2.1 Free-stream disturbances and scaling

Free-stream turbulence is in general of broadband nature. For simplicity, we consider the case of vortical perturbation consisting of a pair of vortical modes with the same frequency (and hence streamwise wavenumber), but opposite spanwise wavenumber $\pm k_{z}^{\ast }$ . In the incompressible analysis of Ricco et al. (Reference Ricco, Luo and Wu2011), this choice of free-stream disturbance has led to good quantitative agreement between the theoretical prediction and wind tunnel experimental data. A similar behaviour is expected in the compressible case. The formulation and computation can be extended to realistic free-stream perturbations, which are of broadband nature as was shown by Zhang et al. (Reference Zhang, Zaki, Sherwin and Wu2011) for incompressible flows.

The velocity field of free-stream convected gusts of the assumed form can be expressed as

(2.3) $$\begin{eqnarray}\boldsymbol{u}-\boldsymbol{i}=\unicode[STIX]{x1D716}\boldsymbol{u}_{\infty }(x-t,y,z)=\unicode[STIX]{x1D716}\left(\hat{\boldsymbol{u}}_{+}^{\infty }\text{e}^{\text{i}k_{z}z}+\hat{\boldsymbol{u}}_{-}^{\infty }\text{e}^{-\text{i}k_{z}z}\right)\text{e}^{\text{i}k_{x}(x-t)+\text{i}k_{y}y}+\text{c.c.},\end{eqnarray}$$

where $\hat{\boldsymbol{u}}_{\pm }^{\infty }=\{\hat{u} _{x,\pm }^{\infty },\hat{u} _{y,\pm }^{\infty },\hat{u} _{z,\pm }^{\infty }\}=O(1)$ is a real vector, $\unicode[STIX]{x1D716}\ll 1$ is a measure of the free-stream perturbation level and c.c. indicates the complex conjugate. From the continuity equation, it follows that

(2.4) $$\begin{eqnarray}k_{x}\hat{u} _{x,\pm }^{\infty }+k_{y}\hat{u} _{y,\pm }^{\infty }\pm k_{z}\hat{u} _{z,\pm }^{\infty }=0.\end{eqnarray}$$

It is appropriate and convenient to take $\unicode[STIX]{x1D706}^{\ast }=1/k_{z}^{\ast }$ , so that $k_{z}=1$ . The characteristic Reynolds number is

(2.5) $$\begin{eqnarray}R_{\unicode[STIX]{x1D706}}\equiv \frac{U_{\infty }^{\ast }\unicode[STIX]{x1D706}^{\ast }}{\unicode[STIX]{x1D708}_{\infty }^{\ast }}\gg 1.\end{eqnarray}$$

Only the components of the free-stream disturbance with $k_{x}\ll 1$ are considered as they have been shown in experiments to be the ones that can penetrate the most into the boundary layer to form streaks.

According to the result of Leib et al. (Reference Leib, Wundrow and Goldstein1999), the velocity perturbation is maximum when $x=O(k_{x}^{-1})$ . The following scaling is thus introduced:

(2.6) $$\begin{eqnarray}\bar{x}\equiv k_{x}x=O(1).\end{eqnarray}$$

As a measure of the ratio between the boundary-layer thickness $\unicode[STIX]{x1D6FF}^{\ast }$ and the spanwise length scale $\unicode[STIX]{x1D706}^{\ast }$ at $\bar{x}=O(1)$ , a scaled spanwise wavenumber is defined as

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D705}\equiv \frac{1}{\unicode[STIX]{x1D706}^{\ast }}\sqrt{\frac{\unicode[STIX]{x1D706}_{x}^{\ast }\unicode[STIX]{x1D708}_{\infty }^{\ast }}{2\unicode[STIX]{x03C0}U_{\infty }^{\ast }}}=\frac{k_{z}}{\sqrt{k_{x}R_{\unicode[STIX]{x1D706}}}}.\end{eqnarray}$$

