2.1 Triple-deck scalings

The streamwise length scale of the roughness is assumed to be comparable with the characteristic wavelength of lower-branch T–S waves, which is of $O(R^{-3/8}L)$ . The mean flow induced by the roughness and T–S waves are both described by the standard triple-deck structure (Smith Reference Smith1973, Reference Smith1979, Reference Smith1989), and so is the scattering of T–S waves (Wu & Hogg Reference Wu and Hogg2006). It is convenient to introduce a small parameter

(2.2) $$\begin{eqnarray}{\it\epsilon}=R^{-1/8},\end{eqnarray}$$

and the rescaled streamwise coordinate (Stewartson Reference Stewartson1974)

(2.3) $$\begin{eqnarray}X={\it\lambda}^{5/4}(1-M^{2})^{3/8}C^{-3/8}(T_{w}/T_{\infty })^{-3/2}x^{\ast }/({\it\epsilon}^{3}L),\end{eqnarray}$$

where superscript ‘ $\ast$ ’ and subscript ‘ $\infty$ ’ indicate dimensional and free-stream quantities, respectively, ${\it\lambda}=0.33206$ is a constant associated with the wall shear of the Blasius profile, $C$ the constant in the Chapman viscosity law and $T_{w}/T_{\infty }=1+(1/2)({\it\gamma}-1)M^{2}$ with ${\it\gamma}$ being the ratio of the specific heats. T–S waves have the characteristic time scale of $O({\it\epsilon}^{2}L/U_{\infty })$ , or equivalently frequency of $O({\it\epsilon}^{-2}U_{\infty }/L)$ (Smith Reference Smith1979, Reference Smith1989), and thus we introduce the rescaled time variable

(2.4) $$\begin{eqnarray}T={\it\lambda}^{3/2}(1-M^{2})^{1/4}C^{-1/4}(T_{w}/T_{\infty })^{-1}U_{\infty }t^{\ast }/({\it\epsilon}^{2}L),\end{eqnarray}$$

and correspondingly the normalized frequency

(2.5) $$\begin{eqnarray}{\it\omega}={\it\lambda}^{-3/2}(1-M^{2})^{-1/4}C^{1/4}(T_{w}/T_{\infty })L{\it\epsilon}^{2}{\it\omega}^{\ast }/U_{\infty }.\end{eqnarray}$$

Among the upper, main and lower decks, the focus will be on the last, which has a thickness of $O({\it\epsilon}^{5}L)$ . Hence, the local transverse coordinate, suitably renormalized, is

(2.6) $$\begin{eqnarray}Y_{1}={\it\lambda}^{3/4}(1-M^{2})^{1/8}C^{-5/8}(T_{w}/T_{\infty })^{-3/2}y^{\ast }/({\it\epsilon}^{5}L),\end{eqnarray}$$

and the velocity and pressure can be written as (Stewartson Reference Stewartson1974)

(2.7) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}u={\it\lambda}^{-1/4}(1-M^{2})^{1/8}C^{-1/8}(T_{w}/T_{\infty })^{-1/2}u^{\ast }/({\it\epsilon}U_{\infty }),\\ v_{1}={\it\lambda}^{-3/4}(1-M^{2})^{-1/8}C^{-3/8}(T_{w}/T_{\infty })^{-1/2}v^{\ast }/({\it\epsilon}^{3}U_{\infty }),\\ p={\it\lambda}^{-1/2}(1-M^{2})^{1/4}C^{-1/4}(T_{w}/T_{\infty })^{-1}(p^{\ast }-p_{\infty })/({\it\epsilon}^{2}{\it\rho}_{\infty }U_{\infty }^{2}).\end{array}\right\}\end{eqnarray}$$

The surface roughness is assumed to have a height of $O(R^{-5/8}L)$ or smaller, and a width of $O(R^{-3/8}L)$ , as was stated earlier. In the coordinate system $(X,Y_{1})$ , the shape of the roughness is specified as

(2.8) $$\begin{eqnarray}Y_{1}=F(X).\end{eqnarray}$$

On the flat portions upstream and downstream of the roughness element, the steady flow is the Blasius profile, which is approximated in the lower deck as

(2.9) $$\begin{eqnarray}(U_{B},V_{B})=(Y_{1},0).\end{eqnarray}$$

In the vicinity of the roughness, the solution can be decomposed as

(2.10) $$\begin{eqnarray}\displaystyle (u(X,Y_{1}),v_{1}(X,Y_{1}),p(X)) & = & \displaystyle (U(X,Y_{1}),V_{1}(X,Y_{1}),P(X))\nonumber\\ \displaystyle & & \displaystyle +\,\tilde{{\it\epsilon}}(\tilde{u} (X,Y_{1}),\tilde{v}_{1}(X,Y_{1}),\tilde{p}(X))\text{e}^{-\text{i}{\it\omega}T}+\text{c.c.},\qquad\end{eqnarray}$$

where the first part of the right-hand side represents the steady mean flow induced by the roughness, while the second part denotes the unsteady (time periodic) perturbation associated with scattering of the oncoming T–S wave. We assume that the T–S wave has a small amplitude $\tilde{{\it\epsilon}}\ll 1$ , which implies that the entire unsteady perturbation is of that order too for $X=O(1)$ .

On the wall surface, both the steady and unsteady flows satisfy the no-slip condition,

(2.11a,b ) $$\begin{eqnarray}U=V_{1}=0,\quad \tilde{u} =\tilde{v}_{1}=0\quad \text{at}~Y_{1}=F(X).\end{eqnarray}$$

In order to solve the flow fields in a rectangular domain, the Prandtl transformation,

(2.12a,b ) $$\begin{eqnarray}Y=Y_{1}(X)-F(X),\quad (v,V,\tilde{v})=(v_{1},V_{1},\tilde{v}_{1})-(u,U,\tilde{u} )\frac{\text{d}F}{\text{d}X},\end{eqnarray}$$

is introduced.

2.2 Governing equations for the mean flow

By inserting (2.7) into the N–S equations and taking into account the scalings (2.3) and (2.6), the decomposition (2.10) as well as the Prandtl transformation (2.12), it can be shown that the steady flow in the lower deck satisfies the classical steady boundary-layer equations

(2.13a,b ) $$\begin{eqnarray}U_{X}+V_{Y}=0,\quad UU_{X}+VU_{Y}=-P_{X}+U_{YY}.\end{eqnarray}$$

They are subject to the boundary and matching conditions (Smith Reference Smith1973)

(2.14a,b ) $$\begin{eqnarray}U=V=0\quad \text{at}~Y=0;\quad U\rightarrow Y+A+F\quad \text{as}~Y\rightarrow \infty ;\end{eqnarray}$$

(2.15a,b ) $$\begin{eqnarray}A\rightarrow 0,\quad U\rightarrow Y\quad \text{as}~X\rightarrow \pm \infty .\end{eqnarray}$$

In the above, $A$ is what is referred to as the displacement function and $P$ is the pressure generated by the viscous motion. They are related by the so-called pressure–displacement (P–D) law

(2.16) $$\begin{eqnarray}P(X)=\frac{1}{{\rm\pi}}\int _{-\infty }^{\infty }\frac{A^{\prime }({\it\xi})\,\text{d}{\it\xi}}{X-{\it\xi}},\end{eqnarray}$$

which is derived by analysing the flow in the main and upper decks (Smith Reference Smith1973). Equation (2.15) refers to the fact that the steady flow sufficiently upstream and downstream remains unperturbed by a localized roughness.

