3.1 The nonlinear development of the critical layer

The singularity must be regularised by a thin critical layer surrounding $y=y_{c}$ . Inside the critical layer we expand

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{u}=c+\unicode[STIX]{x1D716}(\unicode[STIX]{x1D706}_{1}Z+w_{c})+\unicode[STIX]{x1D716}^{2}(\unicode[STIX]{x1D706}_{2}Z^{2}+W_{2}(Z))+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}=\unicode[STIX]{x1D716}(\unicode[STIX]{x1D716}\unicode[STIX]{x1D719}_{1}(x,Z)+\cdots +\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D719}_{2}(x,Z)+\cdots +\unicode[STIX]{x1D716}^{5}\unicode[STIX]{x1D719}_{5}(x,Z)+\cdots \,), & \displaystyle\end{eqnarray}$$

using $Z=\unicode[STIX]{x1D716}^{-1}(y-y_{c})$ . The first set of dots in (3.14) represents a term of order $\unicode[STIX]{x1D716}^{2}\ln \unicode[STIX]{x1D716}$ which matches to a logarithmic part of the outer Rayleigh solution: further analysis of this contribution is not necessary in the arguments that follow. The critical layer induces a core mean flow distortion $\unicode[STIX]{x1D716}w(y)$ where $w(y)$ is continuous with $w_{c}=w(y_{c})$ , but we allow for a discontinuity in its derivative and denote the derivatives just above and beneath the critical layer as $w_{c+}^{\prime }$ and $w_{c-}^{\prime }$ , respectively.

For what follows it is convenient to define the total stream functions $\unicode[STIX]{x1D6F7}_{1}$ and $\unicode[STIX]{x1D6F7}_{2}$ , so that

(3.15a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}_{1Z}=\unicode[STIX]{x1D719}_{1Z}+w_{c}+\unicode[STIX]{x1D706}_{1}Z,\quad \unicode[STIX]{x1D6F7}_{2Z}=\unicode[STIX]{x1D719}_{2Z}+W_{2}+\unicode[STIX]{x1D706}_{2}Z^{2}. & \displaystyle\end{eqnarray}$$

In view of (3.12) we require the following matching conditions as $Z\rightarrow \pm \infty$ :

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{1}\rightarrow b_{0}\cos (\unicode[STIX]{x1D6FC}x), & \displaystyle\end{eqnarray}$$

and

(3.17a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{2}\rightarrow b_{0}\unicode[STIX]{x1D707}_{c}Z\{\ln |Z|\cos (\unicode[STIX]{x1D6FC}x)+J(\unicode[STIX]{x1D6FC}x)\}+a_{0}Z\cos (\unicode[STIX]{x1D6FC}x),\quad W_{2}\rightarrow w_{c\pm }^{\prime }Z, & \displaystyle\end{eqnarray}$$

where

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle J(\unicode[STIX]{x1D6FC}x)=-\unicode[STIX]{x1D6E9}\sin (\unicode[STIX]{x1D6FC}x)+\text{higher harmonics}. & \displaystyle\end{eqnarray}$$

An analysis inside of the critical layer reveals that $J$ is an odd function and experiences a jump across $y=y_{c}$ (see appendix B). At first sight it seems that the higher harmonics appearing in (3.18) contradict the leading-order monochromatic form for $\unicode[STIX]{x1D713}$ in the Rayleigh region (see (3.2)). However, provided the constant $\unicode[STIX]{x1D707}_{c}$ is asymptotically small (i.e. $\overline{u}$ is inflectional at the critical level to leading order), the higher harmonics play a negligible role outside of the critical layer (see also comments in § 3.2).

From the Navier–Stokes equations we find that the leading-order response is

(3.19) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D6F7}_{1Z}\unicode[STIX]{x2202}_{x}-\unicode[STIX]{x1D6F7}_{1x}\unicode[STIX]{x2202}_{Z})\unicode[STIX]{x1D6F7}_{1ZZ}=\unicode[STIX]{x1D6F7}_{1ZZZZ}. & \displaystyle\end{eqnarray}$$

Despite the nonlinearity of this equation, a simple exact solution is possible, namely

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}_{1}=\frac{\unicode[STIX]{x1D706}_{1}}{2}\left(Z+\frac{w_{c}}{\unicode[STIX]{x1D706}_{1}}\right)^{2}+b_{0}\cos (\unicode[STIX]{x1D6FC}x), & \displaystyle\end{eqnarray}$$

which ensures the continuity of $b_{0}$ across the critical layer and provides the key to unlocking the behaviour at higher orders. Also it is important to note here that there is virtually no effect of viscosity at this order because if we differentiate (3.20) four times with respect to $Z$ it vanishes identically. Hence the next-order response

(3.21) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D6F7}_{1Z}\unicode[STIX]{x2202}_{x}-\unicode[STIX]{x1D6F7}_{1x}\unicode[STIX]{x2202}_{Z})\unicode[STIX]{x1D6F7}_{2ZZ}=\unicode[STIX]{x1D6F7}_{2ZZZZ} & & \displaystyle\end{eqnarray}$$

determines the phase jump. The rescaling $\widetilde{Z}=(\unicode[STIX]{x1D706}_{1}/b_{0})^{1/2}(Z+w_{c}/\unicode[STIX]{x1D706}_{1})$ , $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FC}x$ , $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6F7}_{2}(\unicode[STIX]{x1D706}_{1}/b_{0})^{3/2}/2\unicode[STIX]{x1D706}_{2}$ converts this problem to the simplified form

(3.22) $$\begin{eqnarray}\displaystyle & \displaystyle (\widetilde{Z}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}+\sin \unicode[STIX]{x1D709}\unicode[STIX]{x2202}_{\widetilde{Z}})\unicode[STIX]{x1D6F7}_{\widetilde{Z}\widetilde{Z}}=\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6F7}_{\widetilde{Z}\widetilde{Z}\widetilde{Z}\widetilde{Z}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}=(\unicode[STIX]{x1D706}_{1}/b_{0})^{1/2}/(\unicode[STIX]{x1D6FC}b_{0})$ is a measure of inverse amplitude. It can then be shown that the appropriate matching conditions to the Rayleigh solution are

(3.23) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}\sim \frac{\widetilde{Z}^{3}}{6}-\frac{\unicode[STIX]{x1D6E9}}{4\unicode[STIX]{x1D6FE}}\widetilde{Z}^{2}+\widetilde{Z}\{(\ln |\widetilde{Z}|+\text{const.})\cos \unicode[STIX]{x1D709}+J(\unicode[STIX]{x1D709})\},\quad \widetilde{Z}\rightarrow \pm \infty , & \displaystyle\end{eqnarray}$$

where the first term is due to the basic flow and the third term arises from the logarithmic singularity in the Rayleigh expansion (3.12). The second term is the shear jump in the mean flow distortion created by the interaction of the sinusoidal functions in (3.22) and (3.23). Haberman (Reference Haberman1972) solved the above problem and found the phase shift $\unicode[STIX]{x1D703}$ as a function of $\unicode[STIX]{x1D6FE}$ , whilst the fact that $J$ also contains higher harmonics was first recorded by Smith & Bodonyi (Reference Smith and Bodonyi1982a ) (see appendix B). In particular, both sets of authors found that

(3.24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}\rightarrow 0\quad \text{as}\,\unicode[STIX]{x1D6FE}\rightarrow 0, & & \displaystyle\end{eqnarray}$$

(3.25) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}\rightarrow -\unicode[STIX]{x03C0}\quad \text{as}\,\unicode[STIX]{x1D6FE}\rightarrow \infty , & & \displaystyle\end{eqnarray}$$

with the latter result indicating that the phase shift approaches the classical value of $-\unicode[STIX]{x03C0}$ (see appendix B) as we revert back to the linear regime.

3.2 The non-existence of very small-amplitude states in PPF at order one wavenumbers

Now let us consider the physical significance of the phase shift $\unicode[STIX]{x1D703}$ for the behaviour outside of the critical layer, and the implication of it for PPF. The unperturbed flow has $\overline{u}=1-y^{2}$ , with

(3.26a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{1}=-2y_{c},\quad \unicode[STIX]{x1D706}_{2}=-1,\quad y_{c}=-\sqrt{1-c}. & & \displaystyle\end{eqnarray}$$

In addition the symmetry condition $\widehat{\unicode[STIX]{x1D713}}^{\prime }=0$ at $y=0$ , and the impermeability condition $\widehat{\unicode[STIX]{x1D713}}=0$ at $y=-1$ should be applied to (3.1).

