}\unicode[STIX]{x1D6FA}(\text{i}k)^{-2/3}.\end{eqnarray}$$

Use of (4.49) in the boundary condition (4.48) yields

(4.52)$$\begin{eqnarray}\displaystyle & & \displaystyle C_{4}(\text{i}k)^{2/3}\text{Ai}^{\prime }(\unicode[STIX]{x1D709}_{0})-\unicode[STIX]{x03C0}\text{Bi}^{\prime }(\unicode[STIX]{x1D709}_{0})\int _{0}^{\infty }\text{Ai}(\unicode[STIX]{x1D709})\widehat{G}(Y)\,\text{d}Y\nonumber\\ \displaystyle & & \displaystyle \quad =(k^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2})\widehat{p}_{i}+\widehat{F}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})\unicode[STIX]{x1D6FA}[k\widehat{u}_{s}^{\prime }(0)+\widehat{\unicode[STIX]{x1D6FD}}_{T}\widehat{w}_{s}^{\prime }(0)],\end{eqnarray}$$

where $\widehat{\unicode[STIX]{x1D6FD}}_{T}=\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w}$. We integrate (4.49) using boundary condition (4.47) to obtain

(4.53)$$\begin{eqnarray}\widehat{v}_{i}^{\prime }=\int _{0}^{Y}\widehat{S}_{v}\,\text{d}Y+C_{4}\displaystyle \int _{\unicode[STIX]{x1D709}_{0}}^{\unicode[STIX]{x1D709}}\text{Ai}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}+\widehat{F}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})[\text{i}k\widehat{u}_{s}^{\prime }(0)+\text{i}(\widehat{\unicode[STIX]{x1D6FD}}_{s}+\widehat{\unicode[STIX]{x1D6FD}}_{w})\widehat{w}_{s}^{\prime }(0)].\end{eqnarray}$$

After solving (4.43c) and using (4.44), the spanwise velocity satisfying the boundary condition for $\widehat{w}_{i}$ in (4.43d) is found as

(4.54)$$\begin{eqnarray}\widehat{w}_{i}=a_{i}\widehat{w}_{E}(Y;k)-\widehat{F}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})\widehat{w}_{s}^{\prime }(0)+C_{w}\text{Ai}(\unicode[STIX]{x1D709})+\widehat{S}_{w},\end{eqnarray}$$

where

(4.55)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{w}_{E}(Y;k)=-\frac{\text{i}\unicode[STIX]{x03C0}k^{2}\widehat{\unicode[STIX]{x1D6FD}}_{T}(\text{i}k)^{-2/3}}{[\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2}/(M^{2}-1)-k^{2}]^{1/2}}\left[\text{Gi}(\unicode[STIX]{x1D709})-\frac{\text{Gi}(\unicode[STIX]{x1D709}_{0})}{\text{Ai}(\unicode[STIX]{x1D709}_{0})}\text{Ai}(\unicode[STIX]{x1D709})\right], & \displaystyle\end{eqnarray}$$

(4.56)$$\begin{eqnarray}\displaystyle & \displaystyle C_{w}=\unicode[STIX]{x03C0}(\text{i}k)^{1/3}[\text{Bi}(\unicode[STIX]{x1D709}_{0})/\text{Ai}(\unicode[STIX]{x1D709}_{0})]\int _{0}^{\infty }\text{Ai}(\unicode[STIX]{x1D709})\widehat{G}(Y)\,\text{d}Y, & \displaystyle\end{eqnarray}$$

and $\widehat{S}_{w}$ has the same expression as $\widehat{S}_{v}$ shown in (4.50) provided that $\widehat{G}$ is replaced by $\widehat{N}_{2}$. Substituting (4.53) and (4.54) into (4.43a), and making use of (4.44), we obtain the streamwise velocity

(4.57)$$\begin{eqnarray}\widehat{u}_{i}=\widehat{a}_{i}\widehat{u}_{E}(Y;k)-\widehat{F}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})\widehat{u}_{s}^{\prime }(0)+\widehat{S}_{u},\end{eqnarray}$$

where

$$\begin{eqnarray}\widehat{S}_{u}=(\text{i}/k)\int _{0}^{Y}\widehat{S}_{v}\,\text{d}Y-\frac{\text{i}\left[\displaystyle \int _{0}^{\infty }\widehat{S}_{v}\,\text{d}Y+\widehat{v}_{i}^{\prime }(0)\right]}{k\displaystyle \int _{\unicode[STIX]{x1D709}_{0}}^{\infty }\text{Ai}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}}\int _{\unicode[STIX]{x1D709}_{0}}^{\unicode[STIX]{x1D709}}\text{Ai}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}-(\widehat{\unicode[STIX]{x1D6FD}}_{T}/k)C_{w}\text{Ai}(\unicode[STIX]{x1D709})-\widehat{\unicode[STIX]{x1D6FD}}_{T}\widehat{S}_{w}/k,\end{eqnarray}$$

