2.1 Governing equations

We consider the three-dimensional incompressible boundary layer over a surface (of a swept wing or wedge), on which small-amplitude distributed surface roughness is present. The flow is described in the Cartesian coordinate system ( $x^{\ast }$ , $y^{\ast }$ , $z^{\ast }$ ) with its origin at the leading edge, where $x^{\ast }$ , $y^{\ast }$ and $z^{\ast }$ are in the chordwise, vertical and spanwise directions, respectively. The coordinates and velocities are non-dimensionalized by the local boundary-layer displacement thickness $\unicode[STIX]{x1D6FF}_{0}^{\ast }$ and chordwise slip velocity $U_{\infty ,0}^{\ast }$ at the chordwise location $x_{0}^{\ast }$ where the resonant interaction occurs, and $x_{0}^{\ast }$ will be also referred to as the resonant point. The Reynolds number is defined as

(2.1) $$\begin{eqnarray}Re=U_{\infty ,0}^{\ast }\unicode[STIX]{x1D6FF}_{0}^{\ast }/\unicode[STIX]{x1D708}^{\ast },\end{eqnarray}$$

where $\unicode[STIX]{x1D708}^{\ast }$ is the kinematic viscosity.

The flow is governed by the non-dimensional incompressible N–S equations,

(2.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}+\unicode[STIX]{x1D735}p-\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}=0,\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\end{eqnarray}$$

where $\boldsymbol{u}=(u,v,w)$ with $u$ , $v$ and $w$ standing for the chordwise, wall-normal and spanwise velocities respectively, and $p$ denotes the pressure non-dimensionalized by $\unicode[STIX]{x1D70C}^{\ast }U_{\infty ,0}^{\ast 2}$ , with $\unicode[STIX]{x1D70C}^{\ast }$ being the reference density. The velocity and pressure, $(\boldsymbol{u},p)$ , are decomposed into a baseflow $(\boldsymbol{U},P)$ and a disturbance part $(\boldsymbol{u}^{\prime },p^{\prime })$ ,

(2.3a,b ) $$\begin{eqnarray}\boldsymbol{u}=\boldsymbol{U}+\boldsymbol{u}^{\prime },\quad p=P+p^{\prime }.\end{eqnarray}$$

The baseflow is assumed to be spanwise uniform. Its velocity field $\boldsymbol{U}=(U,V,W)$ satisfies the non-dimensional boundary-layer equations,

(2.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle U\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}x}+V\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}y}=U_{\infty }\frac{\text{d}U_{\infty }}{\text{d}x}+\frac{1}{Re}\frac{\unicode[STIX]{x2202}^{2}U}{\unicode[STIX]{x2202}y^{2}}, & \displaystyle\end{eqnarray}$$

(2.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle U\frac{\unicode[STIX]{x2202}W}{\unicode[STIX]{x2202}x}+V\frac{\unicode[STIX]{x2202}W}{\unicode[STIX]{x2202}y}=\frac{1}{Re}\frac{\unicode[STIX]{x2202}^{2}W}{\unicode[STIX]{x2202}y^{2}}, & \displaystyle\end{eqnarray}$$

(2.4c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}y}=0, & \displaystyle\end{eqnarray}$$

(2.5a-c ) $$\begin{eqnarray}U=V=W=0\quad \text{at }y=0;\quad U\rightarrow U_{\infty },\quad W\rightarrow W_{\infty }\quad \text{as }y\rightarrow \infty ,\end{eqnarray}$$

where $U_{\infty }$ and $W_{\infty }$ stand for the non-dimensional slip velocities along the chordwise and spanwise directions, respectively. Substituting the decomposition (2.3) into (2.2) and subtracting out the equations governing the baseflow lead to the nonlinear disturbance equations,

(2.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+(\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}+(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{U}+\unicode[STIX]{x1D735}p-\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}=-(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0.\end{eqnarray}$$

Without ambiguity, the prime symbol for disturbance quantities is omitted for simplicity.

2.2 Baseflow

The baseflow is taken to be the Falkner–Skan–Cooke (FSC) boundary layer, which has frequently been used as a convenient vehicle to investigate key receptivity, instability and transition mechanisms of three-dimensional boundary layers (e.g. Högberg & Henningson Reference Högberg and Henningson1998; Schrader et al. Reference Schrader, Brandt and Henningson2009). The chordwise and spanwise slip velocities are given by

(2.7a,b ) $$\begin{eqnarray}\frac{U_{\infty }^{\ast }(x^{\ast })}{U_{\infty }^{\ast }(x_{0}^{\ast })}=\left(\frac{x^{\ast }}{x_{0}^{\ast }}\right)^{m},\quad W_{\infty }^{\ast }=\text{const.},\end{eqnarray}$$

where the acceleration parameter $m$ is related to the Hartree parameter $\unicode[STIX]{x1D6FD}_{H}$ through the equation $m=\unicode[STIX]{x1D6FD}_{H}/(2-\unicode[STIX]{x1D6FD}_{H})$ . The local sweep angle $\unicode[STIX]{x1D719}_{0}$ at the resonant point $x_{0}^{\ast }$ is defined as $\unicode[STIX]{x1D719}_{0}=\tan ^{-1}(W_{\infty }^{\ast }/U_{\infty }^{\ast }(x_{0}^{\ast }))=\tan ^{-1}(W_{\infty })$ on noting that $U_{\infty }(x_{0})=1$ . The values of $m$ and $\unicode[STIX]{x1D719}_{0}$ (or equivalently $\unicode[STIX]{x1D6FD}_{H}$ and $W_{\infty }$ ) can be chosen such that the profiles are representative of the local flow in the boundary layer over a swept wing.

This boundary-layer flow admits a similarity solution. Introducing a similarity variable $\unicode[STIX]{x1D702}$ and a non-dimensional stream function  $\unicode[STIX]{x1D6F9}$ ,

(2.8a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D702}=\sqrt{\frac{m+1}{2}\frac{U_{\infty }Re}{x}}y,\quad \unicode[STIX]{x1D6F9}=\sqrt{\frac{2}{m+1}\frac{U_{\infty }x}{Re}}f(\unicode[STIX]{x1D702}),\end{eqnarray}$$

we may express the baseflow velocities as $U=\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}/\unicode[STIX]{x2202}y$ , $V=-\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}/\unicode[STIX]{x2202}x$ and $W=W_{\infty }g(\unicode[STIX]{x1D702})$ , substitution of which into (2.4) and (2.5) leads to the ordinary differential equations

(2.9a,b ) $$\begin{eqnarray}f^{\prime \prime \prime }+ff^{\prime \prime }+\unicode[STIX]{x1D6FD}_{H}(1-f^{\prime 2})=0,\quad g^{\prime \prime }+fg^{\prime }=0,\end{eqnarray}$$

and the boundary conditions

(2.10a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle f=f^{\prime }=g=0\quad \text{at }\unicode[STIX]{x1D702}=0;\quad f^{\prime }=g^{\prime }=1\quad \text{at }\unicode[STIX]{x1D702}=\infty , & \displaystyle\end{eqnarray}$$

where the symbol $^{\prime }$ stands for the derivative with respect to  $\unicode[STIX]{x1D702}$ .

By the standard definition, the boundary-layer displacement thickness is expressed as

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{0}^{\ast }=\left(\frac{m+1}{2}\frac{U_{\infty }^{\ast }(x_{0}^{\ast })}{\unicode[STIX]{x1D708}^{\ast }x_{0}^{\ast }}\right)^{-1/2}c,\quad c=\int _{0}^{\infty }(1-f^{\prime })\,\text{d}\unicode[STIX]{x1D702}.\end{eqnarray}$$

Non-dimensionalizing the first of the above equations gives the relation between the non-dimensional resonant point $x_{0}$ and the corresponding Reynolds number $Re$ , $x_{0}=(m+1)Re/(2c^{2})$ (Högberg & Henningson Reference Högberg and Henningson1998).

The dimensionless velocities within the boundary layer and their derivatives (with respect to $y$ ) can be expressed as

(2.12a-d ) $$\begin{eqnarray}U(y)=f^{\prime }(\unicode[STIX]{x1D702}),\quad \frac{\text{d}U}{\text{d}y}=cf^{\prime \prime }(\unicode[STIX]{x1D702});\quad W(y)=W_{\infty }g(\unicode[STIX]{x1D702}),\quad \frac{\text{d}W}{\text{d}y}=W_{\infty }cg^{\prime }(\unicode[STIX]{x1D702}).\end{eqnarray}$$

2.3 Crossflow eigenmodes

According to usual linear stability theory, crossflow eigenmodes, which may be present in the absence of roughness, take the normal-mode form,

(2.13) $$\begin{eqnarray}(u,v,w,p)=\unicode[STIX]{x1D716}(\hat{u} ,\hat{v},{\hat{w}},\hat{p})E+\text{c.c.},\quad E=\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}x+\unicode[STIX]{x1D6FD}z-\unicode[STIX]{x1D714}t)},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the chordwise and spanwise wavenumbers, respectively, $\unicode[STIX]{x1D714}$ is the frequency and $\unicode[STIX]{x1D716}\ll 1$ is a measure of the amplitude. If the Reynolds number $Re$ is taken to be asymptotically large, a variety of long-wavelength regimes (with $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\ll 1$ ) arises (Choudhari Reference Choudhari1995). In the present work, we treat $Re$ as being finite, but make the usual parallel-flow approximation, with which substitution of (2.13) into (2.6) leads to a system of the first-order equations,

(2.14) $$\begin{eqnarray}\frac{\text{d}\hat{\unicode[STIX]{x1D753}}}{\text{d}y}=\unicode[STIX]{x1D647}\hat{\unicode[STIX]{x1D753}},\end{eqnarray}$$

where the vector $\hat{\unicode[STIX]{x1D753}}=[\hat{u} ,\hat{v},{\hat{w}},\text{D}\hat{u} ,\hat{p},\text{D}{\hat{w}}]^{\text{T}}$ , and the operator $\unicode[STIX]{x1D647}$ is given by

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D647}=\left[\begin{array}{@{}cccccc@{}}0\quad & 0\quad & 0\quad & 1\quad & 0\quad & 0\\ -\text{i}\unicode[STIX]{x1D6FC}\quad & 0\quad & -\text{i}\unicode[STIX]{x1D6FD}\quad & 0\quad & 0\quad & 0\\ 0\quad & 0\quad & 0\quad & 0\quad & 0\quad & 1\\ ReS\quad & Re\text{D }U\quad & 0\quad & 0\quad & \text{i}\unicode[STIX]{x1D6FC}Re\quad & 0\\ 0\quad & -S\quad & 0\quad & -\text{i}\unicode[STIX]{x1D6FC}/Re\quad & 0\quad & -\text{i}\unicode[STIX]{x1D6FD}/Re\\ 0\quad & Re\text{D}W\quad & ReS\quad & 0\quad & \text{i}\unicode[STIX]{x1D6FD}Re\quad & 0\end{array}\right],\end{eqnarray}$$

with $S=-\text{i}\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FC}U+\text{i}\unicode[STIX]{x1D6FD}W+(\unicode[STIX]{x1D6FC}^{2}+\unicode[STIX]{x1D6FD}^{2})/Re$ and $D$ standing for $\text{d}/\text{d}y$ . The boundary conditions follow from the requirements that the perturbation velocities vanish at the smooth wall and decay to zero in the free stream, and so they can be expressed as

(2.16a,b ) $$\begin{eqnarray}\hat{u} (y)=\hat{v}(y)={\hat{w}}(y)=0\quad \text{at }y=0;\quad \hat{u} (y),\hat{v}(y),{\hat{w}}(y)\rightarrow 0\quad \text{as }y\rightarrow \infty .\end{eqnarray}$$

By eliminating $\hat{p}$ , $\hat{u}$ and ${\hat{w}}$ , the system (2.14) can be reduced to the well-known O–S equation for $\hat{v}$ ,

(2.17) $$\begin{eqnarray}\left[(-\text{i}\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FC}U+\text{i}\unicode[STIX]{x1D6FD}W)(\text{D}^{2}-\unicode[STIX]{x1D6FC}^{2}-\unicode[STIX]{x1D6FD}^{2})-\text{i}\unicode[STIX]{x1D6FC}U^{\prime \prime }-\text{i}\unicode[STIX]{x1D6FD}W^{\prime \prime }-\frac{1}{Re}(\text{D}^{2}-\unicode[STIX]{x1D6FC}^{2}-\unicode[STIX]{x1D6FD}^{2})^{2}\right]\hat{v}=0,\end{eqnarray}$$

but we will solve the eigenvalue problem (2.14)–(2.16) in the primitive variables, for which an efficient algorithm exists (Malik Reference Malik1990). The present finite-Reynolds-number formulation represents a composite theory valid for all long-wavelength (and possibly viscous) regimes (Choudhari Reference Choudhari1995) except the lower-branch regime, where non-parallelism is a leading-order effect (Butler & Wu Reference Butler and Wu2018a ). We are concerned with spatial stability, and so the chordwise wavenumber $\unicode[STIX]{x1D6FC}$ is calculated for a given spanwise wavenumber $\unicode[STIX]{x1D6FD}$ and frequency $\unicode[STIX]{x1D714}$ . Usually $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{r}+\text{i}\unicode[STIX]{x1D6FC}_{i}$ is complex, with $-\unicode[STIX]{x1D6FC}_{i}$ representing the growth rate. The instability modes are referred to as travelling vortices if $\unicode[STIX]{x1D714}\neq 0$ and stationary vortices if $\unicode[STIX]{x1D714}=0$ .