The interest is in the downstream viscous region where $\unicode[STIX]{x1D6FF}^{\ast }=O(\unicode[STIX]{x1D706}^{\ast })$ so that viscous diffusion effects in the spanwise and wall-normal directions are comparable. This occurs at streamwise locations $x^{\ast }=O(\unicode[STIX]{x1D706}^{\ast }R_{\unicode[STIX]{x1D706}})$ , which, together with (2.6), leads to $k_{x}=O(R_{\unicode[STIX]{x1D706}}^{-1})$ , or, equivalently, $\unicode[STIX]{x1D705}=O(1)$ . As shown by Leib et al. (Reference Leib, Wundrow and Goldstein1999), $O(\unicode[STIX]{x1D716})$ free-stream disturbances can produce $O(\unicode[STIX]{x1D716}/k_{x})$ fluctuations of the streamwise velocity component within the boundary layer. Nonlinear effects become of leading order when $\unicode[STIX]{x1D716}/k_{x}=O(1)$ , i.e. when the turbulent Reynolds number

(2.8) $$\begin{eqnarray}r_{t}\equiv \unicode[STIX]{x1D716}R_{\unicode[STIX]{x1D706}}=O(1),\end{eqnarray}$$

since $k_{x}=O(R_{\unicode[STIX]{x1D706}}^{-1})$ . A schematic illustration of the flow domain and its asymptotic structure is shown in figure 1.

Figure 1. Sketch of the flow configuration representing the asymptotic regions (adapted from Leib et al. Reference Leib, Wundrow and Goldstein1999).

2.2 The inner region: nonlinear unsteady compressible streaks

In the boundary layer the solution is expressed as the superimposition of the unsteady perturbation on the steady laminar compressible boundary layer. The velocities and temperature of the Blasius flow have the similarity solution (Stewartson Reference Stewartson1964)

(2.9a,b ) $$\begin{eqnarray}\left\{U,V\right\}=\left\{F^{\prime }(\unicode[STIX]{x1D702}),\frac{T(\unicode[STIX]{x1D702}_{c}F^{\prime }-F)}{\sqrt{2xR_{\unicode[STIX]{x1D706}}}}\right\},\quad T=T(\unicode[STIX]{x1D702}),\end{eqnarray}$$

where the prime indicates differentiation with respect to the similarity variable $\unicode[STIX]{x1D702}$ ,

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D702}\equiv \sqrt{\frac{R_{\unicode[STIX]{x1D706}}}{2x}}\int _{0}^{y}\unicode[STIX]{x1D70C}(x,\check{y})\,\text{d}\check{y},\end{eqnarray}$$

and $\unicode[STIX]{x1D702}_{c}=T^{-1}\int _{0}^{\unicode[STIX]{x1D702}}T(\check{\unicode[STIX]{x1D702}})\,\text{d}\check{\unicode[STIX]{x1D702}}$ . The $x$ -momentum and the energy equations are:

(2.11a ) $$\begin{eqnarray}\displaystyle & \displaystyle FF^{\prime \prime }+\left(\frac{\unicode[STIX]{x1D707}}{T}F^{\prime \prime }\right)^{\prime }=0, & \displaystyle\end{eqnarray}$$

(2.11b ) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x1D707}T^{\prime }}{T}\right)^{\prime }+PrFT^{\prime }+Pr(\unicode[STIX]{x1D6FE}-1)M_{\infty }^{2}\frac{\unicode[STIX]{x1D707}}{T}{F^{\prime \prime }}^{2}=0, & \displaystyle\end{eqnarray}$$

$Pr=0.7$

(2.12a,b ) $$\begin{eqnarray}\displaystyle F(0)=F^{\prime }(0)=0,\quad T(0)=T_{w}, & & \displaystyle\end{eqnarray}$$

(2.13a,b ) $$\begin{eqnarray}\displaystyle F^{\prime }\rightarrow 1,\quad T\rightarrow 1\quad \text{as}~\unicode[STIX]{x1D702}\rightarrow \infty , & & \displaystyle\end{eqnarray}$$

where $T_{w}$ is the imposed wall temperature (isothermal condition). For $\unicode[STIX]{x1D702}\gg 1$ , $F\rightarrow \bar{\unicode[STIX]{x1D702}}\equiv \unicode[STIX]{x1D702}-\unicode[STIX]{x1D6FD}_{c}$ , where $\unicode[STIX]{x1D6FD}_{c}$ depends on $M_{\infty }$ . Using the equation of state, the density $\unicode[STIX]{x1D70C}$ is given by