(4.58)$$\begin{eqnarray}\widehat{u}_{E}(Y;k)=\displaystyle \frac{\displaystyle \int _{\unicode[STIX]{x1D709}_{0}}^{\unicode[STIX]{x1D709}}\text{Ai}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}}{\displaystyle \int _{\unicode[STIX]{x1D709}_{0}}^{\infty }\text{Ai}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}}-\frac{\unicode[STIX]{x03C0}\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2}(\text{i}k)^{1/3}}{[\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2}/(M^{2}-1)-k^{2}]^{1/2}}\left[-\text{Gi}(\unicode[STIX]{x1D709})+\frac{\text{Gi}(\unicode[STIX]{x1D709}_{0})}{\text{Ai}(\unicode[STIX]{x1D709}_{0})}\text{Ai}(\unicode[STIX]{x1D709})\right].\end{eqnarray}$$

As will transpire later, $\widehat{u}_{E}(Y;k)$ and $\widehat{w}_{E}(Y;k)$ represent the eigenfunctions for the streamwise and spanwise velocities of the instability mode when $k=\widehat{\unicode[STIX]{x1D6FC}}_{T}$.

Inserting (4.53) into (4.43a), and noting (4.43e), we obtain the relation

(4.59)$$\begin{eqnarray}\int _{0}^{Y}\widehat{S}_{v}\,\text{d}Y+C_{4}\int _{\unicode[STIX]{x1D709}_{0}}^{\infty }\text{Ai}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}+\widehat{v}_{i}^{\prime }(0)=\widehat{a}_{i}.\end{eqnarray}$$

From (4.59), (4.52) and (4.44), it follows that

(4.60)$$\begin{eqnarray}\displaystyle \widehat{a}_{i} & = & \displaystyle \frac{[\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2}/(M^{2}-1)-k^{2}]^{1/2}\text{Ai}^{\prime }(\unicode[STIX]{x1D709}_{0})}{k^{2}(k^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2})\unicode[STIX]{x1D6E5}(k,\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)}\left\{\displaystyle \int _{0}^{\infty }K(\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}_{0})\widehat{G}(Y)\,\text{d}Y\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\widehat{F}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})\text{i}(\text{i}k)^{2/3}[k\widehat{u}_{s}^{\prime }(0)+\widehat{\unicode[STIX]{x1D6FD}}_{T}\widehat{w}_{s}^{\prime }(0)](1+\unicode[STIX]{x1D709}_{0}\widetilde{\mathscr{R}})\vphantom{\displaystyle \int _{0}^{\infty }}\right\},\end{eqnarray}$$

where we have introduced the notations

(4.61a,b)$$\begin{eqnarray}K(\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}_{0})=\unicode[STIX]{x03C0}\frac{\text{Ai}^{\prime }(\unicode[STIX]{x1D709}_{0})\text{Gi}(\unicode[STIX]{x1D709})-\text{Gi}^{\prime }(\unicode[STIX]{x1D709}_{0})\text{Ai}(\unicode[STIX]{x1D709})}{\text{Ai}^{\prime }(\unicode[STIX]{x1D709}_{0})},\quad \widetilde{\mathscr{R}}=\int _{\unicode[STIX]{x1D709}_{0}}^{\infty }\text{Ai}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}/\text{Ai}^{\prime }(\unicode[STIX]{x1D709}_{0});\end{eqnarray}$$

here, $\unicode[STIX]{x1D6E5}(k,\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)$ is the dispersion function of the first-mode instability,

(4.62)$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(k,\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)=\int _{\unicode[STIX]{x1D709}_{0}}^{\infty }\text{Ai}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}-\frac{[\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2}/(M^{2}-1)-k^{2}]^{1/2}}{(\text{i}k)^{1/3}(k^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2})}\text{Ai}^{\prime }(\unicode[STIX]{x1D709}_{0}).\end{eqnarray}$$

4.4 The coupling coefficient

The solution in physical space can be obtained by inverting the Fourier transform. For the displacement, we have

(4.63)$$\begin{eqnarray}\widetilde{a}_{i}(X)=\frac{1}{\sqrt{2\unicode[STIX]{x03C0}}}\int _{-\infty }^{\infty }\text{e}^{\text{i}kX}\widehat{a}_{i}(k)\,\text{d}k,\end{eqnarray}$$

where the integrand has poles at the values of $k$ which are roots of the dispersion relation (4.62), i.e.