The eigenvalue problem posed by (2.14) and the boundary conditions (2.16) is solved numerically. The equation is discretized by using the fourth-order compact finite-difference scheme of Malik (Reference Malik1990). The resulting algebraic system has non-zero solutions only when the determinant of its coefficient matrix is equal to zero. For a given $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D714}$ , the determinant is computed by Gaussian elimination, and an appropriate $\unicode[STIX]{x1D6FC}$ making the determinant vanish is found by M $\ddot{\text{u}}$ ller iteration.

2.4 Roughness modes

In general, the wall shape can be represented as a superposition of Fourier components,

(2.18) $$\begin{eqnarray}y_{w}=\mathop{\sum }_{\unicode[STIX]{x1D6FC}_{w},\unicode[STIX]{x1D6FD}_{w}}h(\unicode[STIX]{x1D6FC}_{w},\unicode[STIX]{x1D6FD}_{w})\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{w}x+\unicode[STIX]{x1D6FD}_{w}z)}+\text{c.c.},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{w}$ and $\unicode[STIX]{x1D6FD}_{w}$ are the chordwise and spanwise roughness wavenumbers, respectively. As the roughness height $h^{\ast }$ of interest is approximately several microns, which amounts to a few percent of the boundary-layer thickness $\unicode[STIX]{x1D6FF}_{0}^{\ast }$ , it is appropriate to assume that the non-dimensional height $h\equiv h^{\ast }/\unicode[STIX]{x1D6FF}_{0}^{\ast }$ is small enough that a perturbative mathematical scheme is deemed applicable. Then each Fourier mode is independent, and we can calculate the boundary-layer response to each Fourier component,

(2.19) $$\begin{eqnarray}y_{w}=h\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{w}x+\unicode[STIX]{x1D6FD}_{w}z)}+\text{c.c.},\end{eqnarray}$$

which is to be referred to as single-wavenumber roughness, and a more general form (2.18) will be referred to as a multiple-wavenumber roughness. Random background roughness consists of a wide range of spatial wavelengths, possibly from several microns to millimetres. We will in the present paper identify more dangerous components that affect crossflow vortices more effectively than others do. It turned out that those components have length scales comparable to wavelengths of crossflow instability, which are in millimetres in laboratory conditions.

Under the local parallel-flow assumption, the perturbation induced by (2.19) takes the form,

(2.20) $$\begin{eqnarray}[u_{w},v_{w},w_{w},p_{w}]=h[\hat{u} _{w}(y),\hat{v}_{w}(y),{\hat{w}}_{w}(y),\hat{p}_{w}(y)]\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{w}x+\unicode[STIX]{x1D6FD}_{w}z)}+\text{c.c.},\end{eqnarray}$$

where $\hat{u} _{w}(y)$ and etc. are shape functions. Introducing (2.20) into (2.6) leads to the O–S equation for $\hat{v}_{w}$ ,

(2.21) $$\begin{eqnarray}\left[(\text{i}\unicode[STIX]{x1D6FC}_{w}U+\text{i}\unicode[STIX]{x1D6FD}_{w}W)(\text{D}^{2}-\unicode[STIX]{x1D6FC}_{w}^{2}-\unicode[STIX]{x1D6FD}_{w}^{2})-\text{i}\unicode[STIX]{x1D6FC}_{w}U^{\prime \prime }-\text{i}\unicode[STIX]{x1D6FD}_{w}W^{\prime \prime }-\frac{1}{Re}(\text{D}^{2}-\unicode[STIX]{x1D6FC}_{w}^{2}-\unicode[STIX]{x1D6FD}_{w}^{2})^{2}\right]\hat{v}_{w}=0,\end{eqnarray}$$

or equivalently the first-order system,

(2.22) $$\begin{eqnarray}\frac{\text{d}\hat{\unicode[STIX]{x1D753}}_{w}}{\text{d}y}=\unicode[STIX]{x1D647}_{w}\hat{\unicode[STIX]{x1D753}}_{w},\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D753}}_{w}$ and $\unicode[STIX]{x1D647}_{w}$ have similar expressions as $\hat{\unicode[STIX]{x1D753}}$ and $\unicode[STIX]{x1D647}$ in (2.14)–(2.15) for crossflow eigenmodes. At the wavy wall, the no-slip condition must be satisfied. By performing Taylor expansion about $y=0$ for $h\ll 1$ and using the fact that $U_{B}(0)=W_{B}(0)=V_{B}^{\prime }(0)=0$ for a viscous boundary-layer flow, the boundary is shifted to $y=0$ , leading to the inhomogeneous boundary conditions (cf. Choudhari Reference Choudhari1994),

(2.23a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{u} _{w}(0)=-U^{\prime }(0),\quad \hat{v}_{w}(0)=0,\quad {\hat{w}}_{w}(0)=-W^{\prime }(0). & \displaystyle\end{eqnarray}$$

The boundary conditions in the far field are

(2.24) $$\begin{eqnarray}\hat{u} _{w}(y),\hat{v}_{w}(y),{\hat{w}}_{w}(y)\rightarrow 0\quad \text{as }y\rightarrow \infty .\end{eqnarray}$$

The assumption underlying linearization and the simplified boundary condition (2.23) is now made more precise and further justified. With $h\ll 1$ , the induced motion is primarily within a viscous sublayer layer with an $O(R\text{e}^{-1/3}\unicode[STIX]{x1D6FF}_{0}^{\ast })$ thickness near the wall, where the perturbation is $O(h)$ (Choudhari & Duck Reference Choudhari and Duck1996). The perturbation in the sublayer is linear if $h\ll R\text{e}^{-1/3}$ , but becomes nonlinear when $h=O(R\text{e}^{-1/3})$ . For both linear and nonlinear regimes, the sublayer is governed by the steady boundary-layer equations, the Prandtl transform can be applied to convert the boundary to $y=0$ while keeping the equations exactly invariant, and the induced motion in the main layer is $O(R\text{e}^{-1/3}h)$ and remains linear. For the three-dimensional layers of interest, typically $Re=O(10^{3})$ , and so nonlinearity within the sublayer is expected when $h=O(10^{-1})$ . Our focus here is, however, on $h=O(10^{-2})$ , and so it is reasonable to assume that $h\ll R\text{e}^{-1/3}$ , for which the roughness mode is linear in the entire boundary layer.

Discretization of (2.22) and its boundary conditions (2.23), (2.24) by the fourth-order compact finite-difference scheme leads to an inhomogeneous system of linear algebraic equations, which is solved for given roughness wavenumbers $\unicode[STIX]{x1D6FD}_{w}$ and $\unicode[STIX]{x1D6FC}_{w}$ to obtain the shape functions of the roughness mode.

2.5 Generalized resonance mechanisms

When crossflow eigenmodes and roughness mode are present simultaneously in the boundary layer, interactions between them take place. Of particular interest are interactions of resonant nature, occurring at the quadratic order. These include triadic resonance and Bragg scattering generalized to unstable eigenmodes. Consider two crossflow eigenmodes with carrier waves $E_{1}=\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{1}x+\unicode[STIX]{x1D6FD}_{1}z-\unicode[STIX]{x1D714}_{1}t)}$ and $E_{2}=\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{2}x+\unicode[STIX]{x1D6FD}_{2}z-\unicode[STIX]{x1D714}_{2}t)}$ , and one roughness mode $E_{w}=\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{w}x+\unicode[STIX]{x1D6FD}_{w}z)}$ . One of the eigenmodes interacts with the roughness mode to generate the other and vice versa, and this may take place in the two forms:

(2.25) $$\begin{eqnarray}E_{1}=E_{2}E_{w},\end{eqnarray}$$

and

(2.26) $$\begin{eqnarray}E_{1}=E_{2}^{\ast }E_{w},\end{eqnarray}$$

where the superscript $\ast$ denotes the complex conjugate. For (2.25) and (2.26) to hold, the corresponding wavenumbers and frequencies must satisfy the resonance conditions

(2.27) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\text{Im}\,\unicode[STIX]{x1D6FC}_{1}=\text{Im}\,\unicode[STIX]{x1D6FC}_{2},\quad \unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}_{2},\\ \text{Re}\,\unicode[STIX]{x1D6FC}_{1}-\text{Re}\,\unicode[STIX]{x1D6FC}_{2}=\unicode[STIX]{x1D6FC}_{w},\quad \unicode[STIX]{x1D6FD}_{1}-\unicode[STIX]{x1D6FD}_{2}=\unicode[STIX]{x1D6FD}_{w},\end{array}\right\}\end{eqnarray}$$

and

(2.28) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\text{Im}\,\unicode[STIX]{x1D6FC}_{1}=\text{Im}\,\unicode[STIX]{x1D6FC}_{2},\quad \unicode[STIX]{x1D714}_{1}=-\unicode[STIX]{x1D714}_{2},\\ \text{Re}\,\unicode[STIX]{x1D6FC}_{1}+\text{Re}\,\unicode[STIX]{x1D6FC}_{2}=\unicode[STIX]{x1D6FC}_{w},\quad \unicode[STIX]{x1D6FD}_{1}+\unicode[STIX]{x1D6FD}_{2}=\unicode[STIX]{x1D6FD}_{w},\end{array}\right\}\end{eqnarray}$$

respectively. These two forms of resonance will be referred to as triad difference and triad sum, respectively. Each set of the resonance conditions has two parts: the first specifies the requirements that must be satisfied by the eigenmodes involved in the resonance, and the second determines the wavenumbers of the participating roughness mode. For the triad difference, the growth rates and frequencies of the two eigenmodes must be equal, and the roughness wavenumbers are the differences of those of the two eigenmodes; for the triad sum, the growth rates must also be equal but the frequencies must be of opposite sign, and the roughness wavenumbers are the sums of those of the two eigenmodes. That these conditions can be satisfied will be illustrated below with reference to the dispersion characteristics of crossflow vortices.

There is a special case for the triad sum where the two eigenmodes are identical. The corresponding resonance conditions become

(2.29) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D714}=0,\\ 2\text{Re}\,\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{w},\quad 2\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{w}.\end{array}\right\}\end{eqnarray}$$

This kind of resonance will be referred to as generalized Bragg scattering, which can take place between any stationary eigenmode and the roughness element with wavenumbers twice those of the former.

The triad difference, triad sum and Bragg scattering are referred to as the generalized resonance mechanisms in this paper. A roughness component satisfying one of the resonant conditions (2.27)–(2.29) is of special significance as it produces, through quadrati interactions, an $O(h)$ correction to the growth rates of crossflow vortices. In contrast, non-resonant components influence each eigenmode via interactions at cubic order as follows. At the quadratic level, the roughness mode interacts with itself to generate an $O(h^{2})$ mean-flow distortion, and it also interact with the eigenmode to produce an $O(\unicode[STIX]{x1D716}h)$ forcing on the Fourier components ( $\unicode[STIX]{x1D6FC}_{1}\pm \unicode[STIX]{x1D6FC}_{w},\unicode[STIX]{x1D6FD}_{1}\pm \unicode[STIX]{x1D6FD}_{w}$ ), which are not eigenmodes as none of (2.27)–(2.29) are satisfied, and the responses are thus finite. At the cubic order, the mean-flow distortion and ( $\unicode[STIX]{x1D6FC}_{1}\pm \unicode[STIX]{x1D6FC}_{w},\unicode[STIX]{x1D6FD}_{1}\pm \unicode[STIX]{x1D6FD}_{w}$ ) interact with the eigenmode and the roughness mode, respectively, both reproducing the eigenmode whose magnitude is $O(h^{2}\unicode[STIX]{x1D716})$ . The resulting effect is a much smaller $O(h^{2})$ correction to the growth rate.

