2.3.1. Mean flow distorted by roughness

We assume the shape of the 2-D roughness to be

(2.5)\begin{equation} y=\epsilon hf(\epsilon x)=\epsilon hf(X), \end{equation}

where $\epsilon h$ characterises the dimensionless roughness height $h^{*}/\delta ^{*}$, and $f$ denotes the shape function normalised by its peak (i.e. $\max |\,f(X)|=1$). For demonstration purposes, we select the shape function from Zhao et al. (Reference Zhao, Dong and Yang2019),

(2.6)\begin{equation} f(X)=\left\{\begin{array}{@{}ll} \displaystyle\tfrac 12[1+\cos(2{\rm \pi} X/d)] & \mbox{for }|X| \leqslant d/2,\\ 0 & \mbox{otherwise}, \end{array}\right. \end{equation}

where $d$ denotes the roughness width. The dimensionless width of the roughness is $d^{*}/\delta ^{*}=d/\epsilon$. Of course, our theory also applies for other roughness shapes. It is easy to see that $\int _{-\infty }^{\infty } f(X)\ \textrm {d}X=d/2$. Following the standard triple-deck theory, if $h\ll 1$ or $h=O(1)$, the roughness induces a linear or nonlinear mean-flow distortion in the lower deck; see a detailed analysis in appendix B. The lower-deck flow produces a displacement function $\epsilon A(X)$ to the main deck where $y=O(1)$, leading to an $O(\epsilon )$ correction to the main-deck solution. In the main layer, the velocity field, density, temperature and pressure of the steady base flow are expressed as

(2.7)\begin{align} \bar{\boldsymbol\varphi}&=\left(U_B,0,0,\frac 1{T_B},T_B,\frac 1{\gamma M^{2}}\right)+\epsilon \left( U_1,0,0,R_1,T_1,0\right)\notag\\ &\quad + \varDelta_1X\left(U_2,0,0,R_2, T_2,0\right)+\cdots, \end{align}

where $U_B(y)=\bar {U}_B(0,y)$ and $T_B(y)=\bar {T}_B(0,y)$ represent the streamwise velocity and temperature of the Blasius solution at $\bar {x}=0$, respectively, and $\varDelta _1=\epsilon ^{3}\varDelta ^{2}\ll 1$. Here, $\varDelta _1$ is not necessarily smaller than $\epsilon$, because the $O(\epsilon )$ and $O(\varDelta _1)$ terms do not interact directly. The $O(\epsilon )$ terms are the roughness-induced mean-flow distortion, which are given by (Stewartson Reference Stewartson1969; Messiter Reference Messiter1970; Smith Reference Smith1973)

(2.8a–c)\begin{equation} U_1= A(X)U_B',\quad R_1={-}A(X){T_B'}/{T_B^{2}},\quad T_1=A(X)T_B', \end{equation}

where the prime throughout this paper denotes the derivative with respect to its argument. The $O(\varDelta _1)$ terms are associated with the non-parallelism of the Blasius solution, i.e.

(2.9a–c)\begin{equation} U_2= \bar{U}_{B,\bar{x}}(0,y),\quad R_2={-}\bar{T}_{B,\bar{x}}(0,y)/T_B^{2},\quad T_2=\bar{T}_{B,\bar{x}}(0,y), \end{equation}

where the subscripts $\bar {x}$ denotes the derivative with respect to $\bar {x}$. Note that the transverse velocity induced by the roughness is of $O(\epsilon ^{2})$, and that due to the non-parallelism of the Blasius solution is of $O(\epsilon \varDelta _1)$, both of which are negligible in the expansion (2.7).

2.3.2. Perturbation field

We consider the oncoming perturbation to be an inviscid first or second Mack mode. According to Dong et al. (Reference Dong, Liu and Wu2020), the inviscid mode in the main layer can be described by the Rayleigh equation, and therefore, we introduce $\hat{\boldsymbol\phi}=(\hat {v},\hat {p})^\textrm {T}$ as the eigenfunction of the eigenvalue system. If the roughness is absent, the main-layer perturbation in the neighbourhood of $\bar {x}=0$ is expressed as

(2.10)\begin{align} {\mathcal{E}}[\hat{\boldsymbol\phi}(y;\bar{x}=0)+\varDelta_1 X\hat{\boldsymbol\phi}_{\bar{x}}(y;\bar{x}=0)+\cdots]\exp\left({{\textrm{i}} \left[\epsilon^{{-}1}\int \alpha(X)\ \textrm{d}X+\beta z-\omega t\right]}\right)+\textrm{c.c.}, \end{align}

where $\alpha$, $\beta$, $\omega$ and $\mathcal {E}$ denote the streamwise wavenumber, wall-normal wavenumber, frequency and the amplitude, respectively, and c.c. denotes the complex conjugate. For a spatially evolving mode, $\omega$ and $\beta$ are real, and $\alpha =\alpha _r+{\textrm {i}}\alpha _i$ is complex with $-\alpha _i$ representing the growth rate. For normalisation, we let $\hat {p}(0)=1$.

As the Mack mode approaches the roughness vicinity, the perturbation is deformed by the mean-flow distortion, which can be expressed in terms of an asymptotic series,

(2.11)\begin{align} \tilde{\boldsymbol\phi}(X,y,z,t)&={\mathcal{E}}(\tilde{\boldsymbol\phi}_0+ \epsilon \tilde{\boldsymbol\phi}_1+O(\epsilon^{2})+\epsilon\varDelta_1\tilde{\boldsymbol\phi}_2+\cdots)\nonumber\\ &\quad\times \exp\left({{\textrm{i}}\left[\epsilon^{{-}1}\int_{-\infty}^{X}\alpha \ \textrm{d}X+\beta z-\omega t\right]}\right)+\textrm{c.c.}. \end{align}

Here, the interaction between the oncoming mode and the second-order roughness-induced mean-flow distortion, the $O(\epsilon ^{2})$ term, is neglected, but the interaction between the non-parallelism of the oncoming mode and the leading-order roughness-induced mean-flow distortion, the $O(\epsilon \varDelta _1)$ term, is retained. Although the latter is numerically smaller, it determines how the perturbation evolves from the inner region to the outer region.