Two important features of the Rayleigh solution should be noted:

(i) If $y_{c}\notin [y_{1},y_{2}]$ , then the Wronskian of any two independent solutions of (3.1), say $\widehat{\unicode[STIX]{x1D713}}_{1},\widehat{\unicode[STIX]{x1D713}}_{2}$ , must satisfy

(3.27) $$\begin{eqnarray}\displaystyle & \displaystyle [\widehat{\unicode[STIX]{x1D713}}_{1}\widehat{\unicode[STIX]{x1D713}}_{2y}-\widehat{\unicode[STIX]{x1D713}}_{1y}\widehat{\unicode[STIX]{x1D713}}_{2}]_{y=y_{1}}^{y_{2}}=0. & \displaystyle\end{eqnarray}$$

(ii) Rayleigh solutions do not force the mean flow via a Reynolds stress: it is easy to see this by applying the fact that $\widehat{\unicode[STIX]{x1D713}}_{yy}$ is proportional to a real function of $y$ multiplied by $\widehat{\unicode[STIX]{x1D713}}$ in (2.6c ), and hence the Reynolds stress $(\text{i}\unicode[STIX]{x1D6FC}\widehat{\unicode[STIX]{x1D713}}\widehat{\unicode[STIX]{x1D713}}_{yy}^{\ast }+\text{c.c.})$ is identically zero.

The Frobenius solution suggests that if $\unicode[STIX]{x1D707}_{c}\unicode[STIX]{x1D703}\neq 0$ then across the critical layer the phase of the wave will jump. Here we remark that inviscid equations preserve symmetry in $x$ : for example, the Euler equations are invariant under reflection in $x$ but the Navier–Stokes equations are not. Therefore, the origin of this phase jump is the effect of viscosity within the critical layer. We further note that since $\widehat{\unicode[STIX]{x1D713}}^{(e)}$ and $\widehat{\unicode[STIX]{x1D713}}^{(o)}$ are two independent solutions of (3.1), then from the property (i) mentioned above,

(3.28) $$\begin{eqnarray}\displaystyle & \displaystyle 0=[\widehat{\unicode[STIX]{x1D713}}_{y}^{(e)}\widehat{\unicode[STIX]{x1D713}}^{(o)}-\widehat{\unicode[STIX]{x1D713}}^{(e)}\widehat{\unicode[STIX]{x1D713}}_{y}^{(o)}]_{y=y_{c+}}^{0}+[\widehat{\unicode[STIX]{x1D713}}_{y}^{(e)}\widehat{\unicode[STIX]{x1D713}}^{(o)}-\widehat{\unicode[STIX]{x1D713}}^{(e)}\widehat{\unicode[STIX]{x1D713}}_{y}^{(o)}]_{y=-1}^{y_{c-}}. & \displaystyle\end{eqnarray}$$

If we apply the symmetry condition (i.e. $\widehat{\unicode[STIX]{x1D713}}^{(e)}=\widehat{\unicode[STIX]{x1D713}}_{y}^{(o)}=0$ at $y=0$ , and thus we require $\unicode[STIX]{x1D6E9}_{0}=0$ ) and the impermeability condition for $\widehat{\unicode[STIX]{x1D713}}^{(e)}$ (i.e. $\widehat{\unicode[STIX]{x1D713}}^{(e)}=0$ at $y=-1$ ), (3.28) reduces to

(3.29) $$\begin{eqnarray}\displaystyle & \displaystyle [\widehat{\unicode[STIX]{x1D713}}^{(e)}\widehat{\unicode[STIX]{x1D713}}_{y}^{(o)}]_{y=y_{c-}}^{y_{c+}}=(\widehat{\unicode[STIX]{x1D713}}_{y}^{(e)}\widehat{\unicode[STIX]{x1D713}}^{(o)})|_{y=-1}. & \displaystyle\end{eqnarray}$$

Now it is easy to see that the left-hand side equals $b^{2}\unicode[STIX]{x1D707}_{c}\unicode[STIX]{x1D703}$ , and thus if this quantity is $O(1)$ the impermeability condition for $\widehat{\unicode[STIX]{x1D713}}^{(o)}$ cannot be satisfied. Of course, if the basic flow is inflectional in the vicinity of the critical level, then $\unicode[STIX]{x1D707}_{c}$ is zero to leading order, and there is no difficulty. However, for the linear PPF case $\unicode[STIX]{x1D703}\unicode[STIX]{x1D707}_{c}=-\unicode[STIX]{x03C0}/y_{c}$ , and it therefore follows that the critical layer solution is incompatible with the outer Rayleigh region. Therefore, it appears at first sight that there can be no large-Reynolds-number linear neutral solution for this flow.

Nevertheless, such a structure is still possible provided the critical layer is situated sufficiently close to the wall, as we shall see in detail in the following sections. For the simplest case the critical layer is embedded within a viscous wall layer and the balance (3.29) is no longer relevant. This case corresponds to the lower neutral branch (see figure 5). Even when the critical layer is slightly detached from the wall, neutral solutions can still exist provided the ratio $b/a$ in a near-wall version of the Rayleigh solution is small. In this case the non-zero odd contribution to the stream function, induced by the jump across the critical layer, is small (see (3.29) and note that the size of the right-hand side is of the order of $ab$ ). A small non-zero vertical velocity near the wall generated by the jump can now be reduced to zero within a thin Stokes layer attached to the wall which satisfies no-slip conditions (a sketch of the layer is given later in figure 10). This structure corresponds to a mode on the upper neutral branch, and the nonlinear development of the wave within the thin wall layer is similar to the one discussed here. Notably the structure of the critical layer remains identical near the bifurcation point. However, the associated matched asymptotic expansion becomes more complicated because of the long wavelength that is then in operation.

For larger amplitudes the modulus of the phase shift $|\unicode[STIX]{x1D703}|$ is expected to be asymptotically small, in view of (3.24), and a neutral Rayleigh structure is possible even for PPF (see the sketch later in figure 11). The physical reason for the vanishingly small phase shift is of course that now the convective terms completely dominate the higher-order critical layer structure (3.22): see the discussion just above (3.28) concerning the symmetry that appears in inviscid flows. In this strongly nonlinear case the phase shift occurs at higher order (corresponding to $\unicode[STIX]{x1D719}_{5}$ in (3.14)) due to viscous effects. Despite its now tiny size the phase shift still remains important, as its balance with the Stokes layer phase shift through (3.29) determines the amplitude of the wave, as described in § 5.2.

4.1 Nonlinear double-deck theory

Figure 5.

The nonlinear double-deck structure relevant for travelling wave solutions bifurcating from the lower branch of the linear neutral curve.

The lower-branch neutral wave is driven by near-wall dynamics, and in the majority of the flow field the fluid motion is predominantly inviscid. The wall layer (which is essentially a critical layer attached to the wall) is viscous and adjusts the inviscid flow appropriately in order to satisfy the no-slip condition. We shall see that, as the amplitude of the disturbance increases, nonlinear effects first become evident in the wall layer, whilst the core remains linear to leading order.

Let us estimate the thickness of the wall layer $\unicode[STIX]{x1D6FF}$ and the appropriate sizes of wavenumber and phase speed from an order of magnitude analysis. We denote the size of the stream function and pressure as $O(\unicode[STIX]{x1D713}),O(p)$ in the core and $O(\unicode[STIX]{x1D6F9}),O(P)$ , in the wall layer. Since the size of the stream function must be reduced towards the wall so that the normal velocity vanishes there, $O(\unicode[STIX]{x1D6F9})\sim O(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713})$ . Then, in the wall layer, we require the terms on the left-hand side of (2.6a ) to be in balance. Thus,

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle O(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6F9})\sim O(c\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}^{-1}\unicode[STIX]{x1D6F9})\sim O(\unicode[STIX]{x1D6FC}P). & \displaystyle\end{eqnarray}$$

It should also be noted that the mean flow is $O(\unicode[STIX]{x1D6FF})$ in this layer. Next we examine the normal component of the momentum equation (2.6b ). In the wall layer the pressure gradient term dominates, indicating here that $P$ is independent of the vertical coordinate, and hence the order of magnitude of the pressure in the core and wall layer coincide, i.e. $O(p)\sim O(P)$ . However, in the core the pressure varies in $y$ , and thus the size of the convective term must be increased in order to balance the pressure term. That balance is achieved if $O(\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D713})\sim O(p)$ , i.e. $O(\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D6F9})\sim O(\unicode[STIX]{x1D6FF}P)$ . From this balance and (4.1), we have

(4.2a-c ) $$\begin{eqnarray}\displaystyle O(\unicode[STIX]{x1D6FC}^{2})\sim O(\unicode[STIX]{x1D6FF}),\quad O(c)\sim O(\unicode[STIX]{x1D6FF}),\quad O(P)\sim O(\unicode[STIX]{x1D6FF}^{2}). & & \displaystyle\end{eqnarray}$$

As we shall see in § 5, these balances also hold for the flow structure on the upper branch of the neutral curve. Returning to our investigation of the lower-branch structure, we wish to include the effects of both viscosity and nonlinearity within the wall layer. We therefore assume that the viscous term of $O(R^{-1}\unicode[STIX]{x1D6FF}^{-3}\unicode[STIX]{x1D6F9})$ is in balance with (4.1). This implies $\unicode[STIX]{x1D6FF}\sim (\unicode[STIX]{x1D6FC}R)^{-1/3}$ , from which the wall layer thickness can be determined to be

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}=R^{-2/7}. & \displaystyle\end{eqnarray}$$

With the value of $\unicode[STIX]{x1D6FF}$ established, the orders of magnitude of all the other fluid quantities can be expressed in terms of $R$ . In particular we have $\unicode[STIX]{x1D6FC}\sim R^{-1/7},c\sim R^{-2/7},\unicode[STIX]{x1D6F9}\sim R^{-4/7}$ . A sketch of the asymptotic structure is given in figure 5.