(4.64)$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(\widehat{\unicode[STIX]{x1D6FC}}_{T},\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)=0.\end{eqnarray}$$

The instability waves correspond to the contributions to the integral from these poles, which can be calculated by using the residual theorem. Of interest is the lower-branch unstable mode. Then, from (4.60) and (4.63), it can be derived that

(4.65)$$\begin{eqnarray}\widetilde{a}_{i}(X)=\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})\unicode[STIX]{x1D6EC}(\widehat{\unicode[STIX]{x1D6FC}}_{T},\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)\text{e}^{\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{T}X},\end{eqnarray}$$

where we have put

(4.66)$$\begin{eqnarray}\displaystyle & & \displaystyle \!\!\!\unicode[STIX]{x1D6EC}(\widehat{\unicode[STIX]{x1D6FC}}_{T},\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)=\frac{\text{i}\sqrt{2\unicode[STIX]{x03C0}}[\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2}/(M^{2}-1)-k^{2}]^{1/2}\text{Ai}^{\prime }(\unicode[STIX]{x1D709}_{0})}{k^{2}(k^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2})\unicode[STIX]{x1D6E5}^{\prime }(k,\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)}\nonumber\\ \displaystyle & & \displaystyle \!\!\!\quad \left.\times \left\{\frac{1}{\widehat{F}(k-\widehat{\unicode[STIX]{x1D6FC}}_{s})}\int _{0}^{\infty }K(\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}_{0})\widehat{G}(Y)\,\text{d}Y-\text{i}(\text{i}k)^{2/3}[k\widehat{u}_{s}^{\prime }(0)+\widehat{\unicode[STIX]{x1D6FD}}_{T}\widehat{w}_{s}^{\prime }(0)](1+\unicode[STIX]{x1D709}_{0}\widetilde{\mathscr{R}})\right\}\right|_{k=\widehat{\unicode[STIX]{x1D6FC}}_{T}}\!,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6E5}^{\prime }$ stands for the derivative of $\unicode[STIX]{x1D6E5}$ with respect to $k$. The factor $\unicode[STIX]{x1D6EC}$ is a function of the wavenumbers, $\widehat{\unicode[STIX]{x1D6FC}}_{s}$, $\widehat{\unicode[STIX]{x1D6FD}}_{s}$, $\widehat{\unicode[STIX]{x1D6FD}}_{w}$, frequency $\unicode[STIX]{x1D6FA}$ and Mach number $M$, but independent of the roughness geometry. It measures the efficiency of acoustic waves in exciting instability modes, and is thus referred to as the efficiency function.

The result (4.65) indicates the absolute amplitude of the instability mode excited is proportional to $\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})$. The role of the streamwise scale of the surface roughness transpires when one considers $\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})$ for a Gaussian shape (4.5), the Fourier transform of which is $\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})=\widehat{d}/\sqrt{2}\exp [-\widehat{d}^{2}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})^{2}/4]$. Clearly, $\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})$ attains its maximum when

(4.67)$$\begin{eqnarray}\widehat{d}=\sqrt{2}/[(\widehat{\unicode[STIX]{x1D6FC}}_{Tr}-\widehat{\unicode[STIX]{x1D6FC}}_{s})^{2}-\widehat{\unicode[STIX]{x1D6FC}}_{Ti}^{2}]^{1/2}\equiv \widehat{\unicode[STIX]{x1D706}}_{T},\end{eqnarray}$$

i.e. when $\widehat{d}\sim (\widehat{\unicode[STIX]{x1D6FC}}_{s}-\widehat{\unicode[STIX]{x1D6FC}}_{Tr})^{-1}$ since $\widehat{\unicode[STIX]{x1D6FC}}_{Ti}$ is much smaller than $\widehat{\unicode[STIX]{x1D6FC}}_{Tr}$. The result indicates that the streamwise length scale of the roughness must be comparable with the characteristic wavelength of the instability mode, a conclusion that holds for an isolated roughness of an arbitrary shape. If the roughness variation is over a much longer length scale (i.e. $\widehat{d}\gg (\widehat{\unicode[STIX]{x1D6FC}}_{s}-\widehat{\unicode[STIX]{x1D6FC}}_{Tr})^{-1}$, the amplitude of the excited instability mode could be extremely small, indeed exponentially small for Gaussian and many other shapes.

In order to compare the theoretical predictions with the DNS results, we find from equations (4.57) and (4.65) the streamwise velocity of the instability mode,

(4.68)$$\begin{eqnarray}\widetilde{u}_{i}=\widehat{u}_{E}(Y;\widehat{\unicode[STIX]{x1D6FC}}_{T})\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})\unicode[STIX]{x1D6EC}(\widehat{\unicode[STIX]{x1D6FC}}_{T},\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)\text{e}^{\text{i}\widehat{\unicode[STIX]{x1D6FC}}_{T}X}.\end{eqnarray}$$

Note that only the first terms of the right-hand sides of (4.57) and (4.54) consist of the eigenfunction; all other terms stand for the near-field response, which decays algebraically downstream. It follows from (3.26), (4.6) and (4.15) that the initial amplitude