In the downstream limit ( $X\gg 1$ ), the steady distortion spreads to a region corresponding to (Smith Reference Smith1973)

(2.17) $$\begin{eqnarray}{\it\eta}_{B}=Y/X^{1/3}=O(1),\end{eqnarray}$$

and decays algebraically. The solution takes the form

(2.18a,b ) $$\begin{eqnarray}U=X^{1/3}{\it\eta}_{B}+X^{-7/3}c_{0}G^{\prime }({\it\eta}_{B})+\cdots ,\quad V=X^{-3}c_{0}\left(2G+{\textstyle \frac{1}{3}}{\it\eta}_{B}G^{\prime }\right)+\cdots ,\end{eqnarray}$$

with $G$ being governed by (4.12) of Smith (Reference Smith1973) and

(2.19) $$\begin{eqnarray}c_{0}=\int _{-\infty }^{\infty }F(X)\,\text{d}X=\sqrt{{\rm\pi}}h\,\text{d}.\end{eqnarray}$$

For the present study, it suffices to note that as ${\it\eta}_{B}\rightarrow 0$ , $G\rightarrow (1/2)G^{\prime \prime }(0){\it\eta}_{B}^{2}$ and $V\rightarrow (4/3){\it\lambda}_{w}X^{-3}{\it\eta}_{B}^{2}$ , where

(2.20a,b ) $$\begin{eqnarray}G^{\prime \prime }(0)=5\times 3^{-8/3}/(-\text{Ai}^{\prime }(0))\approx 0.1579,\quad {\it\lambda}_{w}=G^{\prime \prime }(0)c_{0}.\end{eqnarray}$$

It follows that in the lower deck, where $Y=O(1)$ (i.e. ${\it\eta}_{B}=O(X^{-1/3})$ ), the steady flow can be expressed as

(2.21a,b ) $$\begin{eqnarray}U=Y+{\it\lambda}_{w}X^{-8/3}Y,\quad V={\textstyle \frac{4}{3}}{\it\lambda}_{w}X^{-11/3}Y^{2}.\end{eqnarray}$$

2.3 Governing equations for the scattered perturbation field

Following a similar procedure to that for the steady flow, it can be shown that in terms of the shifted coordinate $Y$ , the unsteady perturbation in the lower deck satisfies the unsteady linearized boundary-layer equations,

(2.22) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{u} _{X}+\tilde{v}_{Y}=0, & \displaystyle\end{eqnarray}$$

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}{\it\omega}\tilde{u} +U\tilde{u} _{X}+U_{Y}\tilde{v}+U_{X}\tilde{u} +V\tilde{u} _{Y}+\tilde{p}_{X}-\tilde{u} _{YY}=0. & \displaystyle\end{eqnarray}$$

Note that as in Bodonyi et al. (Reference Bodonyi, Welch, Duck and Tadjfar1989), the coefficients of the equations governing scattering depend on the streamwise as well as the transverse coordinates (cf. Wu & Hogg Reference Wu and Hogg2006). The boundary and matching conditions in the wall-normal direction are

(2.24a,b ) $$\begin{eqnarray}\tilde{u} =\tilde{v}=0\quad \text{at}~Y=0;\quad \tilde{u} \rightarrow \tilde{A}(X)\quad \text{as}~Y\rightarrow \infty ,\end{eqnarray}$$

where again the displacement function $\tilde{A}$ and the pressure $\tilde{p}$ of the unsteady perturbation satisfy the P–D relation

(2.25) $$\begin{eqnarray}\tilde{p}(X)=\frac{1}{{\rm\pi}}\int _{-\infty }^{\infty }\frac{\tilde{A}^{\prime }({\it\xi})\,\text{d}{\it\xi}}{X-{\it\xi}}.\end{eqnarray}$$

At upstream locations, where the base flow is the undistorted Blasius profile (2.9), the perturbation corresponds to the oncoming T–S wave, and it has the behaviour

(2.26) $$\begin{eqnarray}(\tilde{u} ,\tilde{v},\tilde{p},\tilde{A})\rightarrow (\hat{u} (Y),\hat{v}(Y),\hat{p},\hat{A})\text{e}^{\text{i}{\it\alpha}X}\quad \text{as}~X\rightarrow -\infty ,\end{eqnarray}$$

where ${\it\alpha}$ is the rescaled wavenumber of the incident T–S mode, related to the dimensional wavenumber ${\it\alpha}^{\ast }$ via

(2.27) $$\begin{eqnarray}{\it\alpha}={\it\lambda}^{-5/4}(1-M^{2})^{-3/8}C^{3/8}(T_{w}/T_{\infty })^{3/2}{\it\epsilon}^{3}{\it\alpha}^{\ast }L,\end{eqnarray}$$

and ( $\hat{u} ,\hat{v},\hat{p},\hat{A}$ ) is the eigenfunction, among which one may take $\hat{A}=1$ . The remaining quantities are found as

(2.28a-c ) $$\begin{eqnarray}\hat{u} =\hat{q}\int _{{\it\eta}_{0}}^{{\it\eta}}\text{Ai}({\it\eta})\,\text{d}{\it\eta},\quad \hat{v}=-(\text{i}{\it\alpha})^{2/3}\hat{q}\int _{{\it\eta}_{0}}^{{\it\eta}}({\it\eta}-{\it\zeta})\text{Ai}({\it\zeta})\,\text{d}{\it\zeta},\quad \hat{p}={\it\alpha},\end{eqnarray}$$

where

(2.29a,b ) $$\begin{eqnarray}{\it\eta}=(\text{i}{\it\alpha})^{1/3}Y-\text{i}{\it\omega}(\text{i}{\it\alpha})^{-2/3},\quad \hat{q}=\left[\int _{{\it\eta}_{0}}^{\infty }\text{Ai}({\it\eta})\,\text{d}{\it\eta}\right]^{-1}.\end{eqnarray}$$

The frequency ${\it\omega}$ and wavenumber ${\it\alpha}$ satisfy the dispersion relation (Smith Reference Smith1979, Reference Smith1989)

(2.30) $$\begin{eqnarray}{\it\Delta}({\it\omega},{\it\alpha})\equiv (\text{i}{\it\alpha})^{1/3}{\it\alpha}\int _{{\it\eta}_{0}}^{\infty }\text{Ai}({\it\eta})\,\text{d}{\it\eta}-\text{Ai}^{\prime }({\it\eta}_{0})=0\quad ({\it\eta}_{0}\equiv -\text{i}{\it\omega}(\text{i}{\it\alpha})^{-2/3}).\end{eqnarray}$$

As the T–S wave propagates through the vicinity of the roughness, it is scattered by the roughness-induced mean flow and undergoes sudden change. The perturbation then relaxes back to a local T–S mode far downstream, which is referred to as the transmitted wave. It has the same wavenumber as that of the incident wave but acquires a different amplitude (see figure 1). It follows that $(\tilde{u} ,\tilde{v},\tilde{p},\tilde{A})$ takes the form

(2.31) $$\begin{eqnarray}\displaystyle (\tilde{u} ,\tilde{v},\tilde{p},\tilde{A}) & \rightarrow & \displaystyle \mathscr{T}(\hat{u} (Y),\hat{v}(Y),\hat{p},\hat{A})\text{e}^{\text{i}{\it\alpha}X}+\mathscr{T}{\it\lambda}_{w}\{\!X^{-5/3}\mathscr{C}(\hat{u} (Y),\hat{v}(Y),\hat{p},\hat{A})\nonumber\\ \displaystyle & & \displaystyle +\,X^{-8/3}(\hat{u} _{c}(Y),\hat{v}_{c}(Y),\hat{p}_{c},\hat{A}_{c})\!\}\text{e}^{\text{i}{\it\alpha}X}\quad \text{as}~X\rightarrow \infty ,\end{eqnarray}$$

where the constant $\mathscr{T}$ is defined as the transmission coefficient, representing the ratio of the amplitude of the transmitted T–S wave to that of the incident wave ( $\hat{A}=1$ ). The transmission coefficient $\mathscr{T}$ depends on the frequency of the incident T–S wave, as well as on the shape and height of the local roughness. The second and third terms in (2.31) arise due to the transmitted T–S wave being rescattered by the wake. A forcing of the form $\mathscr{T}X^{-8/3}\text{e}^{\text{i}{\it\alpha}X}$ is generated as can be deduced by noting (2.21). Because the forcing is in resonance with the T–S mode, it drives a larger correction, $\mathscr{T}X^{-5/3}\text{e}^{\text{i}{\it\alpha}X}$ .

