2. Free-stream coherent structures in incompressible parallel boundary-layer flows

To provide some context for the discussion of free-stream coherent structures in the compressible ASBL flow, we briefly summarise the results of Deguchi & Hall (Reference Deguchi and Hall2014) for free-stream coherent structures in incompressible ASBL flow.

Incompressible ASBL flow describes viscous, incompressible flow $(u^*, v^*, w^*)$ with respect to Cartesian coordinates $(x^*, y^*, z^*)$, with dynamic viscosity $\mu$ and kinematic viscosity $\nu$, over a flat plate at $y^* = 0$. Uniform flow exists in the free stream, so denoting free-stream values by subscript $\infty$, at the free-stream $(u^*, v^*, w^*) = (u_\infty , -v_\infty , 0)$. The plate is subject to constant suction, so the velocity at the plate is $(u^*, v^*, w^*) = (0, -v_\infty , 0)$. Non-dimensionalising the velocity components on the free-stream speed $u_\infty$ and the coordinates on the length scale $\nu /v_\infty$, and defining the Reynolds number $Re = u_\infty /v_\infty$, the basic flow is given by

(2.1)\begin{equation} (u_b, v_b, w_b) = ( 1- {\mathrm{e}^{{-}y}}, -Re^{{-}1}, 0). \end{equation}

Deguchi & Hall (Reference Deguchi and Hall2014) showed that, at high Reynolds numbers, the incompressible Navier–Stokes equations allow for nonlinear equilibrium solutions taking the form of a roll–wave–streak interaction propagating in a viscous layer at the outer edge of the boundary layer; this layer is termed the production layer and the solutions are known as free-stream coherent structures. The interaction in the production layer is characterised by nonlinear travelling-wave solutions propagating with wave speed $c$; numerical computations suggest that the asymptotic behaviour of the wave speed is $1-c = O(Re^{-1})$, so that the wave propagates downstream with almost free-stream speed. The solutions also have streamwise length scales comparable to the spanwise length scales, and the thickness of production layer is comparable to the boundary-layer thickness. Seeking a solution with these scalings, which is periodic in the streamwise and spanwise directions with respective wavenumbers $\alpha$ and $\beta$, shows that the production layer in ASBL flow is located at $y = \ln Re$.

The solution inside the production layer is $\boldsymbol {U}(X, Y, z) = (U, V, W)$, where $(x, y, z) = (X-ct, {Y}-\ln Re, z)$, $c =1-Re^{-1} c_1$, and is determined by numerically solving the full Navier–Stokes equations at unit Reynolds number as a nonlinear eigenvalue problem for the perturbed wave speed $c_1$ of the travelling wave:

(2.2)$$\begin{gather} ([\boldsymbol{U}+c_1\hat{\boldsymbol{i}}]\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{U} ={-} \boldsymbol{\nabla} P + \nabla^2\boldsymbol{U}, \end{gather}$$

(2.3)$$\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U}= 0. \end{gather}$$

The asymptotic structure of the solution emerging from the lower side of the production layer shows that, below the layer, the disturbance to the streamwise velocity (termed the streak), which occurs as a result of the nonlinear interaction in the production layer, can grow exponentially like $\mathrm {e}^{-Y}$ as $Y \to -\infty$ while the other velocity components decay. Thus moving beneath the production layer,

(2.4)$$\begin{gather} u \to 1 - \mathrm{e}^{{-}y} + \frac{d_0}{Re} + \frac{ J_1}{Re^{\omega_1}}\mathrm{e}^{(\omega_1-1)y} \cos(2\beta z) + \frac{ J_1 K_1}{Re^{2\omega_1}4\omega_1}\textrm{e}^{(2\omega_1-1)y}+ \cdots, \end{gather}$$

(2.5)$$\begin{gather}v \to -\frac{1}{Re} + \frac{ K_1}{Re^{\omega_1+1}} \mathrm{e}^{\omega_1y} \cos(2\beta z) + \cdots, \end{gather}$$

(2.6)$$\begin{gather}{w \to -\frac{K_1 \omega_1}{2\beta Re^{\omega_1+1}} \mathrm{e}^{\omega_1y} \sin(2 \beta z) + \cdots,} \end{gather}$$

where

(2.7a,b)\begin{equation} J_1 = \frac{K_1}{(\omega_1-1)^2 + (\omega_1-1) - 4 \beta^2}, \quad \omega_1 = \frac{ -1 + \sqrt{ 1 + 16\beta^2}}{2} \geqslant 0, \end{equation}

for spanwise wavenumber $\beta$, and where $K_1$ is found as part of the numerical solution of the eigenvalue problem in the production layer. These solutions are valid as the wall layer is approached, i.e. when $1 \ll y \ll \ln Re$. The constant of integration $d_0$ is found by matching with the numerical solution of the eigenvalue problem (2.2)–(2.3) which was computed for a range of spanwise wavenumbers $\beta$ in Deguchi & Hall (Reference Deguchi and Hall2014). Thus the term $d_0/\textit {Re}$ is the next order correction to the mean flow due to the nonlinear interaction in the production layer. Therefore in general the streamwise velocity solution is only given up to a constant, however, the correction does not influence the vortex field which is the quantity of interest. By solving for the induced flow throughout the boundary region between the production layer and the wall, Deguchi & Hall (Reference Deguchi and Hall2014) show that for $\beta < 1/\sqrt {2}$ the streak disturbance grows down to the main part of the boundary layer, before being reduced to zero at the wall to satisfy the boundary conditions.

3. Governing equations for compressible ASBL flow

We now consider the compressible counterpart of ASBL flow. Consider a viscous, compressible perfect gas with density, temperature and dynamic viscosity $\rho ^*$, $\theta ^*$ and $\mu ^*$ respectively, flowing with velocity $\boldsymbol {u}^{\boldsymbol {*}} = (u^*, v^*, w^*)$ with respect to Cartesian coordinates $(x^*, y^*, z^*)$ over an infinitely long flat plate at $y^* =0$. Uniform suction exists at the plate boundary so that, denoting free-stream values by subscript $\infty$, the velocity is $\boldsymbol {u}^{\boldsymbol {*}} = (0, -v_\infty , 0)$ at the plate. Meanwhile, a long way from the plate at the free stream, $\boldsymbol {u}^{\boldsymbol {*}} \to (u_\infty , -v_\infty , 0)$ and ($\rho ^*, \theta ^*, \mu ^*, p^*) \to (\rho _\infty , \theta _\infty , \mu _\infty , p_\infty /\rho _\infty u_\infty ^2)$. The suction at $y^*=0$ does not allow for zero heat transfer over the plate due to the transfer of kinetic energy across it, and therefore we assume the temperature at the plate is fixed so that $\theta ^* = \theta _p$ at $y^* = 0$.