(4.69)$$\begin{eqnarray}A_{u0}=\unicode[STIX]{x1D706}^{1/4}(M^{2}-1)^{-1/8}C^{1/8}T_{w}^{1/2}\unicode[STIX]{x1D716}\widetilde{\unicode[STIX]{x1D716}}HN_{u}|\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})\unicode[STIX]{x1D6EC}(\widehat{\unicode[STIX]{x1D6FC}}_{T},\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)|,\end{eqnarray}$$

where

(4.70a,b)$$\begin{eqnarray}H=\frac{h^{\ast }}{L^{\ast }}\unicode[STIX]{x1D706}^{3/4}(M^{2}-1)^{1/8}C^{-5/8}T_{w}^{-3/2}\unicode[STIX]{x1D716}^{-5},\quad N_{u}=\max _{Y}|\widehat{u}_{E}(Y;\widehat{\unicode[STIX]{x1D6FC}}_{T})|.\end{eqnarray}$$

The relations (4.11) and (4.36) imply that

(4.71)$$\begin{eqnarray}\unicode[STIX]{x1D716}_{u}=\unicode[STIX]{x1D706}^{1/2}(M^{2}-1)^{-1/4}C^{1/4}\unicode[STIX]{x1D716}^{2}\widetilde{\unicode[STIX]{x1D716}}|M\cos \unicode[STIX]{x1D703}P_{I}|,\end{eqnarray}$$

and thus the receptivity coefficient (3.27) is expressed as

(4.72)$$\begin{eqnarray}\displaystyle {\mathcal{K}} & = & \displaystyle A_{u0}/\unicode[STIX]{x1D716}_{u}=\frac{h^{\ast }}{L^{\ast }}\unicode[STIX]{x1D706}^{1/2}(M^{2}-1)^{1/4}C^{-3/4}T_{w}^{-1}R^{3/4}N_{u}\nonumber\\ \displaystyle & & \displaystyle |\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})\unicode[STIX]{x1D6EC}(\widehat{\unicode[STIX]{x1D6FC}}_{T},\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T};M)/(M\cos \unicode[STIX]{x1D703})|.\end{eqnarray}$$

5.1 Reflection coefficient of free-stream acoustic wave

Numerical calculations were performed first to compute the acoustic reflection coefficient $\mathscr{R}$ given by (4.37). The Airy functions and its derivative were evaluated numerically by integrating the equations that they satisfy using Matlab.

Figure 2.

The reflection coefficient $|\mathscr{R}|$ (a,b) and the pressure in the lower deck (c,d) versus $\unicode[STIX]{x1D703}$ at $M=2$ and $R=9.05\times 10^{6}$ for $\widehat{\unicode[STIX]{x1D6FD}}_{s}=0$ (a,c) and $\widehat{\unicode[STIX]{x1D6FD}}_{s}=2.0$ (b,d).

Figure 2(a,b) shows how $|\mathscr{R}|$ varies with the incident angle $\unicode[STIX]{x1D703}$ of the incoming acoustic wave for different frequencies with $\widehat{\unicode[STIX]{x1D6FD}}_{s}=0$ and $\widehat{\unicode[STIX]{x1D6FD}}_{s}=2.0$ at $M=2.0$. For $\widehat{\unicode[STIX]{x1D6FD}}_{s}=0$ and $\unicode[STIX]{x1D6FA}=2,3$, there exists a range of $\unicode[STIX]{x1D703}$ values in which $|\mathscr{R}|>1$, implying that the wave reflected by the viscous boundary layer is stronger than the incident wave. This is the so-called ‘over-reflection’ phenomenon. However, $|\mathscr{R}|<1$ for $\widehat{\unicode[STIX]{x1D6FD}}_{s}=0$ and $\unicode[STIX]{x1D6FA}=1$, and for $\widehat{\unicode[STIX]{x1D6FD}}_{s}=2$ and all three frequencies; in these cases, the boundary layer absorbs the energy of the incident acoustic wave. The figures also show that for $\unicode[STIX]{x1D703}$ close to $\cos ^{-1}(1/M)=60^{\circ }$, higher-frequency acoustic waves have larger $|\mathscr{R}|$, but the trend may reverse when $\unicode[STIX]{x1D703}$ is smaller than a critical angle, as is the case for $\widehat{\unicode[STIX]{x1D6FD}}_{s}=2.0$. In order to characterize the boundary-layer response to the free-stream acoustic wave, figure 2(c,d) displays how $|\widetilde{p}_{w}|$, the pressure amplitude in the lower deck, varies with the incident angle $\unicode[STIX]{x1D703}$ for a selection of frequencies. Lower-frequency acoustic waves have larger $|\widetilde{p}_{w}|$ for $\unicode[STIX]{x1D703}$ close to $60^{\circ }$, and again the trend reverses for smaller $\unicode[STIX]{x1D703}$. For each frequency, $|\widetilde{p}_{w}|$ exhibits a peak at a particular $\unicode[STIX]{x1D703}$. High-frequency waves tend to have larger peak $|\widetilde{p}_{w}|$. For all the cases, as $\unicode[STIX]{x1D703}\rightarrow 60^{\circ }$, $|\mathscr{R}|\rightarrow 1$ and $|\widetilde{p}_{w}|\rightarrow 0$.