The constant $\mathscr{C}$ in (2.31) can be determined, by considering $(\hat{u} _{c},\hat{v}_{c},\hat{p}_{c},\hat{A}_{c})$ , which are found, by substituting (2.21) and (2.31) into (2.22) and (2.23), to satisfy the equations

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}{\it\alpha}\hat{u} _{c}+\hat{v}_{c}^{\prime }(Y)={\textstyle \frac{5}{3}}\mathscr{C}\hat{u} , & \displaystyle\end{eqnarray}$$

(2.33) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}(-{\it\omega}+{\it\alpha}Y)\hat{u} _{c}+\hat{v}_{c}-\hat{u} _{c,YY}=-\text{i}{\it\alpha}\hat{p}_{c}+{\textstyle \frac{5}{3}}\mathscr{C}(\hat{p}+Y\hat{u} )-(\text{i}{\it\alpha}Y\hat{u} +\hat{v}). & \displaystyle\end{eqnarray}$$

These equations reduce to a single equation

(2.34) $$\begin{eqnarray}[\partial ^{2}/\partial Y^{2}-\text{i}({\it\alpha}Y-{\it\omega})]\hat{u} _{c,Y}=(\text{i}{\it\alpha}-{\textstyle \frac{5}{3}}\mathscr{C})Y\hat{u} _{Y}.\end{eqnarray}$$

The solution for $\hat{u} _{c}$ can be expressed as

(2.35) $$\begin{eqnarray}\hat{u} _{c}=\frac{1}{3}\hat{q}\left(1+\frac{5}{3}\text{i}{\it\alpha}^{-1}\mathscr{C}\right)[({\it\eta}-3{\it\eta}_{0})\text{Ai}({\it\eta})+2{\it\eta}_{0}\text{Ai}({\it\eta}_{0})]+q_{c}\int _{{\it\eta}_{0}}^{{\it\eta}}\text{Ai}({\it\eta})\,\text{d}{\it\eta}.\end{eqnarray}$$

The boundary and matching conditions yield

(2.36) $$\begin{eqnarray}\displaystyle & \displaystyle {\textstyle \frac{2}{3}}(\text{i}{\it\alpha})^{2/3}\hat{q}(1+{\textstyle \frac{5}{3}}\text{i}{\it\alpha}^{-1}\mathscr{C})[\text{Ai}^{\prime }({\it\eta}_{0})-{\it\eta}_{0}^{2}\text{Ai}({\it\eta}_{0})]+(\text{i}{\it\alpha})^{2/3}\text{Ai}^{\prime }({\it\eta}_{0})q_{c}=\text{i}{\it\alpha}\hat{p}_{c}-{\textstyle \frac{5}{3}}\mathscr{C}\hat{p},\qquad & \displaystyle\end{eqnarray}$$

(2.37) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{2}{3}\hat{q}\left(1+\frac{5}{3}\text{i}{\it\alpha}^{-1}\mathscr{C}\right){\it\eta}_{0}\text{Ai}({\it\eta}_{0})+\int _{{\it\eta}_{0}}^{\infty }\text{Ai}({\it\eta})\,\text{d}{\it\eta}\,q_{c}=\hat{A}_{c}.\qquad & \displaystyle\end{eqnarray}$$

The relation between $\hat{p}_{c}$ and $\hat{A}_{c}$ follows from use of (A 21) for the displacement asymptote $A_{\infty }\sim \mathscr{T}{\it\lambda}_{w}(X^{-5/3}\mathscr{C}\hat{A}+X^{-8/3}\hat{A}_{c})\text{e}^{\text{i}{\it\alpha}X}$ , which corresponds to the second and third terms in (2.31). At $O(X^{-5/3})$ , we obtain $\hat{p}={\it\alpha}\hat{A}$ , confirming the known P–D relation for the T–S wave, and at $O(X^{-8/3})$ ,

(2.38) $$\begin{eqnarray}\hat{p}_{c}={\it\alpha}\hat{A}_{c}+{\textstyle \frac{5}{3}}\text{i}\mathscr{C}.\end{eqnarray}$$

The same relation can also be derived by considering the pressure in the upper deck.

From (2.36)–(2.38), we find that

(2.39) $$\begin{eqnarray}\mathscr{C}=-\frac{3}{5}\text{i}{\it\alpha}\frac{\text{Ai}^{\prime }({\it\eta}_{0})-{\it\eta}_{0}^{2}\text{Ai}({\it\eta}_{0})-{\it\alpha}(\text{i}{\it\alpha})^{1/3}{\it\eta}_{0}\text{Ai}({\it\eta}_{0})}{2\text{Ai}^{\prime }({\it\eta}_{0})+{\it\eta}_{0}^{2}\text{Ai}({\it\eta}_{0})+{\it\alpha}(\text{i}{\it\alpha})^{1/3}{\it\eta}_{0}\text{Ai}({\it\eta}_{0})},\end{eqnarray}$$

where use has been made of (2.30).

The analysis above shows that the wake of the mean-flow distortion contributes an $O(X^{-5/3}\mathscr{T}\text{e}^{\text{i}{\it\alpha}X})$ perturbation in the far-field downstream. This perturbation itself grows exponentially but remains uniformly smaller than the transmitted T–S wave $\mathscr{T}\text{e}^{\text{i}{\it\alpha}X}$ . It may therefore be inferred that the wake may be cutoff for sufficiently large $X$ without affecting the transmission coefficient to leading order, an assertion also consistent with the expectation on the physical ground that the decaying wake should not affect the leading-order scattering.

For the supercritical case of our interest, the growth rate $-{\it\alpha}_{i}>0$ and hence the displacement function $\tilde{A}$ tends to infinity as $X\rightarrow \infty$ (see (2.31)), which makes the integral in the P–D law (2.25) divergent, as was pointed out by Bodonyi & Duck (Reference Bodonyi and Duck1988). The integral must be interpreted as the finite part in the sense of Hadamard, which requires careful numerical evaluation; see appendix A. Two approaches will be adopted to overcome this difficult thereby establishing the relationship between $\tilde{p}$ and  $\tilde{A}$ .

The first approach involves recasting the P–D law into a modified version by subtracting out the unbounded part of $\tilde{A}$ corresponding to the transmitted wave, and treating its contribution to the pressure analytically. This amounts to introducing a new displacement function

(2.40) $$\begin{eqnarray}\tilde{A}_{1}(X)=\tilde{A}(X)-\mathscr{T}\hat{A}\text{e}^{\text{i}{\it\alpha}X},\end{eqnarray}$$

which decays sufficiently fast as $X\rightarrow \pm \infty$ . The P–D law (2.25) then becomes

(2.41) $$\begin{eqnarray}\tilde{p}=\frac{1}{{\rm\pi}}\int _{-\infty }^{\infty }\frac{\tilde{A}_{1}^{\prime }({\it\xi})}{X-{\it\xi}}\,\text{d}{\it\xi}+\mathscr{T}{\it\alpha}\hat{A}\text{e}^{\text{i}{\it\alpha}X}.\end{eqnarray}$$

Since (2.31) implies that $\tilde{A}_{1}({\it\xi})\rightarrow \mathscr{T}\mathscr{C}{\it\lambda}_{w}{\it\xi}^{-5/3}\text{e}^{\text{i}{\it\alpha}{\it\xi}}$ as ${\it\xi}\rightarrow \infty$ , the integral is taken to be the finite part as defined in appendix A by (A 3), or more precisely by (A 14), which can be approximated asymptotically by (A 5) or equivalently (A 12). The finite part, or its asymptotic approximation (A 5), is to be evaluated numerically by a suitable quadrature in the next section. The second term on the right-hand side of (2.41) represents the exponentially growing transmitted T–S wave. The scattering is described by the boundary-value problem consisting of the linearized boundary-layer equations (2.22) and (2.23), the boundary conditions (2.24), and the upstream and downstream matching conditions (2.26) and (2.31), supplemented by the P–D relation (2.41). In this approach, the transmission coefficient $\mathscr{T}$ is taken as an explicit unknown quantity to be found directly. This is an attractive feature as $\mathscr{T}$ is the central concept reflecting the intrinsic physics in the present theory. In the next section, we will show that $\mathscr{T}$ appears in the discretized system as an eigenvalue.