Because the mean-flow distortion induced by the roughness varies on a longer length scale than the Mack wavelength, the leading-order perturbation can be expressed as

(2.12)\begin{equation} \tilde{\boldsymbol\phi}_0(X,y)=C(X)(\hat{\boldsymbol\phi}(y)+ \varDelta_1 X\hat{\boldsymbol\phi}_{\bar{x}}(y)+\cdots), \end{equation}

where $C$ is expanded in terms of an asymptotic series,

(2.13)\begin{equation} C(X)=C_0(X)+O(\epsilon)+\varDelta_1 C_1(X)+\cdots. \end{equation}

Substituting (2.10) with (2.11)–(2.13) into (2.2) and neglecting the $O({\mathcal {E}}^{2})$ terms, we obtain the governing equations for $\tilde{\boldsymbol\phi}$, $\tilde{\boldsymbol\phi}_1$ and $\tilde{\boldsymbol\phi}_2$,

(2.14)\begin{gather} {\mathcal{L}}_R\hat{\boldsymbol\phi}\equiv(\partial_y-{{\boldsymbol{\mathsf{H}}}}_0)\hat{\boldsymbol\phi}=0, \end{gather}

(2.15)\begin{gather}{\mathcal{L}}_R\tilde{\boldsymbol\phi}_1={-}{\textrm{i}}\frac{\partial {{\boldsymbol{\mathsf{H}}}}_0}{\partial \alpha}C'_0(X)\hat{\boldsymbol\phi} +{{\boldsymbol{\mathsf{H}}}}_1A(X)C_0(X)\hat{\boldsymbol\phi}, \end{gather}

(2.16)\begin{gather}{\mathcal{L}}_R\tilde{\boldsymbol\phi}_2={-}{\textrm{i}}\frac{\partial {{\boldsymbol{\mathsf{H}}}}_0}{\partial \alpha}[C'_1\hat{\boldsymbol\phi}+(C_0'X+C_0)\hat{\boldsymbol\phi}_{\bar{x}}]+{{\boldsymbol{\mathsf{H}}}}_1A(C_1\hat{\boldsymbol\phi}+C_0X\hat{\boldsymbol\phi}_{\bar{x}})+{{\boldsymbol{\mathsf{H}}}}_2C_0(X)\hat{\boldsymbol\phi}, \end{gather}

where

(2.17a,b)\begin{equation} {{\boldsymbol{\mathsf{H}}}}_0=\left(\begin{array}{cc} \displaystyle \dfrac{U_B'}{U_B-c} & \displaystyle \dfrac{-M^{2}S_0^{2}-\tilde\alpha^{2}T_B}{S_0} \\ \displaystyle -\dfrac{S_0}{T_B} & 0 \end{array} \right),\quad {{\boldsymbol{\mathsf{H}}}}_1=\frac{\textrm{d}{{\boldsymbol{\mathsf{H}}}}_0}{\textrm{d} y}, \end{equation}

with $S_0={\textrm {i}}\alpha (U_B-c)$, $c\equiv \omega /\alpha$, $\tilde {\alpha }=\sqrt {\alpha ^{2}+\beta ^{2}}$ and ${\boldsymbol{\mathsf{H}}}_2$ is introduced in (A5). The operator ${\mathcal {L}}_R\equiv (\partial _y-{{\boldsymbol{\mathsf{H}}}}_0)$ is referred to as the Rayleigh operator. It is seen that the viscous terms do not appear in the equations we are concerned with, illustrating the inviscid nature of the main layer. The boundary conditions read

(2.18a,b)\begin{gather} \hat{v}(0)=0,\quad\hat{p}(y)\rightarrow 0\quad\mbox{as }y\rightarrow\infty, \end{gather}

(2.19a,b)\begin{gather} \tilde{v}_{1}(X,0)={-}hf(X)C_{0}(X)\hat{v}_{y}(0),\quad \tilde{p}_{1}(X,y)\rightarrow 0\quad\mbox{as }y\rightarrow\infty, \end{gather}

(2.20a,b)\begin{gather} \tilde{v}_{2}(X,0)={-}hf(X)C_{1}(X)\hat{v}_{y}(0),\quad\tilde{p}_{2}(X,y)\rightarrow 0\quad\mbox{as }y\rightarrow\infty. \end{gather}

Here, the boundary conditions (2.19a) and (2.20a) are obtained by analysing the Stokes-layer behaviour of the perturbation, which is the same as that in § 5.4.1 of Dong et al. (Reference Dong, Liu and Wu2020). From (5.35) of Dong et al. (Reference Dong, Liu and Wu2020), if we let $k=\alpha _a=\alpha$, replace $\hat {p}_{a0}(0)$ by $\hat {p}(0)$ and apply the Rayleigh equation (2.14) at the bottom, then we directly arrive at the conditions (2.19a) and (2.20a). Interesting enough, these boundary conditions obtained by the inhomogeneous forcing from the curved Stokes layer are exactly the same as those obtained by applying the first-order Taylor expansion at the bottom of the main layer. Note that, here, we have assumed $|c-1|<\sqrt {\alpha ^{2}+\beta ^{2}}/(\alpha M)$, such that the instability wave propagates subsonically relative to the main-stream velocity. Otherwise (2.18b) and (2.19b) need to be replaced by the radiating conditions, but the rest of the analysis remains the same.

Following a standard treatment as in Fedorov & Khokhlov (Reference Fedorov and Khokhlov2002), the systems (2.15) and (2.16) with boundary conditions (2.19a,b) and (2.20a,b) can be recast to

(2.21)\begin{align} {\mathcal{L}}_R\tilde{\boldsymbol\phi}_1&={-}{\textrm{i}}\frac{\partial {{\boldsymbol{\mathsf{H}}}}_0}{\partial \alpha}C'_0(X)\hat{\boldsymbol\phi} +{{\boldsymbol{\mathsf{H}}}}_1A(X)C_0(X)\hat{\boldsymbol\phi}+hf(X)C_0(X)\delta(y){\boldsymbol{\mathsf{D}}}, \end{align}

(2.22)\begin{align} {\mathcal{L}}_R\tilde{\boldsymbol\phi}_2&={-}{\textrm{i}}\frac{\partial {{\boldsymbol{\mathsf{H}}}}_0}{\partial \alpha} [C'_1\hat{\boldsymbol\phi}+(C_0'X+C_0)\hat{\boldsymbol\phi}_{\bar{x}}] +{{\boldsymbol{\mathsf{H}}}}_1A(C_1\hat{\boldsymbol\phi}+C_0X\hat{\boldsymbol\phi}_{\bar{x}})+{{\boldsymbol{\mathsf{H}}}}_2C_0 \hat{\boldsymbol\phi}\nonumber\\ &\quad +hfC_1\delta(y){\boldsymbol{\mathsf{D}}}, \end{align}

(2.23)\begin{align} \tilde{v}_1(X,0)&=\tilde{v}_2(X,0)=0,\quad\left(\tilde{p}_1(X,y),\tilde p_2(X,y)\right)\rightarrow 0\quad\mbox{as }y\rightarrow\infty, \end{align}

where $\delta$ is the Dirac delta function, and ${\boldsymbol{\mathsf{D}}}=(-\hat {v}_{y},0)^\textrm {T}$.

2.3.3. The amplitude equation and the transmission coefficient

We now introduce the adjoint vector of the Rayleigh equation, $\hat{\boldsymbol\phi}^{{\dagger} }=(\hat{\phi}^{{\dagger} }_{(1)},\hat{\phi}^{{\dagger} }_{(2)})^\textrm {T}$, which satisfies

(2.24)\begin{gather} (\partial_y+{{\boldsymbol{\mathsf{H}}}}_0^{T})\hat{\boldsymbol\phi}^{{\dagger}ger}=0, \end{gather}

(2.25a,b)\begin{gather} \hat{\phi}^{{\dagger}ger}_{(2)}(0)=0,\quad\hat{\phi}^{{\dagger}ger}_{(1)}(y)\rightarrow 0\quad\mbox{as }y\rightarrow\infty. \end{gather}

For normalisation, we let $\hat{\phi}^{{\dagger} }_{(1)}(0)=1$.

