4.1. The intermediate wavenumber regime

Here we consider the limit $\varGamma \rightarrow \infty$ with $\varGamma ^{-({3}/{4})} \ll k \ll \varGamma ^{3/2}$. This includes the $k=O(1)$ case which in the corresponding Görtler problem corresponds to the inviscid limit and an exact solution exists; see DHS. But the Görtler problem instability is associated with centrifugal effects whereas here the instability is generated by the roughness boundary condition which depends on viscosity for its existence. We see below that the disturbance adjusts to this reality by becoming inviscid over most of the flow apart from a thin wall layer where it is generated.

The thickness of the wall layer can be inferred from the disturbance equations (2.11)–(2.14). Suppose then that the $X$-disturbance velocity is $U(X,y)\exp ({\int ^X\beta (X)\,\textrm {d} X})$, where the growth rate $\beta$ is large. Within the momentum equations a balance between diffusion in $Y$ and advection in $X$ requires $\bar {u}({\partial }/{\partial X})\sim ({\partial ^2}/{\partial y^2}),$ so that

(4.4)\begin{equation} \beta y \sim \frac{1}{y^2}, \end{equation}

on the assumption that $y$ is small. The continuity equation and the roughness boundary condition respectively require

(4.5a,b)\begin{equation} \frac{V}{U}\sim {\beta}y,\quad \frac{V}{U}\sim \varGamma k^{4/3}. \end{equation}

These three balances are consistent if

(4.6a,b)\begin{equation} \beta \sim k^2\varGamma^{3/2},\quad y\sim \varGamma^{-({1}/{2})} k^{-({2}/{3})}. \end{equation}

Within the wall we are therefore led to the expansions

(4.7)$$\begin{gather} U=\exp\left({k^2\varGamma^{3/2}\int^X\tilde{\beta}(X)\,\textrm{d} X}\right)[\tilde{U}_0(X,\bar{Y})+\cdots], \end{gather}$$

(4.8)$$\begin{gather}V=\varGamma k^{4/3}\exp\left({k^2\varGamma^{3/2}\int^X\tilde{\beta}(X)\,\textrm{d} X}\right) [\tilde{V}_0(X,\bar{Y})+\cdots], \end{gather}$$

(4.9)$$\begin{gather}W=\varGamma^{3/2} k \exp\left({k^2\varGamma^{3/2}\int^X\tilde{\beta}(X)\,\textrm{d} X}\right)[\tilde{W}_0(X,\bar{Y})+\cdots], \end{gather}$$

(4.10)$$\begin{gather}P=k^{4/3}\varGamma^{5/2}\exp\left({k^2\varGamma^{3/2}\int^X\tilde{\beta}(X)\,\textrm{d} X}\right)[\tilde{P}_0(X,\bar{Y})+\cdots], \end{gather}$$

where we have defined the wall layer variable $\bar {Y}=(\lambda \varGamma ^{3/2} k^2 \tilde {\beta })^{1/3}y.$ The leading-order equations in the wall layer are

(4.11)$$\begin{gather} \tilde{\beta}\tilde{U}_0+(\lambda \tilde{\beta})^{1/3}\tilde{V}_{0\bar{Y}}+\tilde{W}_0=0, \end{gather}$$

(4.12)$$\begin{gather}\left[\frac{\partial^2}{\partial \bar{Y}^2}- \bar{Y}\right]\tilde{U}_0= \frac{ \lambda^{1/3} }{\tilde{\beta}^{{2}/{3}}} \tilde{V}_0, \end{gather}$$

(4.13)$$\begin{gather}\frac{\partial \tilde{P}_0}{\partial \bar{Y}}=0, \end{gather}$$

(4.14)$$\begin{gather}\left[\frac{\partial^2}{\partial \bar{Y}^2}- \bar{Y}\right]\tilde{W}_0={-}\frac{1}{{\lambda^{2/3}}{\tilde{\beta}^{{2}/{3}}}}\tilde{P}_0. \end{gather}$$

The solution of the above equations satisfying $\tilde {V}_0=\bar {W}_0=0, \bar {Y}=0$ with $\tilde {V}_{0\bar {Y}}=0$ for large $\bar {Y}$ is

(4.15)$$\begin{gather} \tilde{V}_{0}= B \int_0 ^{\bar{Y} } \textrm{d}\phi \int_\infty^\phi Ai({\theta})\,\textrm{d}\theta, \end{gather}$$

(4.16)$$\begin{gather}\tilde{W}_{0}=B\left[-\frac{P _0}{{ \lambda^{2/3} }{\tilde{\beta}^{{2}/{3}}}}{\mathcal{L}}(\bar{Y})+ \frac{\lambda^{1/3} \tilde{\beta}^{1/3} }{ {3Ai (0)}} Ai(\bar{Y})\right], \end{gather}$$

(4.17)$$\begin{gather}\tilde{P}_0=\lambda\tilde{\beta}BAi' (0), \end{gather}$$

Here $Ai, {\mathcal {L}}$ are the Airy and Scorer functions, $B$ is an arbitrary constant and $\tilde {U}_0$ is then found using (4.11). Note that we require a solution for $\tilde {V}_0$ which is finite for large $\bar {Y}$ because the disturbance in the main part of the boundary layer is inviscid but, since that inviscid problem cannot support a neutrally stable disturbance, the normal velocity at the bottom of the main boundary layer remains finite. It remains to satisfy the leading-order approximation to the roughness boundary condition, this takes the form

(4.18)\begin{equation} \tilde{V}_{0\bar{Y}}+\lambda^{1/3}\tilde{U}_{0\bar{Y}}=0,\quad \bar{Y} =0, \end{equation}

and is satisfied if

(4.19)\begin{equation} \tilde{\beta}=\lambda \left[{-}3Ai(0)-\frac{2Ai'(0)}{Ai(0)}\right]^{3/2}\simeq0.246\lambda. \end{equation}

It follows that in the intermediate wavenumber regime the disturbance evolves in a quasi-parallel manner and the local growth rate is

(4.20)\begin{equation} \beta=0.246 \lambda k^2\varGamma^{3/2}. \end{equation}

Above the wall layer for $y=O(1)$, the disturbance velocity is passive and driven by the matching condition at $y=0$ found from the wall layer solution as $\bar {Y} \rightarrow \infty$, but by rescaling $B$ we can take this to be unity. So for $y=O(1)$, the normal velocity component of the disturbance expands as

(4.21)\begin{equation} V= \varGamma k^{4/3}\exp\left({k^2\varGamma^{3/2}\int^X\tilde{\beta}(X)\,\textrm{d} X}\right)[\hat{V}_{0 }(X,y)+\cdots], \end{equation}

and $\widehat {V_{0}}$ satisfies the inhomogeneous Rayleigh equation problem

(4.22)$$\begin{gather} \bar{u}(\hat{V}_0''-k^2\hat{V}_0)- \bar{u}''\hat{V}_0=0, \end{gather}$$