We non-dimensionalise by scaling the coordinates $(x^*, y^*, z^*)$ on the velocity-boundary-layer thickness $\delta = \mu _\infty /\rho _\infty v_\infty$, the velocity components $(u^*, v^*, w^*)$ on $u_\infty$, the pressure on $\rho _\infty u_\infty ^2$ and the quantities $\rho ^*$, $\theta ^*$ and $\mu ^*$ on their free-stream values. We define the Reynolds number $\textit {Re}$ by

(3.1)\begin{equation} \textit{Re} = u_\infty/v_\infty. \end{equation}

Throughout the analysis that follows, we assume the Reynolds number is large. We also define the following physical constants:

1. (i) $c_v$, $c_p,$ are the specific heats at constant volume and constant pressure respectively;

2. (ii) $\gamma = c_p/c_v$ is the ratio of specific heats; for air, $\gamma \approx 1.4$;

3. (iii) $R$ is the molecular gas constant which is approximately $286 \ \textrm {m}^2\ \textrm {s}^{-2}\ \textrm {K}^{-1}$ for air;

4. (iv) $a_\infty = \sqrt {\gamma R \theta _\infty }$ is the speed of sound in the free stream;

5. (v) $M_\infty = u_\infty /a_\infty$ is the free-stream Mach number;

6. (vi) $k$ is the thermal diffusivity of the gas;

7. (vii) $\sigma = \mu _\infty c_p/k$ is the Prandtl number which defines the ratio of momentum diffusivity to thermal diffusivity; for air, $\sigma \approx 0.71$.

We consider values of $u_\infty$ and $a_\infty$ such that we obtain Mach numbers $M_\infty$ in the subsonic and moderate supersonic regimes so that $M_\infty \leqslant 2$. In the moderate supersonic regime we assume that the plate is sufficiently thin so that shocks are not present. We choose parameters $\gamma$ and $\sigma$ that are appropriate for the ideal gas assumption; in particular, this means that $\sigma < 2$, which will become important in the scaling arguments below.

Then, using mixed notation so that $(x_1,x_2,x_3)$ represents $(x,y,z)$, $\boldsymbol {\nabla }=(\partial _{x_1}, \partial _{x_2}, \partial _{x_3})$ and $\boldsymbol {u} = ( u_1, u_2, u_3)$ represents $( u, v, w)$, the Navier–Stokes equations have the form

(3.2) $$\begin{gather} \rho \frac{\mathrm{D} u_i}{\mathrm{D} t } ={-}\frac{\partial p}{\partial x_i} + \frac{1}{\textit{Re}} \left\{ \frac{\partial }{\partial x_i} \left( -\frac{2}{3}\mu \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} \right) + \frac{\partial }{\partial x_j}\left( \mu \frac{\partial u_j}{\partial x_i} \right) + \frac{\partial }{\partial x_j}\left( \mu \frac{\partial u_i}{\partial x_j} \right)\right\}\notag\\ \qquad\qquad\qquad (i, j = 1, 2, 3), \end{gather}$$

(3.3)$$\begin{gather}\frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} ( \rho \boldsymbol{u}) = 0, \end{gather}$$

(3.4)$$\begin{gather}\rho \frac{\mathrm{D} \theta}{\mathrm{D} t } = \frac{(\gamma -1)M_\infty^2}{\textit{Re}} \varPhi + (\gamma -1)M_\infty^2 \frac{\mathrm{D} p}{\mathrm{D} t } + \frac{1}{\textit{Re}}\frac{1}{\sigma}\frac{\partial }{\partial x_i} \left(\mu \frac{\partial \theta}{\partial x_i} \right), \end{gather}$$

(3.5)$$\begin{gather}p = \theta_\infty U_\infty^{{-}2} \rho R \theta, \end{gather}$$

where the dissipation function $\varPhi$ is defined by

(3.6)\begin{equation} \varPhi = \tfrac{1}{2} \mu e_{ij} e_{ij} -\tfrac{2}{3} \mu ( {\boldsymbol{\nabla}} \boldsymbol{\cdot} \boldsymbol{u})^2, \end{equation}

and $e_{ij} = \partial u_i/ \partial x_j + \partial u_j/ \partial x_i$ is the rate of strain tensor.

We close the equations of motion with a power-law viscosity law, so that after non-dimensionalisation

(3.7)\begin{equation} \mu = \theta^\zeta. \end{equation}

The index $\zeta =1$ gives the Chapman–Rubesin viscosity law (Chapman & Rubesin Reference Chapman and Rubesin1949) which is suitable for the subsonic regime; for the moderate supersonic regime a slightly more accurate model has $\zeta = 0.76$ (Cebeci Reference Cebeci2002). If we were to extend the analysis to higher Mach numbers then a more realistic viscosity model, such as Sutherland's law (Sutherland Reference Sutherland1893), would be required.

4. The basic flow

We now solve the equations of motion for the basic boundary-layer flow state. The ASBL flow is steady, two-dimensional and independent of $x$. Therefore we seek a boundary-layer solution in the form

(4.1a)$$\begin{gather} \left(u, v, w, p \right) = \left(\hat u(y), \textit{Re}^{{-}1} \hat v(y), 0 , \hat p(y) \right), \end{gather}$$

(4.1b)$$\begin{gather}\left(\theta, \rho, \mu \right) = \left(\hat \theta(y), \hat \rho(y), \hat \mu(y)\right), \end{gather}$$

where the scaling for the normal velocity arises from the need to retain viscous effects in the boundary layer. The boundary conditions at the plate and the free stream are given by

(4.2a)$$\begin{gather} (\hat{u}, \hat{v}, \hat{w}) = \left(0, -1, 0\right), \quad \hat{\theta}= \theta_p/\theta_\infty \quad \mbox{at} \ y = 0, \end{gather}$$

(4.2b)$$\begin{gather}(\hat{u}, \hat{v}, \hat{w}) \to \left(1, -1 , 0\right), \quad \hat{p} \to p_\infty/ \rho_\infty u_\infty^2 , \quad (\hat{\theta}, \hat{\rho}, \hat{\mu}) \to (1, 1, 1) \quad \mbox{as} \ y \to \infty. \end{gather}$$

We substitute the expansion (4.1) into the governing equations (3.2)–(3.5) and, assuming that the Reynolds number is large, retain leading-order terms. The $y$-momentum equation from (3.2) with $i = 2$ reduces to $\partial \hat {p} /\partial y=0$, which means that the pressure $\hat {p}$ is constant across the boundary layer and equal to its free-stream value of $p_\infty / \rho _\infty {u}_\infty ^2$. It follows that the equation of state (3.5) reduces to $\hat {\rho }\hat {\theta } = 1$. Then the continuity equation (3.3) reduces to $\partial _y(\hat {\rho }\hat {v}) = 0$; integrating and applying free-stream boundary conditions (4.2) gives $\hat {\rho }\hat {v} = -1$ across the boundary layer. Thus, $\hat {v} = -\hat {\theta }$, so in particular, the suction condition at the plate gives $\theta _p/\theta _\infty = 1$.