The crossflow eigenmodes and roughness modes satisfying the resonance conditions of the generalized resonance mechanisms are searched for stationary and non-stationary cases by applying the first part of each set of the resonance conditions. This requires computation of unstable crossflow eigenmodes by linear stability analysis. For illustration, the results for the baseflow parameters of $\unicode[STIX]{x1D719}_{0}=45^{\circ }$ , $m=0.34207$ , $Re=338$ and 2000 are presented here; the value of $m$ is taken to be the same as in Högberg & Henningson (Reference Högberg and Henningson1998) so that our linear stability results can be validated by comparison with theirs. Figure 1 displays the results for the stationary case. As figure 1(a) shows, the variation of $\unicode[STIX]{x1D6FC}_{r}$ against $\unicode[STIX]{x1D6FD}$ is almost linear, which is a remarkable property and will have an important implication for resonance. Figure 1(b) shows that instability exists ( $-\unicode[STIX]{x1D6FC}_{i}>0$ ) in an interval of $\unicode[STIX]{x1D6FD}$ , the ends of which correspond to two neutral eigenmodes, both of which turn out to be relevant for understanding the response of the boundary layer to roughness. Unstable modes to the left/right of the most unstable one will be referred to as left/right branches, respectively. The schematic of resonance for stationary vortices is shown in figure 1(b). Two stationary vortices can have equal growth rate ( $-\unicode[STIX]{x1D6FC}_{i}$ ), and they are denoted as E1 and E2 in the figure. They can interact with each other in the presence of suitable roughness through the triad difference and triad sum, referred to as $1-2$ and $1+2$ , respectively. Furthermore, each mode may interact with a roughness mode through Bragg scattering, denoted by $1+1$ and $2+2$ , respectively. Note that the crossflow eigenmodes are denoted by numbers, and the roughness modes are designated by numbers and symbols ‘ $+$ ’ and ‘ $-$ ’, indicating that the resonance mechanism involved is the triad sum and difference, respectively. Stationary eigenmodes 1 and 2 can also interact with a multiple-wavenumber roughness comprised of the four components, in which case multiple generalized resonance mechanisms take effect at the same time.

Figure 1.

Instability characteristics of stationary vortices for the parameters: $\unicode[STIX]{x1D719}_{0}=45^{\circ }$ , $m=0.34207$ and $Re=338$ . (a) The variation of $\unicode[STIX]{x1D6FC}_{r}$ with $\unicode[STIX]{x1D6FD}$ . (b) The variation of $-\unicode[STIX]{x1D6FC}_{i}$ with $\unicode[STIX]{x1D6FD}$ and the schematic of the resonances, where the letters ‘E’ and ‘R’ stand for eigenmode and roughness mode, respectively. The eigenmodes with the same growth rate are designated by numbers 1 and 2, and the $\pm$ signs signify the triadic difference/sum interactions.

Figure 2.

Contours of growth rate and schematic of the resonances for travelling vortices. The parameters are $\unicode[STIX]{x1D719}_{0}=45^{\circ }$ , $m=0.34207$ , (a) $Re=338$ and (b) $Re=2000$ . The letters ‘E’ and ‘R’ stand for ‘eigenmode’ and ‘roughness mode’, respectively.

Now we turn to resonances involving travelling vortices. Figure 2 shows the contours of the growth rate at a low and a high Reynolds number. For the former, every contour line is convex so that we can locate up to four travelling eigenmodes with equal $|\unicode[STIX]{x1D714}|$ and $-\unicode[STIX]{x1D6FC}_{i}$ , which are labelled as E1, E2, E3 and E4 in figure 2(a). The frequency of travelling vortices 1 and 2 is negative, and that of 3 and 4 is positive. These four travelling vortices can interact with six single-wavenumber components, which are $1+3$ , $1+4$ , $2+3$ and $2+4$ through the triad sum, and $1-2$ and $3-4$ through the triad difference. Figure 2(b) indicates that at a higher $Re$ contour lines of small growth rates can be concave for certain range of $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D714}$ , where up to six travelling eigenmodes with equal $|\unicode[STIX]{x1D714}|$ and $-\unicode[STIX]{x1D6FC}_{i}$ may be located. For simplicity, in this paper we concentrate only on the case of four travelling eigenmodes, but the methodology can be easily extended to the more complex case.

There exist connections between stationary and non-stationary cases. When the frequency of the four travelling eigenmodes is reduced to zero, eigenmodes 3 and 4 coalesce with 1 and 2, respectively. It follows that roughness $3-4$ becomes the same as $1-2$ , while roughnesses $1+4$ and $2+3$ collapse, both becoming equivalent to roughness $1+2$ for the stationary case; see figure 1(b). Roughnesses $1+3$ and $2+4$ become $1+1$ and $2+2$ , respectively, i.e. these triad-sum interactions degenerate into Bragg scattering.

3.1 Interaction of eigenmodes with single-wavenumber roughness: triad difference

The interaction through the triad difference (2.27) involves two eigenmodes $j$ and $k$ , and one single-wavenumber roughness. The disturbance $\unicode[STIX]{x1D719}$ (standing for $u$ , $v$ , $w$ and  $p$ ), comprising of the roughness mode and the eigenmodes, can be represented by

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D719}=h[\hat{\unicode[STIX]{x1D719}}_{w}E_{w}+\text{c.c.}]+\unicode[STIX]{x1D716}[A_{j}(\bar{x})\hat{\unicode[STIX]{x1D719}}_{j0}(y)E_{j}+h\hat{\unicode[STIX]{x1D719}}_{j1}E_{j}+\text{c.c.}]+\unicode[STIX]{x1D716}[A_{k}(\bar{x})\hat{\unicode[STIX]{x1D719}}_{k0}(y)E_{k}+h\hat{\unicode[STIX]{x1D719}}_{k1}E_{k}+\text{c.c.}],\end{eqnarray}$$

where $E_{w}=\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{w}x+\unicode[STIX]{x1D6FD}_{w}z)}$ , $E_{j}=\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{j}x+\unicode[STIX]{x1D6FD}_{j}z-\unicode[STIX]{x1D714}_{j}t)}$ and $E_{k}=\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}_{k}x+\unicode[STIX]{x1D6FD}_{k}z-\unicode[STIX]{x1D714}_{k}t)}$ . $A_{j}$ and $A_{k}$ denote the amplitude functions of the respective eigenmodes. Substituting (3.1) into (2.6), at $O(\unicode[STIX]{x1D716})$ and $O(h)$ we recover respectively the eigenvalue problem and the inhomogeneous problem determining the roughness mode. At $O(\unicode[STIX]{x1D716}h)$ , we obtain the equations for $\hat{\unicode[STIX]{x1D719}}_{j1}$ and $\hat{\unicode[STIX]{x1D719}}_{k1}$ ,

(3.2a ) $$\begin{eqnarray}\displaystyle \frac{\text{d}\hat{\unicode[STIX]{x1D753}}_{j1}}{\text{d}y} & = & \displaystyle \unicode[STIX]{x1D647}_{j}\hat{\unicode[STIX]{x1D753}}_{j1}+\frac{\text{d}A_{j}}{\text{d}\bar{x}}\boldsymbol{f}_{j}+A_{k}\,\boldsymbol{f}_{k-w},\end{eqnarray}$$

(3.2b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}\hat{\unicode[STIX]{x1D753}}_{k1}}{\text{d}y} & = & \displaystyle \unicode[STIX]{x1D647}_{k}\hat{\unicode[STIX]{x1D753}}_{k1}+\frac{\text{d}A_{k}}{\text{d}\bar{x}}\boldsymbol{f}_{k}+A_{j}\,\boldsymbol{f}_{j-w^{\ast }},\end{eqnarray}$$

$\hat{\unicode[STIX]{x1D753}}_{j,k1}$

$\unicode[STIX]{x1D647}_{j,k}$

$\hat{\unicode[STIX]{x1D753}}$

$\unicode[STIX]{x1D647}$

$\unicode[STIX]{x1D6FC}$

$\unicode[STIX]{x1D6FD}$

$\unicode[STIX]{x1D6FC}_{j,k}$

$\unicode[STIX]{x1D6FD}_{j,k}$

(3.3) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\boldsymbol{f}_{j}=[0,-f_{j}^{c},0,Ref_{j}^{x},-\text{D}f_{j}^{c}/Re-f_{j}^{y},Ref_{j}^{z}]^{\text{T}},\\ \boldsymbol{f}_{k-w}=[0,0,0,Ref_{k-w}^{x},-f_{k-w}^{y},Ref_{k-w}^{z}]^{\text{T}},\end{array}\right\}\end{eqnarray}$$

where

(3.4a-d ) $$\begin{eqnarray}f_{j}^{x}=\left(U-\frac{2\text{i}\unicode[STIX]{x1D6FC}_{j}}{Re}\right)\hat{u} _{j0}+\hat{p}_{j0},\quad f_{j}^{y}=\left(U-\frac{2\text{i}\unicode[STIX]{x1D6FC}_{j}}{Re}\right)\hat{v}_{j0},\quad f_{j}^{z}=\left(U-\frac{2\text{i}\unicode[STIX]{x1D6FC}_{j}}{Re}\right){\hat{w}}_{j0},\quad f_{j}^{c}=\hat{u} _{j0},\end{eqnarray}$$

and the Reynolds stresses

(3.5) $$\begin{eqnarray}\displaystyle & f_{k-w}^{x}=\text{i}\unicode[STIX]{x1D6FC}_{w}\hat{u} _{k0}\hat{u} _{w}+\text{i}\unicode[STIX]{x1D6FC}_{k}\hat{u} _{k0}\hat{u} _{w}+\hat{v}_{k0}\text{D}\hat{u} _{w}+\text{D}\hat{u} _{k0}\hat{v}_{w}+\text{i}\unicode[STIX]{x1D6FD}_{w}{\hat{w}}_{k0}\hat{u} _{w}+\text{i}\unicode[STIX]{x1D6FD}_{k}\hat{u} _{k0}{\hat{w}}_{w}, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & f_{k-w}^{y}=\text{i}\unicode[STIX]{x1D6FC}_{w}\hat{u} _{k0}\hat{v}_{w}+\text{i}\unicode[STIX]{x1D6FC}_{k}\hat{v}_{k0}\hat{u} _{w}+\hat{v}_{k0}\text{D}\hat{v}_{w}+\text{D}\hat{v}_{k0}\hat{v}_{w}+\text{i}\unicode[STIX]{x1D6FD}_{w}{\hat{w}}_{k0}\hat{v}_{w}+\text{i}\unicode[STIX]{x1D6FD}_{k}\hat{v}_{k0}{\hat{w}}_{w}, & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & f_{k-w}^{z}=\text{i}\unicode[STIX]{x1D6FC}_{w}\hat{u} _{k0}{\hat{w}}_{w}+\text{i}\unicode[STIX]{x1D6FC}_{k}{\hat{w}}_{k0}\hat{u} _{w}+\hat{v}_{k0}\text{D}{\hat{w}}_{w}+\text{D}{\hat{w}}_{k0}\hat{v}_{w}+\text{i}\unicode[STIX]{x1D6FD}_{w}{\hat{w}}_{k0}{\hat{w}}_{w}+\text{i}\unicode[STIX]{x1D6FD}_{k}{\hat{w}}_{k0}{\hat{w}}_{w}. & \displaystyle\end{eqnarray}$$

The expression for $\boldsymbol{f}_{k}$ follows from $\boldsymbol{f}_{j}$ by changing $j$ to $k$ , and the expression for $\boldsymbol{f}_{j-w^{\ast }}$ from $\boldsymbol{f}_{k-w}$ by the substitutions: $k\rightarrow j$ , $(\unicode[STIX]{x1D6FC}_{w},\unicode[STIX]{x1D6FD}_{w})\rightarrow (-\unicode[STIX]{x1D6FC}_{w},-\unicode[STIX]{x1D6FD}_{w})$ and $\unicode[STIX]{x1D719}_{w}\rightarrow \unicode[STIX]{x1D719}_{w}^{\ast }$ . The corresponding boundary conditions at $y=0$ for (3.2) are given by

(3.8a ) $$\begin{eqnarray}\displaystyle & \hat{u} _{j1}(0)=-A_{k}\hat{u} _{k0}^{\prime }(0),\quad \hat{v}_{j1}(0)=0,\quad {\hat{w}}_{j1}(0)=-A_{k}{\hat{w}}_{k0}^{\prime }(0); & \displaystyle\end{eqnarray}$$

(3.8b ) $$\begin{eqnarray}\displaystyle & \hat{u} _{k1}(0)=-A_{j}\hat{u} _{j0}^{\prime }(0),\quad \hat{v}_{k1}(0)=0,\quad {\hat{w}}_{k1}(0)=-A_{j}{\hat{w}}_{j0}^{\prime }(0). & \displaystyle\end{eqnarray}$$

In order for the inhomogeneous boundary-value problem (3.2a ) with (3.8a ), or (3.2b ) with (3.8b ), to have a solution, a solvability condition must be satisfied. This can be derived by reducing (3.2a ) or (3.2b ) to an inhomogeneous O–S equation, and then using the adjoint eigenfunction. Alternatively, we may discretize (3.2a ) and (3.2b ), and apply Gaussian elimination to the resulting algebraic systems. Either way, we obtain two coupled amplitude equations,

(3.9a ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{j}}{\text{d}\bar{x}}+F_{k-w}A_{k}(\bar{x}) & = & \displaystyle 0,\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{k}}{\text{d}\bar{x}}+F_{j-w^{\ast }}A_{j}(\bar{x}) & = & \displaystyle 0,\end{eqnarray}$$

$F_{k-w}$

$F_{j-w^{\ast }}$

The amplitude equations (3.9) can be rearranged into the matrix form

(3.10) $$\begin{eqnarray}\frac{\text{d}}{\text{d}\bar{x}}[A_{j},A_{k}]^{\text{T}}=\unicode[STIX]{x1D641}[A_{j},A_{k}]^{\text{T}},\end{eqnarray}$$

with $\unicode[STIX]{x1D641}$ being the $2\times 2$ matrix,

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D641}=-\left[\begin{array}{@{}cc@{}}0 & F_{k-w}\\ F_{j-w^{\ast }} & 0\end{array}\right].\end{eqnarray}$$