Based on this order of magnitude analysis we now write $\unicode[STIX]{x1D6FC}_{0}=R^{1/7}\unicode[STIX]{x1D6FC}$ , $X=R^{-1/7}x$ and $c_{0}=R^{2/7}c$ in order to establish a formal asymptotic framework. We substitute the core expansions

(4.4a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}_{0}(X,y)+\cdots \,,\quad p=\unicode[STIX]{x1D6FF}^{2}p_{0}(X,y)+\cdots & \displaystyle\end{eqnarray}$$

into (2.6a ) and (2.6b ), obtaining

(4.5a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{b}\unicode[STIX]{x1D713}_{0Xy}-u_{b}^{\prime }\unicode[STIX]{x1D713}_{0X}=0,\quad u_{b}\unicode[STIX]{x1D713}_{0XX}=p_{0y}, & \displaystyle\end{eqnarray}$$

which has solution

(4.6a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}_{0}=-Au_{b},\quad p_{0}=-A_{XX}\int ^{y}u_{b}^{2}\,\text{d}y_{\ast } & \displaystyle\end{eqnarray}$$

up to an unknown amplitude $A(X)$ , which satisfies

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle A(X)=-A(X+\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}), & \displaystyle\end{eqnarray}$$

in view of the assumed symmetry (2.9) and the evenness of $u_{b}$ in $y$ . Here we have used the fact that $\overline{u}_{0}=u_{b}(y)$ to leading order because the Reynolds stress acting on the leading part of $\overline{u}_{0}$ is determined solely by linear, inviscid fluctuations, and thus the property (ii) remarked upon in § 3.2 holds true here.

In the wall layer we expand

(4.8a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}=\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6F9}_{0}(X,Y)+\cdots \,,\quad \overline{u}=\unicode[STIX]{x1D6FF}\overline{U}_{0}(Y)+\cdots \,,\quad p=\unicode[STIX]{x1D6FF}^{2}P_{0}(X)+\cdots \,, & \displaystyle\end{eqnarray}$$

where $Y=\unicode[STIX]{x1D6FF}^{-1}(y+1)$ . Substituting these expansions into (2.6a ) and (2.6c ) we obtain the leading-order nonlinear problem

(4.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle (\overline{U}_{0}-c_{0}+\unicode[STIX]{x1D6F9}_{0Y})\unicode[STIX]{x1D6F9}_{0XY}-(\overline{U}_{0}^{\prime }+\unicode[STIX]{x1D6F9}_{0YY})\unicode[STIX]{x1D6F9}_{0X}=-P_{0}^{\prime }+\unicode[STIX]{x1D6F9}_{0YYY}, & \displaystyle\end{eqnarray}$$

(4.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\overline{\unicode[STIX]{x1D6F9}_{0X}\unicode[STIX]{x1D6F9}_{0YY}}+\overline{U}_{0}^{\prime \prime }. & \displaystyle\end{eqnarray}$$

(4.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{U}_{0}=\unicode[STIX]{x1D6F9}_{0}=\unicode[STIX]{x1D6F9}_{0Y}=0,\quad \text{at}\,Y=0, & \displaystyle\end{eqnarray}$$

(4.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{U}_{0}^{\prime }\rightarrow 2,\quad \unicode[STIX]{x1D6F9}_{0Y}\rightarrow -2A\quad \text{as}\,Y\rightarrow \infty . & \displaystyle\end{eqnarray}$$

$P_{0}$

$A$

$P_{0Y}=0$

$P_{0}=p_{0}|_{y=-1}$

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle P_{0}(X+\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}_{0})-P_{0}(X)=-{\textstyle \frac{16}{15}}A_{XX}. & \displaystyle\end{eqnarray}$$

In order to solve the nonlinear system (4.9)–(4.11) numerically, it is convenient to adopt the decomposition $\unicode[STIX]{x1D6F9}_{0}=\unicode[STIX]{x1D6F9}_{0}^{(1)}+\unicode[STIX]{x1D6F9}_{0}^{(0)}$ , $P_{0}=P_{0}^{(1)}+P_{0}^{(0)}$ with the same symmetries as in (2.10). Then from (4.10b ), (4.11):

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle P_{0X}^{(1)}=-{\textstyle \frac{4}{15}}\unicode[STIX]{x1D6F9}_{0XXXY}^{(1)}|_{Y\rightarrow \infty }, & \displaystyle\end{eqnarray}$$

whilst applying $\unicode[STIX]{x1D6F9}_{0Y}^{(0)}|_{Y\rightarrow \infty }=0$ to the far-field form of (4.9a ) we have

(4.13) $$\begin{eqnarray}\displaystyle & \displaystyle P_{0X}^{(0)}=(2\unicode[STIX]{x1D6F9}_{0X}^{(0)}-\unicode[STIX]{x1D6F9}_{0Y}^{(1)}\unicode[STIX]{x1D6F9}_{0XY}^{(1)})|_{Y\rightarrow \infty }. & \displaystyle\end{eqnarray}$$

Then we can eliminate $P_{0}$ and $A$ from the governing equations (4.9), (4.10) by substituting (4.12) and (4.13) into (4.9) and using the boundary conditions

(4.14a ) $$\begin{eqnarray}\displaystyle \overline{U}_{0}=\unicode[STIX]{x1D6F9}_{0}^{(0)}=\unicode[STIX]{x1D6F9}_{0Y}^{(0)}=\unicode[STIX]{x1D6F9}_{0}^{(1)}=\unicode[STIX]{x1D6F9}_{0Y}^{(1)}=0,\quad \text{at}\,Y=0, & & \displaystyle\end{eqnarray}$$

(4.14b ) $$\begin{eqnarray}\displaystyle \overline{U}_{0}^{\prime }\rightarrow 2,\quad \unicode[STIX]{x1D6F9}_{0Y}^{(0)}\rightarrow 0,\quad \unicode[STIX]{x1D6F9}_{0YY}^{(1)}\rightarrow 0\quad \text{as}\,Y\rightarrow \infty . & & \displaystyle\end{eqnarray}$$

In order to numerically solve this reduced system we expand

(4.15a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F9}_{0}=\left(\mathop{\sum }_{l=0}^{L}\mathop{\sum }_{n=1}^{N}s_{l,n}F_{l}^{(j)}(Y)\text{e}^{\text{i}n\unicode[STIX]{x1D6FC}_{0}X}\right)+\text{c.c.},\quad \overline{U}_{0}=2Y+\mathop{\sum }_{l=0}^{L}s_{l,0}G_{l}(Y), & \displaystyle\end{eqnarray}$$

where $j=(n\hspace{0.6em}{\rm mod}\hspace{0.2em}2)$ and $[F_{l}^{(0)},F_{l}^{(1)},G_{l}]$ are modified $l$ th Chebyshev polynomials defined in appendix A, satisfying boundary conditions for $[\unicode[STIX]{x1D6F9}_{0}^{(0)},\unicode[STIX]{x1D6F9}_{0}^{(1)},\overline{U}_{0}]$ . Newton’s method can be used to find the spectral coefficients $s_{l,n}$ in the same way as when solving the Navier–Stokes equations in § 2.