We now use the Dorodnitsyn–Howarth transformation (Dorodnitsyn Reference Dorodnitsyn1942; Howarth Reference Howarth1948) given by

(4.3)\begin{equation} \xi = \int_0^y \hat{\rho}(y') \,\mathrm{d} y', \end{equation}

so that $y$-derivatives $\textrm {d}_y$ are replaced by $\hat {\rho }(\xi ) \,\textrm {d}_\xi$. The equations of motion then reduce to

(4.4a,b)\begin{equation} \hat{u}' + (\hat{\theta}^{\zeta-1}\hat{u}')' = 0, \quad \hat{\theta}' + \sigma^{{-}1} (\hat{\theta}^{\zeta-1} \hat{\theta}')' + (\gamma-1)M_\infty^2 (\hat{u}')^2 = 0, \end{equation}

where a prime denotes derivative with respect to $\xi$. In general, these equations must be solved numerically subject to the boundary conditions (4.2). An analytic solution can be found in the special case of the Chapman–Rubesin law when $\zeta = 1$, and is given by

(4.5a,b)$$\begin{gather} \hat{u}(\xi) = 1-\mathrm{e}^{-\xi}, \quad \hat{\theta}(\xi)= 1 +\frac{(\gamma-1)M_\infty^2 \sigma}{2(2-\sigma)} ( \mathrm{e}^{-\sigma \xi} -\mathrm{e}^{{-}2\xi}), \end{gather}$$

(4.6a–c)$$\begin{gather}\hat{v}(\xi) ={-}\hat{\theta}(\xi), \quad \hat{w}(\xi) = 0, \quad \hat{p}(\xi) =p_\infty/ \rho_\infty u_\infty^2, \end{gather}$$

(4.7a,b)$$\begin{gather}\hat{\rho}(\xi) = \left(\hat{\theta}(\xi)\right)^{{-}1}, \quad \hat{\mu}(\xi) = \hat{\theta}(\xi). \end{gather}$$

For large $\xi$, when the temperature and streamwise velocity are approaching their (non-dimensional) free-stream values of $1$, this analytical basic solution can be used regardless of the index in the viscosity law (3.7). We can invert the Dorodnitsyn–Howarth transformation (4.3) as

(4.8)\begin{equation} y = \int_0^\xi \hat{\theta}(\xi') \,\mathrm{d} \xi'. \end{equation}

Thus if $\xi$ is large, then we can approximate the Dorodnitsyn–Howarth variable by

(4.9a,b)\begin{equation} \xi \approx g(y) = y + C_0; \quad C_0= {\frac{(1-\gamma)M_\infty^2}{4}}. \end{equation}

Consequently, in the free stream, we can write the basic flow in terms of the physical variable $y$. For the interior region we find $\xi = g(y)$ by solving the inversion equation (4.8) numerically.

Thus for large $\xi$, i.e. large $y$, the basic streamwise velocity is given by $\hat {u}\approx 1-\mathrm {e}^{-C_0}\mathrm {e}^{-y}$. Thus the streamwise velocity approaches its free-stream form exponentially as a function of distance from the wall. Therefore the free-stream coherent structure theory of Deguchi & Hall (Reference Deguchi and Hall2014) can be applied. The basic solution for the temperature field also approaches its free-stream form exponentially, with the rate of decay being dependent on the value of the Prandtl number. As discussed above in § 3, gases which provide a good approximation to the ideal gas assumption have Prandtl numbers $\sigma < 2$, and therefore the decay of the basic state to its free-stream form will be dominated by the $\exp (-\sigma \xi )$ term in the basic flow (4.5a,b). Hence, the decay will be slower than that of the streamwise velocity field $\hat {u}$ if $\sigma < 1$. Thus the thermal boundary layer is thicker than the velocity boundary layer if $\sigma < 1$, and vice versa if $\sigma > 1$; this is consistent with laminar boundary-layer theory which suggests that the thickness of the thermal boundary layer $\delta _\theta$ scales relative to the thickness of the velocity boundary layer $\delta _v$ as $\delta _\theta \sim \delta _v \sigma ^{-1/3}$ (Schlichting Reference Schlichting1968, p. 307).

5. The production-layer problem for compressible ASBL flow

Using the inversion of the Dorodnitsyn–Howarth transformation for large $\xi$ (4.8), at the production layer we obtain $\xi \approx y + C_0$, and therefore the solution in the production layer can be expressed in terms of the physical variable $y$. To find the location of the production layer and the scalings of the flow components in the layer, following Deguchi & Hall (Reference Deguchi and Hall2014), we seek a travelling-wave solution propagating with almost the free-stream speed with wavelengths comparable to the boundary-layer scalings of § 4 so that $\partial _x = \partial _y = \partial _z = O(1)$. Then, if viscosity is to play a role in the interaction, $v = O(\textit {Re}^{-1})$, and by the continuity equation (3.3), $1-u = w = O(\textit {Re}^{-1})$. To retain convective terms in the $x$-momentum equation (3.2) the $\rho (\partial _t + u \partial _x)$ term must also be $O(\textit {Re}^{-1})$; this defines the wave dependence in the production layer. The pressure must then be $O(\textit {Re}^{-2})$ to stay in play.

The streamwise component of the velocity field in the production layer must include the basic flow component (4.5a,b) for matching. For large $\xi$, the basic flow $\hat {u}$ has the form $1- \hat{u} = \exp (-y-C_0)$, therefore, in the production layer, $\mathrm {e}^{-y-C_0} = \textit {Re}^{-1}$. Thus the location of the production layer is given by $y = y_{PL} = \ln \textit {Re}-C_0$; this allows us to define a production-layer variable $Y = y-\ln \textit {Re}+C_0$. The thickness of the production layer must then be $O(1)$ to ensure that the streamwise velocity $u$ can only vary on an $O(1)$ length scale in the production layer.