The systems (2.21) and (2.22) with boundary conditions (2.23) have non-trivial solutions only if the right-hand sides of (2.21) and (2.22) are orthogonal to the adjoint vector $\hat{\boldsymbol\phi}^{{\dagger} }$ (solvability condition). This leads to the amplitude equations,

(2.26)\begin{gather} {\textrm{i}}C'_0(X)=\left[A(X)K_0+hf(X)K_1\right]C_0(X), \end{gather}

(2.27)\begin{gather}{\textrm{i}}C'_1=\displaystyle\left(AK_0+hfK_1\right)C_1-{\textrm{i}}K_2 C_0'X+K_3AXC_0+(K_4-{\textrm{i}}K_2) C_0, \end{gather}

where the efficiency functions are

(2.28a,b)\begin{gather} K_0=\frac{\langle\hat{\boldsymbol\phi}^{{\dagger}ger},{{\boldsymbol{\mathsf{H}}}}_1\hat{\boldsymbol\phi}\rangle}{ \langle\hat{\boldsymbol\phi}^{{\dagger}ger},({\partial {{\boldsymbol{\mathsf{H}}}}_0}/{\partial \alpha})\hat{\boldsymbol\phi}\rangle}, \quad K_1=\frac{\langle\hat{\boldsymbol\phi}^{{\dagger}ger},\delta(y){\boldsymbol{\mathsf{D}}}\rangle}{ \langle\hat{\boldsymbol\phi}^{{\dagger}ger},({\partial {{\boldsymbol{\mathsf{H}}}}_0}/{\partial \alpha})\hat{\boldsymbol\phi}\rangle}, \end{gather}

(2.28c–e)\begin{gather} K_2=\frac{\langle\hat{\boldsymbol\phi}^{{\dagger}ger},({\partial {{\boldsymbol{\mathsf{H}}}}_0}/{\partial \alpha})\hat{\boldsymbol\phi}_{\bar{x}}\rangle} {\langle\hat{\boldsymbol\phi}^{{\dagger}ger},({\partial {{\boldsymbol{\mathsf{H}}}}_0}/{\partial \alpha})\hat{\boldsymbol\phi}\rangle}, \quad K_3=\frac{\langle\hat{\boldsymbol\phi}^{{\dagger}ger},{ {{\boldsymbol{\mathsf{H}}}}_1}\hat{\boldsymbol\phi}_{\bar{x}}\rangle}{ \langle\hat{\boldsymbol\phi}^{{\dagger}ger},({\partial {{\boldsymbol{\mathsf{H}}}}_0}/{\partial \alpha})\hat{\boldsymbol\phi}\rangle} ,\quad K_4=\frac{\langle\hat{\boldsymbol\phi}^{{\dagger}ger},{{\boldsymbol{\mathsf{H}}}}_2\hat{\boldsymbol\phi}\rangle}{ \langle\hat{\boldsymbol\phi}^{{\dagger}ger},({\partial {{\boldsymbol{\mathsf{H}}}}_0}/{\partial \alpha})\hat{\boldsymbol\phi}\rangle}, \end{gather}

and

(2.29)\begin{equation} \langle{\boldsymbol\phi}^{{\dagger}ger},{\boldsymbol\phi}\rangle\equiv\int_0^{\infty}({\boldsymbol\phi}^{{\dagger}ger})^\textrm{T}\boldsymbol{\phi}\, {\textrm{d} y} \end{equation}

denotes the inner product. Here, $AK_0$ represents the impact of the main-layer mean-flow distortion on the oncoming Mack modes, where the strength of the mean-flow distortion $A$ and the instability property $K_0$ are well separated; $hfK_1$ represents the impact of the wall displacement on the oncoming modes. Terms with $K_2$, $K_3$ and $K_4$ are associated with the non-parallelism of the Blasius base flow. It is proved in appendix C that $K_0=K_1$.

Because the impact of the roughness in the upstream limit is rather weak, we put

(2.30)\begin{equation} C_0(X)\rightarrow 1\quad\mbox{as }X\rightarrow-\infty. \end{equation}

Solving the amplitude equation (2.26) subject to (2.30), we obtain

(2.31)\begin{equation} C_0(X)=\exp\left[-{\textrm{i}}K_0 \int_{-\infty}^{X}\left(A(X)+hf(X)\right)\textrm{d}X\right]. \end{equation}

In the downstream limit, since both $A$ and $f$ approach zero, $C_0(X)$ approaches a constant

(2.32)\begin{equation} {\mathscr{T}}_0\equiv C_0(\infty)=\exp\left(-{\textrm{i}}K_0 ({\mathcal{D}}_1+h{\mathcal{D}}_2)\right), \end{equation}

where

(2.33a,b)\begin{equation} {\mathcal{D}}_1=\int_{-\infty}^{\infty} A(X)\ \textrm{d}X,\quad {\mathcal{D}}_2=\int_{-\infty}^{\infty} f(X)\ \textrm{d}X. \end{equation}

Note that for a linear roughness element, $|h|\ll 1$, ${\mathcal {D}}_1$ would be a linear function of $h$, and from appendix B.2 we know that ${\mathcal {D}}_1+h{\mathcal {D}}_2=0$. However, for a nonlinear hump/indentation, $|h|=O(1)$, ${\mathcal {D}}_1$ increases with $h$ super-/sub-linearly, as will be shown in table 2. Therefore, to leading order, the impacts of the instability property ($K_0$) and the roughness-induced mean-flow distortion (${\mathcal {D}}_1+h{\mathcal {D}}_2$) are readily separated.

Table 2.

Case studies for different roughness shapes, where $R=131820$, $d^{*}/\delta _{99}=8.0$, $d=0.495$ and $D=0.938$ for all cases.