Before we describe the numerical results, we confirm that at small amplitude the asymptotic system yields the well-known lower-branch neutral point. Considering the wall layer problem (4.9)–(4.10) linearised around the basic flow and seeking a small-amplitude response of the normal mode form

(4.16a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{U}_{0}=2Y,\quad (\unicode[STIX]{x1D6F9}_{0},P_{0},A)={\textstyle \frac{1}{2}}(\widehat{\unicode[STIX]{x1D6F9}}_{0}(\unicode[STIX]{x1D701}),\widehat{P}_{0},\widehat{A})\text{e}^{\text{i}\unicode[STIX]{x1D6FC}_{0}X}+\text{c.c.}, & \displaystyle\end{eqnarray}$$

where the new variable $\unicode[STIX]{x1D701}=(2\text{i}\unicode[STIX]{x1D6FC}_{0})^{1/3}(Y-c_{0}/2),$ we find that $\widehat{\unicode[STIX]{x1D6F9}}_{0}$ satisfies

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}\widehat{\unicode[STIX]{x1D6F9}}_{0\unicode[STIX]{x1D701}}-\widehat{\unicode[STIX]{x1D6F9}}_{0}=-\widehat{P}_{0}/2+\widehat{\unicode[STIX]{x1D6F9}}_{0\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}. & \displaystyle\end{eqnarray}$$

The boundary conditions on (4.17) are, from (4.10):

(4.18a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D6F9}}_{0}(\unicode[STIX]{x1D701}_{0})=\widehat{\unicode[STIX]{x1D6F9}}_{0\unicode[STIX]{x1D701}}(\unicode[STIX]{x1D701}_{0})=0,\quad \widehat{\unicode[STIX]{x1D6F9}}_{0\unicode[STIX]{x1D701}}(\infty )=-2\widehat{A}/(2\text{i}\unicode[STIX]{x1D6FC}_{0})^{1/3}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D701}_{0}=-(2\text{i}\unicode[STIX]{x1D6FC}_{0})^{1/3}c_{0}/2$ is the value of $\unicode[STIX]{x1D701}$ on the wall $Y=0.$ Noting that $\widehat{\unicode[STIX]{x1D6F9}}_{0\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}$ satisfies Airy’s equation, the linearised system (4.17), (4.18) may be integrated to yield

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D6F9}}_{0\unicode[STIX]{x1D701}}=\frac{2\widehat{A}}{(2\text{i}\unicode[STIX]{x1D6FC}_{0})^{1/3}}\left(\frac{\unicode[STIX]{x1D705}(\unicode[STIX]{x1D701})}{\unicode[STIX]{x1D705}(\unicode[STIX]{x1D701}_{0})}-1\right),\quad \unicode[STIX]{x1D705}(\unicode[STIX]{x1D701})=\int _{\unicode[STIX]{x1D701}}^{\infty }\text{Ai}(s)\,\text{d}s. & \displaystyle\end{eqnarray}$$

Evaluating (4.17) on the wall, and making use of (4.19), we have

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\widehat{P}_{0}}{2}=-\frac{2\widehat{A}}{(2\text{i}\unicode[STIX]{x1D6FC}_{0})^{1/3}}\frac{\text{Ai}^{\prime }(\unicode[STIX]{x1D701}_{0})}{\unicode[STIX]{x1D705}(\unicode[STIX]{x1D701}_{0})}, & \displaystyle\end{eqnarray}$$

Using (4.6), the core flow can also be written in the normal mode form

(4.21) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}_{0}={\textstyle \frac{1}{2}}\widehat{\unicode[STIX]{x1D713}}_{0}(y)\text{e}^{\text{i}\unicode[STIX]{x1D6FC}_{0}X}+\text{c.c.},\quad \widehat{\unicode[STIX]{x1D713}}_{0}=-\widehat{A}u_{b}, & \displaystyle\end{eqnarray}$$

while, from the pressure–displacement law (4.11):

(4.22) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{P}_{0}=-{\textstyle \frac{8}{15}}\unicode[STIX]{x1D6FC}_{0}^{2}\widehat{A}. & \displaystyle\end{eqnarray}$$

Combining (4.20) and (4.22) we obtain the familiar lower-branch PPF dispersion relation

(4.23) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{0}^{7/3}=-\frac{15}{8}(2\text{i})^{5/3}\frac{\text{Ai}^{\prime }(\unicode[STIX]{x1D701}_{0})}{\unicode[STIX]{x1D705}(\unicode[STIX]{x1D701}_{0})}. & \displaystyle\end{eqnarray}$$

It is a straightforward numerical task to find the values of $\unicode[STIX]{x1D701}_{0},\unicode[STIX]{x1D6FC}_{0}$ from the real and imaginary parts of (4.23), and this yields the well-known result $(\unicode[STIX]{x1D6FC}_{0},c_{0})\approx (2.149,2.826)$ , shown as the lower asymptote on figure 1, which can be compared with that arising from previous linear analyses of PPF (for example, Reid Reference Reid and Holt1965).

4.2 Comparison of nonlinear lower-branch Navier–Stokes and asymptotic computations

We first compare the bifurcation diagram obtained from the computation of the nonlinear lower-branch asymptotic problem (4.9), (4.12)–(4.14) with the Navier–Stokes results. Figure 6 shows the comparison of the solution curves on the lower-branch scale. We can see that at large Reynolds number the Navier–Stokes results converge to the asymptotic result shown by the red dashed curve. The bifurcation is supercritical for $R=10^{5}$ and $10^{6}$ , as Itoh (Reference Itoh1974) predicted from a weakly nonlinear analysis of the Navier–Stokes equations. However, our computations for $R$ larger than $10^{7}$ show that the nature of the bifurcation becomes supercritical when $R$ is sufficiently large. This is consistent with the large-Reynolds-number analysis of Hall & Smith (Reference Hall and Smith1982).

The point (e) corresponds to that in figure 2. At this point $cR^{2/7}=3$ , and for this value of phase speed we compare the Navier–Stokes near-wall flow field to that found from the solution of the asymptotic problem. The comparison of the total vorticity, given in figure 7, shows excellent agreement. In the asymptotic result only the leading-order part is used, namely the terms in the curly bracket in the asymptotic expansion

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}=\{-\unicode[STIX]{x1D6F9}_{0YY}-\overline{U}_{0Y}\}+\cdots \,. & \displaystyle\end{eqnarray}$$

The convergence of the mean flow distortion is also confirmed at this phase speed (see figure 8).

At this disturbance level the flow field is slowly varying in the streamwise direction. However, further away from the bifurcation point (i.e. for larger $cR^{2/7}$ ) the flow develops a sharp structure, as was shown earlier in figure 3(f). We can see how this structure emerges in figure 9: here we plot the wall shear $u_{y}(y=-1)$ versus $\unicode[STIX]{x1D6FC}x$ for the solutions denoted by the filled and open circles in figure 6. It should be noted that the wall shear stress calculated using the asymptotic theory passes through zero in a smooth fashion: this is to be expected since the lower branch equations (4.9)–(4.11) are elliptic in the streamwise direction due to the pressure–displacement law (4.11), and therefore no Goldstein singularity will arise. This phenomenon is similar to that observed in calculations of the interactive boundary layer flow over a surface roughness, where regular separation is encountered as the roughness height parameter approaches a critical value (see, for example, Smith & Walton Reference Smith and Walton1998). We can see from both the Navier–Stokes and asymptotic results that the flow is becoming localised: this seems to suggest that the asymptotic theory will eventually break down as it is based on an assumed slow streamwise development of the flow. The emergence of this sharp structure requires many Fourier modes to fully resolve, and it was not possible to continue the computation of either the Navier–Stokes or the asymptotic problem beyond the end point of the branch shown in figure 6. In a similar investigation of nonlinear lower-branch travelling waves, but in the context of boundary layer flow, Hall (Reference Hall1995) also reported an inability to continue the neutral branch beyond a critical amplitude. In that case the difficulty appeared to be associated with the wall shear approaching zero at a specific streamwise location, whereas in the present scenario, as noted above, we had no difficulty obtaining solutions with negative values of this quantity. In fact, according to figure 9, our localisation appears to be occurring in a region where the wall shear is strongly positive. The difference is likely to be caused by the different forms of the pressure–displacement law relevant to these two cases.

Figure 6.

The bifurcation diagram on the lower-branch scale. The solid black curves are Navier–Stokes results for $R=10^{5},10^{6},10^{7},10^{8}$ . The points indicated by (e) and (f) are those seen in figure 2. The red dashed curve is the asymptotic result. The magenta crosses are the bifurcation points from the linear neutral curve.

Figure 7.

The comparison of total vorticity in the wall layer for $cR^{2/7}=3$ . (a) is the Navier–Stokes result at point (e) in figure 2 ( $R=10^{6}$ ). (b) is the asymptotic result. The colour bar range is $[C_{min},C_{max}]=[-3.5,0.5]$ , and the vertical coordinate is the wall layer variable $Y=R^{2/7}(y+1)$ .

Figure 8.

The comparison of the mean flow distortion for lower-branch solutions in the wall layer. The value of $(\overline{u}-u_{b})^{\prime }$ is shown for $cR^{2/7}=3$ , plotted versus $Y=R^{2/7}(y+1)$ . The solid curves are Navier–Stokes results for $R=10^{6},10^{7},10^{8}$ , while the dashed curve is the asymptotic result.

Figure 9.

The distribution of the wall shear $u_{y}(y=-1)$ . The solid/dashed curves are the Navier–Stokes/asymptotic results indicated by the filled/open circles in the bifurcation diagram (figure 6). For the Navier–Stokes computations the Reynolds number $R=10^{6}$ is chosen. Note that for the undisturbed flow the value of wall shear is equal to 2 for all $x$ .