Instead of treating the P–D law as in (2.41), an alternative approach is to solve numerically the (Laplace) equation governing the displacement-induced pressure in the upper deck simultaneously with the linearized boundary-layer equations controlling the viscous motion in the lower deck. This method was suggested by Bodonyi & Duck (Reference Bodonyi and Duck1988), and the details are given in appendix C.

3.1 Grids and the mean flow

Both the steady and unsteady triple-deck systems, governing the mean-flow distortion and the scattering, respectively, are solved numerically in a truncated rectangular domain, $X_{0}\leqslant X\leqslant X_{I}$ and $Y_{0}\leqslant Y\leqslant Y_{J}$ . A uniform mesh is used in the $X$ direction with the mesh size ${\it\Delta}_{X}=(X_{I}-X_{0})/I$ , while a non-uniform one is used in the $Y$ direction with the width of each interval ${\it\Delta}_{Y_{j}}=Y_{j}-Y_{j-1}$ . The resulting grid points are denoted by $(X_{j},Y_{j})$ with $i\in [0,I]$ and $j\in [0,J]$ .

The steady triple-deck system, consisting of equations (2.13), the boundary and matching conditions (2.14), (2.15) and the P–D law (2.16), is solved numerically first using the method as described in El-Mistikawy (Reference El-Mistikawy1994, Reference El-Mistikawy2010). The details are omitted since the methodology is rather mature.

3.2 Discretization of the boundary-value problem governing scattering

The linear boundary-layer equations (2.22) and (2.23) is first recast, by introducing

(3.1) $$\begin{eqnarray}\tilde{w}\equiv \tilde{u} _{Y},\end{eqnarray}$$

into a first-order system, which is then discretized using a second-order finite-difference scheme centred at midpoints of the grid, $(i-1/2,j-1/2)$ . This leads to the algebraic system

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D63C}_{ij}^{\prime }\tilde{{\it\varphi}}_{i-1,j-1}+\unicode[STIX]{x1D63D}_{ij}^{\prime }\tilde{{\it\varphi}}_{i-1,j}+\unicode[STIX]{x1D63E}_{ij}^{\prime }\tilde{{\it\varphi}}_{i,j-1}+\unicode[STIX]{x1D63F}_{ij}^{\prime }\tilde{{\it\varphi}}_{i,j}+\boldsymbol{h}_{i}^{\prime }(\tilde{p}_{i}-\tilde{p}_{i-1})=0,\end{eqnarray}$$

for $i\in [1,I]$ and $j\in [1,J]$ , where $\tilde{{\it\varphi}}_{i,j}=[\tilde{u} _{i,j},\tilde{v}_{i,j},\tilde{w}_{i,j}]^{\text{T}}$ , $\boldsymbol{h}_{i}^{\prime }=[0,0,1/{\it\Delta}_{X}]^{\text{T}}$ , and the expressions for $\unicode[STIX]{x1D63C}_{ij}^{\prime }$ , $\unicode[STIX]{x1D63D}_{ij}^{\prime }$ , $\unicode[STIX]{x1D63E}_{ij}^{\prime }$ and $\unicode[STIX]{x1D63F}_{ij}^{\prime }$ are given in appendix B.

By using the ‘modified’ trapezoidal rule for the integral, the P–D law (2.41) is discretized as

(3.3) $$\begin{eqnarray}\tilde{p}(X_{i})=\mathop{\sum }_{i^{\prime }=-\unicode[STIX]{x1D610}_{1}}^{\unicode[STIX]{x1D610}_{2}}{\it\beta}_{ii^{\prime }}\tilde{A}_{1,i^{\prime }}+\mathscr{T}{\it\alpha}\hat{A}\text{e}^{\text{i}{\it\alpha}X_{i}}=\mathop{\sum }_{i^{\prime }=-\unicode[STIX]{x1D610}_{1}}^{\unicode[STIX]{x1D610}_{2}}{\it\beta}_{ii^{\prime }}(\tilde{A}_{i^{\prime }}-\mathscr{T}\tilde{A}_{0}\text{e}^{\text{i}{\it\alpha}(X_{i^{\prime }}-X_{0})})+\mathscr{T}\tilde{A}_{0}{\it\alpha}\text{e}^{\text{i}{\it\alpha}(X_{i}-X_{0})},\end{eqnarray}$$

where $\unicode[STIX]{x1D610}_{2}>\unicode[STIX]{x1D644}$ and we have put

(3.4) $$\begin{eqnarray}{\it\beta}_{ii^{\prime }}=\frac{1}{{\rm\pi}}\left[\left(X_{i}-X_{i^{\prime }}+\frac{{\it\Delta}_{X}}{2}\right)^{-1}-\left(X_{i}-X_{i^{\prime }+1}+\frac{{\it\Delta}_{X}}{2}\right)^{-1}\right].\end{eqnarray}$$

The integral is calculated in an extended domain $(-\unicode[STIX]{x1D610}_{1}\leqslant i^{\prime }\leqslant \unicode[STIX]{x1D610}_{2})$ larger than that in which the boundary-layer equations are solved. The values of $\tilde{A}$ outside the latter region, are approximated by the upstream and downstream asymptotes, respectively, namely

(3.5) $$\begin{eqnarray}\tilde{A}_{i^{\prime }}=\left\{\begin{array}{@{}ll@{}}\displaystyle \tilde{A}_{0}\text{e}^{\text{i}{\it\alpha}(X_{i^{\prime }}-X_{0})} & \text{for}~i^{\prime }<0,\\ \displaystyle [(1+{\it\lambda}_{w}\mathscr{C}X_{i^{\prime }}^{-5/3})/(1+{\it\lambda}_{w}\mathscr{C}X_{I}^{-5/3})]\tilde{A}_{I}\text{e}^{\text{i}{\it\alpha}(X_{i^{\prime }}-X_{I})} & \text{for}~i^{\prime }>I.\end{array}\right.\end{eqnarray}$$

It should be pointed out that if viewed purely as a quadrature for the integral in the Hilbert transform, the ‘modified’ trapezoidal rule used in (3.3) differs from the conventional one in that the weighting factor $1/2$ at the end points is absent. However, the resulting expression is actually an appropriate discretization of (A 5) with the second term included rather than just that of $J_{N}$ . The above treatment is equivalent to approximating the finite part by its leading-order asymptotic representation (A 5), and then approximating the integral $J_{N}$ in it using the conventional trapezoidal formula. In appendix A, for two given functions which have the same large- $X$ asymptotes as those of $\tilde{A}$ and $\tilde{A}_{1}$ respectively, we verify this equivalence numerically and demonstrate further that (3.3) and (A 5) both give, for a suitably large $N$ , the same result as (A 14), which is known analytically to be independent of $N$ . The above conclusions were verified also by further calculations for $\tilde{A}$ . The results indicate that the finite part is defined/approximated properly and can be computed to satisfactory accuracy by using a suitably large $N$ . Provided chosen appropriately, the value of $N$ , which is introduced to facilitate the definition of the finite part, does not cause any arbitrariness.

The boundary and matching conditions in (2.24) are expressed as

(3.6a-c ) $$\begin{eqnarray}\tilde{u} _{i,0}=\tilde{v}_{i,0}=0,\quad \tilde{u} _{i,J}=\tilde{A}_{i},\quad \tilde{w}_{i,J}=0.\end{eqnarray}$$

The far downstream behaviour (2.31) implies that

(3.7) $$\begin{eqnarray}(\tilde{u} _{I,j},\tilde{v}_{I,j},\tilde{w}_{I,j},\tilde{p}_{I},\tilde{A}_{I})=\mathscr{T}(1+{\it\lambda}_{w}\mathscr{C}X_{I}^{-5/3}){\it\chi}_{0}(\tilde{u} _{0,j},\tilde{v}_{0,j},\tilde{w}_{0,j},\tilde{p}_{0},\tilde{A}_{0}),\end{eqnarray}$$

for $j=0,1,2,\ldots ,J$ , where ${\it\chi}_{0}=\text{e}^{\text{i}{\it\alpha}(X_{I}-X_{0})}$ . This relation links the perturbation at the exit of the domain to that at the inlet, and plays a role similar to that of a periodic boundary condition in the streamwise direction. Note that if we set $\mathscr{C}=0$ in (3.5) and (3.7), i.e. cutoff the wake of the mean-flow distortion, the Hilbert transform in (2.41) becomes convergent and the finite part interpretation for it is not needed.