The second-order amplitude equation yields

(2.34)\begin{equation} C_1= C_{10}+ b(X), \end{equation}

where

(2.35a,b)\begin{gather} C_{10}={-}{\textrm{i}}{C_0}\left(K_5\int_{-\infty}^{X} AX\ \textrm{d}X-hK_0 K_2\int_{-\infty}^{X}f\kern0.05em X \ \textrm{d}X\right), \quad K_5=K_3-K_0 K_2, \end{gather}

(2.36)\begin{gather} b(X)=C_0 ({-}K_2-{\textrm{i}}K_4)X. \end{gather}

The term $b(X)$ matches with the outer-region solution (A9), which is purely driven by the non-parallelism of the Blasius base flow. On the other hand, $C_{10}$ is associated with the interaction of the roughness-distorted mean flow with the non-parallelism of the perturbation profile. Indeed, the second-order scattering is only linked to $C_{10}$. Because the roughness impact on the oncoming modes in the upstream limit is negligible, we have put $C_{10}\rightarrow 0$ as $X\rightarrow -\infty$. However, the integral $\int _{-\infty }^{X} AX\ \textrm {d}X$ may not converge as $X\rightarrow \infty$, indicating a re-scattering effect induced by the roughness ‘wake’, which is the same as the T–S scattering in Wu & Dong (Reference Wu and Dong2016b). In the latter work, the divergent integral was interpreted as a Hardmard finite-part integral, and a transmission coefficient independent of the computational domain was obtained. In this paper, we are going to compare the asymptotic predictions with the numerical results, for which the amplitude growth of the perturbation in the downstream limit may be slightly different from that of the smooth-surface configuration, as will be shown in figure 15. Such a phenomenon can also be observed in the DNS results of Fong et al. (Reference Fong, Wang and Zhong2014, Reference Fong, Wang and Zhong2015b). Therefore, in the HLNS or DNS calculations, we choose the transmission coefficient from the perturbation amplitude at the outlet of the computational domain. This choice also agrees with the physical situation, for which laminar–turbulent transition or the nonlinear state may occur before the asymptotic state is reached. Therefore, in the asymptotic theory, we define

(2.37)\begin{equation} {\mathscr{T}}_1\equiv C_{10}(X_{HLNS})={-}{\textrm{i}} {\mathcal{B}} {\mathscr{T}}_0, \end{equation}

where $X_{HLNS}\gg 1$ is the converted outlet boundary from the HLNS computational domain (as will be defined in (3.8)),

(2.38)\begin{equation} \left.\begin{gathered} {\mathcal{B}}=K_5{\mathcal{D}}_3-hK_0K_2{\mathcal{D}}_4,\\ {\mathcal{D}}_3=\int_{-\infty}^{X_{HLNS}} A(X)X\ \textrm{d}X,\quad {\mathcal{D}}_4=\int_{-\infty}^{X_{HLNS}} f(X)X\, \textrm{d}X. \end{gathered}\right\} \end{equation}

Note that if the roughness is symmetric along its centreline $X=0$, ${\mathcal {D}}_4$ would be zero as long as $X_{HLNS} \geqslant d/2$, but ${\mathcal {D}}_3$ would not. Admittedly, ${\mathscr {T}}_1$ may vary if the computational domain of the HLNS calculation is changed. However, because ${\mathscr {T}}_1$ has only a second-order impact, the variation of the total transmission coefficient ${\mathscr {T}}$ would be rather mild. More importantly, this variation agrees with the numerical results obtained by the HLNS and DNS approaches, which can well reflect the physical situation.

Following Wu & Dong (Reference Wu and Dong2016b), the impact of the roughness on the oncoming Mack modes can be quantified by a transmission coefficient,

(2.39)\begin{equation} {\mathscr{T}}={\mathscr{T}}_0+O(\epsilon)+\varDelta_1{\mathscr{T}}_1+\cdots, \end{equation}

where the $O(\epsilon )$ terms is caused by the second-order mean-flow distortion induced by the roughness, which is not taken into account in this paper. It is easy to see that the oncoming Mack mode is enhanced by the roughness if $|{\mathscr {T}}|>1$, but is suppressed if $|{\mathscr {T}}|<1$. It is estimated from (2.32) and (2.37) that

(2.40)\begin{gather} |{\mathscr{T}}|=\exp\left[ K_{0,i}({\mathcal{D}}_1+h{\mathcal{D}}_2)\right](1+\varDelta_1{\mathcal{B}}_i), \end{gather}

(2.41)\begin{gather}\arg({\mathscr{T}})={-}K_{0,r}({\mathcal{D}}_1+h{\mathcal{D}}_2)+\arg(1-{\textrm{i}} \varDelta_1{\mathcal{B}}), \end{gather}

where the subscripts $r$ and $i$ denote the real and imaginary parts, respectively. For a relatively low roughness, for which $K_{0,i}({\mathcal {D}}_1+h{\mathcal {D}}_2)\ll 1$, the norm of the transmission coefficient (2.40) is estimated by

(2.42)\begin{equation} |{\mathscr{T}}|=1+ K_{0,i} {\mathcal{D}}_1+hK_{0,i} {\mathcal{D}}_2+\varDelta_1 {\mathcal{B}}_i+\cdots. \end{equation}

It is clearly seen from (2.42) that the local scattering effect is mainly attributed to three factors: (i) $K_{0,i}{\mathcal {D}}_1$, the interaction between the mean-flow distortion and the oncoming Mack mode in the main layer; (ii) $hK_{0,i}{\mathcal {D}}_2$, the inhomogeneous forcing from the curved wall (Stokes layer); (iii) $\varDelta _1{\mathcal {B}}_i$, the interaction of the non-parallelism of the oncoming normal modes with the roughness-induced mean-flow distortion, which is merely the second-order effect. Whether the roughness plays a stabilising or destabilising role depends on their summed effects.

4.3.1. Comparison between the triple-deck and DNS results

Figure 6 compares the mean shear and the streamlines in the near-wall region obtained by the triple-deck theory with those obtained by DNS. The coordinate system and velocity field of the DNS results are converted to the triple-deck scalings through the relations (B1a,b) and (B5a–c). For a hump case with $h^{*}/\delta _{99}=0.20$, shown in figure 6(a,b), both the triple-deck and DNS results exhibit a high-shear region on the top of the hump and two low-shear regions at its two sides. For an indentation case with $h^{*}/\delta _{99}$=-0.20, as figure 6(c,d) indicates, a low-shear region and two high-shear regions appear in the bottom and at the two sides, respectively, and a circulation appears inside the indentation. Quantitative differences between the two calculations exist due to the finite-$R$ effect.

Figure 6.

Contours of the mean shear $\tilde {U}_{\tilde {Y}}$ and the streamlines in the close neighbourhood of the roughness for cases H4 (a,b) and HM4 (c,d), where the left and right columns are for results obtained by triple-deck calculation and DNS, respectively.

Figure 7 compares the wall shear and the wall pressure between the triple-deck and DNS results. For both the two hump heights, $h^{*}/\delta _{99}=0.10$ and 0.20, the two families of curves agree in quality. The mean-flow distortion obtained by DNS is slightly larger in the region $-1 \leqslant \tilde {X} \leqslant 0.4$. The relative error stays at the same level as $H$ increases. Note that this comparison is for a relatively large Reynolds number, and as $R$ decreases, the agreement becomes worse due to the impact of the neglected $O(\epsilon ^{2})$ terms.

Figure 7.

Comparison of the wall shear $\tilde {U}_{\tilde {Y},w}$ and the wall pressure $\tilde {P}$ between triple-deck and DNS results. (a,b) Are for cases H2 and H4, respectively.