5.1 Upper-branch neutral structure for weakly nonlinear disturbances

Figure 10.

The relatively small-amplitude asymptotic structure of the nonlinear solutions bifurcating from the upper branch of the linear neutral curve.

The upper-branch asymptotic structure is more complicated than its lower-branch counterpart. The wall layer is now primarily inviscid in nature: its thickness $\unicode[STIX]{x1D6FF}$ is no longer given by the lower-branch scaling (4.3) but is instead determined by considering the dynamics of the thinner viscous nonlinear critical layer that must now be considered as a separate entity and no longer occupies the entire wall layer, as was the case on the lower branch. The no-slip condition is now satisfied within a Stokes layer, which forms a sublayer of the original wall layer. We denote the thickness of the Stokes layer as $d$ : its size can be found from an unsteady-viscous balance as

(5.1) $$\begin{eqnarray}\displaystyle & \displaystyle O(d)\sim O((c\unicode[STIX]{x1D6FC}R)^{-1/2}). & \displaystyle\end{eqnarray}$$

In the wall layer we decompose the stream function into even and odd functions of $x$ and denote the size of these parts as $\unicode[STIX]{x1D6F9}^{(e)}\text{ and }\unicode[STIX]{x1D6F9}^{(o)}$ . The odd part represents a phase-shift-induced velocity jump across the critical layer. This quantity will assume a non-zero value at the wall (according to (3.29)), with this value reduced to zero within the Stokes layer. As mentioned above, the dynamics in the bulk of the wall layer are inviscid and linear, and hence a Rayleigh equation with basic flow $u_{b}$ governs the evolution of the perturbations. Examining the solution (3.5) for small $z\sim O(\unicode[STIX]{x1D6FF})$ , we see that in order for $\unicode[STIX]{x1D6F9}^{(e)}$ to satisfy the impermeability condition, the ratio $b/a$ in (3.5) must be $O(\unicode[STIX]{x1D6FF})$ . This, together with the fact that $\unicode[STIX]{x1D707}_{c}$ remains $O(1)$ , leads us to the conclusion that

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F9}^{(o)}/\unicode[STIX]{x1D6F9}^{(e)}=O(\unicode[STIX]{x1D6FF}). & \displaystyle\end{eqnarray}$$

This relative velocity jump must be in tune with that produced by the Stokes layer. This can be estimated as follows. Since $\unicode[STIX]{x1D6F9}^{(e)}$ decays proportional to distance from the wall, upon entering the Stokes layer it has reduced in size to $\unicode[STIX]{x1D6F9}^{(e)}d/\unicode[STIX]{x1D6FF}$ . Under the influence of viscosity, the Stokes layer induces an odd contribution to this term (this can be seen explicitly in (5.19) below) and hence

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F9}^{(o)}/\unicode[STIX]{x1D6F9}^{(e)}=O(d/\unicode[STIX]{x1D6FF}). & \displaystyle\end{eqnarray}$$

Reconciling the two estimates (5.2), (5.3) we obtain

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle O(d)\sim O(\unicode[STIX]{x1D6FF}^{2}). & \displaystyle\end{eqnarray}$$

The core flow balances (4.2) are unchanged from the lower-branch case. Putting these together with the order of magnitude estimates (5.1), (5.4), we arrive at the upper-branch scalings

(5.5a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}=R^{-2/11},\quad d=R^{-4/11}, & \displaystyle\end{eqnarray}$$

and $\unicode[STIX]{x1D6FC}\sim O(R^{-1/11}),c\sim O(R^{-2/11})$ . Next we examine the dynamics of the critical layer, of thickness $\unicode[STIX]{x1D716}$ say, in order to determine the size of $\unicode[STIX]{x1D6F9}^{(e)}$ . Here, as per the discussion in § 3, there is a balance between the viscous operator $O(R^{-1}\unicode[STIX]{x1D716}^{-2})$ and the inertial operators $u\unicode[STIX]{x2202}_{x}\sim O(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D716})$ and $v\unicode[STIX]{x2202}_{y}\sim O(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6F9}^{(e)}/\unicode[STIX]{x1D716})$ . As in the aforementioned section, this leads to $\unicode[STIX]{x1D6F9}^{(e)}\sim O(\unicode[STIX]{x1D716}^{2})$ , but now in view of the longer wavelength in operation we have

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}=R^{-10/33}. & \displaystyle\end{eqnarray}$$

A sketch of the asymptotic structure is given in figure 10. The asymptotic structure within the wall layer is similar to that discussed in § 3 but takes a reduced form due to the long-wavelength effect. The structure outside of the wall layer remains the same as the lower-branch case, but with a smaller wave size.

Motivated by this order of magnitude analysis, we can now derive the asymptotic theory for a viscous nonlinear critical layer structure near the upper neutral branch. We introduce the $O(1)$ quantities $\unicode[STIX]{x1D6FC}_{0}=R^{1/11}\unicode[STIX]{x1D6FC},X=R^{-1/11}x,c_{0}=R^{2/11}c$ and in the wall layer we express the stream function $\unicode[STIX]{x1D713}$ as

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle {\textstyle \frac{1}{2}}\unicode[STIX]{x1D716}^{2}\widehat{\unicode[STIX]{x1D6F9}}_{0}(Y)\text{e}^{\text{i}\unicode[STIX]{x1D6FC}_{0}X}+\text{c.c.}+\text{higher harmonics}, & \displaystyle\end{eqnarray}$$

with $y=-1+\unicode[STIX]{x1D6FF}Y$ . Note that $\widehat{\unicode[STIX]{x1D6F9}}_{0}$ contains an $O(\unicode[STIX]{x1D6FF})$ term that contains the phase jump. The higher-order harmonics in (5.7) are of $O(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6FF})$ , and therefore do not invalidate the assumption of a leading-order monochromatic wave in the core (see (5.14) below). In the wall layer, the leading-order part of the mean flow $\overline{u}$ can be calculated solely from the basic PPF as $\unicode[STIX]{x1D6FF}Y(2-\unicode[STIX]{x1D6FF}Y)$ . The critical layer location is represented by $Y=Y_{c}$ , with $Y_{c}$ related to $c_{0}$ via

(5.8) $$\begin{eqnarray}\displaystyle & \displaystyle c_{0}=2Y_{c}, & \displaystyle\end{eqnarray}$$

and

(5.9) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}_{c}=y_{c}^{-1}+\cdots =-1+O(\unicode[STIX]{x1D6FF}). & \displaystyle\end{eqnarray}$$

Since the wall layer flow is governed by Rayleigh’s equation we can make use of the general solution (3.12). However, by exploiting the thinness of this region we can simplify the response.

Upon writing $z=\unicode[STIX]{x1D6FF}(Y-Y_{c}),a_{0}=a_{00}/\unicode[STIX]{x1D6FF}+a_{01}+\cdots \,,b_{0}=b_{00}+\cdots \,$ in (3.12) and using the small $z$ form (3.9) for ${\mathcal{P}}_{a}$ and ${\mathcal{P}}_{b}$ , the expression (3.5) will yield the desired solution $\widehat{\unicode[STIX]{x1D6F9}}_{0}=\widehat{\unicode[STIX]{x1D6F9}}_{0}^{(e)}-\text{i}\widehat{\unicode[STIX]{x1D6F9}}_{0}^{(o)}$ with

(5.10) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D6F9}}_{0}^{(e)}=a_{00}(Y-Y_{c})+b_{00}, & \displaystyle\end{eqnarray}$$

(5.11) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D6F9}}_{0}^{(o)}=\unicode[STIX]{x1D6FF}b_{00}(Y-Y_{c})\unicode[STIX]{x1D6E9}, & \displaystyle\end{eqnarray}$$

to leading order. Since the leading-order term in (5.10) must satisfy the impermeability condition on $Y=0$ we have

(5.12) $$\begin{eqnarray}\displaystyle & \displaystyle b_{00}=a_{00}Y_{c}. & \displaystyle\end{eqnarray}$$

Also, assuming impermeable walls, the boundary pressure is given by

(5.13) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{P}_{0}=c_{0}a_{00}, & \displaystyle\end{eqnarray}$$

recalling that in this predominantly inviscid region, once the stream function is known we can calculate the pressure; see (3.4) where we can find $\widehat{p}|_{y=-1}=c\widehat{\unicode[STIX]{x1D713}}|_{y=-1}^{\prime }$ .