The reflection coefficient and the induced acoustic signature were also obtained by computing the acoustic-branch continuous spectrum of the O–S equation. The results are compared with the triple-deck predictions in figure 2. The agreement is rather satisfactory, especially for the lower-frequency case ($\unicode[STIX]{x1D6FA}=1$).

Figure 3.

The response of the boundary layer to free-stream acoustic waves for different $\widehat{\unicode[STIX]{x1D6FD}}_{s}$ at $M=2.0$ and $R=9.05\times 10^{6}$. (a) The reflection coefficient $|\mathscr{R}|$ versus $\unicode[STIX]{x1D703}$, (b) the streamwise-velocity amplitude $|\widetilde{u}|_{max}$ versus $\unicode[STIX]{x1D703}$, (c) the spanwise velocity amplitude $|\widetilde{w}|_{max}$ versus $\unicode[STIX]{x1D703}$, (d) the wall pressure $|\widetilde{p}_{w}|$ versus $\unicode[STIX]{x1D703}$.

Figure 4.

The response of the boundary layer to free-stream acoustic waves with $\widehat{\unicode[STIX]{x1D6FD}}_{s}=2.0$, $\unicode[STIX]{x1D6FA}=1$ and 3.0 at different $M$ and $R=9.05\times 10^{6}$. (a) The reflection coefficient $|\mathscr{R}|$ versus $\unicode[STIX]{x1D703}$, (b) the streamwise velocity amplitude $|\widetilde{u}|_{max}$ versus $\unicode[STIX]{x1D703}$, (c) the spanwise velocity amplitude $|\widetilde{w}|_{max}$ versus $\unicode[STIX]{x1D703}$, (d) the wall pressure $|\widetilde{p}_{w}|$ versus $\unicode[STIX]{x1D703}$.

The effects of the spanwise wavenumber $\widehat{\unicode[STIX]{x1D6FD}}_{s}$ on the reflection and response are shown in figure 3(a–d). As $\widehat{\unicode[STIX]{x1D6FD}}_{s}$ increases, the reflection of acoustic waves is enhanced, while the response of the boundary layer is reduced. Each of $|\widetilde{u}|_{max}$, $|\widetilde{w}|_{max}$ and $|\widetilde{p}_{w}|$ exhibits a peak at $\unicode[STIX]{x1D703}$ close to $\cos ^{-1}(1/M)=60^{\circ }$. The peaks of $|\widetilde{u}|_{max}$ and $|\widetilde{w}|_{max}$ are aligned with each other, but not with that of $|\widetilde{p}_{w}|$. A notable feature is that the streamwise and spanwise velocities, $|\widetilde{u}|_{max}$ and $|\widetilde{w}|_{max}$, are significantly enlarged, by a factor of $O(R^{1/8})$ according to the triple-deck theory. The peak value of $|\widetilde{u}|_{max}$ turns out to be much greater than that of $|\widetilde{w}|_{max}$. The large acoustic signature in the boundary layer implies that the latter plays the role of sound board. The same features remain at other Mach numbers as is shown in figure 4(a–d). As the Mach number increases, for lower-frequency sound waves, $|\mathscr{R}|$ falls below unity more significantly, and correspondingly, $|\widetilde{u}|_{max}$, $|\widetilde{w}|_{max}$ and $|\widetilde{p}_{w}|$ exhibit more pronounced peaks, i.e. the ‘sound board’ effect of the boundary layer becomes stronger. Interestingly, for high-frequency waves at large Mach numbers, an over-reflection peak emerges, corresponding to which $|\widetilde{u}|_{max}$, $|\widetilde{w}|_{max}$ and $|\widetilde{p}_{w}|$ peak as well, now all nearly aligned. It may be inferred that stronger absorption and over-reflection both lead to a stronger response, that is, the boundary-layer response to free-stream acoustic waves is enhanced when $||\mathscr{R}|-1|$ is large.

Figure 5.

The profiles of the $|\widetilde{u}|$, $|\widetilde{v}|$ and $|\widetilde{w}|$ components of an acoustic signature with $\widehat{\unicode[STIX]{x1D6FD}}_{s}=2.0$ at $M=2$ and $R=9.05\times 10^{6}$ for $(\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D703})=(1.0,57.7^{\circ })$.