The triple-deck theory can also provide a composite solution for the whole boundary-layer profile,

(4.2)\begin{equation} \bar{U}_{com}=\bar{U} +\epsilon{\mathcal{R}}_U^{{-}1}\tilde{U}-U_B'(0)\left(y+\epsilon A(X)\right), \end{equation}

where $\bar {U}$ is the mean flow in the main layer defined in (2.7), and $\tilde {U}$ is the lower-deck solution introduced in appendix B. Figure 8 compares the composite solutions near the roughness with the DNS results for cases H2 and H4. For a relatively low hump, the mean-flow distortion is quite weak, but we can still see that the general trends of the mean-flow distortions obtained by the two approaches agree overall. Slight discrepancies are observed at the two ends of the roughness, i.e. $X\approx \pm d/2$. The discrepancies increase with the roughness height. Interestingly, in the near-wall region, the agreement between the two calculations is much better, which confirms the observations from figure 7.

Figure 8.

Profiles of the mean streamwise velocity $\bar {U}$ at five representative stations near the roughness for cases H2 (a) and H4 (b), where the red solid and blue dashed lines are for the DNS results and triple-deck composite solutions (4.2), respectively. The profiles are shifted horizontally for plotting convenience. The thin solid lines denote the Blasius solution for the smooth-surface case.

4.3.2. Mean-flow distortion for different cases

Figure 9 compares the displacement function $\tilde {A}(\tilde {X})$ and the wall function $H\tilde {f}(\tilde {X})$ for different hump heights or indentation depths. For each hump case, shown in figure 9(a), the displacement function $\tilde {A}$ is negative in the vicinity of the hump and approaches zero in both the upstream and downstream limits. By contrast, $\tilde {A}$ for each indentation, shown in figure 9(b), is positive. Both panels indicate that $\tilde {A}$ distributes in a wider range than the wall shape $H\tilde {f}$ does. The mean-flow distortion, measured by the absolute value of $\tilde {A}$, is greater for a higher hump, but varies only slightly as the indentation depth increases.

Figure 9.

The streamwise distributions of the displacement function $\tilde {A}$ and the shape of the roughness $Hf(\tilde {X})$. (a) Hump cases H1 to H5; (b) indentation cases HM2 to HM5.

As revealed by (2.42), the scattering effect is attributed to three factors associated with the mean-flow distortion, i.e.

(4.3a–c)\begin{equation} {\mathcal{D}}_1={\mathcal{R}}_X^{{-}1}{\mathcal{R}}_Y^{{-}1}\bar{\mathcal{D}}_1,\quad h{\mathcal{D}}_2={\mathcal{R}}_X^{{-}1}{\mathcal{R}}_Y^{{-}1}H\bar{\mathcal{D}}_2,\quad {\mathcal{D}}_3={\mathcal{R}}_X^{{-}2}{\mathcal{R}}_Y^{{-}1}\bar{\mathcal{D}}_3, \end{equation}

and the values of $\bar {\mathcal {D}}_1$, $H\bar {\mathcal {D}}_2$ and $\bar {\mathcal {D}}_3$ for the chosen cases are also listed in table 2. Note that the quantity $\int _{-\infty }^{\infty } \tilde {X}\tilde {f}\ \textrm {d}\tilde {X}$ is always zero in this paper because the shapes of the humps and indentations are all symmetric about $\tilde {X}=0$. For a linear hump or indentation, for which $|H|\ll 1$, $\bar {\mathcal {D}}_1$ is equal to $H{\mathcal {D}}_2$ exactly, as (B18) indicates, which implies that a linear roughness does not have any impact on the oncoming Mack modes to leading-order accuracy. For nonlinear roughness cases, $\bar {\mathcal {D}}_1$ decreases remarkably with the hump height for cases H1 to H5, but increases mildly with the indentation depths for cases HM2 to HM5. Because the hump heights of cases H2 to H5 are opposite to the indentation depths of cases HM2 to HM5, the values of $H\bar {\mathcal {D}}_2$ between each counterpart (H$l$ and HM$l$ with $l=2$ to 5) are exactly opposite; $\bar {\mathcal {D}}_3$ is one order of magnitude smaller than $\bar {\mathcal {D}}_1$ for small hump cases, but increases to the same order of magnitude as the latter for moderate hump and all indentation cases.

4.4.1. Efficiency functions

From (2.28a,b) and (2.35a,b), we calculate the efficiency functions $K_0$ and $K_5$ using the original asymptotic theory in § 2.3 and the improved asymptotic theory in § 2.4, as shown in Figure 10. Note that the efficiency functions obtained by the original asymptotic theory are independent of $R$, but those by the improved theory are not. For the latter calculations, we choose $R=131\,820$. All the panels show that the efficiency functions vary remarkably in the frequency band $1.5<\omega <2.5$, corresponding to most of the second mode and a small portion of the first mode. It is easy to see from (2.40) and (2.41) that a relatively strong scattering effect may appear in this frequency band. For the original asymptotic theory, as $\omega$ increases from 1.5, $K_{0,r}$ falls dramatically and there is a valley at around $\omega =1.8$; it then rises sharply to positive and exhibits a wide bump shape before falling back to zero. As shown in panel (b), $K_{0,i}$ increases with $\omega$ slightly from $\omega =1.5$, but falls drastically at $\omega \approx 1.8$; after reaching its valley at approximately $\omega =1.815$, $K_{0,i}$ increases with $\omega$ again and peaks at $\omega \approx 2.28$; it falls back towards zero as $\omega$ further increases. The frequency $\omega _c=2.10$ for which $K_{0,i}=0$ is marked in panel (b), which, according to (2.40) or (2.42), represents the critical frequency distinguishing the stabilising and destabilising effects of the roughness to leading-order accuracy. This frequency is quite close to the synchronisation frequency, as shown in figure 5, which agrees with the DNS results in Fong et al. (Reference Fong, Wang and Zhong2014, Reference Fong, Wang and Zhong2015b).

Figure 10.

Dependence of the efficiency functions $K_0$ (a,b) and $K_5$ (c,d) on frequency $\omega$. The thin horizontal line in (b) represents the zero line, and $\omega _c=2.10$ and $\bar \omega _c=2.07$ are the frequencies for which $K_{0,i}$ crosses the zero line.