Now place all unknown dependent variables except $\mathscr{T}$ into a high-dimensional vector

(3.8) $$\begin{eqnarray}\tilde{{\bf\phi}}=[\tilde{u} _{00},\tilde{v}_{00},\tilde{w}_{00},\tilde{u} _{01},\ldots \ldots ,\tilde{u} _{IJ},\tilde{v}_{IJ},\tilde{w}_{IJ},\tilde{p}_{0},\tilde{p}_{1},\ldots ,\tilde{p}_{I}]^{\text{T}}.\end{eqnarray}$$

The discrete equations (3.2)–(3.7) can be written as a generalized linear eigenvalue problem

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D63C}\tilde{{\bf\phi}}=\mathscr{T}\unicode[STIX]{x1D63D}\tilde{{\bf\phi}},\end{eqnarray}$$

where $\unicode[STIX]{x1D63C}$ and $\unicode[STIX]{x1D63D}$ are known matrices with their expressions being presented in appendix B. The eigenvalue problem (3.9) can be solved by the Arnoldi iterative method by using the ARPACK library in MATLAB.

The present eigenvalue problem is a global one in the sense that the mean flow is two-dimensional. It is different from conventional local linear stability problems, in which disturbances are assumed to be of the normal mode form in the streamwise direction. With transmission coefficient $\mathscr{T}$ appearing as the eigenvalue, the present eigenvalue formulation differs also from the so-called bi-global instability, in which a local spatial, or global temporal, growth rate appears as the eigenvalue (Theofilis Reference Theofilis2003).

3.3 Transmission coefficient and the $\text{e}^{N}$ -method for boundary-layer transition in the presence of roughness elements

The local scattering theory provides an appropriate framework to account for the effects of abrupt change in terms of a transmission coefficient, whereas traditional methods, such as LST and PSE, remain applicable in relatively flat portions of the boundary layer. The transmission coefficient can readily be combined or integrated with one of these approaches to provide a complete description of linear development of disturbances in the presence of roughness elements.

Suppose that in the absence of any roughness, a T–S wave with an amplitude $A_{0}$ at $X_{0}$ is amplified to $A_{0}\text{e}^{N_{0}}$ at $X$ , where $N_{0}$ is the $N$ -factor. Now if an isolated roughness is located between $X_{0}$ and $X$ , the T–S wave amplitude at $X$ becomes $A=|\mathscr{T}|A_{0}\text{e}^{N_{0}}$ instead so that $A/A_{0}=|\mathscr{T}|\text{e}^{N_{0}}$ . More generally, in the presence of $K$ number of isolated elements with the corresponding transmission coefficients $\mathscr{T}_{k}$ ( $k=1,2,\ldots ,K$ ), then

(3.10) $$\begin{eqnarray}A/A_{0}=\mathop{\prod }_{k=1}^{K}|\mathscr{T}_{k}|\text{e}^{N_{0}}.\end{eqnarray}$$

The accumulated amplification $A/A_{0}$ can be measured by an equivalent $N$ -factor $N$ , introduced by writing $A/A_{0}$ as $\text{e}^{N}$ . It follows that

(3.11) $$\begin{eqnarray}N\equiv \ln (A/A_{0})=N_{0}+\mathop{\sum }_{k=1}^{K}\ln |\mathscr{T}_{k}|,\end{eqnarray}$$

with the $N$ -factor increment

(3.12) $$\begin{eqnarray}{\rm\Delta}N=\mathop{\sum }_{k=1}^{K}\ln |\mathscr{T}_{k}|,\end{eqnarray}$$

characterizing the impact of all local roughness elements. The expression (3.12) generalises (1.2) to the case of multiple roughness elements. When all roughness elements are identical, the relations (3.11) and (3.12) simplify to

(3.13a,b ) $$\begin{eqnarray}N=N_{0}+K\ln |\mathscr{T}|,\quad {\rm\Delta}N=K\ln |\mathscr{T}|.\end{eqnarray}$$

In the absence of roughness, the $\text{e}^{N}$ -method asserts that transition occurs when $N=N_{c}$ , i.e. when the amplitude reaches the threshold $A_{0}\text{e}^{N_{c}}$ . With roughness elements being present, it is reasonable to assume that transition takes place when the disturbance acquires the same threshold, namely, $A_{0}\text{e}^{N}=A_{0}\text{e}^{N_{c}}$ . It follows, on noting (3.11), that

(3.14) $$\begin{eqnarray}N_{0}=N_{c}-\mathop{\sum }_{k=1}^{K}\ln |\mathscr{T}_{k}|,\end{eqnarray}$$

which means that the transition position in the presence of roughness elements may be predicted by a modified $\text{e}^{N}$ -method, which calculates the $N$ -factor for the smooth wall using LST, but the critical $N$ -factor must be adjusted to

(3.15) $$\begin{eqnarray}{\tilde{N}}_{c}=N_{c}-\mathop{\sum }_{k=1}^{K}\ln |\mathscr{T}_{k}|.\end{eqnarray}$$

For identical roughness elements, the above equation becomes

(3.16) $$\begin{eqnarray}{\tilde{N}}_{c}=N_{c}-K\ln |\mathscr{T}|.\end{eqnarray}$$

The present local scattering theory predicts the transmission coefficient and hence the $N$ -factor increment on the basis of first principles. Previously, ${\rm\Delta}N$ was extracted empirically from experimental data (e.g. Wang & Gaster Reference Wang and Gaster2005; Crouch et al. Reference Crouch, Kosorygin, Ng and Govindarajan2006). Theoretically, it was calculated by performing local linear stability analysis of distorted flows (e.g. Nayfeh et al. Reference Nayfeh, Ragab and Almaaitah1988; Cebeci & Egan Reference Cebeci and Egan1989; Masad & Iyer Reference Masad and Iyer1994; Perraud et al. Reference Perraud, Arnal, Seraudie and Tran2004) or by PSE (Wie & Malik Reference Wie and Malik1998). The validity of these approaches was discussed in the introduction, and will be examined farther below.

5.1 The mean-flow distortion induced by roughness and indentation

Calculations will be performed for a local roughness with a Gaussian shape

(5.1) $$\begin{eqnarray}F(X)=h\exp (-X^{2}/d^{2}),\end{eqnarray}$$

where $h$ is the height of the roughness, representing a hump when $h>0$ and an indentation when $h<0$ , and $d$ characterizes its width. A more precise or intuitive measure of the streamwise length scale ${\it\Delta}_{R}$ may be defined as the span of the region in which $F(X)\leqslant 0.1h$ , i.e. $\exp (-{\it\Delta}_{R}^{2}/d^{2})=0.1$ , which gives the relation

(5.2) $$\begin{eqnarray}{\it\Delta}_{R}=3.0349d.\end{eqnarray}$$

For the results to be presented in this subsection, computations are typically performed in the domain, $-5.0\leqslant X\leqslant 5.0$ and $0\leqslant Y\leqslant 7.5$ , with the number of grid points being $201\times 31$ . Resolution checks were carried by enlarging the computational domain and/or doubling the number of grid points.

Figure 2. The pressure $P(X)$ (a) and normalized surface shear stress $U_{Y}(X,0)$ (b) for a hump with $d=0.5$ and different heights $h=0.1$ (solid lines), 0.5 (dashed lines), 1.0 (dot-dashed lines), 2.0 (dot-dot-dashed lines). The thin line in (b) marks $U_{Y}(X,0)=0$ .