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D70C}=\frac{1}{T}.\end{eqnarray}$$

The viscosity $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}(T)$ is assumed to follow a power law,

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D707}=T^{\unicode[STIX]{x1D714}}\quad \text{with}~\unicode[STIX]{x1D714}=0.76.\end{eqnarray}$$

This relation has been proved to be more appropriate than the linear Chapman law ( $\unicode[STIX]{x1D714}=1$ ) in the Mach number range of interest $M_{\infty }<4$ (Stewartson Reference Stewartson1964).

The total boundary-layer flow is decomposed as the sum of the Blasius flow and the perturbation induced by the free-stream disturbance, namely,

(2.16) $$\begin{eqnarray}\displaystyle \left\{u_{tot},v_{tot},w_{tot},p_{tot},\unicode[STIX]{x1D70F}_{tot}\right\} & = & \displaystyle \left\{U,V,0,-\frac{1}{2},T\right\}+r_{t}\left\{\bar{u}(\bar{x},\unicode[STIX]{x1D702},z,t),\sqrt{\frac{2\bar{x}k_{x}}{R_{\unicode[STIX]{x1D706}}}}\bar{v}(\bar{x},\unicode[STIX]{x1D702},z,t),\right.\nonumber\\ \displaystyle & & \displaystyle \left.\frac{k_{x}}{k_{z}}\bar{w}(\bar{x},\unicode[STIX]{x1D702},z,t),\frac{k_{x}}{R_{\unicode[STIX]{x1D706}}}\bar{p}(\bar{x},\unicode[STIX]{x1D702},z,t),\bar{\unicode[STIX]{x1D70F}}(\bar{x},\unicode[STIX]{x1D702},z,t)\vphantom{\sqrt{\frac{2\bar{x}k_{x}}{R_{\unicode[STIX]{x1D706}}}}}\right\},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{tot}$ stands for the temperature. The Blasius flow does not correspond to the mean flow; the latter also consists of the time-independent components generated by the nonlinear interactions, which are included in the perturbation. The scaling (2.8) and the decomposition (2.16) indicate that for low-frequency free-stream disturbances with spanwise wavelength comparable with the boundary-layer thickness and streamwise wavelength of $O(\unicode[STIX]{x1D706}^{\ast }R_{\unicode[STIX]{x1D706}})$ , the streamwise velocity and the temperature of the induced streaks acquire an amplitude of $O(\unicode[STIX]{x1D716}R_{\unicode[STIX]{x1D706}})$ , which is much larger than $O(\unicode[STIX]{x1D716})$ transverse velocity components (and the intensity of free-stream disturbances). The boundary-layer signature therefore bears all hallmarks of streaks observed in experiments. For the assumed simple composition of free-stream disturbance, the seeded oblique-mode pair would be dominant in the earlier stage, and is expected to remain the most significant in the nonlinear stage downstream. However, nonlinear interactions generate harmonics and the mean-flow distortion. The disturbance can be expressed as a Fourier series in time and $z$ ,

(2.17) $$\begin{eqnarray}\left\{\bar{u},\bar{v},\bar{w},\bar{p},\bar{\unicode[STIX]{x1D70F}}\right\}=\mathop{\sum }_{m,n}\left\{\hat{u} _{m,n},\hat{v}_{m,n},{\hat{w}}_{m,n},\hat{p}_{m,n},\hat{\unicode[STIX]{x1D70F}}_{m,n}\right\}\text{e}^{\text{i}mk_{x}t+\text{i}nk_{z}z},\end{eqnarray}$$

where $\{\hat{u} _{m,n},\hat{v}_{m,n},{\hat{w}}_{m,n},\hat{p}_{m,n},\hat{\unicode[STIX]{x1D70F}}_{m,n}\}$ are functions of $\bar{x}$ and $\unicode[STIX]{x1D702}$ . Unless otherwise specified, hereinafter the upper and lower limits of the summations are $\pm \infty$ . As the physical quantities are real, the Fourier coefficients are Hermitian,