In order to describe further the boundary-layer response to the free-stream acoustic wave, the profiles of the acoustic signature inside the boundary layer are displayed in figure 5. The profiles predicted by the O–S and triple-deck approaches are in good agreement in the lower-deck region of interest. Deviation occurs at the top of the lower deck, which is expected since the lower-deck solution cannot of course capture the main-deck acoustic signature, but the latter is unimportant since the dominant interaction takes place in the lower deck. Since the triple-deck theory amounts to a large-Reynolds-number asymptotic approximation of the O–S solution, a slight mismatch of the disturbance profiles is expected.

The comparisons over an extensive range of parameters indicate that the results given by the triple-deck theory and the O–S equation are in good agreement even at moderate Reynolds numbers, especially for relatively low-frequency acoustic waves, which are of main interest to receptivity. The triple-deck theory predicts adequately the impingement and reflection of such acoustic waves. Although not shown here, calculations at higher $R$ confirm that the agreement improves even farther, and $|\mathscr{R}|$ and the acoustic signature predicted by the O–S equation approach the triple-deck solution asymptotically.

5.2 First-mode instability wave

In order to aid the calculation of the amplitude of the instability mode excited, its spatial instability properties are revisited. With $\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T}$ being kept real, we calculate first the eigenvalues $\widehat{\unicode[STIX]{x1D6FC}}_{T}$ from their dispersion relation (4.64). Alternatively, we also solved numerically the homogeneous version of the boundary-value problem (4.43a–e), which is, on letting $\widehat{\unicode[STIX]{x1D719}}=\widehat{\unicode[STIX]{x1D6FC}}_{T}u_{i}+\widehat{\unicode[STIX]{x1D6FD}}_{T}w_{i}$, recast as (cf. Smith Reference Smith1989)

(5.1a)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D719}}_{YYY}+\text{i}(\unicode[STIX]{x1D6FA}-\widehat{\unicode[STIX]{x1D6FC}}_{T}Y)\widehat{\unicode[STIX]{x1D719}}_{Y}=0, & \displaystyle\end{eqnarray}$$

(5.1b)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D719}}_{YY}(0)=\text{i}(\widehat{\unicode[STIX]{x1D6FC}}_{T}^{2}+\widehat{\unicode[STIX]{x1D6FD}}_{T}^{2})p_{i},\quad \widehat{\unicode[STIX]{x1D719}}_{Y}(\infty )=0, & \displaystyle\end{eqnarray}$$

(5.1c)$$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D719}}(\infty )=\widehat{\unicode[STIX]{x1D6FC}}_{T}a_{i}, & \displaystyle\end{eqnarray}$$

$\widehat{\unicode[STIX]{x1D719}}_{Y}$

$\widehat{\unicode[STIX]{x1D719}}_{YY}$

$Y$

$\widehat{\unicode[STIX]{x1D6FC}}_{T}$

$\unicode[STIX]{x1D6FA}$

$\widehat{\unicode[STIX]{x1D6FD}}_{T}$

Figure 6.

Linear instability characteristics of lower-branch first Mack modes at $M=2.0$. (a) The streamwise wavenumber $\widehat{\unicode[STIX]{x1D6FC}}_{Tr}$ versus $\unicode[STIX]{x1D6FA}$, (b) the streamwise growth rate $-\widehat{\unicode[STIX]{x1D6FC}}_{Ti}$ versus $\unicode[STIX]{x1D6FA}$.

The real and imaginary parts of $\widehat{\unicode[STIX]{x1D6FC}}_{T}$ are plotted in figures 6(a) and 6(b) against $\unicode[STIX]{x1D6FA}$ for a number of values of the spanwise wavenumber $\widehat{\unicode[STIX]{x1D6FD}}_{T}$ at $M=2.0$. For small $\widehat{\unicode[STIX]{x1D6FD}}_{T}$, modes are damped in a small low-frequency band, and unstable modes come into existence when $\unicode[STIX]{x1D6FA}$ exceeds a certain cutoff. As $\widehat{\unicode[STIX]{x1D6FD}}_{T}$ increases, the stable band shrinks and eventually disappears. The results of growth rate $-\widehat{\unicode[STIX]{x1D6FC}}_{Ti}$ are in excellent agreement with those of Smith (Reference Smith1989). Of interest are unstable lower-branch modes, and so the range of frequencies to study is chosen to be $1.0\leqslant \unicode[STIX]{x1D6FA}\leqslant 4.0$. However, the low-frequency case ($\unicode[STIX]{x1D6FA}=1.0$), pertinent to the generation of modes near the lower-branch neutral curve, is of more relevance and importance. Excitation near the lower branch is of most relevance since once excited a mode will enter the unstable regime because with its dimensional frequency and spanwise wavenumber fixed, $\unicode[STIX]{x1D6FA}$ and $\widehat{\unicode[STIX]{x1D6FD}}_{T}$ increase as the boundary-layer thickens.