Thus since $C_0 < 0$ for $\gamma =1.4$, a key feature of the compressible problem is that the location of the production layer where the waves and rolls are concentrated moves further away from the wall as both the Reynolds number and the Mach number increase. Since $C_0 \propto M_\infty ^2$, it is anticipated that the Mach number may have a strong influence on the hypersonic (large Mach number) production-layer problem; this is discussed further in the conclusion. However, our choice of parameters means that ${|C_0|} \ll \ln \textit {Re}$. Therefore the values of $\sigma$ and $M_\infty$ do not strongly influence the location of the production layer.

Under the scalings described above, the basic states for the streamwise velocity and temperature (4.5a,b) in the production layer are given by

(5.1a,b)\begin{equation} \hat{u} = 1-\frac{1}{\textit{Re}}\mathrm{e}^{{-}Y}, \quad \hat{\theta}= 1 +\lambda\left( \frac{1}{\textit{Re}^{\sigma}}\mathrm{e}^{-\sigma Y} - \frac{1}{\textit{Re}^{2}}\mathrm{e}^{{-}2Y}\right), \end{equation}

where

(5.2)\begin{equation} \lambda = \frac{(\gamma-1)M_\infty^2 \sigma}{2(2-\sigma)}. \end{equation}

Thus the largest deviation of the temperature field from its free-stream value at the production layer is controlled by the value of the Prandtl number $\sigma$. In particular, if $\sigma < 1$, then the deviation of the temperature field from its free-stream value is greater than the streamwise velocity deviation; this is again due to the relative thickness of the thermal and velocity boundary layers as discussed in § 4.

It is also important to stress that, although the $\exp (-\sigma \xi )$ exponential in the basic temperature state (4.5a,b) dominates the decay of the basic state to its free-stream value, upon exiting the production layer towards the wall as $Y \to -\infty$, any growing temperature disturbances will be dominated by the $\exp (-2Y)$ term in (5.1) and thus both exponentials need to be retained in the production-layer scalings.

Based on the discussion above, in the production layer we seek a solution of the Navier–Stokes equations in the form

(5.3a–e)\begin{equation} \left.\begin{gathered} (X, Y, z) = (x-ct, y-\ln \textit{Re}+C_0, z); \quad c = 1 - \textit{Re}^{{-}1}c_1 + \dots, \\ \boldsymbol{u} = (1, 0, 0) + \textit{Re}^{{-}1}\bar{\boldsymbol{u}} (X, Y, z)+ \dots, \quad p = p_\infty/ \rho_\infty u_\infty^2 + \textit{Re}^{{-}2} \bar{p}(X, Y, z) + \dots, \\ (\theta, \rho, \mu) = 1 + \textit{Re}^{-\sigma}(\bar{\theta}_1, \bar{\rho}_1, \bar{\mu}_1)(X, Y, z) + \textit{Re}^{{-}2}(\bar{\theta}_2, \bar{\rho}_2, \bar{\mu}_2)(X, Y, z). \end{gathered}\right\} \end{equation}

We substitute these scalings into the Navier–Stokes equations (3.2)–(3.5) and, at leading order, we obtain the production-layer problem

(5.4)$$\begin{gather} \mathcal{L}\bar{\boldsymbol{u}} ={-}\boldsymbol{\nabla} \bar{p} + \nabla^2\bar{\boldsymbol{u}}, {\ \mbox{at order} \ \textit{Re}^{{-}1},} \end{gather}$$

(5.5)$$\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \bar{\boldsymbol{u}}= 0, {\ \mbox{at order} \ \textit{Re}^{{-}1}, } \end{gather}$$

(5.6)$$\begin{gather}\mathcal{L}\bar{\theta}_1 = \sigma^{{-}1} \nabla^2 \bar{\theta}_1, \ \mbox{at order} \ \textit{Re}^{-(\sigma +1)}, \end{gather}$$

(5.7)$$\begin{gather}\mathcal{L}\bar{\theta}_2 = (\gamma-1)M_\infty^2 \mathcal{L} \bar{p} + (\gamma-1)M_\infty^2 \bar{\varPhi} + \sigma^{{-}1} \nabla^2 \bar{\theta}_2, \ \mbox{at order} \ \textit{Re}^{{-}3}, \end{gather}$$

(5.8)$$\begin{gather}\bar{\rho}_1 + \bar{\theta}_1 = 0, \ \mbox{at order} \ \textit{Re}^{-\sigma}, \end{gather}$$

(5.9)$$\begin{gather}\bar{p} = R\theta_\infty u_\infty^{{-}2} (\bar{\rho}_2 + \bar{\theta}_2), \ \mbox{at order} \ \textit{Re}^{{-}2}, \end{gather}$$

(5.10)$$\begin{gather}\bar{\mu}_1 = \zeta \bar{\theta}_1, \ \mbox{at order} \ \textit{Re}^{-\sigma}, \end{gather}$$

(5.11)$$\begin{gather}\bar{\mu}_2 = \zeta \bar{\theta}_2, \ \mbox{at order} \ \textit{Re}^{{-}2}, \end{gather}$$

where the operator $\mathcal {L} = ([\bar {\boldsymbol {u}}+c_1\hat {\boldsymbol {i}}]\boldsymbol {\cdot } \boldsymbol {\nabla })$, $\boldsymbol {\nabla } = (\partial _X, \partial _Y, \partial _z)$ and the dissipation function $\bar \varPhi$ is found by substituting the production-layer scalings into (3.6).

We see that the production-layer equations for the velocity field $\bar {\boldsymbol {u}}$ (5.4)–(5.5) are the same as the equations (2.2)–(2.3) for the incompressible production-layer problem in Deguchi & Hall (Reference Deguchi and Hall2014), which describe a unit-Reynolds-number eigenvalue problem for the wave speed $c_1$. The only difference in the compressible problem is that the equations are solved at a slightly different value of $y$. Therefore, the solution to the incompressible eigenvalue problem, which was calculated in Deguchi & Hall (Reference Deguchi and Hall2014), can now also be used for the compressible problem. The velocity field then drives the temperature field through the heat equations (5.6)–(5.7); (5.6), which is obtained at $O(Re^{-\sigma })$, is dominant in the production layer, but we require the solution of the equation at $O(Re^{-2})$ as the production layer is exited towards the wall.