In order to probe the intrinsic mechanism, we plot separately the numerator $K_{01}=\langle \hat{\boldsymbol\phi},{{\boldsymbol{\mathsf{H}}}}_1\hat{\boldsymbol\phi}\rangle$ and the denominator $K_{02}=\langle \hat{\boldsymbol\phi},(\partial {{\boldsymbol{\mathsf{H}}}}_0/\partial \alpha )\hat{\boldsymbol\phi}\rangle$, as shown in figure 11(a). The numerator $K_{01}$ is almost pure imaginary, which can be predicted analytically as follows. As is proven in appendix C, the numerator $K_{01}$ is equal to $\langle \hat{\boldsymbol\phi}^{{\dagger} },\delta (y)\boldsymbol {D}\rangle$. The latter can be expressed as

(4.4)\begin{equation} -\hat{\phi}^{{\dagger}ger}_{(1)}(0)\hat{v}_y(0)={\textrm{i}}\left({-}M^{2}\omega+\frac{(\alpha_r^{2}-\alpha_i^{2}+\beta^{2})T_w}{ \omega}\right) -\frac{2\alpha_r\alpha_iT_w}{\omega}. \end{equation}

Since $|\alpha _i|\ll \alpha _r$, the real part of the right-hand side of (4.4) is much smaller than its imaginary part. Therefore, $K_{0,i}$ crosses zero when $K_{02,r}\approx 0$, which appears at $\omega \approx 2.1$, as shown by the filled circle in figure 11(a). Figure 12 plots the wall-normal profiles of the real part of the integrand of $K_{02}$, $[(\hat{\boldsymbol\phi}^{{\dagger} })^\textrm {T}(\partial {\boldsymbol{\mathsf{H}}}_0/\partial \alpha )\hat{\boldsymbol\phi}]_r$, for three representative frequencies. These profiles are all concentrated around the edge of the boundary layer, and their positive and negative parts contribute oppositely to the integral $K_{02}$. For the critical frequency, $\omega \approx 2.1$, these opposite contributions cancel out, leading to $K_{02,r}\approx 0$. For higher and lower frequencies, the integrals become negative and positive, respectively, agreeing with figure 11(a).

Figure 11.

Variation of the numerator and denominator of $K_0\equiv K_{01}/K_{02}$ on $\omega$ in the second-mode frequency band obtained by the original (a) and improved (b) asymptotic theories.

Figure 12.

Wall-normal profiles of the real part of the integrand of $K_{02}$ for three representative frequencies.

For the improved asymptotic theory, as shown by the green dashed lines in figures 10(a) and 10(b), the overall trend is the same as that for the original asymptotic theory, but the curves vary more gently, especially in the region around $\omega\approx 1.8$ and shift slightly towards the lower-frequency direction. A direct consequence is that the critical frequency $\omega _c$ is shifted to $\bar {\omega }_c=2.07$. Additionally, the denominator $K_{02}$ predicted by the original asymptotic theory, shown in figure 11(a), approaches zero as $\omega$ approaches 1.8, the intersection frequency of the first and second modes, indicating that the efficiency function $K_0$ obtained by the original theory tends to blow up at the frequency. However, for the improved theory, both $K_0$ and $K_{02}$ tend to be finite at $\omega \approx 1.8$, which is of course more reasonable.

The second-order scattering effect is linked to $K_5$ as shown in figures 10(c) and 10(d). Both the real and imaginary parts of $K_5$ vary drastically in a narrow band $1.7<\omega <1.85$, which corresponds to the local peak frequency of the phase speed in figure 5. This is due to the small denominators of $K_2$ and $K_3$, ${\langle \hat{\boldsymbol\phi}^{{\dagger} },({\partial {{\boldsymbol{\mathsf{H}}}}_0}/{\partial \alpha })\hat{\boldsymbol\phi}\rangle }$, near $\omega =1.8$. Note that, in this narrow frequency band, the second-order efficiency function is almost one order of magnitude greater than the first-order efficiency function, indicating a potentially strong scattering effect contributing from the second order.

4.4.2. Comparison with the HLNS results for a relatively large Reynolds number

In order to verify the accuracy of the asymptotic theories, we perform DNS and HLNS to calculate the evolution of the Mack modes. Here, we choose case H4 and $R=131\,820$. Figure 13 shows the contours of the instantaneous perturbation pressure and temperature for $\omega =1.78$ obtained by DNS. The value of $\tilde {p}$ peaks at the wall, while $\tilde {T}$ peaks at the edge of the boundary layer. The perturbation profiles undergo rapid distortions in the vicinity of the hump and a Mach wave is radiated to the far field, as the pressure perturbation in panel (a) indicates. Figure 14 further plots the amplitude evolution of the wall pressure $p_w$ normalised by the value at the inlet boundary for four representative frequencies. For DNS calculations, the normalised wall pressure amplitude $p_w$ is defined as

(4.5)\begin{equation} p_w(x)=\frac{\displaystyle\max_{t>t_0}|\tilde{p}(x,\eta=0,t)|}{\displaystyle\max_{t>t_0}|\tilde{p}(x_0,\eta=0,t)|}, \end{equation}

where $t_0$ has been defined in (3.3), and for HLNS calculations,

(4.6)\begin{equation} p_w(x)=|\breve{p}(x,\eta=0)\,{\textrm{e}}^{{\textrm{i}}\alpha_0 x}|. \end{equation}

Note that $x$ is the same as $\xi$. The amplitude curves grow exponentially, but each curve exhibits a kink at the roughness. The HLNS calculations agree perfectly with the DNS results.

Figure 13.

Contours of the instantaneous perturbation pressure (a) and temperature (b) obtained by DNS for case H4, where $\omega =1.78$ and $R=131\,820$.

Figure 14.

Amplitude evolution of the normalised perturbation pressure at the wall obtained by the DNS (continuous curves) and HLNS (circles) approaches for case H4, where $R=131\,820$.

In order to quantify the scattering effect, we further normalise the perturbation amplitude by that of the smooth-surface configuration,

(4.7)\begin{equation} {\mathcal{A}}(x)=\frac{p_w|_{roughness}}{p_w|_{smooth-surface}}, \end{equation}

which is comparable with the amplitude evolution $C$ defined in (2.12). Figure 15 shows the evolution of $\mathcal {A}$ for two representative frequencies for case H4. For $\omega =1.86$, the roughness plays a destabilising role. The normalised amplitude at the outlet boundary reflects the transmission coefficient, which is approximately 3.2 from DNS and HLNS calculations, although the amplitude curves decrease slightly in the downstream region. The latter is due to the re-scattering effect of the roughness ‘wake’, as discussed in § 2.3.3. The asymptotic prediction $C$ obtained by the improved theory agrees well with the numerical results downstream of the roughness, and the mild decrease of the normalised amplitude $\mathcal {A}$ with $\omega$ is clearly reproduced by the asymptotic theory. The leading-order prediction $C_0$ yields a small amount of error, which increases with $x$. In the close neighbourhood of the roughness, the numerical results exhibit a kink, which is over-predicted by the asymptotic theory appreciably. The discrepancy is due to the finite-$R$ effect. For a higher frequency, $\omega =2.20$, the roughness plays a stabilising role, but the asymptotic theory under-predicts this effect quantitatively. The transmission coefficient obtained by HLNS is approximately 0.35, but that predicted by the asymptotic theory is approximately 0.54. For the asymptotic theory, the second-order impact is rather weak because $K_5$ is quite small at this frequency. The kink predicted by the asymptotic theory near the roughness is not obvious in the numerical results. In summary, the asymptotic theory can predict the overall destabilising or stabilising effect of the roughness, although there exists an amount of quantitative error.

Figure 15.

Amplitude evolution of the perturbations with $\omega =1.86$ (a) and 2.20 (b) for case H4 obtained by HLNS, DNS and the improved asymptotic theory, where $C_0$ is the leading-order prediction.