Seeking solutions of the exponential form,

(3.12) $$\begin{eqnarray}[A_{j},A_{k}]^{\text{T}}=\boldsymbol{a}\text{e}^{\unicode[STIX]{x1D706}\bar{x}}=[a_{j},a_{k}]^{\text{T}}\text{e}^{\unicode[STIX]{x1D706}\bar{x}},\end{eqnarray}$$

leads to an eigenvalue problem

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D706}\boldsymbol{a}=\unicode[STIX]{x1D641}\boldsymbol{a}.\end{eqnarray}$$

The physical meanings of the eigenvalue $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{r}+\text{i}\unicode[STIX]{x1D706}_{i}$ and eigenvector $\boldsymbol{a}$ are now analysed. In terms of $\boldsymbol{a}$ and $\unicode[STIX]{x1D706}$ , the chordwise disturbance velocity $u_{j}$ and $u_{k}$ at the leading order can be written as

(3.14a ) $$\begin{eqnarray}\displaystyle A_{j}\hat{u} _{j0}E_{j} & = & \displaystyle a_{j}\hat{u} _{j0}\text{e}^{(\unicode[STIX]{x1D706}_{r}h-\text{Im}\,\unicode[STIX]{x1D6FC}_{j})x}\text{e}^{\text{i}(\text{Re}\,\unicode[STIX]{x1D6FC}_{j}x+\unicode[STIX]{x1D706}_{i}hx+\unicode[STIX]{x1D6FD}_{j}z-\unicode[STIX]{x1D714}_{j}t)},\end{eqnarray}$$

(3.14b ) $$\begin{eqnarray}\displaystyle A_{k}\hat{u} _{k0}E_{k} & = & \displaystyle a_{k}\hat{u} _{k0}\text{e}^{(\unicode[STIX]{x1D706}_{r}h-\text{Im}\,\unicode[STIX]{x1D6FC}_{k})x}\text{e}^{\text{i}(\text{Re}\,\unicode[STIX]{x1D6FC}_{k}x+\unicode[STIX]{x1D706}_{i}hx+\unicode[STIX]{x1D6FD}_{k}z-\unicode[STIX]{x1D714}_{k}t)},\end{eqnarray}$$

$(\unicode[STIX]{x1D706}_{r}h-\text{Im}\,\unicode[STIX]{x1D6FC}_{j})$

$j$

$(\unicode[STIX]{x1D706}_{r}h-\text{Im}\,\unicode[STIX]{x1D6FC}_{k})$

$k$

$j$

$k$

$\unicode[STIX]{x1D706}_{r}h$

$h$

$(-\unicode[STIX]{x1D6FC}_{i},\unicode[STIX]{x1D714})$

$\unicode[STIX]{x1D706}_{r}$

$-\unicode[STIX]{x1D6FC}_{i}$

$\unicode[STIX]{x1D714}$

(3.15) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{r}=\unicode[STIX]{x1D706}_{r}(-\unicode[STIX]{x1D6FC}_{i},\unicode[STIX]{x1D714}).\end{eqnarray}$$

The eigenvector $\boldsymbol{a}$ represents the ratio of the amplitudes of the two interacting eigenmodes. The amplitude ratio, defined as

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D70C}(-\unicode[STIX]{x1D6FC}_{i},\unicode[STIX]{x1D714})=|a_{j}/a_{k}|,\end{eqnarray}$$

is also a function of $-\unicode[STIX]{x1D6FC}_{i}$ and $\unicode[STIX]{x1D714}$ . It transpires that the two eigenmodes are coupled by small-amplitude roughness, whereas they are independent of each other for the smooth-wall case in the linear regime.

3.2 Interaction of eigenmodes with single-wavenumber roughness: triad sum

The disturbance $\unicode[STIX]{x1D719}$ remains of the form (3.1). By a similar procedure as that for the triad difference (2.28), the equations at $O(\unicode[STIX]{x1D716}h)$ are given as

(3.17a ) $$\begin{eqnarray}\displaystyle \frac{\text{d}\hat{\unicode[STIX]{x1D753}}_{j1}}{\text{d}y}=\unicode[STIX]{x1D647}_{j0}\hat{\unicode[STIX]{x1D753}}_{j1}+\frac{\text{d}A_{j}}{\text{d}\bar{x}}\boldsymbol{f}_{j}+A_{k}^{\ast }\boldsymbol{f}_{k^{\ast }-w}, & & \displaystyle\end{eqnarray}$$

(3.17b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}\hat{\unicode[STIX]{x1D753}}_{k1}}{\text{d}y}=\unicode[STIX]{x1D647}_{k0}\hat{\unicode[STIX]{x1D753}}_{k1}+\frac{\text{d}A_{k}}{\text{d}\bar{x}}\boldsymbol{f}_{k}+A_{j}^{\ast }\boldsymbol{f}_{j^{\ast }-w}, & & \displaystyle\end{eqnarray}$$

$\boldsymbol{f}_{j}$

$\boldsymbol{f}_{k}$

$\boldsymbol{f}_{k^{\ast }-w}$

(3.18) $$\begin{eqnarray}\boldsymbol{f}_{k^{\ast }-w}=[0,0,0,Ref_{k^{\ast }-w}^{x},-f_{k^{\ast }-w}^{y},Ref_{k^{\ast }-w}^{z}]^{\text{T}},\end{eqnarray}$$

with the Reynolds stresses

(3.19) $$\begin{eqnarray}\displaystyle & f_{k^{\ast }-w}^{x}=\text{i}\unicode[STIX]{x1D6FC}_{w}\hat{u} _{k0}^{\ast }\hat{u} _{w}-\text{i}\unicode[STIX]{x1D6FC}_{k}^{\ast }\hat{u} _{k0}^{\ast }\hat{u} _{w}+\hat{v}_{k0}^{\ast }\text{D}\hat{u} _{w}+\text{D}\hat{u} _{k0}^{\ast }\hat{v}_{w}+\text{i}\unicode[STIX]{x1D6FD}_{w}{\hat{w}}_{k0}^{\ast }\hat{u} _{w}-\text{i}\unicode[STIX]{x1D6FD}_{k}\hat{u} _{k0}^{\ast }{\hat{w}}_{w}, & \displaystyle\end{eqnarray}$$

(3.20) $$\begin{eqnarray}\displaystyle & f_{k^{\ast }-w}^{y}=\text{i}\unicode[STIX]{x1D6FC}_{w}\hat{u} _{k0}^{\ast }\hat{v}_{w}-\text{i}\unicode[STIX]{x1D6FC}_{k}^{\ast }\hat{v}_{k0}^{\ast }\hat{u} _{w}+\hat{v}_{k0}^{\ast }\text{D}\hat{v}_{w}+\text{D}\hat{v}_{k0}^{\ast }\hat{v}_{w}+\text{i}\unicode[STIX]{x1D6FD}_{w}{\hat{w}}_{k0}^{\ast }\hat{v}_{w}-\text{i}\unicode[STIX]{x1D6FD}_{k}\hat{v}_{k0}^{\ast }{\hat{w}}_{w}, & \displaystyle\end{eqnarray}$$

(3.21) $$\begin{eqnarray}\displaystyle & f_{k^{\ast }-w}^{z}=\text{i}\unicode[STIX]{x1D6FC}_{w}\hat{u} _{k0}^{\ast }{\hat{w}}_{w}-\text{i}\unicode[STIX]{x1D6FC}_{k}^{\ast }{\hat{w}}_{k0}^{\ast }\hat{u} _{w}+\hat{v}_{k0}^{\ast }\text{D}{\hat{w}}_{w}+\text{D}{\hat{w}}_{k0}^{\ast }\hat{v}_{w}+\text{i}\unicode[STIX]{x1D6FD}_{w}{\hat{w}}_{k0}^{\ast }{\hat{w}}_{w}-\text{i}\unicode[STIX]{x1D6FD}_{k}{\hat{w}}_{k0}^{\ast }{\hat{w}}_{w}.\quad & \displaystyle\end{eqnarray}$$

The expression for $\boldsymbol{f}_{j^{\ast }-w}$ follows from $\boldsymbol{f}_{k^{\ast }-w}$ by replacing $k$ by $j$ . The corresponding boundary conditions at $y=0$ are

(3.22a ) $$\begin{eqnarray}\displaystyle \hat{u} _{j1} & = & \displaystyle -A_{k}^{\ast }\hat{u} _{k0}^{\ast ^{\prime }}(0),\quad \hat{v}_{j1}=0,\quad {\hat{w}}_{j1}=-A_{k}^{\ast }{\hat{w}}_{k0}^{\ast ^{\prime }}(0);\end{eqnarray}$$

(3.22b ) $$\begin{eqnarray}\displaystyle \hat{u} _{k1} & = & \displaystyle -A_{j}^{\ast }\hat{u} _{j0}^{\ast ^{\prime }}(0),\quad \hat{v}_{k1}=0,\quad {\hat{w}}_{k1}=-A_{j}^{\ast }{\hat{w}}_{j0}^{\ast ^{\prime }}(0),\end{eqnarray}$$

Applying solvability conditions to (3.17a ) with (3.22a ) and (3.17b ) with (3.22b ) leads to two coupled amplitude equations

(3.23a ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{j}}{\text{d}\bar{x}}+F_{k^{\ast }-w}A_{k}^{\ast } & = & \displaystyle 0,\end{eqnarray}$$

(3.23b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{k}}{\text{d}\bar{x}}+F_{j^{\ast }-w}A_{j}^{\ast } & = & \displaystyle 0.\end{eqnarray}$$

(3.24) $$\begin{eqnarray}\frac{\text{d}}{\text{d}\bar{x}}[\text{Re}\,A_{j},\text{Im}\,A_{j},\text{Re}\,A_{k},\text{Im}\,A_{k}]^{\text{T}}=\unicode[STIX]{x1D641}[\text{Re}\,A_{j},\text{Im}\,A_{j},\text{Re}\,A_{k},\text{Im}\,A_{k}]^{\text{T}},\end{eqnarray}$$

where the $4\times 4$ coefficient matrix

(3.25) $$\begin{eqnarray}\unicode[STIX]{x1D641}=-\left[\begin{array}{@{}cccc@{}}0\quad & 0\quad & \text{Re}\,F_{k^{\ast }-w}\quad & \text{Im}\,F_{k^{\ast }-w}\\ 0\quad & 0\quad & \text{Im}\,F_{k^{\ast }-w}\quad & -\text{Re}\,F_{k^{\ast }-w}\\ \text{Re}\,F_{j^{\ast }-w}\quad & \text{Im}\,F_{j^{\ast }-w}\quad & 0\quad & 0\\ \text{Im}\,F_{j^{\ast }-w}\quad & -\text{Re}\,F_{j^{\ast }-w}\quad & 0\quad & 0\end{array}\right]\end{eqnarray}$$

is real-valued. Equation (3.24) admits solutions of the exponential form,

(3.26) $$\begin{eqnarray}[\text{Re}\,A_{j},\text{Im}\,A_{j},\text{Re}\,A_{k},\text{Im}\,A_{k}]^{\text{T}}={\textstyle \frac{1}{2}}(\boldsymbol{a}\text{e}^{\unicode[STIX]{x1D706}\bar{x}}+\text{c.c.}),\end{eqnarray}$$

substituting of which into (3.24) yields the eigenvalue problem, $\unicode[STIX]{x1D706}\boldsymbol{a}=\unicode[STIX]{x1D641}\boldsymbol{a}$ . The real part of $\unicode[STIX]{x1D706}$ , $\unicode[STIX]{x1D706}_{r}$ , represents the growth-rate correction coefficient, the same as that for the triad difference. However, the definition of the amplitude ratio $\unicode[STIX]{x1D70C}$ for the triad sum is different. Note that the eigenvector $\boldsymbol{a}$ can be expressed in the polar form,

(3.27) $$\begin{eqnarray}\boldsymbol{a}=[a_{j1},a_{j2},a_{k1},a_{k2}]^{\text{T}}=[|a_{j1}|\text{e}^{\text{i}\unicode[STIX]{x1D703}_{j1}},|a_{j2}|\text{e}^{\text{i}\unicode[STIX]{x1D703}_{j2}},|a_{k1}|\text{e}^{\text{i}\unicode[STIX]{x1D703}_{k1}},|a_{k2}|\text{e}^{\text{i}\unicode[STIX]{x1D703}_{k2}}]^{\text{T}}.\end{eqnarray}$$

According to (3.24), (3.26) and (3.27), the modulus of $A_{j,k}$ can be expressed as