(2.18) $$\begin{eqnarray}\hat{q}_{-m,-n}=\hat{q}_{m,n}^{\star },\end{eqnarray}$$

where $\hat{q}$ indicates any of $\{\hat{u} ,\hat{v},{\hat{w}},\hat{p},\hat{\unicode[STIX]{x1D70F}}\}$ and the symbol $^{\star }$ denotes the complex conjugate. The total density is decomposed as $\unicode[STIX]{x1D70C}_{tot}=\unicode[STIX]{x1D70C}+r_{t}\bar{\unicode[STIX]{x1D70C}}$ , where $\unicode[STIX]{x1D70C}$ is given by (2.14). Substituting the total flow into the equation of state, one finds

(2.19) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D70C}}=\frac{\unicode[STIX]{x1D6FE}M_{\infty }^{2}k_{x}}{TR_{\unicode[STIX]{x1D706}}}\bar{p}-\frac{\bar{\unicode[STIX]{x1D70F}}}{T^{2}}-r_{t}\frac{\bar{\unicode[STIX]{x1D70C}}\bar{\unicode[STIX]{x1D70F}}}{T}.\end{eqnarray}$$

It follows that the total density is:

(2.20) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{tot}=\frac{1}{T}+r_{t}\left(\frac{\unicode[STIX]{x1D6FE}M_{\infty }^{2}k_{x}}{TR_{\unicode[STIX]{x1D706}}}\bar{p}-\frac{\bar{\unicode[STIX]{x1D70F}}}{T^{2}}\right)-r_{t}^{2}\frac{\bar{\unicode[STIX]{x1D70C}}\bar{\unicode[STIX]{x1D70F}}}{T}.\end{eqnarray}$$

The total viscosity is expressed by applying (2.15) to the total flow and by expanding it using the binomial formula as

(2.21) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{tot}=(T+r_{t}\bar{\unicode[STIX]{x1D70F}})^{\unicode[STIX]{x1D714}}=\mathop{\sum }_{j=0}^{\infty }\frac{(\unicode[STIX]{x1D714})_{j}}{j!}T^{\unicode[STIX]{x1D714}-j}r_{t}^{\,\,j}\bar{\unicode[STIX]{x1D70F}}^{j}=\unicode[STIX]{x1D707}+r_{t}\unicode[STIX]{x1D707}^{\prime }\bar{\unicode[STIX]{x1D70F}}+r_{t}^{\,2}\bar{\unicode[STIX]{x1D707}},\end{eqnarray}$$

where $(\unicode[STIX]{x1D714})_{j}=\unicode[STIX]{x1D714}(\unicode[STIX]{x1D714}-1)(\unicode[STIX]{x1D714}-2)\ldots (\unicode[STIX]{x1D714}-j+1)$ , $\unicode[STIX]{x1D707}^{\prime }=\text{d}\unicode[STIX]{x1D707}/\text{d}T$ and $\bar{\unicode[STIX]{x1D707}}$ is the nonlinear part of the viscosity perturbation, which is decomposed as

(2.22) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D707}}=\mathop{\sum }_{m,n}\hat{\unicode[STIX]{x1D707}}_{m,n}(\bar{x},\unicode[STIX]{x1D702})\text{e}^{\text{i}mk_{x}t+\text{i}nk_{z}z}.\end{eqnarray}$$

The nonlinearity of the viscosity comes from the power law exponent $\unicode[STIX]{x1D714}\neq 1$ and the turbulent Reynolds number $r_{t}=O(1)$ . The nonlinear part $\bar{\unicode[STIX]{x1D707}}$ is null when either (i)  $\unicode[STIX]{x1D714}=1\;\forall \,r_{t}$ or (ii)  $r_{t}=0\;\forall \unicode[STIX]{x1D714}$ in (2.21). As long as $|r_{t}\bar{\unicode[STIX]{x1D70F}}|<1$ , which applies to all the cases considered, the series in (2.21) is absolutely convergent.

By inserting (2.16) and (2.17) into the continuity, momentum and energy equations, using (2.20)–(2.22), and taking the limits $k_{x}\ll k_{z}$ , $R_{\unicode[STIX]{x1D706}}\gg 1$ with $k_{x}R_{\unicode[STIX]{x1D706}}=O(1)$ , the nonlinear unsteady compressible boundary-region equations are found as follows.