For each roughness height listed in table 2, we can repeat the above calculations for a wide band of frequencies, and obtain the dependence of the transmission coefficients on $\omega$, as shown in figure 16. The HLNS results are confirmed to be sufficiently accurate as compared with the DNS results for case H4, as shown in (figure 16g). The agreement of both the norms and phase angles of the transmission coefficients obtained by the original asymptotic theory and HLNS is reasonable for most of the frequencies, except at the intersection frequency of the first and second modes, $\omega \approx 1.8$. In (figure 16j), the phase angle around this frequency has too many wiggles and so is not shown. The predictions of the improved asymptotic theory are more accurate around this frequency, as compared with the HLNS results. Both the original and improved theories under-predict slightly the stabilising effect for supercritical frequencies, $\omega >\omega _c$. Surprisingly, even when the roughness height is increased to $0.25\delta _{99}$, the improved asymptotic theory can still do a reasonably good job, although the strongest destabilising effect is over-predicted quantitatively.

Figure 16.

Dependence of the norm (a,c,e,g,i) and phase angle (b,d,f,h,j) of the transmission coefficient on frequency $\omega$ for cases H1 (a,b), H2 (c,d), H3 (e,f), H4 (g,h) and H5 (i,j). The DNS results for case H4 are shown in panel (g).

As indicated by (2.39), the transmission coefficient is contributed to by two orders, which are calculated by (2.32) and (2.37), respectively. Figure 17 plots the budget of the two contributors, $|{\mathscr {T}}_0|-1$ and $\varDelta _1|{\mathscr {T}}_1|$, for two representative cases obtained by the improved asymptotic theory. For a small hump height, shown in panel (a), the transmission coefficient is close to unity, indicating that (2.42) is also a good approximation of $\mathscr {T}$. Therefore, the leading-order contributor, $|{\mathscr {T}}_0|-1$, is approximated by $K_{0,i}({\mathcal {D}}_1+h{\mathcal {D}}_2)$. The implication is that the trend of the dependence of $|{\mathscr {T}}_0|-1$ on $\omega$ is proportional to $K_{0,i}$, which can be seen by comparing with the green dashed line in figure 10(b). Its strength is determined by the level of the mean-flow distortion. For both panels, the second-order contributor, $\varDelta _1{\mathscr {T}}_1$, is always small except at $\omega \approx 1.8$, where the leading-order contributor also peaks. Thus, the second-order contributor may only have a limited impact in a narrow frequency band around this frequency.

Figure 17.

Leading- and second-order contributors to the transmission coefficient obtained by the improved asymptotic theory, where (a,b) are for cases H1 and H4, respectively.

The comparison between the improved asymptotic predictions and HLNS calculations for indentation cases are shown in figure 18. For relatively shallow indentations, as shown in figure 18(a), the two results agree reasonably well, although the asymptotic theory under-predicts slightly the stabilising effect for supercritical frequencies. On the other hand, for deeper indentations ($|h| \geqslant 0.20\delta _{99}$), the agreement becomes poorer, i.e. the improved theory over-predicts the destabilising effect in a narrow frequency band around $\omega =1.8$. The critical frequency calculated by the HLNS approach shows an appreciable decrease trend as the indentation depth increases, but that obtained by the asymptotic theory stays almost unchanged. This may be attributed to the impact of the second-order mean-flow distortion that is neglected (2.7), which could be relatively greater for deeper indentations.

Figure 18.

Dependence of the norm of the transmission coefficient on frequency $\omega$ for indentation cases (cases HM2 to HM5), where curves and symbols are for the improved asymptotic solutions and the HLNS calculations, respectively.

The displacement function $A$ or $\tilde {A}$, representing the mean-flow distortion, is negative for humps, but positive for indentations, as shown in figure 9. However, both humps and indentations produce the same scattering effect, i.e. the stabilising effect for supercritical frequencies and destabilising effect for subcritical frequencies, which has been listed as the unexplained phenomenon (i) in the Introduction. Indeed, the asymptotic theory is able to explain this phenomenon clearly. From (2.32) we know that the leading-order scattering is determined by the product of $K_0$ and ${\mathcal {D}}_1+h{\mathcal {D}}_2$. The former is independent of the mean-flow distortion, and therefore is the same for the hump and indentation cases. The latter is proportional to $\bar {\mathcal {D}}_1+H\bar {\mathcal {D}}_2$, whose values for different cases can be found in table 2. It is easy to see that $\bar {\mathcal {D}}_1$ and $H\bar {\mathcal {D}}_2$ have opposite signs, and the scattering effect is determined by their comprehensive effects. For nonlinear hump cases, $\bar {\mathcal {D}}_1$ is negative, and $|\bar {\mathcal {D}}_1|>|H\bar {\mathcal {D}}_2|$, implying that $\bar {\mathcal {D}}_1+H\bar {\mathcal {D}}_2<0$. For nonlinear indentation cases, $\bar {\mathcal {D}}_1$ is positive but smaller than $|H \bar {\mathcal {D}}_2|$, leading to $\bar {\mathcal {D}}_1+H\bar {\mathcal {D}}_2<0$ also. Consequently, the destabilising and stabilising effects would be observed in the same frequency band. $|\bar {\mathcal {D}}_1+H\bar {\mathcal {D}}_2|$ is greater for a hump than that for an indentation with the same $|h|$ or $|H|$, implying that the scattering effect of a nonlinear indentation is always weaker.

4.4.3. Comparison with the HLNS results for a moderate Reynolds number

The error of the asymptotic theories is mainly attributed to two factors: (a) the discrepancy of the mean-flow distortion between the triple-deck theory and DNS calculations, and (b) the discrepancy of the (improved) Rayleigh and O–S operators. Both factors lead to a greater error as $R$ decreases. In this subsection, we choose the Reynolds number $R=26\,364$, which is the same as that of case 1 in Zhao et al. (Reference Zhao, Dong and Yang2019). Note that, in that paper, the Reynolds number $Re$ was defined as $U_\infty L/\nu _\infty$, which is related to the present Reynolds number $R$ via $Re=R^{2}/\varDelta ^{2}$.

Figure 11(a,b) of Zhao et al. (Reference Zhao, Dong and Yang2019) shows the dependence of the transmission coefficient on frequency for different heights $h^{*}/\delta _{99}$ and widths $d^{*}/\delta _{99}$, respectively. We carry out calculations by the improved asymptotic theory for the same parameters, and the comparisons are shown in figure 19. Both families of curves exhibit a critical frequency at $\omega \approx 2$, which distinguishes the stabilising and destabilising effects of the humps. The overall trends agree well, despite the fact that the asymptotic theory over-predicts quantitatively the destabilising effect in the frequency band $1.6< \omega < 2$ and under-predicts the stabilising effect in supercritical frequencies.

Figure 19.