(3.28) $$\begin{eqnarray}|A_{j,k}|^{2}=\text{e}^{2\unicode[STIX]{x1D706}_{r}\bar{x}}|a_{j,k}|^{2}=\text{e}^{2\unicode[STIX]{x1D706}_{r}\bar{x}}[|a_{j,k1}|^{2}\cos ^{2}(\unicode[STIX]{x1D706}_{i}\bar{x}+\unicode[STIX]{x1D703}_{j,k1})+|a_{j,k2}|^{2}\cos ^{2}(\unicode[STIX]{x1D706}_{i}\bar{x}+\unicode[STIX]{x1D703}_{j,k2})].\end{eqnarray}$$

For the triad sum, with the coefficient matrix $\unicode[STIX]{x1D641}$ given by (3.25), the elements of the eigenvector $\boldsymbol{a}$ satisfy

(3.29a,b ) $$\begin{eqnarray}a_{j1}=\text{i}a_{j2},\quad a_{k1}=\text{i}a_{k2},\end{eqnarray}$$

from which it follows $|a_{j1}|=|a_{j2}|$ , $|a_{k1}|=|a_{k2}|$ , $\unicode[STIX]{x1D703}_{j1}=\unicode[STIX]{x1D703}_{j2}+\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x1D703}_{k1}=\unicode[STIX]{x1D703}_{k2}+\unicode[STIX]{x03C0}/2$ . The amplitude ratio $\unicode[STIX]{x1D70C}$ for the triad sum is thus defined as

(3.30) $$\begin{eqnarray}\unicode[STIX]{x1D70C}(-\unicode[STIX]{x1D6FC}_{i},\unicode[STIX]{x1D714})=|a_{j1}|/|a_{k1}|.\end{eqnarray}$$

For the interaction between the eigenmodes and multiple-wavenumber roughness, which will be discussed later, the relations in (3.29) may not hold. Using trigonometric identities, $a_{j,k}$ can be expressed as

(3.31) $$\begin{eqnarray}|a_{j,k}|^{2}=G_{j,k}\cos (2\unicode[STIX]{x1D706}_{i}\bar{x})-H_{j,k}\sin (2\unicode[STIX]{x1D706}_{i}\bar{x})+(|a_{j,k1}|^{2}+|a_{j,k2}|^{2})/2,\end{eqnarray}$$

where $G_{j,k}=|a_{j,k1}|^{2}\cos (2\unicode[STIX]{x1D703}_{j,k1})/2+|a_{j,k2}|^{2}\cos (2\unicode[STIX]{x1D703}_{j,k2})/2$ and $H_{j,k}=|a_{j,k1}|^{2}\sin (2\unicode[STIX]{x1D703}_{j,k1})/2+|a_{j,k2}|^{2}\sin (2\unicode[STIX]{x1D703}_{j,k2})/2$ . The maximum value of $|a_{j,k}|$ is given by

(3.32) $$\begin{eqnarray}|a_{j,k}|_{max}^{2}=\sqrt{G_{j,k}^{2}+H_{j,k}^{2}}+(|a_{j,k1}|^{2}+|a_{j,k2}|^{2})/2.\end{eqnarray}$$

Now the amplitude ratio is defined as

(3.33) $$\begin{eqnarray}\unicode[STIX]{x1D70C}=|a_{j}|_{max}/|a_{k}|_{max},\end{eqnarray}$$

from which we can recover (3.30) with $|a_{j,k}|_{max}=|a_{j,k1}|$ following from the relations in (3.29). Note that for the triad sum the frequencies of the two eigenmodes are of opposite sign. We use the positive one as the control parameter for $\unicode[STIX]{x1D706}_{r}$ and $\unicode[STIX]{x1D70C}$ .

For stationary vortices, the triad difference and sum both operate, but an additional interaction, Bragg scattering, exists. Following a similar procedure as for the triad sum, the amplitude equation can be derived as

(3.34) $$\begin{eqnarray}\frac{\text{d}A}{\text{d}\bar{x}}+F_{w}A^{\ast }=0,\end{eqnarray}$$

which can be rewritten into a $2\times 2$ system for $[\text{Re}\,A,\text{Im}\,A]^{\text{T}}$ . The growth-rate correction coefficient for Bragg scattering can be obtained by the same method as that for the triad sum. Since only one stationary eigenmode is involved in Bragg scattering, the issue of amplitude ratio does not arise.

3.3 Interaction of crossflow eigenmodes with multiple-wavenumber roughness

The analysis is now extended to the interactions between crossflow eigenmodes and multiple-wavenumber roughness. Stationary and non-stationary cases must be considered separately due to different forms of resonance.

Two stationary vortices with the same growth rate can in general interact simultaneously with four roughness modes: namely, the triad sum and difference interactions with $(\unicode[STIX]{x1D6FC}_{w_{1\pm 2}},\unicode[STIX]{x1D6FD}_{w_{1\pm 2}})$ , and Bragg scattering by $(\unicode[STIX]{x1D6FC}_{w_{1+1}},\unicode[STIX]{x1D6FD}_{w_{1+1}})$ and $(\unicode[STIX]{x1D6FC}_{w_{2+2}},\unicode[STIX]{x1D6FD}_{w_{2+2}})$ . The disturbance comprised of these modes takes the form,

(3.35) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719} & = & \displaystyle h(\hat{\unicode[STIX]{x1D719}}_{w_{1-2}}E_{w_{1-2}}+\text{c.c.})+h(\hat{\unicode[STIX]{x1D719}}_{w_{1+2}}E_{w_{1+2}}+\text{c.c.})\nonumber\\ \displaystyle & & \displaystyle +\,h(\hat{\unicode[STIX]{x1D719}}_{w_{1+1}}E_{w_{1+1}}+\text{c.c.})+h(\hat{\unicode[STIX]{x1D719}}_{w_{2+2}}E_{w_{2+2}}+\text{c.c.})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D716}[A_{1}(\bar{x})\hat{\unicode[STIX]{x1D719}}_{10}(y)E_{1}+h\hat{\unicode[STIX]{x1D719}}_{11}E_{1}+\text{c.c.}]+\unicode[STIX]{x1D716}[A_{2}(\bar{x})\hat{\unicode[STIX]{x1D719}}_{20}(y)E_{2}+h\hat{\unicode[STIX]{x1D719}}_{21}E_{2}+\text{c.c.}].\qquad\end{eqnarray}$$

Following similar steps as before, we can derive the two coupled amplitude equations,

(3.36a ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{1}}{\text{d}\bar{x}}+{\mathcal{F}}_{2-w_{1-2}}A_{2}(\bar{x})+{\mathcal{F}}_{2^{\ast }-w_{1+2}}A_{2}^{\ast }(\bar{x})+{\mathcal{F}}_{1^{\ast }-w_{1+1}}A_{1}^{\ast }(\bar{x}) & = & \displaystyle 0,\end{eqnarray}$$

(3.36b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{2}}{\text{d}\bar{x}}+{\mathcal{F}}_{1-w_{1-2}^{\ast }}A_{1}(\bar{x})+{\mathcal{F}}_{1^{\ast }-w_{1+2}}A_{1}^{\ast }(\bar{x})+{\mathcal{F}}_{2^{\ast }-w_{2+2}}A_{2}^{\ast }(\bar{x}) & = & \displaystyle 0.\end{eqnarray}$$

$\unicode[STIX]{x1D706}\boldsymbol{a}=\unicode[STIX]{x1D641}\boldsymbol{a}$

$\unicode[STIX]{x1D641}$

$4\times 4$

For travelling vortices, resonances can take place among four travelling eigenmodes (with equal $\unicode[STIX]{x1D6FC}_{i}$ and $|\unicode[STIX]{x1D714}|$ ) and six roughness modes. The disturbance consisting of these components takes the form,

(3.37) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719} & = & \displaystyle h(\hat{\unicode[STIX]{x1D719}}_{w_{1-2}}E_{w_{1-2}}+\text{c.c.})+h(\hat{\unicode[STIX]{x1D719}}_{w_{1+3}}E_{w_{1+3}}+\text{c.c.})+h(\hat{\unicode[STIX]{x1D719}}_{w_{1+4}}E_{w_{1+4}}+\text{c.c.})\nonumber\\ \displaystyle & & \displaystyle +\,h(\hat{\unicode[STIX]{x1D719}}_{w_{2+3}}E_{w_{2+3}}+\text{c.c.})+h(\hat{\unicode[STIX]{x1D719}}_{w_{2+4}}E_{w_{2+4}}+\text{c.c.})+h(\hat{\unicode[STIX]{x1D719}}_{w_{3-4}}E_{w_{3-4}}+\text{c.c.})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D716}[A_{1}(\bar{x})\hat{\unicode[STIX]{x1D719}}_{10}(y)E_{1}+h\hat{\unicode[STIX]{x1D719}}_{11}E_{1}+\text{c.c.}]+\unicode[STIX]{x1D716}[A_{2}(\bar{x})\hat{\unicode[STIX]{x1D719}}_{20}(y)E_{2}+h\hat{\unicode[STIX]{x1D719}}_{21}E_{2}+\text{c.c.}]\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D716}[A_{3}(\bar{x})\hat{\unicode[STIX]{x1D719}}_{30}(y)E_{3}+h\hat{\unicode[STIX]{x1D719}}_{31}E_{3}+\text{c.c.}]+\unicode[STIX]{x1D716}[A_{4}(\bar{x})\hat{\unicode[STIX]{x1D719}}_{40}(y)E_{4}+h\hat{\unicode[STIX]{x1D719}}_{41}E_{4}+\text{c.c.}].\qquad\end{eqnarray}$$

Following similar steps as before, we can derive the four coupled amplitude equations,

(3.38a ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{1}}{\text{d}\bar{x}}+F_{2-w_{1-2}}A_{2}+F_{3^{\ast }-w_{1+3}}A_{3}^{\ast }+F_{4^{\ast }-w_{1+4}}A_{4}^{\ast } & = & \displaystyle 0,\end{eqnarray}$$

(3.38b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{2}}{\text{d}\bar{x}}+F_{1-w_{1-2}^{\ast }}A_{1}+F_{3^{\ast }-w_{2+3}}A_{3}^{\ast }+F_{4^{\ast }-w_{2+4}}A_{4}^{\ast } & = & \displaystyle 0,\end{eqnarray}$$

(3.38c ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{3}}{\text{d}\bar{x}}+F_{1^{\ast }-w_{1+3}}A_{1}^{\ast }+F_{2^{\ast }-w_{2+3}}A_{2}^{\ast }+F_{4-w_{3-4}}A_{4} & = & \displaystyle 0,\end{eqnarray}$$

(3.38d ) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{4}}{\text{d}\bar{x}}+F_{1^{\ast }-w_{1+4}}A_{1}^{\ast }+F_{2^{\ast }-w_{2+4}}A_{2}^{\ast }+F_{3-w_{3-4}^{\ast }}A_{3} & = & \displaystyle 0.\end{eqnarray}$$

$\unicode[STIX]{x1D706}\boldsymbol{a}=\unicode[STIX]{x1D641}\boldsymbol{a}$

$\unicode[STIX]{x1D641}$

$8\times 8$

For multiple-wavenumber cases, the growth-rate correction coefficient $\unicode[STIX]{x1D706}_{r}$ can be defined as the largest real part of the eigenvalues, and the amplitude ratio $\unicode[STIX]{x1D70C}$ is computed by a procedure described earlier; see (3.31)–(3.33).

The mathematical description of the resonant interactions assumes that the surface roughness is comprised of one or several Fourier components that have suitable wavenumbers to facilitate the resonances between the crossflow vortices. However, given the fact that it is not uncommon that instability modes satisfying resonant conditions develop out of broadband and even random background disturbances (Medeiros & Gaster Reference Medeiros and Gaster1999; Borodulin, Kachanov & Koptsev Reference Borodulin, Kachanov and Koptsev2002), we think that the restriction to such spatially quasiperiodic patterned roughness may be unnecessary. The mechanisms could operate even if such patterns are submerged in the background surface roughness consisting of a broad range of scales, because the crossflow vortices having the same growth rate could select the required components from all those that are present.

4.1 Resonant interactions of stationary vortices with roughness modes

Before presenting the growth-rate correction coefficients and amplitude ratios, a parametric study is performed to monitor the eigenmodes and roughness modes including their wavenumbers and shape functions, as the information is helpful for understanding the sensitive role of roughness. For brevity, the results are presented for $m=0.5$ ; our calculations indicate that those for other values of $m$ are similar.