(4.23)$$\begin{gather}\hat{V}_0=1,\quad y=0,\quad \hat{V}_0\rightarrow 0,\quad y \rightarrow \infty. \end{gather}$$

We see then that in the intermediate wavenumber regime the disturbance is inviscid almost everywhere but with the instability sustained by the roughness within a viscous wall layer. The upper range of validity of the above structure occurs when diffusion in $z$ is comparable with diffusion in $y$. This occurs when ${\partial }/{\partial y}\sim k$ which gives $\varGamma \sim k^{2/3}$, and this is the right-hand branch regime. Hence, for $O(1)$ spanwise wavenumbers, the growth rate is of size $\varGamma ^{3/2}$ and rises to O($\varGamma ^{9/2})$ when $k$ increases to $O(\varGamma ^{3/2})$. At small values of $k$ the breakdown is more subtle and is associated with the small $k$ behaviour of (4.22). We can see from (4.22) that for small $k$, a double layer structure develops with layers of thickness $O(1), O(k^{-1}).$ Initially the behaviour remains passive but, for sufficiently small $k$, these two layers and the wall layer interact and there develops a three-layer interactive structure akin to that discussed by CHS for Görtler vortices. That structure will be described in Appendix A. Now let us examine the right-hand branch regime where the intermediate wavenumber problem breaks down.

4.2. Large wavenumbers: the fastest growing mode and the neutral curve

We saw in the previous section that asymptotic suction flow has a right-hand branch with $\varGamma \sim k^{2/3}$ and an examination of the calculated growth rates suggests the fastest growing mode occurs in the same regime. In the intermediate wavenumber regime we saw that at large wavenumbers the structure fails when $\varGamma \sim k^{2/3}$; this also suggests the right-hand branch and fastest growing mode both occur where $\varGamma \sim k^{2/3}.$ We define the wall layer variable $Y$ by

(4.24)\begin{equation} Y=\varGamma^{3/2}y, \end{equation}

and note that for spatially growing modes, the operator $\bar {u}({\partial }/{\partial X})$ is comparable with the viscous operator ${\partial ^2}/{\partial y^2}-k^2$ if ${\partial }/{\partial X}\sim \varGamma ^{9/2}$. Thus, we seek a solution in the wall layer of the form

(4.25)$$\begin{gather} U=[\tilde{U}(Y,X)+\cdots]\exp\left({\varGamma^{9/2}\int^X\beta(X)\,\textrm{d} X} \right), \end{gather}$$

(4.26)$$\begin{gather}V=[\varGamma^3 \tilde{V}(Y,X)+\cdots]\exp\left({\varGamma^{9/2}\int^X\beta(X)\,\textrm{d} X}\right), \end{gather}$$

(4.27)$$\begin{gather}W=[\varGamma^3 \tilde{W}(Y,X)+\cdots]\exp\left({ \varGamma^{9/2}\int^X\beta(X)\,\textrm{d} X}\right), \end{gather}$$

(4.28)$$\begin{gather}P=[\varGamma^{9/2} \tilde{P}(Y,X)+\cdots]\exp\left({\varGamma^{9/2}\int^X\beta(X)\,\textrm{d} X}\right), \end{gather}$$

so that the local growth rate at station $X$ is $\varGamma ^{9/2}\beta (X)$. If we then write $k=\bar {k}\varGamma ^{3/2}$ then the leading-order approximation to (2.11)–(2.14) is

(4.29)$$\begin{gather} \beta \tilde{U}+\tilde{V}_Y+\bar{k}\tilde{W} =0, \end{gather}$$

(4.30)$$\begin{gather}\left[\frac{\partial^2}{\partial Y^2}-\beta \lambda Y-\bar{k}^2\right]\tilde{U}=\tilde{V}\lambda, \end{gather}$$

(4.31)$$\begin{gather}\left[\frac{\partial^2}{\partial Y^2}-\beta \lambda Y-\bar{k}^2\right]\tilde{V}=\tilde{P}_Y, \end{gather}$$

(4.32)$$\begin{gather}\left[\frac{\partial^2}{\partial Y^2}-\beta \lambda Y-\bar{k}^2\right]\tilde{W}=\bar{k}\tilde{P}, \end{gather}$$

which must be solved subject to

(4.33)\begin{equation} \tilde{U}=\tilde{V}= \frac{\partial}{\partial Y}[\tilde{V}+\lambda^{1/3}\bar{k}^{4/3}\tilde{U} ]=0,\quad Y=0,\enspace \tilde{\boldsymbol{U}} \rightarrow 0,\enspace Y \rightarrow \infty. \end{equation}

Here $\lambda (X)=\overline {u}_y(X,0)$ is the wall shear associated with the $X$ velocity component of the basic flow and can be scaled out of (4.29)–(4.33) by making the transformations

(4.34a–d)\begin{equation} Y\rightarrow \lambda^{{-}2}Y,\quad \bar{k}\rightarrow \lambda^2 \bar{k},\quad \beta \rightarrow \lambda^5 \beta,\quad \tilde{V}\rightarrow \lambda^3 \tilde{V}. \end{equation}

Therefore, we need only solve (4.29)–(4.33) with $\lambda =1$ to find the growth rate dependences on wavenumber and roughness parameter $\varGamma$ for an arbitrary boundary layer. However, we first observe that the neutral value of $\bar {k}$, which corresponds to the right-hand branch of the neutral curve, is readily found from (4.29)–(4.33) by setting $\beta =0$. The system (4.29)–(4.33) can then be solved analytically to give the neutral wavenumber $\bar {k}={\lambda ^2}/{8}$ which corresponds to $\varGamma \lambda ^{4/3}=4 k^{2/3}$.

The eigenvalue problem (4.29)–(4.33) is solved numerically by first eliminating $\tilde {W}, \tilde {P}$ from (4.31 and (4.32) to obtain a fourth-order ordinary differential equation for $\tilde {V}$. That equation is then solved using finite differences subject to $\tilde {V}$ vanishing at $Y=0, \infty$ with the normalisation $\tilde {V}'(0)=1.$ The solution for $\tilde {V}$ is then substituted into (4.30) which is then solved using finite differences subject to $\tilde {U}=0$ at $Y=0, \infty$. We then iterate on $\beta$ at each $\bar {k}$ until the roughness boundary condition is satisfied. Our computations found a single unstable mode in the interval $0<{\bar {k}}/{\lambda ^2}<\frac {1}{8}$. Figure 7 shows $\beta _1={\beta }/{\lambda ^5}$ as a function of $k_1={\bar {k}}/{\lambda ^2}$. The maximum of $\beta _1$ as a function of $k_1$ occurs at $k_1=0.079$. For small values of $k_1$, we observe that the growth rate goes to zero, more precisely, the calculations show that, for small $k_1$, the growth rate $\beta _1$ goes to zero like $k_1^2$, and, therefore, is consistent with the limiting large wavenumber form of the growth rate in the intermediate wavenumber range. Note that the small $\bar {k}$ limit of (4.29)–(4.33) shows that the disturbance here develops the double layer structure found in the intermediate wavenumber regime. Also shown in figure 7 for the smaller values of $k_1$ is the growth rate predicted by the intermediate wavenumber solution, we see that, for small enough $k_1$, the results coincide.