The continuity equation

(2.23) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x1D702}_{c}}{2\bar{x}}\frac{T^{\prime }}{T}\hat{u} _{m,n}+\frac{\unicode[STIX]{x2202}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\bar{x}}-\frac{\unicode[STIX]{x1D702}_{c}}{2\bar{x}}\frac{\unicode[STIX]{x2202}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}-\frac{T^{\prime }}{T^{2}}\hat{v}_{m,n}+\frac{1}{T}\frac{\unicode[STIX]{x2202}\hat{v}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}+\text{i}n{\hat{w}}_{m,n}-\left(\frac{\text{i}m}{T}+\frac{FT^{\prime }}{2\bar{x}T^{2}}\right)\hat{\unicode[STIX]{x1D70F}}_{m,n}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{F^{\prime }}{T}\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}_{m,n}}{\unicode[STIX]{x2202}\bar{x}}+\frac{F}{2\bar{x}T}\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}=r_{t}\hat{{\mathcal{C}}}_{m,n},\end{eqnarray}$$

the $x$ -momentum equation

(2.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(\text{i}m-\frac{\unicode[STIX]{x1D702}_{c}}{2\bar{x}}F^{\prime \prime }+\unicode[STIX]{x1D705}^{2}n^{2}T\unicode[STIX]{x1D707}\right)\hat{u} _{m,n}+F^{\prime }\frac{\unicode[STIX]{x2202}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\bar{x}}-\frac{1}{2\bar{x}}\left(F+\frac{\unicode[STIX]{x1D707}^{\prime }T^{\prime }}{T}-\frac{\unicode[STIX]{x1D707}T^{\prime }}{T^{2}}\right)\frac{\unicode[STIX]{x2202}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}-\frac{\unicode[STIX]{x1D707}}{2\bar{x}T}\frac{\unicode[STIX]{x2202}^{2}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{F^{\prime \prime }}{T}\hat{v}_{m,n}+\left(\frac{FF^{\prime \prime }-\unicode[STIX]{x1D707}^{\prime }F^{\prime \prime \prime }-\unicode[STIX]{x1D707}^{\prime \prime }F^{\prime \prime }T^{\prime }}{2\bar{x}T}+\frac{\unicode[STIX]{x1D707}^{\prime }T^{\prime }F^{\prime \prime }}{2\bar{x}T^{2}}\right)\hat{\unicode[STIX]{x1D70F}}_{m,n}-\frac{\unicode[STIX]{x1D707}^{\prime }F^{\prime \prime }}{2\bar{x}T}\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}=r_{t}\hat{{\mathcal{X}}}_{m,n},\end{eqnarray}$$