In order to describe the high-Reynolds-number characteristics of eigenmodes and roughness modes, the concept of a critical level for the stationary crossflow eigenmode and roughness mode is introduced. For the former, the O–S equation (2.17) in the high-Reynolds-number limit reduces to the Rayleigh equation

(4.1) $$\begin{eqnarray}(\text{D}^{2}-\unicode[STIX]{x1D6FC}^{2}-\unicode[STIX]{x1D6FD}^{2})\hat{v}=\frac{\unicode[STIX]{x1D6FC}U^{\prime \prime }+\unicode[STIX]{x1D6FD}W^{\prime \prime }}{\unicode[STIX]{x1D6FC}U+\unicode[STIX]{x1D6FD}W}\hat{v}.\end{eqnarray}$$

The singularity of this equation occurs at the critical level $y_{c}$ , which coincides with an inflection point, and thus

(4.2a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}U(y_{c})+\unicode[STIX]{x1D6FD}W(y_{c})=0,\quad \unicode[STIX]{x1D6FC}U^{\prime \prime }(y_{c})+\unicode[STIX]{x1D6FD}W^{\prime \prime }(y_{c})=0.\end{eqnarray}$$

By eliminating the wavenumbers, we obtain the equation

(4.3) $$\begin{eqnarray}U(y_{c})/U^{\prime \prime }(y_{c})=W(y_{c})/W^{\prime \prime }(y_{c}),\end{eqnarray}$$

which allows us to compute $y_{c}$ from the baseflow before solving the eigenvalue problem. In the case of $\unicode[STIX]{x1D719}_{0}=45^{\circ }$ and $m=0.5$ , it is found that $y_{c}=1.9915$ .

A roughness mode satisfies the forced Rayleigh equation in the inviscid limit, and it also has a critical level $y_{cw}$ , which is given by

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{w}U(y_{cw})+\unicode[STIX]{x1D6FD}_{w}W(y_{cw})=0.\end{eqnarray}$$

In figure 3, the roughness wavenumbers $(\unicode[STIX]{x1D6FD}_{w},\unicode[STIX]{x1D6FC}_{w})$ satisfying the resonance conditions of the triad sum, triad difference and Bragg scattering for stationary vortices are compared with $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ , where $\unicode[STIX]{x1D6FD}_{s}^{r}$ and $\unicode[STIX]{x1D6FC}_{s}^{r}$ are the spanwise and chordwise wavenumbers of the right-branch neutral stationary eigenmode, respectively. We note that $(\unicode[STIX]{x1D6FD}_{w},\unicode[STIX]{x1D6FC}_{w})$ in the triad sum are close to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ for all growth rates; this closeness occurs also in Bragg scattering around the largest growth rate, and in the triad difference near the smallest growth rate ( $-\unicode[STIX]{x1D6FC}_{i}=0$ ). These features of roughness wavenumbers come from the linear instability characteristics of stationary vortices shown in figure 1(a): the wavenumbers of the left-branch neutral eigenmode are much smaller than those of the right-branch neutral eigenmode, and $\unicode[STIX]{x1D6FC}_{r}$ varies with $\unicode[STIX]{x1D6FD}$ almost linearly. Therefore, the sum and difference of their wavenumbers are close to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ when $\unicode[STIX]{x1D6FC}_{i}=0$ . The sum of the spanwise wavenumbers of unstable eigenmodes 1 and 2 with equal growth rate turns out to be $\unicode[STIX]{x1D6FD}_{s}^{r}$ approximately, i.e. $\unicode[STIX]{x1D6FD}_{1}+\unicode[STIX]{x1D6FD}_{2}\approx \unicode[STIX]{x1D6FD}_{s}^{r}$ , for all $-\unicode[STIX]{x1D6FC}_{i}$ . The linear relation implies that the second resonant condition, $\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D6FC}_{2}=\unicode[STIX]{x1D6FC}_{s}^{r}$ , is automatically satisfied. Since the triad sum degenerates to Bragg scattering at the largest growth rate, where eigenmodes 1 and 2 are identical, the roughness wavenumbers in Bragg scattering around the largest growth rate are close to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ . As the growth rate decreases, the roughness wavenumbers in Bragg scattering deviate from $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ , whereas the opposite is true for the triad difference. With the Reynolds number increased, the roughness wavenumbers in the triad sum become closer to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ generally, as indicated by figure 3(b,d).

Figure 3.

Variations of the roughness wavenumbers satisfying the resonance conditions of the triad sum ( $1+2$ ), triad difference ( $1-2$ ) and Bragg scattering ( $1+1$ and $2+2$ ) with the growth rate for stationary vortices at $m=0.5$ . The long dashed lines stand for $\unicode[STIX]{x1D6FD}_{s}^{r}$ and $\unicode[STIX]{x1D6FC}_{s}^{r}$ , the wavenumbers of the right-branch neutral stationary eigenmodes. (a,c) $Re=338$ ; (b,d) $Re=2000$ .

In Bragg scattering, there is the relation $(\unicode[STIX]{x1D6FD}_{w},\unicode[STIX]{x1D6FC}_{w})=(2\unicode[STIX]{x1D6FD},2\unicode[STIX]{x1D6FC}_{r}(\unicode[STIX]{x1D6FD}))$ . On the other hand, the crossflow eigenmode $(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD}))$ generates the harmonic $(2\unicode[STIX]{x1D6FC},2\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD}))$ . Since $\unicode[STIX]{x1D6FC}_{i}$ is numerically small, the harmonic and the roughness mode are likely to be fairly close to each other. Furthermore, the nearly linear relation between $\unicode[STIX]{x1D6FC}_{r}$ and $\unicode[STIX]{x1D6FD}$ implies that $\unicode[STIX]{x1D6FC}_{r}(2\unicode[STIX]{x1D6FD})\approx 2\unicode[STIX]{x1D6FC}_{r}(\unicode[STIX]{x1D6FD})$ . It follows that the roughness mode is probably also close to the eigenmode $(2\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FC}(2\unicode[STIX]{x1D6FD}))$ if $\unicode[STIX]{x1D6FD}\leqslant \unicode[STIX]{x1D6FD}_{s}^{r}/2$ .

Figure 4.

The eigenfunctions of the eigenmodes on the right branch (a,b) and left branch (c,d): (a,c) $Re=338$ ; (b,d) $Re=2000$ .

The shape functions of the stationary vortices and roughness modes in a resonant triad are monitored for the five cases marked by the thin vertical lines in figure 3. The growth rates and wavenumbers of these eigenmodes are listed in table 1. Among these cases, the stationary vortices are nearly most unstable in case 1 and neutral in case 5. The exactly most unstable stationary eigenmode is not chosen because the corresponding difference of its wavenumbers is zero, in which case the equations governing the roughness-induced perturbation degenerate (and non-parallelism needs to be accounted for in order to resolve this degeneracy). The eigenfunctions of the eigenmodes 1 located on the right branch are shown in figure 4(a,b) for $Re=338$ and 2000. Note that eigenfunctions are normalized by the maximum value of $|\hat{u} |$ . For brevity, only the chordwise disturbance velocity $|\hat{u} |$ is displayed since the spanwise disturbance velocity $|{\hat{w}}|$ closely resembles $|\hat{u} |$ . At $Re=338$ , the vertical position $y_{m}$ of the maximum value of the eigenfunction for the neutral eigenmode of case 5 is closest to the critical level $y_{c}$ among all cases. Although the eigenmodes of cases 1–4 are unstable, their $y_{m}$ are also fairly close to $y_{c}$ . As the Reynolds number increases, the eigenfunctions in cases 2–5 feature a two-layer structure. Inspecting, for example, the eigenfunctions at $Re=2000$ of eigenmodes with low growth rates, one observes a wall layer, where viscosity cannot be ignored. Towards the outer edge of the wall layer, the eigenfunctions tend to constants. The perturbation concentrates in the so-called critical layer surrounding $y_{c}$ . The positions of the maxima of the eigenfunctions are extremely close to $y_{c}$ . For the nearly most unstable eigenmode in case 1, the wall layer and the critical layer are not distinct, which is expected since the critical-layer concept is not attainable. The above observations suggest that right-branch stationary eigenmodes are essentially inviscid. The critical-layer concept, which originates from inviscid instability for neutral stationary modes, is applicable to right-branch unstable stationary eigenmodes provided that their growth rates are not too large, and the Reynolds number is high enough. The eigenfunctions of the eigenmodes 2 located on the left branch for the five cases are shown in figure 4(c,d). They do not exhibit a two-layer structure as right-branch eigenmodes do. Nevertheless, the positions of their maxima are fairly close to $y_{c}$ as well.

Figure 5.

Distribution of the roughness modes in the triad sum ( $1+2$ ) interaction: (a)  $Re=338$ ; (b) $Re=2000$ .

Table 1.

Wavenumbers of stationary vortices for $m=0.5$ , $Re=338$ and 2000.

Figure 6.

Comparison of the roughness modes in the triad sum of cases 1, 3 and 5 at $Re=2000$ with the eigenfunction of the right-branch neutral eigenmode $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ .

The shape functions of the roughness modes for the chosen cases are now presented. Figure 5 shows the results for the roughness modes interacting with stationary vortices through the triad sum. For all the cases, the roughness modes share the same boundary value at the wall surface, and the difference of the shape functions is due to the roughness wavenumbers. Relative to the size at the wall surface, the peak values of the roughness modes are amplified significantly in the critical layer. The reason for this is that the corresponding roughness wavenumbers in the triad sum are very close to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ , as shown in figure 3 – that is, the roughness modes are in near-resonance with the right-branch neutral stationary eigenmode. At $Re=338$ , as the growth rate increases from case 5 to case 2, and the roughness wavenumbers $(\unicode[STIX]{x1D6FD}_{w},\unicode[STIX]{x1D6FC}_{w})$ in the triad sum become closer to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ (figure 3 a,c), the peak values of the corresponding shape functions become larger and their vertical positions move closer to $y_{c}$ , as displayed in figure 5(a). At $Re=2000$ , a similar trend can be observed from case 1 to case 3 (figure 5 b), and at higher Reynolds numbers, the peak values of the roughness modes are greater in general, because the corresponding roughness wavenumbers are closer to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ ; see figure 3(b,d).

Figure 7.

Distribution of the roughness modes in the Bragg scattering for the (a,b) right-branch and (c,d) left-branch eigenmodes: (a,c) $Re=338$ ; (b,d) $Re=2000$ .

Figure 8.

Distribution of the roughness modes in the triad difference ( $1-2$ ) interaction: (a) $Re=338$ ; (b) $Re=2000$ .

In figure 6, the shape functions of the roughness modes in the triad sum of cases 1, 3 and 5 at $Re=2000$ are compared with those of the right-branch neutral stationary eigenmode $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ . Note that the roughness-mode shape functions are normalized by the maximum value of $|\hat{u} _{w}|$ . The transverse distributions of the roughness modes almost overlap with the eigenfunction of $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ . For case 1, where the growth rate is almost the largest, the resemblance between the shape functions is not so remarkable as that for lower growth rates. Nevertheless, in the critical layer the roughness mode is still similar to the right-branch neutral stationary eigenmode.

Figure 9.

Distribution of the Reynolds stresses induced by eigenmode 2 interacting with the roughness modes at $Re=338$ : solid lines, roughness $1+2$ ; dashed lines, roughness $1-2$ . (a) Case 1; (b) case 3; (c) case 5.

Figure 10.

Distribution of the Reynolds stresses induced by eigenmode 2 interacting with the roughness modes at $Re=2000$ : solid lines, roughness $1+2$ ; dashed lines, roughness $1-2$ . (a) Case 1; (b) case 3; (c) case 5.

Figure 7 shows the roughness modes in Bragg scattering for the right-branch and left-branch stationary eigenmodes. Among the five chosen cases, only the roughness mode of case 1 exhibits a critical-layer amplification for both the right-branch and left-branch eigenmodes. The reason is that only in this case are the wavenumbers $(\unicode[STIX]{x1D6FD}_{w},\unicode[STIX]{x1D6FC}_{w})$ of the roughness mode sufficiently close to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ ; see figure 3. At $Re=2000$ , the roughness mode of case 1 in Bragg scattering displays a two-layer structure as those in the triad sum. As the growth rate decreases, the roughness wavenumbers deviate from $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ (figure 3), which means that the corresponding response is non-resonant. There is no critical-layer amplification for cases 2–5, but there still exists a local peak near the critical level. By comparing the maximum values of the roughness modes of case 1 in Bragg scattering at $Re=338$ and 2000, one finds that the maximum values at a lower Reynolds number are much larger than those at a higher Reynolds number, especially for the left-branch eigenmode. Bragg scattering is therefore expected to have a greater effect on the nearly most unstable stationary vortices at lower Reynolds numbers. This will be demonstrated directly after computing the growth-rate corrections.

Figure 8 shows the shape functions of the roughness modes in the triad difference. Just as those in Bragg scattering, the roughness modes in the triad difference fall into two categories: near-resonant and non-resonant. The roughness modes of case 5 have large critical-layer amplification, and are therefore near-resonant; this is expected since their wavenumbers are very close to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ , as shown in figure 3. Increase of the Reynolds number leads to greater critical-layer amplification (figure 8 b). The roughness modes in cases 1–4 are non-resonant since their wavenumbers differ appreciably from $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ , and their shape functions do not show amplification in the critical layer.

The main difference between the triad sum and difference for the same pair of stationary vortices is the roughness mode involved. The Reynolds stresses induced by the interaction between the eigenmode and the roughness mode can be very different in these two mechanisms. Figure 9 presents the distribution of the Reynolds stresses induced by the eigenmodes 2 interacting with the roughness mode at $Re=338$ ; the Reynolds stresses induced by the eigenmodes 1 interacting with the roughness mode exhibit similar features and are not shown. For cases 1 and 3, the Reynolds stresses in the triad sum are significantly larger than those in the triad difference. A similar feature is also observed at a higher Reynolds number $Re=2000$ ; see figure 10. The huge disparity in Reynolds stresses is due to the fact that the corresponding roughness modes of cases 1 and 3 in the triad sum are near-resonant, while those in the triad difference are non-resonant. For case 5, where neutral eigenmodes are involved, the roughness modes for the triad sum and difference are both near-resonant, and so the Reynolds stresses are comparable.