Figure 7.

The growth rate $\beta _1={\beta }/{\lambda ^5}$ of the fastest growing mode as a function of the wavenumber $k_1={\bar {k}}/{\lambda ^2}$. The growth rate passes through zero at ${k_1}=\frac {1}{8}$. Also shown is the prediction from the intermediate wavenumber problem.

Figure 8 shows the eigenfunctions $U_1=\tilde {U},\ V_1={\tilde {V}}/{\lambda ^3}$ as functions of $Y_1=\lambda ^2 Y.$ Note that the eigenfunctions decay to zero more slowly as the wavenumber decreases so that there is non-zero vortex activity over a bigger region adjacent to the wall. The spreading of the eigenfunctions to large values of the scaled variable $Y$ again enables a matching with the large $k$ intermediate wavenumber solution of the previous subsection.

Figure 8.

The eigenfunctions $U_1=\tilde {U},\ V_1={\tilde {V}}/{\lambda ^3}$ of the fastest growing mode eigenvlaue problem (4.29)–(4.33) for $k_1= 0.125, 0.079, 0.004$ in (a,b), (c,d) and (e,f), respectively.

Now let us determine how the intermediate wavenumber solution develops at small wavenumbers. As pointed out already, the breakdown occurs when the originally passive layer occupying the main part of the boundary layers splits into two layers and eventually couples the wall layer with the two upper layers. Thus, the solution for small $k$ becomes interactive and the situation is similar to that first described by Rozhko & Ruban (Reference Rozhko and Ruban1987). The structure in the interactive regime is relatively straightforward but plays the crucial role of connecting the most unstable mode, which is of course the only unstable mode, in the left-hand branch regime with the only unstable mode at higher wavenumbers. That connection problem was recently described for the Görtler problem in H1, that connection was overlooked in the analysis of Wu et al. (Reference Wu, Zhao and Luo2011). Rather than discuss the interactive regime at this stage we now consider the left-hand branch regime $\varGamma \sim k^{-({4}/{3})}$. This regime is the only one which remains intrinsically non-parallel at large values of the roughness parameter and plays the crucial role in selecting the size of the vortex to be amplified as the disturbance moves downstream into higher local wavenumber regimes.

4.3. The non-parallel evolution in the left-hand branch regime

Here we will concern ourselves with the asymptotic solution of (2.11)–(2.14) in the limit $k\rightarrow 0$ with $\varGamma =O(k^{-({4}/{3})})$. The latter distinguished limit defines the left-hand branch for asymptotic suction flow and we anticipate that it is also relevant to growing boundary layers. We write

(4.35)\begin{equation} \varGamma=\varGamma_1k^{-({4}/{3})}, \end{equation}

and look for solution of the equations of motion with the disturbance varying on the same length scale as the unperturbed boundary layer, so we are assuming the disturbance develops in a non-parallel manner. The appropriate expansions for the disturbance are

(4.36)$$\begin{gather} U=U_0(X,y)+\cdots, \end{gather}$$

(4.37)$$\begin{gather}V=V_0(X,y)+\cdots, \end{gather}$$

(4.38)$$\begin{gather}W=k^{{-}1}W_0(X,y)+, \end{gather}$$

(4.39)$$\begin{gather}P= k^{{-}1} P_0(X,y)+\cdots. \end{gather}$$

Here the relative size of the velocity components is fixed by the equation of continuity whilst the size of $P$ is fixed by the fact that $V$ cannot be constrained to the main part of the boundary layer and so there is a pressure driven roll field in an outer layer of depth $O(k^{-1})$. Note that Luchini (Reference Luchini1996), in an analysis for the flat plate case without roughness, claimed $V$ cannot be reduced to zero within the context of (2.11)–(2.14). However, that is only the case for the reduced system in the long-wave limit. The leading-order problem is found from substituting the above expansions into (2.11)–(2.14) and the corresponding boundary conditions to give

(4.40)$$\begin{gather} \bar{u}U_{0X}+\bar{v}U_{0y}+U_0\overline{u}_X+ {V_0}\overline{u}_y=U_{0yy}, \end{gather}$$

(4.41)$$\begin{gather}P_{0y}=0, \end{gather}$$

(4.42)$$\begin{gather}\bar{u}W_{0X}+\bar{v}W_{0y}= W_{0yy}, \end{gather}$$

(4.43)$$\begin{gather}U_{0X}+V_{0y}+ W_0=0, \end{gather}$$

(4.44)$$\begin{gather}U_0,\ V_0,\ \frac{\partial}{\partial y}[V_{0 } +\varGamma_1 \lambda^{1/3} U_{0 }]=0,\quad y=0, \end{gather}$$

(4.45)$$\begin{gather}U_0,\ \frac {\partial V_0}{\partial y},\quad W_0 \rightarrow 0,\enspace y\rightarrow \infty. \end{gather}$$

Note that the above system can only be solved subject to $V_{0y}$ tending to zero at large $y$ and so outside the boundary layer there exists a passive outer region defined by the stretched variable $\psi =ky$, where

(4.46a–c)\begin{equation} V=V_{00}(X, \psi)+\cdots,\quad W=W_{00}(X, \psi)+\cdots,\quad P=k^{{-}1}P_{00}(X, \psi)+\cdots, \end{equation}

and it is readily shown that

(4.47)\begin{equation} V_{00}=V_{0}(X,\infty)\ \textrm{e}^{-\psi}. \end{equation}

Equations (4.40)–(4.45) must be solved subject to a disturbance imposed at an initial value of $X$, and then the disturbance in the free stream responds with the above normal velocity component in an outer layer where it decays to zero. For the Görtler problem, Bassom & Hall (Reference Bassom and Hall1993) found that the full Görtler equations subject to an initial disturbance imposed close to the leading edge do not support a left-hand branch to the neutral curve because disturbances grow algebraically even without curvature. This was explained by Luchini (Reference Luchini1996) who showed that, for Blasius flow, the system (4.40)–(4.45) with $\varGamma _1=0$ supports an algebraically growing disturbance with $U_0\sim X^{\omega }$ with $\omega \simeq .213$. It would seem likely that more general boundary layers might support similar algebraic growth of disturbances. In order to check for that possibility, and to see how the roughness boundary condition impacts on possible algebraic growth, we seek algebraically growing solutions of (4.40)–(4.45) with $\varGamma_{1}=0$ for Falkner–Skan boundary layers.