The production-layer problem (5.4)–(5.11) is solved subject to boundary conditions specifying that the flow exiting the production layer on either side must match asymptotically onto the basic solution (4.5a,b),

(5.12)$$\begin{gather} \bar{\boldsymbol{u}} \to (0, -1, 0), \quad \bar{\theta}_1 \to \lambda \mathrm{e}^{-\sigma Y} , \quad \bar{\theta}_2 \to -\lambda \mathrm e^{{-}2 Y} \quad \mbox{as} \ Y \to \infty, \end{gather}$$

(5.13)$$\begin{gather}\bar{\boldsymbol{u}} \to (- \mathrm{e}^{{-}Y}, -1, 0), \quad \bar{\theta}_1 \to \lambda \mathrm{e}^{-\sigma Y}, \quad \bar{\theta}_2 \to -\lambda \mathrm{e}^{{-}2 Y} \quad \mbox{as} \ Y \to -\infty, \end{gather}$$

and periodicity conditions; defining $\alpha$ and $\beta$ as the streamwise and spanwise wavenumbers, respectively,

(5.14)$$\begin{gather} (\bar{\boldsymbol{u}}, \bar{\theta}_{1,2})(X, Y, z) = ( \bar{\boldsymbol{u}}, \bar{\theta}_{1,2})(X + 2{\rm \pi}/\alpha, Y, z), \end{gather}$$

(5.15)$$\begin{gather}(\bar{\boldsymbol{u}}, \bar{\theta}_{1,2})(X, Y, z) = ( \bar{\boldsymbol{u}}, \bar{\theta}_{1,2})(X , Y, z+ 2{\rm \pi}/\beta). \end{gather}$$

Thus, boundary condition (5.13) allows for the streamwise velocity disturbance $\bar {u}$ to grow exponentially beneath the production layer. However, it also allows for the disturbances to the temperature field $\bar {\theta }_1, \bar {\theta }_2$ to grow exponentially, and at a faster rate than the streamwise velocity disturbance. Coming out of the production layer $\bar {\theta }_2$ is dominant, however, $\bar {\theta }_1$, which satisfies a homogeneous equation, must be retained as it is needed at the wall. All disturbances must be reduced to zero at the wall and therefore, as in the incompressible problem, the maximum value of the disturbances will occur in a layer between the wall and the production layer where the basic flow adjusts to accommodate the disturbance.

6. The adjustment-layer problem

Below the production layer, the flow returns to the unperturbed boundary-layer flow (4.5a,b)–(4.7a,b) at leading order. However, the nonlinear production-layer interaction produces exponentially growing disturbances to the streamwise velocity and temperature fields that interact with the basic flow beneath the production layer. The flow between the production layer and the wall adjusts to accommodate the disturbances; we thus refer to this region as the adjustment layer. The solution in the upper part of this layer is dominated by the solution exiting the production layer. Then as the wall is approached, the solution is described by the boundary-region equations.

6.1. The solution exiting the production layer

Firstly, above the production layer as $Y \to \infty$, the velocity must eventually return to its free-stream form $\bar {\boldsymbol {u}} = (0, -1, 0)$. As in Deguchi & Hall (Reference Deguchi and Hall2014), the decay of the streamwise velocity $u$ will be proportional to ${\mathrm {e}^{-Y-C_0}}$, however, the nonlinear interaction in the production layer gives a constant of proportionality which differs from unity. Thus the production-layer interaction can give at most an $O(1)$ effect on the amplitude of the streamwise velocity displacement. Since the temperature field in the production layer is entirely driven by the equations for the velocity (5.4)–(5.5), any temperature disturbances will also decay above the production layer as there is no interaction to sustain them.

We now consider $Y \to -\infty$. To analyse the flow beneath the production layer, we decompose the velocity disturbance $\bar {\boldsymbol {u}}$ into vortex and wave components. The wave is associated with the $X$-dependent components of the velocity field. The $X$-independent components of the velocity are split into a roll flow, which is associated with the components $\bar {v}$ and $\bar {w}$, and the streak, which is the downstream velocity component $\bar {u}$. The combination of the roll and streak constitutes a streamwise vortex. At leading order, the flow must satisfy the basic ASBL flow given by (5.13), and therefore we split the streak into a mean in $z$ and a $z$-dependent component (there is no mean in $z$ of the roll flow due to symmetry). In addition to the $z$-dependent components, we allow the $z$-independent term to grow exponentially in the adjustment layer as $Y \to -\infty$, but it must eventually at leading order reduce to $-\mathrm {e}^{-Y}$ in order to match onto the unperturbed basic flow at the wall.

We decompose the temperature disturbances $\bar {\theta }_1$ and $\bar {\theta }_2$ in the same way. Following the nomenclature outlined in Ricco & Wu (Reference Ricco and Wu2007), we refer to the $X$-independent component of the temperature disturbance as a ‘thermal streak’ and the corresponding streamwise velocity disturbance shall be termed a ‘velocity streak’. Hence, in the adjustment layer, we seek a solution in the form

(6.1)$$\begin{gather} \bar{\boldsymbol{u}} = (\bar{u}_s(Y), -1, 0) + (u_s(Y, z), v_r(Y, z), w_r(Y, z)) + \boldsymbol{u}_w(X, Y, z), \end{gather}$$

(6.2)$$\begin{gather}\bar{\theta}_{1,2} = {\bar{\theta}_{s_{1,2}}(Y) + \theta_{s_{1,2}}(Y, z)} + \theta_{w_{1,2}}(X, Y, z), \end{gather}$$

where subscripts $s$, $r$ and $w$ refer to streak, roll and wave components respectively.

As in the incompressible ASBL study of Deguchi & Hall (Reference Deguchi and Hall2014), outside of the production layer the roll flow decays as there is no longer any forcing from the Reynolds stresses associated with the wavefield to sustain it. The wave $\boldsymbol {u}_w$ also decays faster than the roll; this can be seen through a balance of advection–diffusion terms and is confirmed by the numerical results of Deguchi & Hall (Reference Deguchi and Hall2014). Since the temperature field is driven entirely by the velocity field, the same is true of the corresponding temperature components $\theta _{w_{1, 2}}$ and $\theta _{w_{1,2}}$. However, the velocity streak ${\bar {u}_{s} + u_s}$ can grow exponentially through interaction with the roll. The growth or decay of the velocity streak depends on the spanwise wavenumber $\beta$ through the periodicity conditions (5.15). The new feature for the compressible problem is that the interaction of the roll flow with the temperature field drives the growth of the thermal streak.