Figure 11.

Variation of $\unicode[STIX]{x1D706}_{r}$ with $-\unicode[STIX]{x1D6FC}_{i}$ for stationary vortices interacting with single-wavenumber roughness through Bragg scattering (a,b), triad sum (c,d) and triad difference (e,f). (a,c,e) $Re=338$ ; (b,d,f) $Re=2000$ .

Figure 11 shows the variation of the growth-rate correction coefficient $\unicode[STIX]{x1D706}_{r}$ with the leading-order growth rate ( $-\unicode[STIX]{x1D6FC}_{i}$ ) for stationary vortices interacting with single-wavenumber roughness through Bragg scattering, triad sum and triad difference for the three acceleration parameters and two Reynolds numbers. For Bragg scattering (figure 11 a,b), the curve representing the correction coefficient has two branches, because the left and right-branch modes are affected independently. The two branches merge at the maximum growth rate. The correction coefficients are larger for all left-branch modes at $Re=338$ (figure 11 a), but at $Re=2000$ , larger $\unicode[STIX]{x1D706}_{r}$ is observed for the left-branch eigenmodes with relatively large $-\unicode[STIX]{x1D6FC}_{i}$ , and this switches to the right-branch modes when $-\unicode[STIX]{x1D6FC}_{i}$ is small (figure 11 b). The correction coefficient is highly peaked around the maximum growth rate, acquiring much larger values there than those elsewhere because the roughness modes involved are near-resonant. This means that the most or nearly most unstable stationary vortices can become even more unstable due to Bragg scattering. At $Re=338$ , the peak value of $\unicode[STIX]{x1D706}_{r}$ through Bragg scattering is extremely large (approximately 45, much greater than that at $Re=2000$ ). This is attributed to the fact that the corresponding roughness wavenumbers are almost equal to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ , and so the forcing is almost in exact resonance with the right-branch neutral stationary eigenmode. For suitable baseflow parameters and stationary eigenmode, an exact resonance may occur and Bragg scattering leads to an infinitely large $\unicode[STIX]{x1D706}_{r}$ , in which case the problem has to be regularized by introducing appropriate physical effects, which can be the non-parallelism of the baseflow.

The destabilizing effect of the Bragg scattering has implications for the application of DRE technology to delay transition. In order to suppresses the targeted most amplified stationary vortices, the spanwise wavelength of DREs is required to be smaller than that of the target mode (Saric & Reed Reference Saric and Reed2003). DREs are streamwise isolated, and the control eigenmode excited is different from the ‘roughness mode’: the former amplifies then attenuates downstream of DREs, whereas the ‘roughness mode’ is sustained by extended streamwise distributed roughness, and does not undergo amplification–attenuation. The spanwise wavelength of DREs is typically taken to be $2/3$ that of the target mode, in which case Bragg scattering cannot occur (Wassermann & Kloker Reference Wassermann and Kloker2002; Saric & Reed Reference Saric and Reed2003). However, if DREs have twice the spanwise wavenumber of the target eigenmode, the ‘control mode’ excited will also have twice streamwise wavenumber because of the (nearly) linear relation between $\unicode[STIX]{x1D6FC}_{r}$ and $\unicode[STIX]{x1D6FD}$ . The Bragg scattering as described here would take place between the target and control modes in the vicinity of the location where the latter becomes neutral (like a ‘roughness mode’) and acquires its maximum amplitude. The consequence is an increased growth rate of the target mode. Based on these theoretical considerations, we believe that DREs with half the spanwise wavelength of the target eigenmode could potentially be counterproductive, and should better be avoided. Control modes such wavelengths were investigated by Li et al. (Reference Li, Choudhari, Chang, Streett and Carpenter2011) and Malik et al. (Reference Malik, Liao, Li and Choudhari2013) using PSE. However, no enhanced growth was observed, and the control modes still suppressed the target mode. This is rather surprising, but may be due to the fact that the control mode is in the highly nonlinear stage, where a significant spanwise uniform mean-flow distortion is generated, and acts to stabilize the target mode. The PSE computations mentioned above indeed indicate that the suppression operates in the region where the control mode has attenuated to a very low level whilst the mean-flow distortion persists. Messing & Kloker (Reference Messing and Kloker2010) carried out a DNS study of the effect of a control mode with twice the wavenumber of the target mode, both being excited by suction through a single row of spanwise periodic and chordwise compact holes. A resonant destabilizing effect was not observed. This is most probably because in the region considered the control mode is linearly unstable $(-\unicode[STIX]{x1D6FC}_{i}\neq 0)$ , and so the resonance condition is not satisfied. The nonlinearly generated mean-flow distortion again suppresses the target mode.

Figure 12.

Comparisons of $\unicode[STIX]{x1D706}_{r}$ for stationary vortices interacting with multiple-wavenumber and single-wavenumber roughness for $m=0.34207$ (a,b), $m=0.5$ (c,d) and $m=1.0$ (e,f). (a,c,e) $Re=338$ ; (b,d,f) $Re=2000$ .

Figure 13.

Variation of $\unicode[STIX]{x1D70C}$ with $-\unicode[STIX]{x1D6FC}_{i}$ for stationary vortices interacting with single-wavenumber roughness through triad sum (a,b), triad difference (c,d) and with multiple-wavenumber roughness (e,f). (a,c,e) $Re=338$ ; (b,d,f) $Re=2000$ .

The correction coefficient due to the triad sum is shown in figure 11(c,d). At $Re=338$ , the correction coefficients for higher growth rates are generally larger than those for lower growth rates, whereas at $Re=2000$ the opposite is true. The correction is rather substantial for the majority of ( $-\unicode[STIX]{x1D6FC}_{i}$ ) values, which is because the roughness wavenumbers are close to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ . As $m$ increases, $\unicode[STIX]{x1D706}_{r}$ becomes larger. Since the case with $m=1$ is representative of the flow condition near the leading edge, the above trend is consistent with the experimental observation that roughness in that region plays a more sensitive role. Figure 11(e,f) displays the correction coefficient due to the triad difference. The correction coefficient attains its maximum at $-\unicode[STIX]{x1D6FC}_{i}=0$ since the roughness mode is in near-resonance with the right-branch neutral stationary crossflow vortices. For the majority of ( $-\unicode[STIX]{x1D6FC}_{i}$ ) values, the correction coefficient due to the triad difference is merely of $O(10^{-2})$ , while that due to the triad sum is of $O(10^{-1})$ ; the reason for this remarkable disparity was given above. As we will show later, the triad sum can be a very efficient mechanism, through which micron-sized distributed roughness creates a more-or-less $O(1)$ correction to the leading-order growth rate of stationary vortices.

The above results are for single-wavenumber roughness. As we noted earlier, a pair of stationary crossflow modes with equal growth rate can interact with four roughness components. Calculations were carried out for this extended case assuming that all roughness components have the same amplitude. The resulting correction coefficients are compared in figure 12 with those for single-wavenumber roughness. A main feature is that the $\unicode[STIX]{x1D706}_{r}$ for the multiple-wavenumber interaction is similar to that for the single-wavenumber interaction that dominates among the four interactions. At $Re=338$ , interactions $2+2$ and $1+2$ dominate at high and low growth rates, respectively; at $Re=2000$ , the dominant one is always $1+2$ . These results indicate the special importance of Bragg scattering for stationary eigenmodes with high growth rates at low Reynolds numbers, and the strong effect of the triad sum on the majority of stationary eigenmodes. For larger $m$ , the overall correction coefficient increases.

Besides the correction to the growth rates, distributed roughness determines the amplitude ratio of the eigenmodes involved in the resonance. Figure 13 shows the amplitude ratio $\unicode[STIX]{x1D70C}$ for single-wavenumber and multiple-wavenumber roughnesses. Note that $\unicode[STIX]{x1D70C}$ is taken as the ratio of the amplitude of the right-branch eigenmode to that of the left-branch eigenmode. For the triad sum $\unicode[STIX]{x1D70C}>1$ (figure 13 a,b), which means that the right-branch stationary eigenmode is stronger than its left-branch counterpart. As the acceleration parameter $m$ increases, so does $\unicode[STIX]{x1D70C}$ . For the triad difference (figure 13 c,d), the amplitude ratio at intermediate growth rates is larger than that at low and high growth rates. The amplitude ratio of stationary vortices interacting with multiple-wavenumber roughness is shown in figure 13(e,f). Interestingly, although the correction coefficient is determined by the dominant interaction, $\unicode[STIX]{x1D70C}$ is not; minor interactions such as the triad difference appear to affect the amplitude ratio substantially.

4.2 Resonant interactions of travelling vortices with roughness modes

Figure 14.

The wavenumbers of the roughness modes in the triad-sum interactions with travelling vortices for $m=1.0$ : (a) $Re=338$ ; (b) $Re=2000$ . The long dashed lines stand for $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ .

Four travelling crossflow eigenmodes can interact with six roughness modes with suitable wavenumbers (figure 2 a). A parametric study is to be performed for $m=0.34207$ and $m=1.0$ at $Re=338$ and 2000. As with stationary vortices, the wavenumbers of the roughness participating in the interactions are presented first with the particular attention to their ‘distances’ to the wavenumbers $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ of the right-branch neutral stationary eigenmode. Then the shape functions of the roughness modes are displayed in order to monitor their near-resonant and non-resonant nature. Finally, the results on the growth-rate correction coefficient and amplitude ratio are presented. For brevity, the results on the roughness modes are shown only for $m=1.0$ since those for other values of $m$ turned out to be similar.

Figure 14 shows the roughness wavenumbers in the triad sum for the four chosen frequencies of travelling vortices with $m=1.0$ at $Re=338$ and 2000. Among four possible triad-sum interactions, the wavenumbers of roughnesses $2+3$ and $1+4$ are fairly close to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ for a considerable range of $(-\unicode[STIX]{x1D6FC}_{i})$ , whereas those of $1+3$ and $2+4$ differ significantly from $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ except at the maximum growth rate, where roughnesses $1+3$ and $2+3$ have the same wavenumbers, and so do roughnesses $2+4$ and $1+4$ because eigenmodes 1 and 2 coalesce. The result suggests that the roughness modes $2+3$ and $1+4$ are mainly near-resonant, whereas $1+3$ and $2+4$ are mainly non-resonant. The interactions $2+3$ and $1+4$ are therefore of primary importance. As the Reynolds number increases, the wavenumbers of roughness $2+3$ become closer to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ . As was mentioned earlier, when the frequency of travelling vortices tends to zero roughnesses $2+3$ and $1+4$ both become the roughness $1+2$ for the stationary vortices, governed by the triad sum, whereas roughnesses $1+3$ and $2+4$ become $1+1$ and $2+2$ , governed by Bragg scattering. The near-resonant or non-resonant nature of the roughnesses in the triad sum for travelling vortices is consistent with that for stationary vortices.

As for stationary vortices, the roughness wavenumbers $(\unicode[STIX]{x1D6FD}_{w},\unicode[STIX]{x1D6FC}_{w})$ in the triad difference for the four chosen frequencies differ significantly from $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ except near $-\unicode[STIX]{x1D6FC}_{i}=0$ . With the frequency increased, the roughness wavenumbers deviate substantially from $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ even for $-\unicode[STIX]{x1D6FC}_{i}\approx 0$ . The results are not shown in order to limit the length of the paper. The roughness modes $1-2$ and $3-4$ in the triad-difference interactions of travelling vortices are both non-resonant except for small $\unicode[STIX]{x1D714}$ and $-\unicode[STIX]{x1D6FC}_{i}\approx 0$ . It is expected that the triad-difference interactions are in general less important than the triad-sum interactions.

Figure 15.

Distribution of the roughness modes in the triad-sum interactions with travelling vortices for $m=1.0$ : (a) $Re=338$ and $\unicode[STIX]{x1D714}=0.035$ ; (b) $Re=2000$ and $\unicode[STIX]{x1D714}=0.029$ .

The shape functions of the roughness modes in the different interactions are now examined. The chosen frequencies for $Re=338$ and 2000 are 0.035 and 0.029, respectively, which are approximately the frequencies of the corresponding most unstable travelling vortices. For each $Re$ and $\unicode[STIX]{x1D714}$ , three cases with different growth rates are chosen; see table 2. Among these, the growth rates of eigenmodes 1 and 2 in case 1 are close to the largest, and the eigenmodes in case 3 are neutral. Figure 15 shows the roughness modes in the triad sum at $Re=338$ and 2000. The perturbation induced by roughness $2+3$ is appreciably stronger than that induced by $1+3$ . This is a consequence of the fact that the wavenumbers of roughness $2+3$ are closer to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ than those of $1+3$ (figure 14). The same trend can also be found for roughnesses $1+4$ and $2+4$ . There exists a local peak near the critical level $y_{c}$ . As the growth rate decreases from case 1 to case 3, the peak value rises for roughness $2+3$ and falls for $1+3$ ; the peak value for roughness $1+4$ rises first and then falls, whereas that for roughness $2+4$ falls. These behaviours are consistent with the closeness of the roughness wavenumbers to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ in the triad sum as shown in figure 14. At $Re=2000$ , the near-resonant roughness modes display high-Reynolds-number features: the two-layer structure and the larger amplifications in the critical layer.