Suppose then that, for large $y$, the streamwise velocity $\bar {u}$ approaches the free-stream speed $u_e=X^n$, we define a similarity variable $\eta$ by

(4.48)\begin{equation} \eta=\sqrt{\frac{n+1}{2}}yX^{{(n-1)}/{2}}, \end{equation}

and let

(4.49a,b)\begin{equation} \bar{u}=X^nf'(\eta),\quad \bar{v}={-}\frac{1}{\sqrt{2[n+1]}}([n+1]f(\eta)+[n-1]\eta f'(\eta))X^{{(n-1)}/{2}}, \end{equation}

where $f(\eta )$ satisfies

(4.50)$$\begin{gather} f'''+\beta_H[1-f'^2]+ff''=0, \end{gather}$$

(4.51)$$\begin{gather}f(0)=f'(0)=0,\quad f'(\infty)=1. \end{gather}$$

Here $\beta _H={2n}/{(n+1)}$ is the Hartree parameter. We now write (4.40)–(4.45) in terms of $X, \eta$ to give

(4.52)$$\begin{gather} \left[\frac{\partial^2}{\partial \eta^2}-2\frac{f'}{n+1} X \frac{\partial}{\partial X}+f\frac{\partial}{\partial \eta}\right]U_0= \frac{2nf'+[n-1]\eta f''}{n+1}U_0+\sqrt{\frac{2}{n+1}}f''V_0X^{{(n+1)}/{2}}, \end{gather}$$

(4.53)$$\begin{gather}\left[\frac{\partial^2}{\partial \eta^2}-2\frac{f'}{n+1} X\frac{\partial}{\partial X}+f \frac{\partial}{\partial \eta}\right]W_0=0, \end{gather}$$

(4.54)$$\begin{gather}\frac{\partial U_0}{\partial X}+\frac{n-1}{2X}\eta \frac{\partial U_0}{\partial \eta}+ \sqrt{\frac{n+1}{2}} X^{{(n-1)}/{2}} \frac{\partial V_0}{\partial \eta}+ W_0 =0, \end{gather}$$

(4.55a,b)$$\begin{gather}U_0=V_0=0,\quad V_{0\eta}(0)+ \varGamma_1 X^{{(3n-1}/{6})}\left[f''(0) \sqrt{\frac{n+1}{2}}\right]^{1/3}U_{0\eta}(0) =0, \end{gather}$$

(4.56)$$\begin{gather}U_0,\ W_0 \rightarrow 0,\quad y\rightarrow \infty. \end{gather}$$

In the zero roughness case, $\varGamma _1=0$, the above system supports the similarity solution

(4.57)\begin{equation} {\boldsymbol{U_0}}=(X^\omega U_{00}(\eta),\ X^{\omega-{[n+1]}/{2}} V_{00}(\eta),\ X^ {\omega-1}W_{00}(\eta)), \end{equation}

where $\omega$ satisfies the ordinary differential equation eigenvalue problem found by substituting the above into (4.52)–(4.56). The similarity solution is an exact solution of the long-wave disturbance equations and generalizes the Blasius case investigated by Luchini (Reference Luchini1996) who found a single positive unstable eigenvalue $\omega =0.213$. It is important to note that Luchini's solution, and the generalization above, are similarity solutions for all $X >0$ of the long-wave equations for streamwise vortices in the case $\varGamma _1=0.$ But the normal velocity $V_0$ associated with the similarity solution does not vanish for large $\eta$ and so it drives an outer roll in the region $y=O(k^{-1}).$ The latter flow is not self-similar and it produces a correction to the flow in the wall layer which turns out to be of relative size $O(k_X)$ smaller, so that the long-wave similarity solution is the first term in a series expansion of the Navier–Stokes equations in terms of the small local wavenumber $k_X$.

If $\varGamma _1\ne 0$ the above similarity solution of the roughness-free problem fails. However, we can instead construct a small $X$ series solution with the leading-order terms in the velocity and the pressure given by the $\varGamma _1=0$ similarity solution. The next order terms in the series expansions for the velocity and pressure are each of size $O(X^{n+{1}/{3}})$ smaller than the leading-order terms so that the roughness-free similarity solution valid for all $X$ now becomes the leading-order part of a small $X$ solution when $\varGamma _1 \ne 0.$ We conclude then that in the presence of roughness disturbances imposed on the flow sufficiently close to the leading edge are modified only at second order by roughness and, thus, develop initially as if there is no roughness.

The eigenvalue $\omega$ is determined by the ordinary differential equation eigenvalue problem found by substituting for $\boldsymbol {U_{0}}$ into (4.52)–(4.56) with $\varGamma _1=0.$ Rather than solve the eigenvalue problem we solved the partial differential system (4.52)–(4.56) numerically by marching in the $X$ direction from an initial disturbance with $\varGamma _1=0$. We found that any initial perturbation with $W_0\ne 0$ quickly evolved into the similarity form and we then extracted $\omega$ from the solution. Notice that if $W_0$ is zero for the initial perturbation then $W_0$ remains zero for all $X$ and $U_0,V_0$ decay algebraically as Libby–Fox eigenfunctions.

Whilst it is common in transient growth problems to define the flow as being unstable when the parameter $\omega$ is positive, it is perhaps not the best measure of growth. We can measure the algebraic growth relative to the algebraically growing unperturbed flow but, following Hall (Reference Hall1983), it is perhaps physically more relevant to look at the disturbance energy

(4.58)\begin{equation} E(X)=\int_0^\infty U_0^2\,\textrm{d} y. \end{equation}

For a streamwise vortex flow, the spanwise and normal velocity components of the disturbance are of relative size $O({1}/{Re})$ smaller than the streamwise component and so the energy is predominantly in the streamwise velocity component. Based on the energy of the disturbance a local spatial growth rate $\beta _L(X)$ can be defined by

(4.59)\begin{equation} \beta_L=\frac{E'}{ 2E}, \end{equation}

and, for the exact similarity solution, we obtain

(4.60)\begin{equation} X\beta_L= \omega+\frac{1-n}{4}. \end{equation}

Thus, the energy of the disturbance is growing if

(4.61)\begin{equation} \omega>\frac{n-1}{4}=\frac{ \beta_H-1}{2[2-\beta_H]}=\omega_c(\beta_H). \end{equation}

The eigenvalue $\omega$ was calculated as indicated above for $0<\beta _H<1.3$ and the results are shown in figure 9. Also shown is the curve $\omega _c=\omega _c(\beta _H)$ and algebraic growth occurs when the later curve is below the curve $\omega =\omega (\beta _H)$. The switch from algebraic growth to decay occurs when $\beta _H\simeq 0.57$ whereas $\omega$ becomes negative when $\beta _H\simeq 0.37$. Therefore, in terms of $n$ the algebraic eigensolution is unstable only for $n<0.39$. Figure 9 indicates that the energy growth of the exact algebraic solutions is a maximum for Blasius flow. Figure 10 shows the functions $U_{00}(\eta ), V_{00}(\eta ),W_{00}(\eta )$, for $n=0,\frac {1}{3}, 1.$ We see that these functions are qualitatively similar as $n$ varies but with the sizes of $V_{00}, W_{00}$ increasing relative to $U_{00}$ as $n$ increases.