We substitute the decomposition of the disturbances (6.1)–(6.2) into the production-layer equations (5.4)–(5.11). After introducing the roll-flow streamfunction $\psi$ such that $\partial _z \psi = {v_r}$ and $\partial _y \psi = -{w_r}$, the resulting equations for the roll-velocity-streak flow are given by

(6.3)$$\begin{gather} \left(\frac{\partial^2 }{\partial Y^2} + \frac{\partial }{\partial Y} + \frac{\partial^2 }{\partial z^2}\right){u_s} = \textrm{e}^{{-}Y}{v_r}, \end{gather}$$

(6.4)$$\begin{gather}\left(\frac{\partial^2 }{\partial Y^2} + \frac{\partial }{\partial Y} + \frac{\partial^2 }{\partial z^2}\right) \left(\frac{\partial^2 }{\partial Y^2} + \frac{\partial^2 }{\partial z^2} \right) \psi= 0, \end{gather}$$

(6.5)$$\begin{gather}\left( \frac{\mathrm{d} }{\mathrm{d} Y} + \frac{\mathrm{d}^2 }{\mathrm{d} Y^2} \right) {\bar{u}_s} = \frac{\beta}{2{\rm \pi}} \frac{\mathrm{d} }{\mathrm{d} Y} \int_{z = 0}^{2{\rm \pi}/\beta} {(u_s v_r)} \,\mathrm{d} z, \end{gather}$$

where the final equation for the mean velocity streak disturbance ${\bar {u}_s}$ has been found by taking the mean in $z$ of the production-layer $x$-momentum equation (5.4). It is important to note the $\partial _Y$ terms in the equations above which arise from the suction in the flow. It is these terms that allow the interaction of the mean part of the basic flow with the roll flow to produce growth.

The roll-velocity-streak equations (6.3)–(6.5) are solved together with the equations for the thermal streak,

(6.6)$$\begin{gather} {\left(\frac{\partial^2 }{\partial Y^2} + \frac 1 \sigma \frac{\partial }{\partial Y} + \frac 1 \sigma \frac{\partial^2 }{\partial z^2}\right)}{\theta_{s_1}} ={-}\sigma \lambda \mathrm{e}^{-\sigma Y} {v_r}, \end{gather}$$

(6.7)$$\begin{gather}{\left(\frac{\partial^2 }{\partial Y^2} + \frac 1 \sigma \frac{\partial }{\partial Y} + \frac 1 \sigma \frac{\partial^2 }{\partial z^2}\right)} {\theta_{s_2}} ={-}2 \lambda {v_r} \mathrm{e}^{{-}2 Y} -2 (\gamma-1)M_\infty^2 \mathrm{e}^{{-}Y} {\frac{\partial u_s}{\partial Y}}, \end{gather}$$

(6.8)$$\begin{gather}{\left( \frac{\mathrm{d} }{\mathrm{d} Y} + \frac 1 \sigma \frac{\mathrm{d}^2 }{\mathrm{d} Y^2} \right)} {\bar{\theta}_{s_1}} = \frac{\beta}{2{\rm \pi}} \frac{\mathrm{d} }{\mathrm{d} Y} \int_{z = 0}^{2{\rm \pi}/\beta} {(v_r \theta_{s_1}) }\,\mathrm{d} z, \end{gather}$$

(6.9)$$\begin{gather}{\left( \frac{\mathrm{d} }{\mathrm{d} Y} + \frac 1 \sigma \frac{\mathrm{d}^2 }{\mathrm{d} Y^2} \right) }{\bar{\theta}_{v_2}} = \frac{\beta}{2{\rm \pi}} \frac{\mathrm{d} }{\mathrm{d} Y} \int_{z = 0}^{2{\rm \pi}/\beta} \left( {v_r \theta_{s_2}} - (\gamma-1)M_\infty^2 {\varPhi_v} \right) \mathrm{d} z , \end{gather}$$

where the dissipation function $\varPhi _v$ associated with the vortex flow is

(6.10)\begin{align} \varPhi_v &= \frac{4}{3}\left( \frac {\partial v_r}{\partial Y} \right) ^{2}+\frac{4}{3} \left( {\frac {\partial { w_r} }{\partial z}} \right) ^{2}+ \left( {\frac {\mathrm{d} {\bar{u}_s} }{ \textrm{d}Y}} +{\frac {\partial {u_s} }{\partial Y}} \right) ^{2}+ \left( {\frac {\partial {u_s} }{ \partial z}} \right) ^{2}+ \left( {\frac { \partial {v_r} }{\partial z}} \right) ^{2} \nonumber\\ &\quad +2\, {\frac {\partial {v_r} }{\partial z}} {\frac {\partial {w_r} }{\partial Y}} + \left( {\frac {\partial {w_r} }{\partial Y}} \right) ^{2}-\frac{4}{3} {\frac {\partial {v_r} }{\partial Y}}{\frac {\partial {w_r} }{\partial z}}. \end{align}

These equations are solved by Fourier expansion in $z$. The numerical results of Deguchi & Hall (Reference Deguchi and Hall2014) show that the vortex wavelength is half that of the wave part of the flow, and therefore the wavelength of the vortex is ${\rm \pi} /\beta$, which sets the wavenumbers of the Fourier expansion. Therefore, we seek a solution for $\psi$ in the form

(6.11)\begin{equation} \psi = \sum_{n=0}^\infty a_n \cos(2n\beta z) + \sum_{n=1}^\infty b_n \sin(2n\beta z). \end{equation}

The roll-velocity-streak equations (6.3)–(6.5) are the same as those for the incompressible equation in Deguchi & Hall (Reference Deguchi and Hall2014), with $Y=y-\ln \textit {Re}+C_0$ where $C_0=0$ (corresponding to $M_\infty = 0$) in the incompressible problem. Thus, the solution of (6.3)–(6.5) is the same as that for the incompressible problem; the incompressible solution with $C_0=0$ is given in (2.4)–(2.6). Thus upon exiting the production layer in the compressible problem, the leading order solution of (6.3)–(6.5) in original boundary-layer coordinates $(x, y, z)$ and associated flow quantities $(u, v, w)$ is

(6.12)\begin{gather} u \to 1 - \exp({-(y+C_0)}) + \frac{d_0}{\textit{Re}} + \frac{ J_1}{\textit{Re}^{\omega_1}}\exp({(\omega_1-1)(y+C_0)})\cos(2\beta z) \nonumber\\ \quad + \frac{ J_1 K_1}{\textit{Re}^{2\omega_1}4\omega_1}\exp({(2\omega_1-1)(y+C_0)})+ \cdots, \end{gather}

(6.13)$$\begin{gather} v \to -\frac {1} {\textit{Re}} + \frac{ K_1}{\textit{Re}^{\omega_1+1}} \exp({\omega_1(y+C_0)}) \cos(2\beta z) + \cdots, \end{gather}$$