The signature of the roughness modes in the triad-difference interactions with travelling vortices is also calculated (but not shown here). According to the sizes of the roughness modes in the critical layer, only $1-2$ of case 3 is near-resonant, and its peak value rises with the Reynolds number. As the growth rate increases from case 3 to case 2, the perturbation induced by roughness $1-2$ decreases. The perturbations induced by roughness $3-4$ for all three cases are all non-resonant. These results are also consistent with the ‘distances’ of the roughness wavenumbers $(\unicode[STIX]{x1D6FD}_{w},\unicode[STIX]{x1D6FC}_{w})$ to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ in the triad difference.

Table 2.

Growth rates for the chosen cases of travelling vortices at $m=1.0$ .

Figure 16 shows the growth-rate correction coefficient for travelling vortices in the triad sum and difference interactions at $Re=338$ . For a fixed frequency of eigenmodes, the correction coefficients of roughnesses $2+3$ and $1+3$ are equal at the maximum growth rate because eigenmodes 1 and 2 coalesce there. The same remarks apply to roughnesses $1+4$ and $2+4$ . The correction coefficient due to roughness $2+3$ is larger than that due to $1+3$ . This can be attributed to the fact that the perturbations induced by the former are larger than those induced by the latter, as shown in figure 15(a). For the same reason, the correction coefficient due to $1+4$ is larger than that due to $2+4$ . For roughness $2+3$ at Travelling vortices with high frequencies, the correction coefficient due to $2+3$ interaction increases as the growth rate decreases. For roughness $1+4$ at $\unicode[STIX]{x1D714}=0.005$ , there exists a prominent peak located close to the maximum growth rate for each acceleration parameter. The corresponding roughness wavenumbers are almost equal to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ , as shown in figure 14(a), which means that the roughness mode is almost in exact resonance with the right-branch neutral stationary eigenmode. An exact resonance may take place in the non-stationary case under suitable conditions, as it does in the stationary case. The correction coefficients caused by $2+3$ and $1+4$ interactions in the triad sum are of $O(10^{-1})$ , whereas those by $1+3$ and $2+4$ interactions are much smaller. The correction coefficients due to the triad difference are of $O(10^{-2})$ . The reason for this significant difference is that the perturbations induced by roughnesses $2+3$ and $1+4$ are mainly near-resonant, while those induced by roughnesses $1+3$ , $2+4$ , $1-2$ and $3-4$ are non-resonant, as was demonstrated earlier. Overall, as $m$ increases, so does the correction coefficient. The correction coefficient at $Re=2000$ is shown in figure 17. The correction coefficient due to roughness $2+3$ for low growth rates and intermediate frequencies can be extraordinarily large, owing again to the fact that the corresponding roughness wavenumbers are close to $(\unicode[STIX]{x1D6FD}_{s}^{r},\unicode[STIX]{x1D6FC}_{s}^{r})$ (figure 14 b) and the roughness modes are amplified hugely in the critical layer (figure 15 b). This result suggests that the leading-order growth rate cannot capture the spatial instability of travelling vortices in the presence of roughness, and the effect of roughness must be considered.

Figure 16.

Variation of $\unicode[STIX]{x1D706}_{r}$ versus $-\unicode[STIX]{x1D6FC}_{i}$ for travelling vortices of different frequency at $Re=338$ : (a) $m=0.34207$ ; (b) $m=1.0$ .

Figure 17.

Variation of $\unicode[STIX]{x1D706}_{r}$ versus $-\unicode[STIX]{x1D6FC}_{i}$ for travelling vortices of different frequency at $Re=2000$ : (a) $m=0.34207$ ; (b) $m=1.0$ .

Figure 18.

Comparisons of $\unicode[STIX]{x1D706}_{r}$ for travelling vortices interacting with multiple-wavenumber and single-wavenumber roughness at $Re=338$ for $\unicode[STIX]{x1D714}=0.005$ (a,b) and $\unicode[STIX]{x1D714}=0.05$ (c,d). (a,c) $m=0.34207$ ; (b,d) $m=1.0$ .

The correction coefficients for travelling vortices interacting with multiple-wavenumber and single-wavenumber roughnesses are compared in figures 18 and 19 for $Re=338$ and 2000, respectively. The multiple-wavenumber roughness is comprised of the six Fourier components all having equal amplitude. Of the six resonant interactions, there exists a dominant one, which makes the largest contribution for each set of parameters. Broadly speaking, the dominant roughness is $2+3$ or $1+4$ , each driving a perturbation in a near-resonant manner. Specifically, at $Re=338$ the dominant single-wavenumber roughnesses are $1+4$ and $2+3$ for low and high frequencies, respectively. At a high Reynolds number ( $Re=2000$ ), for low frequencies ( $\unicode[STIX]{x1D714}=0.005$ ) there exists an intersection between the correction coefficients due to roughnesses $1+4$ and $2+3$ , as shown in figure 19(a,b); roughnesses $2+3$ and $1+4$ play a dominant role at low and high growth rates respectively. For an intermediate frequency ( $\unicode[STIX]{x1D714}=0.029$ ) (figure 19 c,d), roughness $2+3$ dominates at almost all growth rates. For a high frequency ( $\unicode[STIX]{x1D714}=0.05$ ) (figure 19 e,f), the dominant single-wavenumber roughness is always $2+3$ . The correction coefficient $\unicode[STIX]{x1D706}_{r}$ due to multiple-wavenumber roughness is of a similar size to that due to the dominant single-wavenumber roughness, and typically $\unicode[STIX]{x1D706}_{r}=O(10^{-1})$ , which implies that micron-sized roughness with multiple wavenumbers can create an $O(1)$ correction to the growth rates of travelling vortices, as we will show later. For larger values of $m$ , the correction coefficient due to multiple-wavenumber roughness increases. The correction coefficient at low and high Reynolds numbers exhibits different features: for low-frequency vortices, large corrections at $Re=338$ occur at high growth rates, while those at $Re=2000$ occur at low growth rates; for high-frequency vortices, the correction coefficient at $Re=338$ decreases with the growth rate, while that at $Re=2000$ generally increases with the growth rate.

Figure 19.

Comparisons of $\unicode[STIX]{x1D706}_{r}$ for travelling vortices interacting with multiple-wavenumber and single-wavenumber roughness at $Re=2000$ for $\unicode[STIX]{x1D714}=0.005$ (a,b), $\unicode[STIX]{x1D714}=0.029$ (c,d) and $\unicode[STIX]{x1D714}=0.05$ (e,f). (a,c,e) $m=0.34207$ ; (b,d,f) $m=1.0$ .

Figure 20.

Variation of $\unicode[STIX]{x1D70C}$ versus $-\unicode[STIX]{x1D6FC}_{i}$ for travelling vortices of different frequency at $m=1.0$ : (a) $Re=338$ ; (b) $Re=2000$ .

Figure 21.

Comparison of the amplitude ratio in the multiple-wavenumber case with that in the dominant single-wavenumber case at $m=1.0$ for $\unicode[STIX]{x1D714}=0.005$ (a,b) and $\unicode[STIX]{x1D714}=0.05$ (c,d). (a,c) $Re=338$ ; (b,d) $Re=2000$ .

The other effect of distributed roughness on crossflow eigenmodes through the generalized resonance mechanisms is that it renders the eigenmodes involved in the resonance to obey a fixed amplitude ratio. The variation of the amplitude ratio $\unicode[STIX]{x1D70C}$ with the growth rate for travelling vortices of different frequency is shown in figure 20 for $Re=338$ and $Re=2000$ . Note that for roughness $i+j$ or $i-j$ , $\unicode[STIX]{x1D70C}$ is taken as the ratio of the amplitude of eigenmode $i$ to that of eigenmode $j$ . At $Re=338$ , for dominant interactions $\unicode[STIX]{x1D70C}$ takes moderate values in the range of $0.5\leqslant \unicode[STIX]{x1D70C}\leqslant 2$ , but for certain non-dominant interactions, e.g. $3-4$ , $\unicode[STIX]{x1D70C}$ can be quite large. At $Re=2000$ , $\unicode[STIX]{x1D70C}$ can be quite large as well for dominant interactions of relatively low-frequency vortices.

As noted earlier, among all possible interactions, one is dominant in the sense that it more-or-less dictates the growth-rate correction. One may ask whether the same behaviour extends to the amplitude ratio. This question is answered in figure 21 for $Re=338$ and 2000. At $Re=338$ , $\unicode[STIX]{x1D70C}$ in the multiple-wavenumber case follows that in the single-wavenumber case for low frequencies at high growth rates and for high frequencies at low growth rates. At $Re=2000$ , there is a close agreement. The discontinuity of $\unicode[STIX]{x1D70C}$ with the growth rate for $\unicode[STIX]{x1D714}=0.005$ (figure 21 b) occurs where the dominant interaction changes as shown in figure 19(a,b). Therefore, we can conclude that in the multiple-wavenumber case, generally there exists a dominant interaction in the sense that the corresponding growth-rate correction coefficient and amplitude ratio are both marginally affected by other roughness components.

Our numerical results show that the growth-rate correction coefficient $\unicode[STIX]{x1D706}_{r}$ caused by the resonant interactions between the crossflow eigenmodes and roughness modes is approximately 0.3 in the majority of the parameter space. The correction to the leading-order growth rate is $\unicode[STIX]{x1D706}_{r}h$ with $h=h^{\ast }/\unicode[STIX]{x1D6FF}_{0}^{\ast }$ . The typical value of $h$ for a micron-sized roughness is now estimated by referring to the experimental data of Radeztsky et al. (Reference Radeztsky, Reibert and Saric1999). In their experiment, a $12~\unicode[STIX]{x03BC}\text{m}$ isolated roughness element was placed at $x^{\ast }/c^{\ast }=0.023$ , and the non-dimensionalized height is defined as $h^{\ast }/\unicode[STIX]{x1D6FF}_{0.99}^{\ast }=0.014$ with $\unicode[STIX]{x1D6FF}_{0.99}^{\ast }$ representing the transverse position where $U=0.99$ . It follows that $h^{\ast }/\unicode[STIX]{x1D6FF}_{0}^{\ast }=(h^{\ast }/\unicode[STIX]{x1D6FF}_{0.99}^{\ast })y_{0.99}$ , where $y_{0.99}=\unicode[STIX]{x1D6FF}_{0.99}^{\ast }/\unicode[STIX]{x1D6FF}_{0}^{\ast }$ is given by $U(y_{0.99})=0.99$ . Since $x^{\ast }/c^{\ast }=0.023$ is fairly close to the leading edge, the local boundary layer can be approximated by the FSC solution with $m=1.0$ . For the FSC boundary layer with $m=1.0$ , our computation gives $y_{0.99}=3.6725$ , and hence $h^{\ast }/\unicode[STIX]{x1D6FF}_{0}^{\ast }=0.0514$ and $\unicode[STIX]{x1D6FF}_{0}^{\ast }=0.2334~\text{mm}$ . The chord Reynolds number $Re_{c}=U_{\infty ,s}^{\ast }c^{\ast }/\unicode[STIX]{x1D708}^{\ast }=2.6\times 10^{6}$ , where $U_{\infty ,s}^{\ast }$ is the streamwise free-stream velocity, and the airfoil chord length $c^{\ast }=1.83~\text{m}$ (Carrillo et al. Reference Carrillo, Reibert and Saric1997). The sweep angle in the experiment is set to $45^{\circ }$ . The Reynolds number based on $\unicode[STIX]{x1D6FF}_{0}^{\ast }$ at $x^{\ast }/c^{\ast }=0.023$ is therefore estimated as $Re=(Re_{c}\unicode[STIX]{x1D6FF}_{0}^{\ast })/(\sqrt{2}c^{\ast })=234.5$ , fairly close to the low Reynolds number (338) in our computation. The Reynolds number would be equal if the roughness is present somewhat downstream where $\unicode[STIX]{x1D6FF}_{0}^{\ast }=0.2334\times 338/234.5\approx 0.34~\text{mm}$ . Hence, the relative height ( $h=h^{\ast }/\unicode[STIX]{x1D6FF}_{0}^{\ast }$ ) of the micron-sized roughness (typically $h^{\ast }\approx 10~\unicode[STIX]{x03BC}\text{m}$ for unpolished surfaces) at the lower Reynolds number ( $Re=338$ ) in the experimental setting is approximately 0.03. The correction to the growth rate of both stationary and travelling vortices is roughly 0.01, which is comparable to the leading-order growth rates ( $O(10^{-2})$ ).