Figure 9.

The local spatial growth rate of the algebraic eigensolutions of the roughness-free problem. Also shown is the curve $\omega _c=\omega (\beta _H)$ and the index $j$ associated with the large $X$ exponentially growing solutions proportional to $\textrm {e}^{cx^j}$.

Figure 10.

The functions $U_{00}(\eta ), V_{00}(\eta ),W_{00}(\eta )$, for $n=0,\frac {1}{3}, 1$ from (a–i).

For large values of $X$, algebraic solutions can be found but these all decay with increasing $X$. Those solutions are of no physical interest because we now show that there exists a rapidly growing exponential solution available for large $X$. We shall once again consider Falkner–Skan profiles and uncover subtle differences in the large $X$ downstream growth of the disturbances as the Hartree parameter varies. In fact, the exponential growth we find applies to all boundary layers with shear $\lambda \gg 1$, and we will see that the local growth rate is proportional to $\lambda$.

Consider then (4.52)–(4.56) the disturbance equations in terms of $X$ and $\eta$ the similarity variable for an external flow $\sim X^n.$ For the Görtler problem, there are in fact two types of exponential solutions: firstly a family of solutions occupying the main part of the boundary layer and a single mode, the most unstable one, localized in a thin layer near the wall. As mentioned earlier, Wu et al. (Reference Wu, Zhao and Luo2011) overlooked that mode. Here the roughness boundary condition precludes the possibility of exponentially growing solutions occupying the main part of the boundary layer so we seek solutions localized at $\eta =0.$

Suppose then that (4.52)–(4.56) support an exponentially growing solution localized in a layer of thickness $X^{-k}$ near the wall. For large $X$, we anticipate that the disturbance will be proportional to $\textrm {e}^{\varPhi (X)}$ where the phase function $\varPhi (X)\sim X^j$. If the first two terms inside the operator acting on the left-hand sides of (4.52) and (4.53) are in balance in the wall layer with $X$ large, we require $3k=j$. The largest terms proportional to $U_0, V_0$ in (4.52) balance if $X^{2k}U_0\sim X^{{(n+1)}/{2}}V_0,$ and the roughness condition in (4.55a,b) gives $V_0\sim X^{{(3n-1)}/{6}}U_0$. It follows that $k={n}/{2}+\frac {1}{6}$ and $j=3k$. For convenience, we now take the phase function $\varPhi$ in the form

(4.62)\begin{equation} \varPhi=\frac{2\omega_0}{3n+1}X^{{(3n+1)}/{2}}+\cdots. \end{equation}

The phase function satisfies $\varPhi '(x)= \omega _0 X^{{(3n-1)}/{2}}$ so the local growth rate associated with $\varPhi$ is proportional to the shear stress at the wall. Moreover, for $n<\frac {1}{3}$, the exponential growth is slower than the more usual case $\varPhi \sim X$ and, for $n=1$, the disturbance is growing like $\textrm {e}^{\omega _0 X^2}$. Thus, the roughness induces growth like $\exp ({2\omega _0 X^{1/2}})$ for a flat plate whilst flows past a right-angled wedge and against a flat plate have disturbance growth like $\textrm {e}^{\omega _0 X }$, $\exp ({{\omega _0 X^2}/{2}})$, respectively.

We define a wall layer variable ${\bar {\xi }}=X^k \eta$ and seek a solution in the wall layer of the form

(4.63)$$\begin{gather} U_0=[U_{00}(X,{\bar{\xi}})+\cdots]\ \textrm{e}^\varPhi, \end{gather}$$

(4.64)$$\begin{gather}V_0=X^{{n}/{2}-{1}/{6}}[V_{00}(X,{\bar{\xi}})+\cdots]\ \textrm{e}^\varPhi, \end{gather}$$

(4.65)$$\begin{gather}W_0=X^{{(3n-1)}/{2}}[W_{00}(X,{\bar{\xi}})+\cdots]\ \textrm{e}^\varPhi, \end{gather}$$

and the leading-order approximation to the equations of motion in the wall layer is

(4.66)$$\begin{gather} \left[\frac{\partial^2}{\partial {\bar{\xi}}^2}-\frac{2\omega_0 f''(0)}{n+1}\bar{\xi}\right]U_{00}= \sqrt\frac{2}{n+1}f''(0)V_{00}, \end{gather}$$

(4.67)$$\begin{gather}\left[\frac{\partial^2}{\partial {\bar{\xi}}^2}-\frac{2\omega_0 f''(0)}{n+1}{\bar{\xi}}\right]W_{00}=0, \end{gather}$$

(4.68)$$\begin{gather}\omega_0 U_{00}+\sqrt\frac{n+1}{2}\frac{\partial V_{00}}{\partial {\bar{\xi}}}+W_{00}=0. \end{gather}$$

The solution of the above equations must satisfy $V_{00}=U_{00}=0, {\bar {\xi }}=0$ with $V_{00\xi }, W_{00}$ bounded for ${\bar {\xi }} \rightarrow \infty$. We eliminate $U_{00}, W_{00}$ from (4.66)–(4.68) to obtain Airys equation for $V_{00{\bar {\xi }}{\bar {\xi }}}$. The solution of that equation together with $W_{00}$ obtained from (4.67) can then be substituted into (4.68) to give $U_{00}$. If $U_{00}$ is to vanish at the wall, we find the required solution of (4.66)–(4.68) is

(4.69)$$\begin{gather} V_{00}=D{\bar{\xi}},\quad U_{00}={-}\frac{D}{\omega_0 Ai(0)}\sqrt{\frac{n+1}{2}} \left[Ai(0)-Ai \left(\left[(\frac{{2}\omega_0f''(0)}{{{n+1}}}\right]^{1/3}{\bar{\xi}}\right)\right], \end{gather}$$

(4.70)$$\begin{gather}W_{00}={-}\frac{D}{ Ai(0)}\sqrt{\frac{n+1}{2}} Ai\left(\left[(\frac{{2}\omega_0f''(0)}{{{n+1}}}\right]^{1/3}{\bar{\xi}}\right). \end{gather}$$

Here $D$ is a constant and the roughness boundary condition is then satisfied if

(4.71)\begin{equation} \omega_0={\left[{\frac{n+1}{2}}\right]}^{1/2}f''(0) \left[-\frac{\varGamma_1Ai'(0)}{Ai(0)}\right] ^{3/2} \simeq 0.44 f''(0){\varGamma_1}^{3/2}(n+1)^{1/2}. \end{equation}