the $y$ -momentum equation

(2.25) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{1}{4\bar{x}^{2}}\left[FT+\unicode[STIX]{x1D702}_{c}(FT^{\prime }-TF^{\prime })-\unicode[STIX]{x1D702}_{c}^{2}F^{\prime \prime }T\right]\hat{u} _{m,n}+\frac{\unicode[STIX]{x1D707}^{\prime }T^{\prime }}{3\bar{x}}\frac{\unicode[STIX]{x2202}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\bar{x}}-\frac{\unicode[STIX]{x1D707}}{6\bar{x}}\frac{\unicode[STIX]{x2202}^{2}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}\unicode[STIX]{x2202}\bar{x}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{1}{12\bar{x}^{2}}\left(\unicode[STIX]{x1D707}+\unicode[STIX]{x1D702}_{c}T^{\prime }\unicode[STIX]{x1D707}^{\prime }-\frac{\unicode[STIX]{x1D707}T^{\prime }\unicode[STIX]{x1D702}_{c}}{T}\right)\frac{\unicode[STIX]{x2202}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}+\frac{\unicode[STIX]{x1D702}_{c}\unicode[STIX]{x1D707}}{12\bar{x}^{2}}\frac{\unicode[STIX]{x2202}^{2}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left(\text{i}m+\frac{\unicode[STIX]{x1D702}_{c}}{2\bar{x}}F^{\prime \prime }+\frac{F^{\prime }}{2\bar{x}}-\frac{FT^{\prime }}{2\bar{x}T}+\unicode[STIX]{x1D705}^{2}n^{2}\unicode[STIX]{x1D707}T\right)\hat{v}_{m,n}+F^{\prime }\frac{\unicode[STIX]{x2202}\hat{v}_{m,n}}{\unicode[STIX]{x2202}\bar{x}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{1}{\bar{x}}\left(\frac{F}{2}+\frac{2\unicode[STIX]{x1D707}^{\prime }T^{\prime }}{3T}-\frac{2\unicode[STIX]{x1D707}T^{\prime }}{3T^{2}}\right)\frac{\unicode[STIX]{x2202}\hat{v}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}-\frac{2\unicode[STIX]{x1D707}}{3\bar{x}T}\frac{\unicode[STIX]{x2202}^{2}\hat{v}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}+\text{i}n\frac{\unicode[STIX]{x1D707}^{\prime }T^{\prime }}{3\bar{x}}{\hat{w}}_{m,n}-\text{i}n\frac{\unicode[STIX]{x1D707}}{6\bar{x}}\frac{\unicode[STIX]{x2202}{\hat{w}}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{1}{4\bar{x}^{2}}\left[\unicode[STIX]{x1D702}_{c}\left((FF^{\prime })^{\prime }-T\left(\frac{\unicode[STIX]{x1D707}^{\prime }F^{\prime \prime }}{T}\right)^{\prime }\right)-FF^{\prime }-\frac{F^{2}T^{\prime }}{T}-\unicode[STIX]{x1D707}^{\prime }F^{\prime \prime }+\frac{4}{3}\left(\frac{\unicode[STIX]{x1D707}^{\prime }T^{\prime }F}{T}\right)^{\prime }\right]\hat{\unicode[STIX]{x1D70F}}_{m,n}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{\unicode[STIX]{x1D707}^{\prime }F^{\prime \prime }}{2\bar{x}}\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}_{m,n}}{\unicode[STIX]{x2202}\bar{x}}+\left(\frac{\unicode[STIX]{x1D707}^{\prime }T^{\prime }F}{3\bar{x}^{2}T}-\frac{\unicode[STIX]{x1D702}_{c}\unicode[STIX]{x1D707}^{\prime }F^{\prime \prime }}{4\bar{x}^{2}}\right)\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}+\frac{1}{2\bar{x}}\frac{\unicode[STIX]{x2202}\hat{p}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}=r_{t}\hat{{\mathcal{Y}}}_{m,n},\end{eqnarray}$$

the $z$ -momentum equation

(2.26) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{i}n\unicode[STIX]{x1D705}^{2}\frac{\unicode[STIX]{x1D702}_{c}\unicode[STIX]{x1D707}^{\prime }T^{\prime }T}{2\bar{x}}\hat{u} _{m,n}-\text{i}n\unicode[STIX]{x1D705}^{2}\frac{\unicode[STIX]{x1D707}T}{3}\frac{\unicode[STIX]{x2202}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\bar{x}}+\text{i}n\unicode[STIX]{x1D705}^{2}\frac{\unicode[STIX]{x1D702}_{c}\unicode[STIX]{x1D707}T}{6\bar{x}}\frac{\unicode[STIX]{x2202}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}-\text{i}n\unicode[STIX]{x1D705}^{2}\unicode[STIX]{x1D707}^{\prime }T^{\prime }\hat{v}_{m,m}-\text{i}n\unicode[STIX]{x1D705}^{2}\frac{\unicode[STIX]{x1D707}}{3}\frac{\unicode[STIX]{x2202}\hat{v}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left(\text{i}m+\frac{4n^{2}\unicode[STIX]{x1D705}^{2}\unicode[STIX]{x1D707}T}{3}\right){\hat{w}}_{m,n}+F^{\prime }\frac{\unicode[STIX]{x2202}{\hat{w}}_{m,n}}{\unicode[STIX]{x2202}\bar{x}}-\frac{1}{2\bar{x}}\left(F+\frac{\unicode[STIX]{x1D707}^{\prime }T^{\prime }}{T}-\frac{\unicode[STIX]{x1D707}T^{\prime }}{T^{2}}\right)\frac{\unicode[STIX]{x2202}{\hat{w}}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{\unicode[STIX]{x1D707}}{2\bar{x}T}\frac{\unicode[STIX]{x2202}^{2}{\hat{w}}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}-\frac{\text{i}n\unicode[STIX]{x1D705}^{2}FT^{\prime }\unicode[STIX]{x1D707}^{\prime }}{3\bar{x}}\hat{\unicode[STIX]{x1D70F}}_{m,n}+\text{i}n\unicode[STIX]{x1D705}^{2}T\hat{p}_{m,n}=r_{t}\hat{{\mathcal{Z}}}_{m,n},\end{eqnarray}$$