Dependence of the norm of the transmission coefficients on frequency $\omega$ for $R=26\,364$. (a) Fixed $d^{*}/\delta _{99}=2$ with $h^{*}/\delta _{99}=0.05$ (red), 0.1 (green), 0.15 (blue), 0.2 (purple) and 0.25 (black); (b) fixed $h^{*}/\delta _{99}=0.25$ with $d^{*}/\delta _{99}=2$ (black), 4 (red), 6 (green) and 8 (blue). Solid lines: the asymptotic predictions; circles: the HLNS calculations.

Admittedly, the error due to the finite-$R$ effect should be greater than that in the previous subsection, and the narrow width $d^{*}/\delta _{99}=2$ may also lead to a poorer prediction of the mean-flow distortion. However, the asymptotic theory can still predict the HLNS results with a satisfactory accuracy. The reason is that, whether the roughness plays a stabilising or destabilising role is indeed determined by the Rayleigh operator governing the property of the perturbation, which is independent of the mean-flow distortion and is affected weakly by the finite-$R$ and narrow-roughness effects, as is indicated by figure 5. On the other hand, the intensity of the scattering effect is related to the mean-flow calculation, the error of which may be attributed to both the finite-$R$ effect and the narrow-roughness effect. However, the two effects seem to be in the opposite directions, and due to the cancellation effect, the error of the improved asymptotic theory does not increase to an unacceptable state.

One may query why the scattering effect seems to be unchanged as the roughness width increases in figure 19(b), which is the unexplained phenomenon (iii) as listed in the Introduction. According to the triple-deck formalism, the roughness heights for the cases in figure 19(b) are all converted to $H=1.24$, while their widths are converted to $D=0.351$, 0.702, 1.05 and 1.40, respectively. Figure 20 shows $\bar {\mathcal {D}}_1$ and $H\bar {\mathcal {D}}_2$, and their summation. As $D$ increases, $\bar {\mathcal {D}}_1$ decreases, while $H\bar {\mathcal {D}}_2$ increases. Interestingly, the sum of the two stays almost unchanged for different $D$, and so the leading-order scattering effect is almost independent of the roughness width.

Figure 20.

Variations of $\bar {\mathcal {D}}_1+H\bar {\mathcal {D}}_2$ (red solid line), $\bar {\mathcal {D}}_1$ (green dashed line) and $H\bar {\mathcal {D}}_2$ (blue dot-dashed line) with roughness width $D$, where $H=1.24$.

Now let us compare the transmission coefficient for the same dimensionless roughness size, but different Reynolds numbers. The results in figure 16(i) and the blue curves in figure 19(b) are both for $h^{*}/\delta _{99}=0.25$ and $d^{*}/\delta _{99}=8$, but the Reynolds numbers are $R=131\,820$ and 26 364, respectively. For a fixed dimensionless frequency, the scattering effect, measured by $|{\mathscr {T}}-1|$, for the larger $R$ is much greater than that for the smaller $R$, which is listed as the unexplained phenomenon (ii) in the introduction part. For the same frequency, the factor $K_0$ is the same for the two Reynolds numbers, and the scattering effect essentially depends on ${\mathcal {D}}_1+ h{\mathcal {D}}_2$. According to (B2), ${\mathcal {R}}_X$ and ${\mathcal {R}}_Y$ are also the same for the same oncoming condition, then the key factor is $\bar {\mathcal {D}}_1+ H\bar {\mathcal {D}}_2$. Table 3 compares the key parameters for the two Reynolds numbers. For the same $h^{*}/\delta _{99}$ and $d^{*}/\delta _{99}$, as $R$ increases five times, $h$ and $H$ increase $5^{1/4}$ times, while $d$ and $D$ decrease $5^{-1/4}$ times. As a consequence, $HD$ does not change, neither does $H\bar {\mathcal {D}}_2$. However, $|\bar {\mathcal {D}}_1|$ increases remarkably with $R$, leading to a much greater mean-flow distortion. This is why the scattering effect increases with $R$.

Table 3.

Comparison of the key parameters for the two Reynolds numbers.

4.4.4. Impact of Mach number on the efficiency function

In fact, the scattering efficiency is affected by the Mach number, wall temperature, Reynolds number and frequency of the oncoming Mack mode. The above analysis has revealed the effects of the latter two factors by choosing $(M,T_w)=(5.92,6.95)$. In this subsection, we select another group of control parameters, i.e. $(M,T_w)=(4.5,4.4)$, and study the dependence of the scattering effect on $\omega$ for $R=10\,400$ and 104 000. These parameters are selected from Ma & Zhong (Reference Ma and Zhong2003a,b) and case 1 of Dong et al. (Reference Dong, Liu and Wu2020), and the wall temperature is close to the adiabatic temperature at this $M$.

Figure 21 compares the phase speeds and growth rates obtained by the improved Rayleigh equations (symbols) and the O–S equations (curves). The phase speed calculated by the two approaches agree well, but the growth rates show apparent discrepancy. Increase of $R$ leads to a reduction of the difference, which agrees with the observation for $M=5.92$. Being different from the case for $M=5.92$, the first and second modes for $M=4.5$ are separated in two different frequency bands. Therefore, near the first-mode upper-branch and second-mode lower-branch neutral frequencies, the Rayleigh equations become singular. In order to avoid the singularity, we employed the standard method as in Dong et al. (Reference Dong, Liu and Wu2020), namely, the integration path around the critical point, where the mean-flow velocity is equal to the phase speed, is detoured by adding a negative imaginary part to $y$ and extending the mean flow to the complex plane. In panel (a), the synchronisation frequencies for the two Reynolds numbers are also marked, which are $\omega _c\approx 1.87$ and 1.93, respectively.

Figure 21.

Dependence of the phase speed $c_r$ (a) and the growth rate $-\alpha _i$ (b) of 2-D Mack modes on the frequency $\omega\ {\rm for}\ (M,T_w=(4.5,4.4))$. Red solid lines: O–S solution for $R=10\,400$; blue dashed lines: O–S solution for $R=104\,000$; red circles: improved Rayleigh solution for $R=10\,400$; blue squares: improved Rayleigh solution for $R=104\,000$.

Figure 22 shows the efficiency frequency $K_0$ for the two Reynolds numbers. The overall behaviour is the same as that in figure 10(a,b), i.e. $K_0$ shows a remarkable variation in the frequency band $1.5<\omega <2.5$, which includes the second-mode band and a portion of the stable-mode band; $K_{0,i}$ crosses zero at a frequency $\bar {\omega }_c$ which is close to the synchronisation frequency; $K_{0,i}<0$ for subcritical frequencies and $>0$ for supercritical frequencies. As $R$ increases, the maximum of $|K_{0,i}|$ becomes greater and $\bar {\omega }_c$ becomes higher. Comparing figure 22(b) with figure 10(b), we obtain that for similar Reynolds numbers (${\approx }10^{5}$), as $M$ decreases, the maximum of $|K_{0,i}|$ increases, indicating a stronger scattering effect.

Figure 22.

Dependence of the efficiency function $K_0$ on frequency for $(M,T_w=(4.5,4.4))$. The thin horizontal line in (b) represents the zero line, and $\bar \omega _c$ represents the critical frequency for which $K_{0,i}$ crosses zero.