In the main part of the boundary layer the disturbance matches onto a simple displacement flow with $U_0\sim f''(\eta )$ which then drives a roll flow in a passive upper layer of depth $O(k^{-1})$ where it decays to zero. Thus, the disturbance now has a triple-layer structure but the upper two layers are passive with the eigenrelation determined completely by the flow in the wall layer. The key point about this large $X$ regime is that the instability problem has now become quasi-parallel, and indeed the problem remains quasi-parallel as the disturbance moves further downstream. Therefore, at high values of $\varGamma$, non-parallelism occurs only in the initial stages of the left-hand branch regime where algebraic growth is converted into exponential growth. In fact, the large $X$ structure described above applies to any boundary layer having large wall shear $\lambda$. In terms of the phase function $\varPhi$, (4.71) gives $\varPhi '(X)=[-({\varGamma _1Ai'(0)}/{Ai(0)})]^{3/2}\lambda (X)$ and an analysis similar to that above for any boundary layer, not necessarily a self-similar one, based on $\lambda (X)\gg 1$ as the expansion parameter yields the same result. Now consider the local spatial growth rate $\beta _L={E'}/{2E}$ for the large $X$ solution. Here we find that

(4.72)\begin{equation} \beta_L\simeq \frac{1-n}{4X}+ \omega_0 X^{{(3n-1)}/{2}}. \end{equation}

The first term is due to the growth of the boundary layer and the second arises from the exponential growth of the streamwise velocity component. The growth rate is therefore dominated by the second term so for Blasius flow the growth rate $\sim X^{-({1}/{2})}$. When $n=\frac {1}{3}$, the second term is constant so a $90^{\circ }$ wedge gives the more usual exponential growth found in parallel flow stability problems. For stagnation point flow, the second term behaves like $X$ and the disturbance energy is growing like $\textrm {e}^{cX^{2}}$, where $c$ is a constant. The $90^{\circ }$ wedge also has the unique property that as a disturbance moves downstream it remains indefinitely in the right-hand branch regime where the maximum growth rate occurs. In contrast to that behaviour, Tollmien–Schlichting waves or Görtler vortices over walls of constant curvature remain in the unstable region for only a finite distance. This result suggests that roughness on a $90^{\circ }$ wedge is potentially much more likely to cause transition than roughness on other wedges.

The next distinguished limit in the $k$–$\varGamma$ plane can be inferred from the large $X$ form described above or from the small wavenumber limit of the solution in the intermediate regime. The crucial stage which arises as $X$ increases is when the two passive layers on top of the wall layer support a pressure perturbation which enters the wall layer problem at leading order. This is also the manner in which the intermediate wavenumber solution discussed earlier falls as the wavenumber becomes small. It is found that the wavenumber regime $k\sim \varGamma ^{-({3}/{10})}$ leads to the interactive structure involving all three layers. Taking the small or large wavenumber limit of the solution there yields the exponentially growing structure just described above or the small wavenumber limit of the intermediate solution. The interactive structure closely follows that used in the Görtler vortex problem by CHS, the details of the solution in that layer can be found in Appendix A.

Let us summarize the previous discussion concerning solutions of the long-wave system (4.52)–(4.56) for Falkner–Skan flows at small and large $X$. In the absence of roughness, we extended the analysis of Luchini (Reference Luchini1996) to show that the exact algebraically growing similarity solution found by the latter author to show that, in the absence of roughness, Falkner–Skan flows with $0< n<0.39$ also have algebraically growing solutions. The growth is associated with the streamwise velocity component which grows like $X^\omega$ and $\omega$ begins at $\omega =0.213$ for Blasius and decreases monotonically to zero at $n=0.39$. At higher values of $n$ the similarity solution exists but it decays algebraically.

If roughness is present the similarity solution now persists only as a small $X$ solution relevant to disturbances initiated sufficiently close to the leading edge. For small enough $X$, the similarity solution, whether growing or decaying, is corrected at higher order by roughness effects. Thus, a disturbance initiated very close to the leading edge is initially unaffected by roughness and can grow as a roughness-free similarity solution. But as $X$ increases, the higher-order terms due to roughness come into play and destroy self-similarity. The roughness then dominates the evolution of a disturbance and, for large $X$, disturbances grow exponentially like $\textrm {e}^{\varPhi (X)}$ with $\varPhi \sim X^{{(3n+1)}/{2}}$ so the rate of growth increases with $n$. Moreover, for $n>\frac {1}{3}$, that growth is faster than the usual spatial growth where $\varPhi$ is linear in $X$. Thus, in summary, our analysis suggests that arbitrary disturbances initiated sufficiently close to the leading edge in the first instance evolve into the algebraically growing or decaying self-similar solution modified at higher order by roughness. However, further downstream as roughness becomes important disturbances take on the three-layer structure associated with the exponentially growing solutions. The local growth rate at that stage is proportional to the wall shear associated with the basic flow.

We observed at the beginning of this section that a disturbance moving downstream moves on the path $\varGamma _X=Ck_X^{{2[5n-1]}/{3[1-n]}}$ as $X$ increases. Hence, if $n>-\frac {1}{3}$ a disturbance beginning at a sufficiently small $X$ always passes through the left-hand branch regime $\varGamma _X \sim k_X^{-({4}/{3})}$, through the interactive regime and into the small $k$ limit of the intermediate wavenumber regime. Thus, all Falkner–Skan flows of physical interest have the latter behaviour. The development beyond the small $k$ regime then depends crucially on $n$, we restrict attention to the case $n>0$. Firstly, we note that if $0\le n < \frac {1}{5}$ the path taken by a disturbance has $\varGamma _X$ tending to zero and so the right-hand regime is not reachable. For $\frac {1}{5}< n<\frac {1}{3}$, the right-hand branch regime can be reached but ultimately the path takes the disturbance into the stable area to the right of the neutral curve. The case $n=\frac {1}{3}$ has already been commented on; in this case the disturbance ultimately moves on a path maximising the downstream growth. Finally, for $n>\frac {1}{3}$, a disturbance originating in the small wavenumber regime moves on a path steeper than $\varGamma _X \sim k_X^{2/3}$ and grows indefinitely but not at the fastest possible rate.

4.4. Numerical solution of the long-wave evolution equation

The long-wave evolution equations (4.52)–(4.56) are solved numerically using finite differences in the $X, \eta$ directions. Assuming that $U_0, V_0$ and $W_0$ are known at say $X=\bar {X}$, we first step (4.52), (4.53) forward to the new $X$ location using an implicit finite difference scheme to find $U_0, W_0$. We then integrate the continuity equation (4.54) with respect to $\eta$ to find $V_0$ at the new location. The energy of the disturbance is monitored and the local growth rate $\beta _L={E'}/{2E}$ calculated. The calculations are initiated at a sufficiently small value of $X$ using the least stable algebraic eigensolution discussed previously in this section. The initial $X$ is chosen to be sufficiently small that the roughness gives only a second-order correction to the eigensolution.