(6.14)$$\begin{gather}w \to -\frac{K_1 \omega_1}{2\beta\textit{Re}^{\omega_1+1}} \exp({\omega_1 (y+ C_0)}) \sin(2 \beta z) + \cdots, \end{gather}$$

where

(6.15a,b)\begin{equation} J_n = \frac{K_n}{(\omega_n-1)^2 + (\omega_n-1) - 4 n^2\beta^2}, \quad \omega_n = \frac{ -1 + \sqrt{ 1 + 16n^2\beta^2}}{2} \geqslant 0, \end{equation}

for $n \geqslant 1$. The terms represented by ‘$\cdots$’ represent more slowly growing harmonics in $z$, with constants $J_n$, $K_n$ and $\omega _n$ for $n > 1$. The constants $d_0$ and $K_1$ are found as part of the nonlinear eigenvalue production-layer problem; $K_1$ was reported for a range of $\beta$ in Deguchi & Hall (Reference Deguchi and Hall2014). Thus we only give the full streamwise velocity solution up to a constant $d_0/\textit {Re}$, but this constant does not affect the streaks. As required, the flow returns to its unperturbed basic state at leading order, with exponentially growing disturbances that can become larger than the velocities involved in the nonlinear interaction in the production layer where the disturbances originated.

The solutions for $u_s$, $v_r$ and $\bar {u}_s$ are then used as forcing for the equations (6.6)–(6.9) for the thermal streak. In the original boundary-layer variables, $\theta = 1 + \textit {Re}^{-\sigma }\bar {\theta }_1 + \textit {Re}^{-2}\bar {\theta }_2$, we find that upon exiting the production layer,

(6.16)\begin{align} &\theta \to 1 + \lambda \exp({-\sigma (y+C_0)}) - \lambda \exp({-2(y+C_0)}) + \frac{d_1}{\textit{Re}^{\sigma}} + \frac{d_2}{\textit{Re}^2} \nonumber\\ &\quad + \frac{1}{\textit{Re}^{\omega_1}}\left( L_1 \exp({(\omega_1-\sigma)(y+C_0)}) + Q_1 \exp({(\omega_1 -2)(y+C_0)})\right)\cos(2\beta z) \nonumber\\ &\quad + \frac{1}{\textit{Re}^{2\omega_1}} \left( \frac{ L_1 K_1 \sigma}{4 \omega_1} \exp({(2\omega_1-\sigma)(y+C_0)}) + R_1 \exp({(2\omega_1-2)(y+C_0)})\right) + \cdots, \end{align}

where again the terms represented by ‘$\cdots$’ denote more slowly growing harmonics in $z$, with constants $K_n$, $J_n$, $L_n$, $Q_n$, $R_n$ and $\omega _n$ for $n > 1$ and where

(6.17a)$$\begin{gather} L_n = \frac{-K_n \lambda \sigma}{(\omega_n - \sigma) + \sigma^{{-}1} (\omega_n-\sigma)^2 - \sigma^{{-}1}4n^2\beta^2}, \end{gather}$$

(6.17b)$$\begin{gather}Q_n = \frac{-2 \lambda K_n - 2(\gamma-1)M_\infty^2 J_n (\omega_n-1) }{(\omega_n - 2) + \sigma^{{-}1} (\omega_n-2)^2 - \sigma^{{-}1}4n^2\beta^2}, \end{gather}$$

(6.17c)\begin{gather} R_n ={-}2\,{\frac {\sigma \left( \frac 1 4M_\infty^2{{J_n}}^{2} \left( \gamma-1 \right) {{\omega_n}}^{3}-\frac{1}{2} M_\infty^2{{J_n}}^{2} \left( \gamma-1 \right) {{\omega_n}}^{2}-\frac{1}{4}{J_n}M_\infty^2{K_n} \left( \gamma -1 \right) \right) }{{\omega_n} \left( \sigma+2\,{\omega_n}-2 \right)}} \nonumber\\ \quad -2\,{\frac {\sigma \left( M_\infty^2 \left( {\beta}^{2}{n}^{2}+\frac 1 4 \right) \left( \gamma-1 \right) {{J_n}}^{2}+\frac 1 2 {J_n}M_\infty^2{K_n} \left( \gamma-1 \right) -\frac 1 4{Q_n}{K_n} \right)}{\sigma+2\,{\omega_n}-2}}. \end{gather}

The constants $d_1$ and $d_2$ are constants of integration; again, we only find the solution for the temperature field up to a constant, but this constant does not affect the growth of the thermal streaks beneath the production layer.

The asymptotic solution (6.12)–(6.13) for $u$ and $v$ beneath the production layer shows that the roll flow always decays as the wall layer is approached, whereas the mean part of the velocity streak flow (6.12) can grow beneath the production layer if $2\omega _1 < 1$, corresponding to values of $\beta < \sqrt {3}/4$. The $z$-dependent part of the velocity streak can grow if $\omega _1 < 1$, corresponding to values of $\beta < 1/\sqrt 2$, and therefore these latter modes are the fastest growing. If $\beta > 1/\sqrt 2$, then the velocity streak disturbance decays exponentially, and the nonlinear interaction in the production layer simply produces an $O(Re^{-1})$ correction to the flow.

Meanwhile, the asymptotic solution for the temperature (6.16) beneath the production layer shows that the thermal streaks can grow if $\omega _1 < \sigma$ or if $\omega _1 < 2$. For the range of values of Prandtl number $\sigma <2$ considered, the modes proportional to $\exp ( (\omega _1-2) (y+C_0))$ will dominate the growth, and therefore the nonlinear interaction in the production layer will always produce growing temperature disturbances for $\beta < \sqrt {3}/\sqrt {2}$.

The structure of the solution with varying $\beta$ is summarised in table 1. The asymptotic results suggest that there exists a case where the thermal streaks can grow while the velocity streak decays. However, solutions of the production-layer problem (5.4)–(5.5) have not been found for values of $\beta \gtrsim 0.47$ (Deguchi & Hall Reference Deguchi and Hall2014, Reference Deguchi and Hall2015), and therefore cases $(c)$ and $(d)$ are possibly not relevant.

Table 1.

The growth and decay of the disturbances for different values of the spanwise wavenumber $\beta$. Growth is represented by ‘G’ and decay by ‘D’. The growth and decay is shown for the both the mean in $z$ and the $z$-dependent parts of the flow.

We see that, in all cases, a nonlinear interaction in the production layer of size $O(\textit {Re}^{-1})$, which drives $O(\textit {Re}^{-2}), O(\textit {Re}^{-\sigma })$ temperature perturbations, can induce much larger changes to the velocity and temperature fields of ${O(\textit {Re}^{-\omega _1})}$ in the main part of the boundary layer. We now consider the solution as it approaches the wall layer, where all disturbances are eventually reduced to zero to satisfy the wall boundary conditions.