Figure 2 displays the streamwise distributions of the pressure $P(X)$ and the normalized surface shear stress $U_{Y}(X,0)$ , for different hump heights with a fixed width $d=0.5$ . In the vicinity of $X=0$ , a hump produces a favourable pressure gradient, followed by a region of an adverse pressure gradient. The distortion becomes greater as the height increases. For $h=2.0$ , the minimum value of the surface shear stress exhibits a slightly negative value near $X\approx 0.7$ , indicating an incipient separation zone induced by the adverse pressure gradient. Contours of $U_{Y}$ (shades) and streamlines are shown in figure 3. Curved streamlines reflect distortion of the mean flow by the hump, and a separation zone emerges for large hump heights ( $h\geqslant 2$ ).

Figure 3. Contours of $U_{Y}(X,Y_{1})$ and streamlines for a hump with $d=0.5$ and different heights: (a) $h=0.1$ , (b) $h=0.5$ , (c) $h=1.0$ , (d) $h=2.0$ .

Figure 4. Contours of $U_{Y}(X,Y_{1})$ and streamlines for an indentation with $d=0.5$ and different depths: (a) $h=-0.1$ , (b) $h=-0.5$ , (c) $h=-1.0$ , (d) $h=-2.0$ .

Figure 4 displays contours of $U_{Y}$ and streamlines for indentations of different depth. Similar to the case of humps, streamlines curve more significantly as the depth of indentation increases. For the largest depth, $h=-2.0$ , the adverse pressure gradient produces a separation zone.

5.2 Scattering of T–S waves and transmission coefficients

In order to specify appropriate oncoming T–S waves, we calculate first the eigenvalues from their dispersion relation (2.30). The real and imaginary parts of the wavenumber ${\it\alpha}$ are plotted in figure 5 for a range of frequencies. The frequencies for the lower-branch neutral and the most unstable T–S waves are found to be 2.3 and 7.25, respectively, with corresponding wavenumbers of 1.0 and 2.48. The greatest growth rate is 0.305. Of interest are unstable lower-branch T–S waves, and so the range of frequency to be studied is chosen to be $2.3\leqslant {\it\omega}\leqslant 10.0$ .

Figure 5. The dispersion property of oncoming T–S waves: (a) the streamwise wavenumber ${\it\alpha}_{r}$ versus frequency, (b) the streamwise growth rate $-{\it\alpha}_{i}$ versus frequency. The neutral and most unstable modes are marked by a circle and square respectively.

To ensure fidelity and accuracy of our computations, we first perform necessary checks of numerical resolution. The $\mathscr{C}X_{I}^{-5/3}$ terms in (3.5) and (3.7) are neglected by setting $\mathscr{C}=0$ , and calculations were performed with values of $I$ , $J$ and ${\it\Delta}_{X}$ being doubled or halved to ensure that the results are identical at least to graphical precision.

Alternatively, truncation of the integral may be justified more intuitively by noting that cutting off the wake of the mean-flow distortion for sufficiently large $X$ does not affect scattering as the analysis in the previous section implies, whilst $\tilde{A}_{1}$ would vanish beyond a large $X$ . It follows that the integral can be approximated in a finite domain. For higher-frequency T–S waves, a smaller mesh in streamwise direction is found to be necessary since their wavelengths are shorter. Roughly speaking, if ${\it\omega}$ (more precisely ${\it\alpha}$ ) doubles ${\it\Delta}_{X}$ should be halved.

As part of our efforts to validate the present direct method of eigenvalue formulation, the scattering problem was also solved using the alternative approach, which solves numerically the Laplace equation for the displacement-induced pressure in the upper deck simultaneously with the lower-deck boundary-layer equations; see appendix C. Numerical results for $\tilde{A}$ and $\mathscr{A}$ from these two different methods, on a mesh of the same size, turn out to be in complete agreement with each other. However, the eigenvalue approach is found to be more efficient and robust.

5.2.1 Scattering by linear mean-flow distortion: $h=\pm 0.1$

In order to monitor the asymptotic behaviour far downstream, we introduce the ‘normalized displacement’,

(5.3) $$\begin{eqnarray}\mathscr{A}\equiv \tilde{A}\text{e}^{-\text{i}{\it\alpha}X}.\end{eqnarray}$$

As $X\rightarrow \infty$ , $\mathscr{A}$ is expected to approach a constant, which is the transmission coefficient $\mathscr{T}$ . Figure 6(a) shows $\mathscr{A}$ and $\mathscr{T}$ for $d=0.5$ , $h=\pm 0.1$ and different ${\it\omega}$ . The streamwise distributions of $\mathscr{A}$ for hump and indentation are almost symmetric about the value of unity, i.e. $|\mathscr{A}|-1$ have opposite signs, implying that the effect on the T–S wave is linear. However, this effect is completely confined within the vicinity of the roughness because sufficiently downstream $|\mathscr{A}|$ approaches a constant, i.e. $|\mathscr{T}|$ , which is nearly unity, indicating that the T–S wave is almost fully recovered, as though the roughness were non-existent. The transmission coefficients for different frequencies are all approximately unity as is shown in figure 6(b). According to the linear analysis of Wu & Hogg (Reference Wu and Hogg2006), $\mathscr{T}=1$ to leading order. Extending the triple-deck analysis to next order gives an $O(R^{-1/8}h)$ correction, i.e. $\mathscr{T}=1+O(R^{-1/8}h)$ , but that effect is not accounted for in the present formulation. That $|\mathscr{T}|$ is slightly greater than unity is due to weak nonlinearity of the mean-flow distortion.

Figure 6. Impact of roughness/indentation with a small height/depth $h=\pm 0.1$ , and $d=0.5$ . (a) Modified displacement function $\mathscr{A}$ versus $X$ for ${\it\omega}=3.0$ (solid lines), 4.0 (dashed lines), 5.0 (dot-dashed lines), 6.0 (dot-dot-dashed lines). (b) Transmission coefficient versus  ${\it\omega}$ .

In figure 6(a), one may identify an interaction region in which $\mathscr{A}$ varies significantly but beyond which $\mathscr{A}$ approaches a plateau representing $\mathscr{T}$ . As the frequency of the T–S wave is increased, the interaction region becomes smaller (and so the computational domain could be shortened correspondingly).

5.2.2 Scattering by nonlinear mean-flow distortion: $h=O(1)$

The main interest of the present study is in scattering by nonlinear mean-flow distortions occurring for $h=O(1)$ . Figure 7 shows the normalized displacement functions $\mathscr{A}$ for $h=\pm 1.0$ , $d=0.5$ and two different values of ${\it\omega}$ . The perturbation exhibits rapid variations within the interaction region, especially in the case of indentation. In the present nonlinear case, $\mathscr{A}$ for $\pm h$ is no longer symmetric, in contrast to the linear case for small $h$ . After interacting with the roughness, the perturbation relaxes back to a growing a T–S wave sufficiently downstream, which acquires an appreciably higher amplitude than it would if the roughness were absent. Calculations were also performed with the $\mathscr{C}X_{I}^{-5/3}$ terms in (3.5) and (3.7) being included. The close agreement suggests that it suffices to set $\mathscr{C}=0$ . The predictions from solving the Laplace equation for the pressure are also displayed. The good agreement indicates that both approaches are capable of predicting scattering by a nonlinear hump/indentation with acceptable accuracy.

Figure 7. Computed $|\mathscr{A}(X)|$ for a hump ((a)  $h=1$ ) and an indentation ((b)  $h=-1$ ) with $d=0.5$ and T–S wave frequency ${\it\omega}=3.0$ , 5.0. Solid lines: prediction by the eigenvalue formulation with $\mathscr{C}=0$ ; dot-dashed lines: prediction with the correction $\mathscr{C}X_{I}^{-5/3}$ in (3.5) and (3.7) included; dashed lines: prediction by solving the Laplace equation for the pressure.

The variation of the transmission coefficient with the frequency is shown in figure 8. The eigenvalue formulation exhibits clear advantages over the method of solving the Laplace equation in terms of computational costs and robustness. While for low-frequency T–S waves (e.g. ${\it\omega}=3.0$ ), the computational cost is comparable, the latter method takes approximately 10 times as long time as the former does for relatively high-frequency cases ( ${\it\omega}=5.0$ ), and fails to give a convergent solution for even higher frequencies, e.g. the most unstable frequency ${\it\omega}=7.25$ .