The core flow solution is identical to the lower-branch case except for its size. Since the magnitude of the core stream function is reduced by an $O(\unicode[STIX]{x1D6FF})$ amount towards the wall layer and matches to the $O(\unicode[STIX]{x1D716}^{2})$ wall layer solution, in the core we have

(5.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}=\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6FF}^{-1}\unicode[STIX]{x1D713}_{0}+\cdots & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D713}_{0}$ given in (4.21). Matching to the wall layer solution (5.10) requires

(5.15) $$\begin{eqnarray}\displaystyle & \displaystyle a_{00}=-2\widehat{A}. & \displaystyle\end{eqnarray}$$

Eliminating $a_{00}$ and $\widehat{P}_{0}$ from (5.13), (5.15) and using the core pressure prediction in (4.22) we find a first relationship between $\unicode[STIX]{x1D6FC}_{0}$ and $c_{0}$ , namely

(5.16) $$\begin{eqnarray}\displaystyle & \displaystyle c_{0}={\textstyle \frac{4}{15}}\unicode[STIX]{x1D6FC}_{0}^{2}. & \displaystyle\end{eqnarray}$$

In order to complete the analysis and find a second relation between these two quantities we need to consider the Stokes layer in more detail. Here the leading-order expansion is

(5.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}=\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6FF}\left\{{\textstyle \frac{1}{2}}\widehat{\unicode[STIX]{x1D711}}({\mathcal{Y}})\text{e}^{\text{i}\unicode[STIX]{x1D6FC}_{0}X}+\text{c.c.}\right\},\quad \widehat{\unicode[STIX]{x1D711}}=\widehat{\unicode[STIX]{x1D711}}^{(e)}-\text{i}\widehat{\unicode[STIX]{x1D711}}^{(o)}, & \displaystyle\end{eqnarray}$$

with the $\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6FF}$ factor implied by the arguments leading to (5.4), and ${\mathcal{Y}}=(y+1)/\unicode[STIX]{x1D6FF}^{2}$ representing the $O(1)$ normal coordinate within the Stokes layer. From substitution into the stream function form of the Navier–Stokes equations (2.7), we find that

(5.18) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x2202}_{{\mathcal{Y}}}^{2}+\text{i}\unicode[STIX]{x1D6FC}_{0}c_{0})\widehat{\unicode[STIX]{x1D711}}_{{\mathcal{Y}}{\mathcal{Y}}}=0. & \displaystyle\end{eqnarray}$$

It is easy to show that the solution satisfying the no-slip condition on ${\mathcal{Y}}=0$ and appropriately matching to the even part of the wall layer solution (5.10) is

(5.19a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D711}}=a_{00}({\mathcal{Y}}+m^{-1}(\text{e}^{-m{\mathcal{Y}}}-1)),\quad m=m_{0}(1-\text{i}),\quad m_{0}=(\unicode[STIX]{x1D6FC}_{0}c_{0}/2)^{1/2}. & \displaystyle\end{eqnarray}$$

From this solution it is evident that as ${\mathcal{Y}}\rightarrow \infty$ the exponential part diminishes and $\widehat{\unicode[STIX]{x1D711}}^{(o)}\rightarrow a_{00}/2m_{0}$ . Thus we see explicitly that the Stokes layer solution possesses a non-zero odd contribution. This must match to the odd part of the Rayleigh solution (5.11) as $Y\rightarrow 0$ , and thus we conclude

(5.20) $$\begin{eqnarray}\displaystyle & \displaystyle -b_{00}Y_{c}\unicode[STIX]{x1D703}=a_{00}/2m_{0}. & \displaystyle\end{eqnarray}$$

Using (5.19) for $m_{0}$ , (5.12) for $b_{00}$ , and (5.8) for $Y_{c}$ , we arrive at

(5.21) $$\begin{eqnarray}\displaystyle & \displaystyle -c_{0}^{2}\unicode[STIX]{x1D703}=2^{3/2}(\unicode[STIX]{x1D6FC}_{0}c_{0})^{-1/2}. & \displaystyle\end{eqnarray}$$

For given $\unicode[STIX]{x1D703}$ , we can compute $\unicode[STIX]{x1D6FC}_{0}$ and $c_{0}$ from (5.16) and (5.21); for example substitution of $\unicode[STIX]{x1D703}=-\unicode[STIX]{x03C0}$ yields the coefficients $(\unicode[STIX]{x1D6FC}_{0},c_{0})=(1.789,0.8536)$ that describe the asymptotic development of the upper branch of the linear neutral curve shown in figure 1. The critical layer analysis given in appendix C describes how to derive an equation that relates $\unicode[STIX]{x1D703}$ and a normalised amplitude $\unicode[STIX]{x1D6FC}_{0}b_{00}^{3/2}$ .

5.2 Strongly nonlinear critical layer regime

Figure 11.

The asymptotic structure of the upper-branch modes at relatively large disturbance amplitudes. Note that the critical layer is now an $O(1)$ distance from the channel wall.

Now we consider the modification of this structure that occurs at higher amplitude. In this regime the wavenumber and phase speed of the disturbance are $O(1)$ quantities. This shortening of the wavelength leads to a change in the structure within the core region. In particular the perturbed streamwise pressure gradient plays a more active role, with the result that the balances in (4.5) are replaced by a Rayleigh equation for the stream function. This equation is driven by the PPF profile $u_{b}$ , since the mean flow distortion is still small. In effect we now have $\unicode[STIX]{x1D6FF}\sim O(1)$ , with the inviscid wall layer of the previous section enlarging to fill the entire channel. However, the Stokes layer remains essentially as before; although in view of the $O(1)$ size of $\unicode[STIX]{x1D6FC}$ and $c$ , its thickness $d$ is now given, from (5.1), by

(5.22) $$\begin{eqnarray}\displaystyle & \displaystyle d=R^{-1/2}. & \displaystyle\end{eqnarray}$$

Once again an odd part of the stream function is induced within this region via the effects of viscosity. Denoting the sizes of the even and odd parts of the stream function by $\unicode[STIX]{x1D713}^{(e)},\unicode[STIX]{x1D713}^{(o)}$ we have (since $\unicode[STIX]{x1D6FF}\sim O(1)$ ):

(5.23) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}^{(o)}/\unicode[STIX]{x1D713}^{(e)}=O(d); & \displaystyle\end{eqnarray}$$

compare with (5.3). The $O(\unicode[STIX]{x1D716})$ thick critical layer is now sited an $O(1)$ distance from the wall in view of the largeness of the phase speed. As in § 5.1 there is a balance in this region between the inertial operators $u\unicode[STIX]{x2202}_{x}\sim O(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D716})$ and $v\unicode[STIX]{x2202}_{y}\sim O(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D713}^{(e)}/\unicode[STIX]{x1D716})$ , representing advection by the basic flow and perturbation, respectively. Once again this leads to $\unicode[STIX]{x1D713}^{(e)}=O(\unicode[STIX]{x1D716}^{2})$ , but now viscosity is a higher-order effect, with its presence inducing an odd part in the stream function via the inertial-viscous balance $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D713}^{(o)}/\unicode[STIX]{x1D716}\sim R^{-1}\unicode[STIX]{x1D716}^{-2}$ . Hence, since $\unicode[STIX]{x1D6FC}\sim O(1)$ , we have

(5.24) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}^{(o)}/\unicode[STIX]{x1D713}^{(e)}=O(R^{-1}\unicode[STIX]{x1D716}^{-3}). & \displaystyle\end{eqnarray}$$

Reconciling (5.23) and (5.24) we find that the critical layer has increased in thickness to

(5.25) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}=R^{-1/6}. & \displaystyle\end{eqnarray}$$

A sketch of the new strongly nonlinear stage is given in figure 11.

Guided again by our order-of-magnitude analysis we can now write down the core expansions

(5.26) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}^{(e)}=\unicode[STIX]{x1D716}^{2}\widehat{\unicode[STIX]{x1D713}}_{0}(y)\cos (\unicode[STIX]{x1D6FC}x)+\cdots \,, & \displaystyle\end{eqnarray}$$

(5.27) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}^{(o)}=\unicode[STIX]{x1D716}^{5}\{\widehat{\unicode[STIX]{x1D713}}_{1}(y)\sin (\unicode[STIX]{x1D6FC}x)+\text{higher harmonics}\}+\cdots \,, & \displaystyle\end{eqnarray}$$

(5.28) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{u}=(1-y^{2})+\unicode[STIX]{x1D716}w(y)+\cdots \,. & \displaystyle\end{eqnarray}$$

Here the leading-order part of $\unicode[STIX]{x1D713}^{(e)}$ contains only one Fourier component because a neutral mode solution of the Rayleigh equation can be excited. The presence of a non-trivial mean flow distortion of size $O(\unicode[STIX]{x1D716})$ is due to critical layer effects, as we shall see shortly.