Figure 7.

Comparisons of the asymptotic and finite-$R$ predictions for the lower-branch instability characteristics: (a) the streamwise wavenumber $\widehat{\unicode[STIX]{x1D6FC}}_{Tr}$ versus $\unicode[STIX]{x1D6FA}$; (b) the streamwise growth rate $-\widehat{\unicode[STIX]{x1D6FC}}_{Ti}$ versus $\unicode[STIX]{x1D6FA}$ at $M=2.0$ and $\widehat{\unicode[STIX]{x1D6FD}}_{T}=2.0$.

Figure 8.

Eigenfunction profiles of an instability mode, normalized by the amplitude of the wall pressure. (a) Streamwise velocity $R^{-1/8}|\widehat{u}|$, (b) spanwise velocity $R^{-1/8}|\widehat{w}|$. Solid lines: composite solution from triple-deck theory (see appendix A); dashed lines: prediction by the O–S equation.

By setting the mean flow as the Blasius solution, we solve the O–S equations numerically and obtain the eigenvalues and eigenfunctions, which are compared with predictions by the triple-deck theory. Figure 7 shows the dependence of the wavenumber $\widehat{\unicode[STIX]{x1D6FC}}_{Tr}$ and growth rate $-\widehat{\unicode[STIX]{x1D6FC}}_{Ti}$ on frequency $\unicode[STIX]{x1D6FA}$ for $M=2.0$ and $\widehat{\unicode[STIX]{x1D6FD}}_{T}=2.0$. The profiles of the $|\widehat{u}|$ and $|\widehat{w}|$ components of the instability mode, $(\unicode[STIX]{x1D6FA},\widehat{\unicode[STIX]{x1D6FD}}_{T})=(1.0,2.0)$, are displayed in figure 8, where all the quantities are normalized by the amplitude of the wall pressure. Here, the asymptotic solution in each of the three decks is used to construct a composite solution valid in the entire boundary layer, the detail of which is given in appendix A). As illustrated, the results given by the finite-Reynolds-number linear stability theory (i.e. the O–S approach) tend to those by the triple-deck approach as $R$ increases. The latter captures the overall qualitative features of the instability characteristics and modal shape, but appreciable quantitative difference exists due to the finite-Reynolds-number effect, which is stronger for higher frequencies, especially when $\unicode[STIX]{x1D6FA}>3$. Larger errors arise in the wavenumber $\widehat{\unicode[STIX]{x1D6FC}}_{Tr}$ for modes with $\unicode[STIX]{x1D6FA}>3$, and this reduces the accuracy of the receptivity prediction. Compared with its performance in predicting the boundary-layer response to acoustic waves, the triple-deck theory does not perform as satisfactorily in describing instability. Despite this, this approach turns out to predict rather accurately the receptivity of the lower-branch modes with moderate frequencies, and in particular, the rather poor accuracy of the streamwise growth rate given by the triple-deck theory does not affect the prediction of the receptivity, as will be demonstrated next.

5.3 Receptivity coefficient

As was shown earlier, for roughness of Gaussian shape (4.5) the maximum receptivity is obtained when the roughness width satisfies (4.67). In this section, we set $\widehat{d}=\widehat{\unicode[STIX]{x1D706}}_{T}$.

Figure 9.

The efficiency function $|\unicode[STIX]{x1D6EC}|$ and receptivity coefficient $|{\mathcal{K}}|$ for $\widehat{\unicode[STIX]{x1D6FD}}_{s}=2.0$ and $\widehat{\unicode[STIX]{x1D6FD}}_{w}=0$ at $M=2$ and $R=9.05\times 10^{6}$. (a) $|\unicode[STIX]{x1D6EC}|$ versus $\unicode[STIX]{x1D703}$, (b) $|\unicode[STIX]{x1D6EC}|$ versus $c_{s}$, (c) $|{\mathcal{K}}|$ versus $\unicode[STIX]{x1D703}$, (d) $|{\mathcal{K}}|$ versus $c_{s}$.