Figure 11 shows $\vert \beta _L\vert$ for four representative values of $n$. Unstable solutions are represented by continuous curves, stable ones are represented by dashed curves. The red curves denote the large $X$ prediction of growth rates given by (4.72). For the first two values $n=0,\frac {1}{3}$, the algebraic solution is unstable and so we observe the disturbance is unstable for all $X$ shown. We see in both cases the solution initially follows a line of slope $-1$ corresponding to the algebraic eigensolution before switching to the exponential solution. In other words, the algebraic eigensolution, which in the presence of roughness is valid only at small $X$, deforms into the large $X$ exponentially growing solution as it moves downstream. Note also that, for $n=\frac {1}{3}$, the exponential solution varies like $\textrm {e}^{c_0X}$, where $c_0$ is a constant.

Figure 11.

The log of the absolute value of the local growth rate $\beta _L$ as a function of $\log X$ for $n=0,\frac {1}{3}$, (a,b) and $n=\frac {2}{3},1$ (c,d). Note that in the top row $\beta _L$ is always positive whilst in the second row it is positive only to the right of the cusp. Continuous curves denote positive values of $\beta _L$ (unstable) whilst dashed curves denote negative values (stable). The calculation for $n=1$ is for $\varGamma _1=1,$ and the other cases all have $\varGamma _1=0.00125$. The red curves correspond to the large $X$ prediction of the growth rate.

The other two cases correspond to $n=\frac {2}{3}, 1$ and here the algebraic solution is stable so the growth rate is initially negative before going through zero and increasing until it asymptotes onto the exponentially growing solution. Thus, for $n>0.39$, we can define a neutral point and, hence, neutral curve, associated with the zero of $\beta _L$. Since we are plotting $\log \vert \beta _L \vert$, the cusps correspond to zeros of the growth rate.

The neutral configuration is independent of the choice of $\varGamma _1$ used in the calculations provided that we express the neutral configuration in terms of the local values of $\varGamma _X,k_X$. We recall that the latter are as defined in (4.1) and (4.2) from which we deduce that

(4.73)\begin{equation} {\varGamma_X}{ k_X^{4/3}}=\varGamma_1 X^{n-({1}/{3})}=\varGamma_c(n), \end{equation}

so that for a given $n>0.39$, the neutral curve associated with disturbances originating near the leading edge has a left-hand branch defined by the above equation with $\varGamma _c$ fixed by the choice of $\varGamma _1$ and the value of $X$ where the growth rate changes sign. The fact that $\varGamma _c$ above is independent of the choice of $\varGamma _1$ is readily seen from (4.52)–(4.56) by rescaling $X$ to set $\varGamma _1=1$. Figure 12 shows $\varGamma _c$ as a function of $n$ and we see it increases monotonically with $n$ from $n=0.39$ where it can be first defined. For $n<.39$, disturbances initiated close to the leading edge evolve into algebraically growing similarity solutions before roughness comes into play. The algebraic growth is then converted into exponential growth as the effect of roughness increases, that growth continues as the disturbance enters the interactive regime and then into the intermediate wavenumber regime. Beyond that stage the behaviour will depend on $n$ as to whether the disturbance stabilises reaching the right-hand branch regime.

Figure 12.

The constant $\varGamma _c$ which defines the left-hand branch of the neutral curve as a function of $n$.

It follows from the above discussion that if $n<0.39$, a left-hand branch of the neutral curve cannot be defined since the algebraic solution is itself already unstable when it is initiated. But at finite values of the roughness parameter we would expect that the algebraically growing solution might be stabilised and a neutral curve might be defined by where the ultimate exponential growth of the disturbance begins. We will return to that issue later.

It is also of interest to compare the left-hand branch predictions with the large $k$ asymptotic predictions for Falkner–Skan flows. We saw earlier that, for a boundary layer, the right-hand branch neutral curve is given by $\varGamma \lambda ^{4/3}=4k^{2/3}$, where $\lambda$ is the base flow wall shear. In terms of local values of $\varGamma _X, k_X$ for Falkner–Skan flow we obtain

(4.74)\begin{equation} \frac{\varGamma_X}{k_X^{2/3}}=4\left[\frac{2}{n+1}\right]^{2/3}[F''(0)]^{-({4}/{3})}, \end{equation}

which defines the right-hand branch of the neutral curve for $n>0$.

Figure 13 shows the left- and right-hand branches of the neutral curves when they exist for the Falkner–Skan flows with $n=0,\frac {1}{3},\frac {2}{3},1$. For large values of the roughness parameter $\varGamma$, we anticipate that the right-hand branches shown will give good approximations to the curves to be obtained in the next section by integrating the full equations without any assumption on the size of $k$. Likewise we might expect that the left-hand branches for the two larger values of $n$ accurately predict the results of the full calculations to be discussed in the next section.

Figure 13.

Plots (a,b) show the right-hand branches of neutral curves for $n=0, \frac {1}{3}$, no left-hand branch exists. Plots (c,d) show left and right-hand branches for $n=\frac {2}{3},1$.

6.1. Short-scale roughness

From the discussion in § 3 we know that, when the roughness parameter and wavenumber are large, the neutral configuration and the fastest growing mode have $\varGamma \sim k^{2/3}$. Moreover, the disturbance localizes near the wall and its only dependence on the mean flow is through the wall shear of the streamwise mean flow velocity. In that case we might anticipate that the parameterization of the instability might be more informatively expressed in terms of basic flow properties near the wall rather than global ones. With that in mind, we define the friction velocity $U^*$ by

(6.1)\begin{equation} u_\tau=\sqrt{\frac{\tau^*}{\rho}}, \end{equation}

where $\rho$ is the fluid density and $\tau ^*$ is the unscaled wall shear stress. For the basic flow (2.4), we have

(6.2)\begin{equation} u_\tau=\sqrt{\frac{\mu U_0 R_e \lambda}{\rho L}}=Re^{3/2}\frac{\nu}{L}\lambda^{1/2}. \end{equation}

Suppose that the dimensional wall undulation wavelength and amplitude are $b$ and $h$, respectively. It follows that

(6.3a,b)\begin{equation} k= \frac{2 {\rm \pi}L}{b mR_e},\quad 2\epsilon= \frac{h R_e}{L}, \end{equation}

where $m={\alpha }/{k}$. We define the friction Reynolds number $R_f={u_\tau b}/{\nu }$ based on the friction velocity and the roughness wavelength so that

(6.4)\begin{equation} R_e= \left[{\frac{L R_f}{b}}\right]^{2/3}\lambda^{-({1}/{3})}. \end{equation}

From our discussion about the $\varGamma \sim k^{2/3}$ wavenumber regime, we know that if the roughness wavelength is small compared with the boundary layer thickness then the flow is unstable if $\varGamma$ satisfies