The stream function $\widehat{\unicode[STIX]{x1D713}}_{0}$ is the Rayleigh solution for $\overline{u}=u_{b}$ but now, since the contribution from $\unicode[STIX]{x1D713}^{(o)}$ is a higher-order effect, the leading-order phase shift is zero. Thus we can write

(5.29) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D713}}_{0}=a_{0}\,z\,{\mathcal{P}}_{a}(z)+b_{0}({\mathcal{P}}_{b}(z)+\unicode[STIX]{x1D707}_{c}\,z\,\ln |z|{\mathcal{P}}_{a}(z)), & \displaystyle\end{eqnarray}$$

for both positive and negative $z=y-y_{c}$ , where ${\mathcal{P}}_{a},{\mathcal{P}}_{b}$ can be computed from (3.8) for given $(\unicode[STIX]{x1D6FC},c)$ . Applying the boundary conditions $\widehat{\unicode[STIX]{x1D713}}_{0y}=0$ at $y=0$ and $\widehat{\unicode[STIX]{x1D713}}_{0}=0$ at $y=-1$ to (5.29) we obtain an expression of the form

(5.30) $$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}cc@{}}H_{0a} & H_{0b}\\ H_{1a} & H_{1b}\end{array}\right]\left[\begin{array}{@{}c@{}}a_{0}\\ b_{0}\end{array}\right]=0, & & \displaystyle\end{eqnarray}$$

where the constants $H_{0a},H_{1b},H_{0b},H_{1a}$ can be found by using the values of ${\mathcal{P}}_{a}$ and ${\mathcal{P}}_{b}$ and their derivatives at $y=0,-1$ . Thus we can compute the determinant

(5.31) $$\begin{eqnarray}\displaystyle & \displaystyle D=H_{0a}H_{1b}-H_{0b}H_{1a} & \displaystyle\end{eqnarray}$$

as a function of $\unicode[STIX]{x1D6FC},c$ and then find the zero locus in order to determine an implicit form of the dispersion relation

(5.32) $$\begin{eqnarray}\displaystyle & \displaystyle D(c,\unicode[STIX]{x1D6FC})=0, & \displaystyle\end{eqnarray}$$

as shown by the dashed line in figure 12. Also the value of

(5.33) $$\begin{eqnarray}\displaystyle & \displaystyle B(\unicode[STIX]{x1D6FC})\equiv b_{0}^{-1}\widehat{\unicode[STIX]{x1D713}}_{0y}|_{y=-1} & \displaystyle\end{eqnarray}$$

found from the computation along (5.32) is to be used later to fix the undetermined wave amplitude $b_{0}$ .

Figure 12.

The phase speed $c$ and coefficient $C$ as functions of wavenumber $\unicode[STIX]{x1D6FC}$ in the strongly nonlinear critical layer regime.

As mentioned above, the logarithmic phase shift now occurs at higher order than in the lower-amplitude regime considered previously, and thus $\widehat{\unicode[STIX]{x1D713}}_{1}$ can be taken to be zero above the critical layer and non-zero beneath it. The non-zero higher-order flow needs to be reduced in the Stokes layer. The appropriate expansions in the Stokes layer are

(5.34) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}=\unicode[STIX]{x1D716}^{5}\left\{{\textstyle \frac{1}{2}}\widehat{\unicode[STIX]{x1D711}}({\mathcal{Y}})\text{e}^{\text{i}\unicode[STIX]{x1D6FC}x}+\text{c.c.}\right\}+\cdots \,, & \displaystyle\end{eqnarray}$$

(5.35) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{u}=\unicode[STIX]{x1D716}^{3}2{\mathcal{Y}}+\unicode[STIX]{x1D716}^{4}{\mathcal{W}}({\mathcal{Y}})+\cdots \,, & \displaystyle\end{eqnarray}$$

with ${\mathcal{Y}}=\unicode[STIX]{x1D716}^{-3}(y+1)$ . The function $\widehat{\unicode[STIX]{x1D711}}$ satisfies (5.18) with $(\unicode[STIX]{x1D6FC}_{0},c_{0})=(\unicode[STIX]{x1D6FC},c)$ and the appropriate solution is

(5.36) $$\begin{eqnarray}\displaystyle \widehat{\unicode[STIX]{x1D711}}=(\widehat{\unicode[STIX]{x1D713}}_{0y}|_{y=-1})\{{\mathcal{Y}}+m^{-1}(\text{e}^{-m{\mathcal{Y}}}-1)\},\quad m=m_{0}(1-\text{i}), & & \displaystyle\end{eqnarray}$$

similar to (5.19) but now $m_{0}=(\unicode[STIX]{x1D6FC}c/2)^{1/2}$ . Matching to the core solution we then have

(5.37) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D713}}_{1}|_{y=-1}=(2m_{0})^{-1}\widehat{\unicode[STIX]{x1D713}}_{0y}|_{y=-1}. & \displaystyle\end{eqnarray}$$

The required phase balance follows from (3.29) as

(5.38) $$\begin{eqnarray}\displaystyle b_{0}[\widehat{\unicode[STIX]{x1D713}}_{1y}]_{y=y_{c-}}^{y_{c+}}=\frac{(\widehat{\unicode[STIX]{x1D713}}_{0y}|_{y=-1})^{2}}{2m_{0}}, & & \displaystyle\end{eqnarray}$$

which is analogous to the weakly nonlinear result (5.20).

The critical layer analysis proceeds much along the lines of that presented in § 3, and the details are given in appendix D, where it is shown that the total stream function $\unicode[STIX]{x1D713}+\int \overline{u}\,\text{d}y$ possesses an asymptotic expansion $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6F7}_{1}(x,Z)+\cdots \,$ , with $\unicode[STIX]{x1D6F7}_{1}$ given in (3.20), and where $Z=\unicode[STIX]{x1D716}^{-1}(y-y_{c})$ is the $O(1)$ normal variable in the critical layer with the thickness (5.25). The corresponding total vorticity is

(5.39) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}=2y_{c}-\unicode[STIX]{x1D716}K(\unicode[STIX]{x1D701})+\cdots \,;\quad \unicode[STIX]{x1D701}=\unicode[STIX]{x1D6F7}_{1}/b_{0} & \displaystyle\end{eqnarray}$$

with the function $K$ governed by inviscid dynamics. The streamlines of constant $\unicode[STIX]{x1D701}$ form a familiar Kelvin’s cats-eye pattern with the separatrices

(5.40) $$\begin{eqnarray}\displaystyle & \displaystyle Z=Z_{\pm }(x)\equiv w_{c}/2y_{c}\pm (-b_{0}/y_{c})^{1/2}[1-\cos (\unicode[STIX]{x1D6FC}x)]^{1/2}, & \displaystyle\end{eqnarray}$$

obtained by setting $\unicode[STIX]{x1D701}=1$ , defining the edge of the eye. Within the eye all the streamlines are closed. A plot of the streamlines is given in figure 13.

Figure 13.

Streamlines of constant $\unicode[STIX]{x1D701}$ ( $=\unicode[STIX]{x1D6F7}_{1}/b_{0}$ ) with $\unicode[STIX]{x1D6F7}_{1}$ given in (3.20). The dashed curves are the edge of the cats eye $\unicode[STIX]{x1D701}=1$ , namely the separatrices (5.40).

In order to determine the precise form of $K$ we need to proceed to higher order to include viscous effects. The analysis in appendix D shows that

(5.41) $$\begin{eqnarray}\displaystyle K(\unicode[STIX]{x1D701})=\left\{\begin{array}{@{}ll@{}}\displaystyle K_{0}\pm 2^{1/2}\unicode[STIX]{x03C0}C\int _{1}^{\unicode[STIX]{x1D701}}\frac{1}{I(\unicode[STIX]{x1D701}_{\ast })}\,\text{d}\unicode[STIX]{x1D701}_{\ast }, & \unicode[STIX]{x1D701}>1,\\ K_{0} & \unicode[STIX]{x1D701}<1,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where

(5.42) $$\begin{eqnarray}\displaystyle & \displaystyle I(\unicode[STIX]{x1D701})=\int _{0}^{2\unicode[STIX]{x03C0}}(\unicode[STIX]{x1D701}-\cos \unicode[STIX]{x1D709})^{1/2}\,\text{d}\unicode[STIX]{x1D709} & \displaystyle\end{eqnarray}$$

and the $\pm$ sign corresponds to the upper/lower regions wherein $Z>Z_{+}\text{ and }Z<Z_{-}$ . The fact that inside of the cats eye $K$ is a constant is a consequence of the Prandtl–Batchelor theorem (Batchelor Reference Batchelor1956). The explicit form of the coefficient $C(b_{0})$ can be found by matching to be

(5.43) $$\begin{eqnarray}\displaystyle & \displaystyle C=-(-2b_{0}/y_{c})^{1/2}, & \displaystyle\end{eqnarray}$$

which follows from the analysis in appendix D leading to (D 7).