6.2. Boundary-layer analysis

The solutions exiting the production layer, (6.12)–(6.14) and (6.16), do not satisfy the wall boundary conditions. We now find the solution for the induced flow which is valid all the way down to the wall. This solution should also match onto the solution exiting the production layer given by (6.12)–(6.14) and (6.16). An examination of this solution shows that in the boundary layer, disturbances can grow exponentially. The $z$-dependent part of the disturbance grows faster than the $z$-independent part; therefore, to match onto the solution exiting the production layer, the boundary-region solution will have $z$-dependence in the form of $\cos (2\beta z)$. However, the solution must also satisfy conditions (4.2a), and therefore any disturbances must ultimately be reduced to zero at the wall.

The solution in the boundary region is described in terms of the Dorodnitsyn–Howarth variable $\xi$; since disturbances are always small compared with the basic flow, the definition of the variable in (4.3) is valid throughout the flow, and in particular, $\partial _y = \hat \rho (\xi ) \partial _\xi$. The inversion of the Dorodnitsyn–Howarth transformation for $y(\xi )$ is given in (4.8), however, unlike near the wall and in the production layer, we cannot generally find an explicit relationship for $\xi (y)$ as it cannot be assumed that the exponential terms involving $\xi$ are smaller than the linear terms. Therefore, to find the solution for the physical variable $y$, we first solve the boundary-region equations in terms of $\xi$, and then use the monotonic relationship $y(\xi )$ in (4.8) to plot the solutions for each corresponding value of $y$.

Based on this discussion, in the boundary region we seek a solution in terms of the fundamental harmonics of the solution exiting the production layer (6.12)–(6.14), (6.16) in the form

(6.18a)$$\begin{gather} u = \hat{u}(\xi) + { \textit{Re}^{-\omega_1} } \tilde{u} (\xi)\cos(2\beta z), \end{gather}$$

(6.18b)$$\begin{gather}{v}(\xi) = \textit{Re}^{{-}1} \hat{v}(\xi) + {\textit{Re}^{-(1+\omega_1)}} \tilde{v}(\xi)\cos(2\beta z), \end{gather}$$

(6.18c)$$\begin{gather}w = {\textit{Re}^{-(1+\omega_1)}} \tilde{w}(\xi)\sin(2\beta z), \quad p = \hat{p}(\xi) + {\textit{Re}^{-(2+\omega_1)}} \tilde{p}(\xi)\cos(2\beta z), \end{gather}$$

(6.18d)$$\begin{gather}\left(\theta, \rho, \mu \right) = \left( \hat{\theta}(\xi), \hat{\rho}(\xi), \hat{\mu}(\xi) \right) + {\textit{Re}^{-\omega_1} } \left( \tilde{\theta}(\xi), \tilde{\rho}(\xi), \tilde{\mu}(\xi) \right)\cos(2\beta z), \end{gather}$$

where the basic solution (hat quantities) is given by the solution of (4.4a,b). We use the same velocity streak and thermal streak terminology to refer to the disturbances to the streamwise velocity and temperature fields respectively, and again the roll flow is associated with the disturbances to the $(v, w)$ components of the velocity.

We substitute this expansion into the Navier–Stokes equations (3.2)–(3.5), which leads to a set of ordinary differential equations in $\xi$ for the leading-order disturbance amplitudes (tilde quantities). Following Hall (Reference Hall1983), we eliminate the pressure $\tilde {p}$ and the spanwise disturbance velocity $\tilde {w}$; then we also eliminate the viscosity $\tilde {\mu }$ and the density $\tilde {\rho }$ using the equation of state (3.5) and the linearised power-law viscosity law (3.7). We are then left with three coupled differential equations for $\tilde {u}$ (from the $x$-momentum equation), $\tilde {v}$ (from the $y$-momentum equation) and $\tilde {\theta }$ (from the temperature equation)

(6.19)$$\begin{gather} A_1 \tilde{u} + A_2 \tilde{u}' + A_3 \tilde{u}'' = A_4 \tilde{v} + A_5 \tilde{\theta} + A_6 \tilde{\theta}', \end{gather}$$

(6.20)$$\begin{gather}B_1 \tilde{v} + B_2 \tilde{v}' + B_3 \tilde{v}'' + B_4 \tilde{v}^{(3)} + B_5 \tilde{v}^{(4)} = B_6 \tilde{\theta} + B_7 \tilde{\theta}' + B_8 \tilde{\theta}'' + B_9 \tilde{\theta}^{(3)} + B_{10} \tilde{\theta}^{(4)} , \end{gather}$$

(6.21)$$\begin{gather}C_1 \tilde{\theta} + C_2 \tilde{\theta}' + C_3 \tilde{\theta}'' = C_4 \tilde{u}' + C_5 \tilde{v}. \end{gather}$$

Here, the superscripts represent derivatives in the usual way. The coefficients $A_k$, $B_k$ and $C_k$ depend on the basic solution and are too long to write here; details are available from the authors on request. These coupled equations are solved subject to zero-disturbance and no-slip boundary conditions at the wall, and matching to the solution exiting the production layer (6.12), (6.13), (6.16) at $\xi = \xi _{PL} = \ln \textit {Re}$, so that

(6.22a)$$\begin{gather} \tilde{u}(0) =0, \quad \tilde{u}(\xi_{PL}) = J_1 \exp({(\omega_1-1)\xi_{PL}}), \end{gather}$$

(6.22b)$$\begin{gather}\tilde{v}(0) =\tilde{v}'(0) = 0, \quad \tilde{v}(\xi_{PL}) = K_1 \exp({\omega_1\xi_{PL}}), \quad \tilde{v}'(\xi_{PL}) = K_1 \omega_1 \exp({\omega_1\xi_{PL}}), \end{gather}$$

(6.22c)$$\begin{gather}\tilde{\theta}(0) = 0, \quad \tilde{\theta}(\xi_{PL}) = \left( L_1 \exp({(\omega_1-\sigma)\xi_{PL}}) + Q_1 \exp({(\omega_1-2)\xi_{PL}})\right). \end{gather}$$

The reduced boundary-region equations are discretised on a grid with $N$ interior points and we use second-order accurate centred finite differences to approximate the derivatives with step size $\Delta \xi$; see Appendix A for details. We then solve the resulting matrix equation for $\tilde {u}$, $\tilde {v}$ and $\tilde {\theta }$.