The receptivity coefficient is defined by (4.72). We calculate first $N_{u}$. Rather than evaluating (4.70a,b) using the Airy functions, it is more convenient to solve (5.1) numerically. For definitiveness, the efficiency function $|\unicode[STIX]{x1D6EC}|$ was computed with $P_{I}=1$, and then the receptivity coefficient $|{\mathcal{K}}|$ is obtained from (4.72). Figures 9(a) and 9(b) display the dependence of the efficiency function $|\unicode[STIX]{x1D6EC}|$ on the incident angle $\unicode[STIX]{x1D703}$ and phase speed $c_{s}$ of the acoustic wave for a selection of $\unicode[STIX]{x1D6FA}$ at $M=2.0$ and $R=9.05\times 10^{6}$ with $\widehat{\unicode[STIX]{x1D6FD}}_{s}=2.0$ and $\widehat{\unicode[STIX]{x1D6FD}}_{w}=0$ (three-dimensional acoustic wave and two-dimensional roughness). As illustrated, $|\unicode[STIX]{x1D6EC}|$ exhibits a sharp peak at a particular incident angle close, but not equal, to $\cos ^{-1}(1/M)$, or correspondingly at a small value of $c_{s}$. Since $c_{s}$ decreases with the incident angle $\unicode[STIX]{x1D703}$, the axis of $c_{s}$ is in the direction from larger values to small ones so that the two sets of curves show the same trend and look similar. As $\unicode[STIX]{x1D6FA}$ is increased, the incident angle for the maximum efficiency function decreases and the peak becomes broader. In order to compare the results given by the asymptotic theory and DNS in the next section, we calculated the receptivity coefficient ${\mathcal{K}}$ for $h^{\ast }/\unicode[STIX]{x1D6FF}^{\ast }=0.02$ and $\widehat{d}=\widehat{\unicode[STIX]{x1D706}}_{T}$. Figures 9(c) and 9(d) show that $|{\mathcal{K}}|$ varies with the incident angle and phase speed of the acoustic wave in a similar manner as $|\unicode[STIX]{x1D6EC}|$ does. However, the peak position is shifted. This is due to the fact that, for a fixed $\unicode[STIX]{x1D6FA}$, an increase of the incident angle $\unicode[STIX]{x1D703}$ leads to a larger $\widehat{\unicode[STIX]{x1D6FC}}_{s}$ or $\widehat{\unicode[STIX]{x1D6FC}}_{s}-\widehat{\unicode[STIX]{x1D6FC}}_{T}$. Since $\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})\propto \widehat{d}\propto (\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})^{-1}$, $\widehat{F}(\widehat{\unicode[STIX]{x1D6FC}}_{T}-\widehat{\unicode[STIX]{x1D6FC}}_{s})$ decreases with $\unicode[STIX]{x1D703}$, and therefore the maximum of $|{\mathcal{K}}|$ is shifted slightly to a smaller $\unicode[STIX]{x1D703}$ compared with the maximum of $|\unicode[STIX]{x1D6EC}|$. The peak receptivity coefficient increases slightly with $\unicode[STIX]{x1D6FA}$.

Figures 10(a) and 10(b) show how $|\unicode[STIX]{x1D6EC}|$ varies with $\unicode[STIX]{x1D703}$ and $c_{s}$ for a selection of $\unicode[STIX]{x1D6FA}$ with the same parameter values as those in figures 9(a) and 9(b) except that $\widehat{\unicode[STIX]{x1D6FD}}_{s}=0$ and $\widehat{\unicode[STIX]{x1D6FD}}_{w}=2.0$ (planar acoustic waves and three-dimensional roughness). The variation of the efficiency function with $\unicode[STIX]{x1D703}$ or $c_{s}$ exhibits a similar trend.

Figure 10.

The efficiency function $|\unicode[STIX]{x1D6EC}|$ for $\widehat{\unicode[STIX]{x1D6FD}}_{s}=0$ and $\widehat{\unicode[STIX]{x1D6FD}}_{w}=2.0$ at $M=2$ and $R=9.05\times 10^{6}$. (a) $|\unicode[STIX]{x1D6EC}|$ versus $\unicode[STIX]{x1D703}$, (b) $|\unicode[STIX]{x1D6EC}|$ versus $c_{s}$.

Figure 11.

The receptivity coefficient $|{\mathcal{K}}|$ and the amplitude of the acoustic wave $|\widetilde{u}|_{max}$ versus $\unicode[STIX]{x1D703}$ for (a) different spanwise numbers $\widehat{\unicode[STIX]{x1D6FD}}_{s}$ with $M=2.0$, and (b) different Mach numbers $M$ with $\widehat{\unicode[STIX]{x1D6FD}}_{s}=2.0$. The other parameter values are $\unicode[STIX]{x1D6FA}=1.0$, $\widehat{\unicode[STIX]{x1D6FD}}_{w}=0$ and $R=9.05\times 10^{6}$.

Figures 11(a) and 11(b) display $|{\mathcal{K}}|$ for different $\widehat{\unicode[STIX]{x1D6FD}}_{s}$ and $M$. The features revealed by figure 9(c,d) persist. An increase of the spanwise wavenumber or Mach number leads to weaker or stronger receptivity, respectively. It is interesting to note that $|{\mathcal{K}}|$ and $|\widetilde{u}|_{max}$ exhibit very similar dependence on $\unicode[STIX]{x1D703}$, suggesting that strong receptivity is due to large $|\widetilde{u}|_{max}$, which is associated with the ‘sound board’ effect of the boundary layer.