With the function $K$ determined, we can now calculate the jumps in velocity and mean shear across the critical layer. These are shown in appendix D to take the form

(5.44) $$\begin{eqnarray}\displaystyle & \displaystyle [\widehat{\unicode[STIX]{x1D713}}_{1y}]_{y=y_{c-}}^{y_{c+}}=-\frac{4C\unicode[STIX]{x1D6EC}}{\unicode[STIX]{x1D6FC}b_{0}}, & \displaystyle\end{eqnarray}$$

(5.45a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle w_{y}|_{y=y_{c+}}=-C\unicode[STIX]{x1D6EC}+K_{0},\quad w_{y}|_{y=y_{c-}}=C\unicode[STIX]{x1D6EC}+K_{0}, & \displaystyle\end{eqnarray}$$

where the constant $\unicode[STIX]{x1D6EC}\approx 1.379$ . Substituting (3.26), (5.33), (5.43) and (5.44) into (5.38), we can write down an explicit formula for the amplitude $b_{0}$ :

(5.46) $$\begin{eqnarray}\displaystyle & \displaystyle b_{0}=\left(\frac{c}{\unicode[STIX]{x1D6FC}\sqrt{1-c}}\right)^{1/3}\left(\frac{8\unicode[STIX]{x1D6EC}}{B^{2}}\right)^{2/3}. & \displaystyle\end{eqnarray}$$

Once the value of $B$ is obtained from the Rayleigh solution, the amplitude obtained above can be used to calculate $C$ through (5.43) as a function of $\unicode[STIX]{x1D6FC}$ , as shown in figure 12. As we shall see shortly, the mean flow distortion has vanishing shear for $y\in (y_{c},0]$ . Therefore, from (5.45), we can fix the value of the uniform vorticity as $K_{0}=C\unicode[STIX]{x1D6EC}$ .

5.3 Comparison of nonlinear upper-branch Navier–Stokes and asymptotic computations

Figure 14.

The bifurcation diagram on the viscous nonlinear critical layer scale. The black solid curves are Navier–Stokes results for $R=10^{6},10^{7},10^{8},10^{9}$ . The filled circle at (a) is the same point that appears in figure 2. The red dashed curve is the asymptotic result (5.16), (5.21). The magenta crosses are the bifurcation points.

After the bifurcation from the upper branch of the neutral curve, the travelling wave first develops a viscous nonlinear critical layer and thus the asymptotic theory of § 5.1 is relevant here. Figure 14 shows the bifurcation diagram scaled appropriately to this asymptotic regime. It is known that the convergence towards this limit is rather slow (see, for example, figure 1), and thus here we plot the Navier–Stokes results up to $R=10^{9}$ as the black solid curves. The red dashed curve is calculated using the asymptotic result (5.16) and (5.21), varying the phase shift $\unicode[STIX]{x1D703}$ from $-\unicode[STIX]{x03C0}$ to zero. As mentioned earlier, the linear critical point $(\unicode[STIX]{x1D6FC}_{0},c_{0})=(1.789,0.8536)$ corresponds to the case $\unicode[STIX]{x1D703}=-\unicode[STIX]{x03C0}$ . As the amplitude of the wave instability is increased, the modulus of phase $|\unicode[STIX]{x1D703}|$ is reduced and the values of $\unicode[STIX]{x1D6FC}_{0}=R^{1/11}\unicode[STIX]{x1D6FC}$ and $c_{0}=R^{2/11}c$ will increase. Eventually the wavenumber and phase speed of the travelling wave states become $O(1)$ , and thus we enter the strongly nonlinear critical layer regime where the majority of the flow is governed by the Rayleigh equation.

Figure 15.

The bifurcation diagram on the strongly nonlinear critical layer scale. The black solid curves are Navier–Stokes results for $R=10^{5},10^{6},10^{7}$ . The filled circle at (b) is the same point that appears in figure 2. The red dashed curve is the result from figure 12, obtained by solving the Rayleigh equation with zero phase shift. The magenta crosses are the bifurcation points.

Figure 16.

The flow near the critical level $y=y_{c}$ for $c=0.27$ . In the figures we subtract the basic shear contribution $2y_{c}$ from the total vorticity, and rescale by dividing by $\unicode[STIX]{x1D716}=R^{-1/6}$ . Colour shows $\unicode[STIX]{x1D716}^{-1}(\unicode[STIX]{x1D714}-2y_{c})$ , and its range is $[C_{min},C_{max}]=[-1,3.6]$ . The vertical coordinate is $Z=\unicode[STIX]{x1D716}^{-1}(y-y_{c})$ . (a) shows the Navier–Stokes results at point (b) in figure 2 with $R=10^{6}$ . (b) is the theoretical result $-K$ , where $K(\unicode[STIX]{x1D701})$ is given in (5.41).

Figure 17.

The scaled mean flow distortion $\unicode[STIX]{x1D716}^{-1}(\overline{u}-u_{b})$ for $c=0.27$ , with $\unicode[STIX]{x1D716}=R^{-1/6}$ . The black solid curves are Navier–Stokes results for $R=10^{5},10^{6},10^{7}$ . The dashed curve is the asymptotic result (5.47).

The Navier–Stokes solution curves plotted in figure 15 show excellent agreement with the dispersion relation (5.32) arising from the solution of Rayleigh’s equation (taken from figure 12), and shown here by the red dashed curve. At the point (b) the phase speed is close to $c=0.27$ . The flow in the critical layer is expected to develop a Kelvin’s cats-eye pattern, so in the subsequent figures we provide a detailed visual comparison of the Navier–Stokes and asymptotic results at this value of $c$ . The vorticity in the neighbourhood of the critical level is compared in figure 16. In these plots we subtract the contribution of vorticity due to the basic flow, $2y_{c}$ , in view of (5.39), and then divide by $\unicode[STIX]{x1D716}$ . The asymptotic and Navier–Stokes results are in good agreement and a cats-eye pattern is indeed observed.

We can also compare the flux of the Navier–Stokes and asymptotic solutions. For the strongly nonlinear critical layer case, the mean flow in the core expands as $\overline{u}=u_{b}(y)+R^{-1/6}w(y)+\cdots \,$ . In view of the lack of forcing there, see remark (ii) in § 3, the mean flow distortion in the core must assume a piecewise linear form

(5.47) $$\begin{eqnarray}\displaystyle w(y)=\left\{\begin{array}{@{}ll@{}}M(y_{c}+1) & \text{if}\,y\in (y_{c},0],\\ M(y+1) & \text{if}\,y\in [-1,y_{c}),\end{array}\right. & & \displaystyle\end{eqnarray}$$

with the constant $M$ determined from the jump condition (5.45) as

(5.48) $$\begin{eqnarray}\displaystyle & \displaystyle M=2C\unicode[STIX]{x1D6EC}. & \displaystyle\end{eqnarray}$$

Here it should be noted that the shear stress associated with the mean flow distortion must vanish on the wall, because the imposed constant pressure gradient is in balance with the basic shear on the wall. In fact, if the Reynolds stress in (2.6c ) is calculated using (5.36), then the mean flow distortion in the Stokes layer can be determined explicitly as

(5.49) $$\begin{eqnarray}\displaystyle \displaystyle {\mathcal{W}}({\mathcal{Y}}) & = & \displaystyle M{\mathcal{Y}}-\frac{b_{0}^{2}B^{2}}{4c}\left\{2(m_{0}{\mathcal{Y}}+2)\text{e}^{-m_{0}{\mathcal{Y}}}\cos (m_{0}{\mathcal{Y}})\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,2(m_{0}{\mathcal{Y}}-1)\text{e}^{-m_{0}{\mathcal{Y}}}\sin (m_{0}{\mathcal{Y}})-\text{e}^{-2m_{0}{\mathcal{Y}}}-3\right\},\end{eqnarray}$$

upon application of the no-slip condition on ${\mathcal{Y}}=0$ and the matching to (5.47) as ${\mathcal{Y}}\rightarrow \infty$ . Using this expression we can confirm analytically that ${\mathcal{W}}_{{\mathcal{Y}}}=0$ at ${\mathcal{Y}}=0$ , which provides an important check on the asymptotic analysis. The convergence of the Navier–Stokes solution with increasing Reynolds number to the asymptotic state can be seen in the comparison of the corresponding mean flow distortions, shown in figure 17 with the asymptotic solution represented by a dashed line. We can establish a similar mean flow distortion for the viscous nonlinear critical layer regime, as shown in appendix C. These asymptotic forms for the distortion can be used to establish how the leading-order flux scales with Reynolds number. We find

(5.50) $$\begin{eqnarray}\displaystyle & \displaystyle 1-Q\approx -R^{-16/33}\frac{3M_{0}c_{0}}{4}, & \displaystyle\end{eqnarray}$$

with $M_{0}$ given in (C 10) for the viscous nonlinear critical layer regime, and

(5.51) $$\begin{eqnarray}\displaystyle & \displaystyle 1-Q\approx -R^{-1/6}\frac{3Mc}{4}, & \displaystyle\end{eqnarray}$$

with $M$ given in (5.48) for the strongly nonlinear critical layer regime. The comparison with the Navier–Stokes results again shows a good agreement, as can be seen in figure 18.