Figure 8. Transmission coefficient versus ${\it\omega}$ for $h=\pm 1.0$ and $d=0.5$ . Solid lines: computed by the eigenvalue formulation; dashed lines: computed by solving the Laplace equation for the pressure.

As stated above, the transmission coefficient $\mathscr{T}$ depends on the frequency ${\it\omega}$ , the height $h$ and width ${\it\Delta}_{R}$ (or $d$ ) of the roughness. In order to shed light on the physical mechanism and salient features of the local scattering process, a systematic parametric study is conducted. The eigenvalue formulation is used because it is more efficient and robust.

Figure 9 shows the variation of the transmission coefficient with ${\it\omega}$ for several fixed values of $h$ and $d$ . The transmission coefficient $|\mathscr{T}|>1$ for both humps and indentations, indicating that they enhance T–S waves. Their effect becomes stronger as ${\it\omega}$ or $h$ increases. For roughness with a height $h=1$ , the effect is rather moderate with $\mathscr{T}\approx 1.3$ . As $h$ is increased further, $|\mathscr{T}|$ increases rapidly, and the transition point would shift upstream substantially. When $h=2.0$ , for which an incipient separation zone emerges, $|\mathscr{T}|\approx 10$ for sufficiently high frequencies, and transition might take place at the roughness site. Between a hump and indentation of identical height and depth, the former produces a stronger effect; the difference increases with the frequency ${\it\omega}$ .

Figure 9. Transmission coefficients $|\mathscr{T}|$ versus ${\it\omega}$ for different $h$ and $d=0.5$ (a), 1.0 (b). Solid lines with filled symbols are for humps; dashed lines with open symbols are for indentations.

Variations of the transmission coefficient with the width of roughness/indentation are shown in figure 10 for several fixed values of $h$ and ${\it\omega}$ . In most cases, the transmission coefficient attains its maximum value when ${\it\Delta}_{R}/{\it\lambda}_{TS}=O(1)$ , indicating that the interaction between a local roughness and a T–S wave is the strongest when their scales match. For fairly high humps, e.g. $h=2.0$ , the maximum $|\mathscr{T}|$ occurs at a much shorter roughness width. It should be noted that for roughness with ${\it\Delta}_{R}/{\it\lambda}_{TS}=O(1)$ or smaller, local stability analysis is invalid.

Figure 10. Transmission coefficients $|\mathscr{T}|$ versus $d$ , or ${\it\Delta}_{R}/{\it\lambda}_{TS}$ (the ratio the roughness width to the wavelength of the T–S wave) for different $h$ and ${\it\omega}=3.0$ (a), 6.0 (b). Solid lines with filled symbols are for humps ( $h>0$ ); dashed lines with open symbols are for indentations ( $h<0$ ).

Figure 11 displays contours of the eigenfunction near the roughness for ${\it\omega}=6.0$ , $d=0.5$ , and $h=1.0$ and 2.0 in order to illustrate how T–S waves are scattered by a local hump. The wall-normal derivative of the streamwise velocity, representing the intensity of the unsteady perturbation, elevates to higher levels after scattering. This effect is stronger for higher humps.

Figure 11. Contours of $\text{Re}(\tilde{u} _{Y})$ (solid lines) and $\text{Im}(\tilde{u} _{Y})$ (dashed lines) of the eigenfunction for the case of ${\it\omega}=6.0$ and $d=0.5$ : (a) $h=1.0$ , (b) $h=2.0$ .

Figure 12. Variation of the transmission coefficient $|\mathscr{T}|$ with $h$ for $d=0.5$ , ${\it\omega}=3.0$ and 6.0. Solid lines: numerical solution; dashed lines: quadratic fitting (5.4) with $c_{2}=0.08$ and 0.20 for ${\it\omega}=3.0$ and 6.0 respectively.

The variation of the transmission coefficient with the roughness height $h$ , which is of primary concern in applications, is shown in figure 12 for a fixed $d=0.5$ and two frequencies ${\it\omega}=3$ and 6. It appears that $\mathscr{T}$ exhibits a quadratic dependence on $h$ :

(5.4) $$\begin{eqnarray}\mathscr{T}-1=c_{2}({\it\omega},d)h^{2},\end{eqnarray}$$

where $c_{2}$ is a constant depending on ${\it\omega}$ and $d$ . The above scaling and the constant $c_{2}$ can be derived by a regular perturbation procedure in the asymptotic limit $h\ll 1$ . Interestingly, the scaling (5.4) appears to hold up to $h\approx 1$ , with the value of $c_{2}$ being determined by fitting with the computational result. For $h>1$ , the increase of $\mathscr{T}$ with $h$ is much faster than $h^{2}$ . Corresponding to (5.4), the $N$ -factor increment

(5.5) $$\begin{eqnarray}{\rm\Delta}N=\ln [1+c_{2}({\it\omega},d)h^{2}].\end{eqnarray}$$

This ${\rm\Delta}N{-}h$ relation is rather different from power-law functions that have been proposed in the literature. In the limit $h\ll 1$ , ${\rm\Delta}N\approx c_{2}h^{2}$ , in agreement with the finding of Wie & Malik (Reference Wie and Malik1998). However, it should be stressed that for $h=O(1)$ or larger, which must be the case in practice, a power-law scaling for ${\rm\Delta}N$ is not tenable. We note further that if the $O(R^{-1/8})$ correction in the asymptotic expansion is included as was done in Wu & Hogg (Reference Wu and Hogg2006), one would obtain a more general result $\mathscr{T}-1=c_{1}R^{-1/8}h+c_{2}({\it\omega},d)h^{2}$ with $c_{1}$ being an $O(1)$ constant. However, the extra linear term is practically irrelevant because in reality $h\gg O(R^{-1/8})$ , for which the quadratic term is dominant.

Figure 13. Local wavenumber ${\it\alpha}_{r}$ and growth rate $-{\it\alpha}_{i}$ versus $X$ for $h=2.0$ and ${\it\omega}=3.0$ . Solid lines with filled symbols: $\tilde{{\it\alpha}}_{r}$ and $-\tilde{{\it\alpha}}_{i}$ computed by the local scattering theory; dashed lines with open symbols: ${\it\alpha}_{L,r}$ and $-{\it\alpha}_{L,i}$ predicted by LST; dot-dashed lines without symbol: ${\it\alpha}_{N,r}$ and $-{\it\alpha}_{N,i}$ predicted by NPLST. (a,b)  $d=0.5$ , (c,d)  $d=1.0$ , (e,f)  $d=4.0$ , (g,h)  $d=8.0$ .

5.3 Comparison with the local linear stability analysis

The LST and NPLST problems of the distorted mean flow, formulated in § 4, are solved numerically. The predicted local complex wavenumbers are denoted by ${\it\alpha}_{L}={\it\alpha}_{L,r}+\text{i}{\it\alpha}_{L,i}$ and ${\it\alpha}_{N}={\it\alpha}_{N,r}+\text{i}{\it\alpha}_{N,i}$ , respectively, with $-{\it\alpha}_{L,i}$ and $-{\it\alpha}_{N,i}$ representing local growth rates. On the other hand, using $\tilde{A}(X)$ given by the local scattering theory, we can calculate, at each streamwise location, the local wavenumber and growth rate,

(5.6a,b ) $$\begin{eqnarray}\tilde{{\it\alpha}}_{r}(X)=\text{Im}(\tilde{A}^{\prime }/\tilde{A}),\quad -\tilde{{\it\alpha}}_{i}(X)=\text{Re}(\tilde{A}^{\prime }/\tilde{A}).\end{eqnarray}$$

In figure 13, we compare these local wavenumbers and growth rates, yielded by different approaches, for ${\it\omega}=3.0$ , $h=2.0$ and a range of roughness width $d$ . When $d$ is small, the local wavenumbers and growth rates given by (NP)LST differ substantially from those predicted by the local scattering theory, especially when $d<1.0$ , for which qualitative differences exist. LST and NPLST tend to overpredict the stabilising/destabilizing effects in the upstream/downstream zones of the roughness element. The discrepancies decrease as $d$ is increased, and appear to become negligible when $d=8$ . The above trend is expected since (NP)LST is invalid for $d=O(1)$ due to strong non-parallelism. However, when ${\it\Delta}_{R}/{\it\lambda}_{TS}\gg 1$ , non-parallelism would be weak so that the local parallel-flow approximation in (NP)LST could be justified. Further comparison for a T–S wave with a higher frequency ${\it\omega}=6$ (not shown) indicates that LST and NPLST are invalid for $d<4$ .