(6.5)\begin{equation} \varGamma>4\lambda^{-({4}/{3})}k^{2/3}. \end{equation}

We recall that $\varGamma$ is defined in terms of $R_e$ by (2.39), (2.45) so, defining $c=c(m)={2^{7/4} {\rm \pi}^{1/4} {(1+m^2) }^{3/8}}/{5^{5/8} m^{3/4} {\vert Ai '(0) \vert }^{3/8}}$, the above condition for instability can be written as

(6.6)\begin{equation} R_f>c(m) \frac{{\left[\dfrac{b}{h}\right]}^{3/4} }{{\left[\log \dfrac{b}{h}\right]}^{3/8}}, \end{equation}

and the lowest value of $c$ over $m$ occurs for $m\rightarrow \infty$. Therefore, (6.6) with ${c}=c(\infty )\simeq 24.45$ gives a simple universal instability criterion for two-dimensional boundary layers in terms of the friction Reynolds number and the amplitude to wavelength ratio of the undulations. The universality of (6.6) is possible because at high wavenumbers the vortex concentrates near the wall and, therefore, depends only on the wall shear rather than the shape of the mean flow throughout the boundary layer. The dependence on the wall shear is then built into the condition (6.6) through the friction Reynolds number. As an alternative to the friction Reynolds number, the roughness Reynolds number $R_r$ is sometimes used, this is defined to be the Reynolds number based on the flow speed of the unperturbed streamwise velocity at a distance $h$ from the wall and $h$ as the length scale. Thus,

(6.7)\begin{equation} R_r=R^2_f \frac{h^2}{b^2}, \end{equation}

and instability occurs when

(6.8)\begin{equation} R_r>{ {c}}^2\frac{{\left[\dfrac{h}{b}\right]}^{1/2} } {{\left[\log \dfrac{b}{h}\right]}^{3/4}}. \end{equation}

6.2. Long-scale roughness

Suppose now that the wavelength of the roughness is large compared with the boundary layer thickness.This corresponds to $k\ll 1, \varGamma \gg 1$ and the crucial scaling here is $\varGamma \sim k^{-({4}/{3})}$. In this case we saw that, for Falkner–Skan flows, there can be algebraic growth or decay depending on the Falkner–Skan index $n$, but that always there will be a transition to exponential growth. The position where the transition from algebraic variation to exponential variation occurs is not well defined but we know that it occurs in this parameter regime. Thus, we know that exponential growth presumably leading to transition to turbulence will occur for

(6.9)\begin{equation} \epsilon^2 R^{4/3}_e \log R_e \sim k^{-({4}/{3})}, \end{equation}

and rewriting this in terms of $b$ we obtain

(6.10)\begin{equation} \frac{h}{L}\sim \frac{\left[\dfrac{b}{L}\right]^{2/3}}{R_e\sqrt{\log Re}}. \end{equation}

An experimental investigation related to the above result was given by Fage (Reference Fage1943) who looked at the effect of a bulge on transition on an aerofoil. Further discussion of Fage's results and comparable flight test results can be found in Carmichael, Whites & Pfenninger (Reference Carmichael, Whites and Pfenninger1957) and Carmichael (Reference Carmichael1959). A direct comparison between the theory and Fage's experiment is not possible because the experiment had only a finite length of corrugation and the experiments also depended on ${\rm \Delta} L$ the distance from the leading edge where the corrugation occurred. However, from experimental observations over a range of ${\rm \Delta} L$ Fage deduced that instability occurs when

(6.11)\begin{equation} \frac{h}{L} \sim \left[\frac{b}{L}\right]^{1/2}, \end{equation}

whereas (6.10) has

(6.12)\begin{equation} \frac{h}{L}\sim {\left[\frac{b}{L}\right]^{2/3}}. \end{equation}

The result (6.11) was confirmed by Carmichael et al. (Reference Carmichael, Whites and Pfenninger1957) based on Lockheed F94A Starfire flight tests. It was suggested by the experimentalists that the above criterion was associated with the onset of flow separation induced by wall waviness since such a separation would render the boundary layer unstable to rapidly growing inviscid waves. By way of contrast, the criterion (6.12), which is slightly weaker than (6.11) since ${b}/{L}$ is small, corresponds to the onset of exponential growth of streamwise vortex instabilities. But we know from Hall & Horseman (Reference Hall and Horseman1991) and subsequent authors that streamwise vortices are also highly unstable to inviscid waves and so the outcome would be similar to that associated with separation. There is in fact evidence from pipe flow that it is the roughness instability described here rather than separation-induced instability that occurs first. Thus, Loh & Blackburn (Reference Loh and Blackburn2011) and Hall & Ozcakir (Reference Hall and Ozcakir2021) show that at small amplitudes instability occurs before the onset of flow separation. Thus, though the results (6.11), (6.12) differ by a relatively small factor of $[{b}/{L}]^{1/6}$, it may well be that (6.12) is relevant to the experiments.

We close this section with some comments on how the model for wall roughness used here can be made more realistic; a more complete discussion in the context of pipe flows is given in Hall & Ozcakir (Reference Hall and Ozcakir2021). Suppose then that the wall is given by

(6.13)\begin{equation} y=2\epsilon F(X)=2\epsilon \sum_{n=1}^{n=\infty} a_n \cos n \alpha X+b_n \sin n \alpha X. \end{equation}

The key point is that each term in the above Fourier series only contributes to the roughness condition (2.42) by a self-interaction. Thus, in the analysis leading to (2.42) we simply generalize each expansion to have a Fourier series in $X$ rather than a single term. We find that the condition linking $U_y, V_y$ at the wall in (2.42) is then modified to give

(6.14)\begin{equation} \frac{\partial}{\partial y}\left[\frac{10\kappa\alpha^{4/3} \lambda^{1/3}k^2\mu Ai'^2(0)}{\alpha^2+k^2}JU +V\right]=0,\quad y=0. \end{equation}

Here $J$ is defined by

(6.15)\begin{equation} J=\sum_{n=1}^{n=\infty} \frac{n^{4/3}(1+m^2)(a_n^2+b_n^2)}{1+m^2n^2}, \end{equation}

where $m={\alpha }/{k}$. We deduce from the above roughness condition that a disturbance evolving over a pure sine wave wall with $a_1=1,\ b_1=0$ at say $\kappa =\kappa _s$ has exactly the same behaviour as a disturbance evolving over the more general undulation (6.14) if $\kappa ={\kappa _s}/{J}$. Thus, the size of $J$ determines how unstable a given wall is compared with the pure sine wave wall. An estimate of how the particular shape of the wall modifies the stability problem for pipe flow by considering sawtooth, step function, rectified sine wave and triangular shape roughnesses was given by Hall & Ozcakir (Reference Hall and Ozcakir2021). It was found that the step function wall was the most unstable but in fact $J$ was found to vary between roughly $0.5$ and $2$ so the particular shape of the roughness is not crucial for pipe flows. The same result also follows for boundary layer flows.