For all the cases shown in figure 13, T–S waves are stabilized slightly upstream of the roughness but destabilized significantly downstream, and hence the overall effect is destabilizing, corresponding to $|\mathscr{T}|>1.0$ . A similar trend was found by Wörner et al. (Reference Wörner, Rist and Wagner2003) and Park & Park (Reference Park and Park2013) based on their DNS and PSE calculations.

Figure 14. Comparison of the streamwise velocity profiles $|\tilde{u} |$ at different streamwise locations for $d=0.5$ , $h=2.0$ and two frequencies: (a–f) ${\it\omega}=3.0$ ; (g–l) ${\it\omega}=6.0$ . Dashed lines: the incident T–S wave; solid lines: prediction by the local scattering theory; filled/open symbols: LST/NPLST predictions. (a,g)  $X=-1$ , (b,h)  $X=-0.5$ , (c,i)  $X=0$ , (d,j)  $X=1$ , (e,k)  $X=2$ , (f,l)  $X=6$ .

The streamwise velocity profiles of the local eigenfunction of (NP)LST are compared in figure 14 with the global eigenfunction predicted by the local scattering theory. At the far upstream and downstream locations, where the perturbation is a usual T–S wave, the profiles given by three approaches all overlap as expected. However, in the vicinity of the hump ( $-1\leqslant X\leqslant 2$ ), where scattering takes place, the (NP)LST predictions deviate appreciably from the ‘exact solution’ of the local scattering theory. Interestingly, the NPLST prediction turns to be worse than LST, suggesting that including only part of non-parallelism in an ad hoc manner does not necessarily lead to a better prediction.

Figure 15. Comparison of transmission coefficients versus $d$ , as predicted by three methods for $h=\pm 2$ and ${\it\omega}=3.0$ (a,c), 6.0 (b,d). Solid lines: $|\mathscr{T}|$ computed by the local scattering theory; dashed lines: $|\mathscr{T}_{L}|$ predicted by LST; dot-dashed lines: $|\mathscr{T}_{N}|$ predicted by NPLST. Curves with filled symbols are for $h=2$ ; lines with open symbols are for $h=-2$ .

By integrating the growth rates $-{\it\alpha}_{L,i}$ and $-{\it\alpha}_{N,i}$ , which are obtained by LST and NPLST respectively, the amplitude of the T–S wave can be found as

(5.7a,b ) $$\begin{eqnarray}A_{L}=\exp \left\{-\int _{X_{0}}^{X_{I}}{\it\alpha}_{L,i}(X)\,\text{d}X\right\},\quad A_{N}=\exp \left\{-\int _{X_{0}}^{X_{I}}{\it\alpha}_{N,i}(X)\,\text{d}X\right\}.\end{eqnarray}$$

Given that in the case of a smooth surface the downstream amplitude is

(5.8) $$\begin{eqnarray}A_{0}=\exp \{-{\it\alpha}_{i}(X_{I}-X_{0})\},\end{eqnarray}$$

where $-{\it\alpha}_{i}$ is the growth rate of the oncoming T–S wave, we may define the transmission coefficients, as predicted by LST and NPLST, as

(5.9) $$\begin{eqnarray}[\mathscr{T}_{L},\mathscr{T}_{N}]=[A_{L},A_{N}]/A_{0}.\end{eqnarray}$$

Figure 15 compares the transmission coefficients, as a function of $d$ , obtained by the three methods. In general, LST and NPLST predictions are both significantly different from that by the local scattering theory for $O(1)$ and small values of $d$ . The difference decreases as the width of the roughness increases, which is expected because of weaker non-parallelism. For $d>2$ , or equivalently ${\it\Delta}_{R}/{\it\lambda}_{TS}>2$ , (NP)LST appears to give a reasonable approximation. This is so despite that the local behaviour is still badly predicted by LST as figure 13 indicates. The case of $h=-2$ and ${\it\omega}=6.0$ , shown in figure 15(c), seems an exception in that the LST result exhibits a certain degree of agreement with that of the local scattering theory even for $d<1$ . This occurs probably due to a rather fortuitous cancellation of the overpredicted stabilizing and destabilizing effects.

To conclude, we observe that LST (or NPLST) for the distorted flow does not correctly account for the impact of an isolated roughness on instability and transition when its length scale is comparable with, or shorter than, the characteristic wavelength of the instability wave; in that case, the local scattering theory is the only appropriate framework. LST however serves as a useful guide and approximation when the roughness contour varies over a length scale that is twice the wavelength or longer.

5.4 Scattering by two adjacent roughness elements

We now consider the case of two adjacent roughness elements. The purpose is to address the question of how large the distance must be in order for each of them to behave as if they were isolated. As an illustration, calculations will be performed for two elements of Gaussian shape, separated by a distance $2l$ , that is

(5.10) $$\begin{eqnarray}F(X)=\frac{h}{2}[\text{e}^{-(X-l)^{2}/d^{2}}+\text{e}^{-(X+l)^{2}/d^{2}}],\end{eqnarray}$$

with $h$ and $d$ being taken to be 2.0 and 0.5 respectively.

Figure 16 shows the streamwise distribution of the normalized displacement function $\mathscr{A}$ for two different $l$ and ${\it\omega}$ . For a small $l$ , the effect of one hump influences that of the other due to the elliptic nature of the scattering problem. The two humps behave as one roughness element with two peaks, leading to a rather complex near field. For $l=3$ , the scattered field near each of the two humps exhibits a similar pattern, indicating that the two humps scatter the T–S wave independently as is expected for sufficiently large $l$ .

Figure 16. $|\mathscr{A}(X)|$ versus $X$ for $h=2.0$ , $d=0.5$ and $l=0.7$ (a), $3.0$ (b). Solid lines: ${\it\omega}=3.0$ ; dashed lines: ${\it\omega}=6.0$ .

Figure 17. Transmission coefficient versus $l$ for $h=2.0$ , $d=0.5$ and ${\it\omega}=3.0$  (a), 6.0 (b). The dashed lines represent the transmission coefficient in the limit $l\rightarrow \infty$ .

Figure 17 displays the transmission coefficient as a function of $l$ for two different ${\it\omega}$ . When $l=0$ , the transmission coefficient is actually for a single hump with $h=2.0$ . As $l$ increases, the combined hump becomes lower and wider and as a result, $\mathscr{T}$ decreases rapidly. After reaching its minimum, $\mathscr{T}$ increases slightly, and finally saturates at a constant (asymptotic) level when the distance $2l$ exceeds a certain value of approximately 4.0, which is eight times the width $d=0.5$ . The T–S wave then tends to interact with the two humps separately and consecutively, which means that the transmission coefficient should tend to be the square of that for the single hump case with a height $h/2=1.0$ , i.e. $\mathscr{T}(h)=\mathscr{T}^{2}(h/2)$ . This is found to be the case. For ${\it\omega}=3.0$ , $\mathscr{T}(1)=1.08784$ and $\mathscr{T}(2)\rightarrow 1.0878^{2}=1.1833$ ; for ${\it\omega}=6.0$ , $\mathscr{T}(1)=1.1952$ and $\mathscr{T}(2)\rightarrow 1.1952^{2}=1.4285$ . The above result suggests that two consecutive roughness elements can be regarded as scattering the oncoming T–S wave separately when the distance between them exceeds approximately 8 times the width of each element. In the presence of an array of such roughness elements, transition may be predicted by the $\text{e}^{N}$ -method with an adjusted critical $N$ -factor (3.15), or (3.16) if elements are identical.