and the energy equation

(2.27) $$\begin{eqnarray}\displaystyle & & \displaystyle -\frac{\unicode[STIX]{x1D702}_{c}T^{\prime }}{2\bar{x}}\hat{u} _{m,n}-\frac{M_{\infty }^{2}(\unicode[STIX]{x1D6FE}-1)\unicode[STIX]{x1D707}F^{\prime \prime }}{\bar{x}T}\frac{\unicode[STIX]{x2202}\hat{u} _{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}+\frac{T^{\prime }}{T}\hat{v}_{m,n}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left[\text{i}m+\frac{FT^{\prime }}{2\bar{x}T}-\frac{1}{2Pr\bar{x}}\left(\frac{\unicode[STIX]{x1D707}^{\prime }T^{\prime }}{T}\right)^{\prime }-\frac{M_{\infty }^{2}(\unicode[STIX]{x1D6FE}-1)\unicode[STIX]{x1D707}^{\prime }(F^{\prime \prime })^{2}}{2\bar{x}T}+\frac{\unicode[STIX]{x1D707}n^{2}\unicode[STIX]{x1D705}^{2}T}{Pr}\right]\hat{\unicode[STIX]{x1D70F}}_{m,n}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,F^{\prime }\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}_{m,n}}{\unicode[STIX]{x2202}\bar{x}}-\frac{1}{2\bar{x}}\left(F+\frac{2\unicode[STIX]{x1D707}^{\prime }T^{\prime }}{PrT}-\frac{\unicode[STIX]{x1D707}T^{\prime }}{PrT^{2}}\right)\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}-\frac{\unicode[STIX]{x1D707}}{2Pr\bar{x}T}\frac{\unicode[STIX]{x2202}^{2}\hat{\unicode[STIX]{x1D70F}}_{m,n}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}=r_{t}\hat{{\mathcal{E}}}_{m,n},\qquad\end{eqnarray}$$

where the terms $\hat{{\mathcal{C}}}_{m,n}$ , $\hat{{\mathcal{X}}}_{m,n}$ , $\hat{{\mathcal{Y}}}_{m,n}$ , $\hat{{\mathcal{Z}}}_{m,n}$ , $\hat{{\mathcal{E}}}_{m,n}$ arise due to nonlinearity and are given in appendix A. The right-hand sides of (2.23)–(2.27), where the nonlinear terms are collected, vanish as $r_{t}\rightarrow 0$ so that the linearized compressible unsteady boundary-region equations of Ricco & Wu (Reference Ricco and Wu2007) are recovered. Note that the above fully nonlinear equations are to be used to predict the entire development of streaks even though the disturbance evolves through a linear stage near the leading edge.

2.3 The outer flow

The outer-flow dynamics is influenced by the displacement of the underlying boundary-layer flow. The displacement effect becomes of leading order at downstream distances where $\bar{x}=O(1)$ and the streamwise velocity fluctuations acquire an $O(1)$ amplitude. The disturbance in the outer region consists of the three-dimensional vortical perturbation convected from upstream and of the two-dimensional disturbance arising from the boundary-layer displacement effect due to the nonlinear interactions in the boundary layer. The two-dimensional component attenuates over a wall-normal distance $O(\unicode[STIX]{x1D706}^{\ast }R_{\unicode[STIX]{x1D706}})$ , and thus depends on the relatively slow wall-normal variable $\bar{y}=k_{x}y=O(1)$ . The three-dimensional component depends on $y$ and, a priori, also on $\bar{y}$ because its governing equations involve the two-dimensional component as we will discuss later. Following the approach of Wundrow & Goldstein (Reference Wundrow and Goldstein2001) and Ricco et al. (Reference Ricco, Luo and Wu2011), the outer-region solution is expanded as:

(2.28) $$\begin{eqnarray}\left\{u_{out},v_{out},w_{out},p_{out},\unicode[STIX]{x1D70F}_{out}\right\}=\boldsymbol{Q}+\unicode[STIX]{x1D716}\bar{\boldsymbol{q}}_{0}(\bar{x},\bar{y},\bar{t})+\unicode[STIX]{x1D716}\boldsymbol{q}_{0}(\bar{x},y,\bar{y},z,\bar{t})+\unicode[STIX]{x1D716}^{2}\boldsymbol{q}_{1}(\bar{x},y,\bar{y},z,\bar{t})+\cdots \,,\end{eqnarray}$$

where $\bar{t}\equiv k_{x}t=O(1)$ , $\boldsymbol{Q}$ is the uniform mean flow, and $\bar{\boldsymbol{q}_{0}}$ and $\boldsymbol{q}_{i}$ ( $i=0,1,\ldots$ ) indicate the two-dimensional and three-dimensional disturbances, respectively.

2.3.1 Linearized inviscid subsonic and supersonic flows

The two-dimensional inviscid part of the perturbation is considered first: the subsonic and supersonic regimes are analysed by extending the approach of Ricco et al. (Reference Ricco, Luo and Wu2011) to take compressibility into account. The transonic case is however beyond the scope of the present analysis. The two-dimensional terms $\{\bar{u}_{0},\bar{v}_{0},\bar{p}_{0},\bar{\unicode[STIX]{x1D70F}}_{0},\bar{\unicode[STIX]{x1D70C}}_{0}\}$ satisfy the linearized unsteady compressible Euler equations:

(2.29a-c ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\bar{u}_{0}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x2202}\bar{v}_{0}}{\unicode[STIX]{x2202}\bar{y}}+M_{\infty }^{2}\left(\frac{\unicode[STIX]{x2202}\bar{p}_{0}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}\bar{p}_{0}}{\unicode[STIX]{x2202}\bar{x}}\right)=0,\quad \frac{\unicode[STIX]{x2202}\bar{u}_{0}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}\bar{u}_{0}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x2202}\bar{p}_{0}}{\unicode[STIX]{x2202}\bar{x}}=0,\quad \frac{\unicode[STIX]{x2202}\bar{v}_{0}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}\bar{v}_{0}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x2202}\bar{p}_{0}}{\unicode[STIX]{x2202}\bar{y}}=0,\end{eqnarray}$$

where the density has been eliminated from the continuity equation by using the energy equation and the equation of state,

(2.30a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70F}}_{0}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70F}}_{0}}{\unicode[STIX]{x2202}\bar{x}}-(\unicode[STIX]{x1D6FE}-1)M_{\infty }^{2}\left(\frac{\unicode[STIX]{x2202}\bar{p}_{0}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}\bar{p}_{0}}{\unicode[STIX]{x2202}\bar{x}}\right)=0,\quad \bar{\unicode[STIX]{x1D70C}}_{0}=\unicode[STIX]{x1D6FE}M_{\infty }^{2}\bar{p}_{0}-\bar{\unicode[STIX]{x1D70F}}_{0}.\end{eqnarray}$$

The displacement effect is associated with the transpiration velocity, i.e. the spanwise-averaged wall-normal velocity component at the boundary-layer outer edge. The continuity equation,

(2.31) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{tot}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}_{tot}u_{tot})}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}_{tot}v_{tot})}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}_{tot}w_{tot})}{\unicode[STIX]{x2202}z}=0,\end{eqnarray}$$

is integrated first with respect to $z$ over a spanwise period $\unicode[STIX]{x1D706}_{z}=2\unicode[STIX]{x03C0}/k_{z}$ and with respect to $y$ from $0$ to $\infty$ . For the term $\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}_{tot}w_{tot})/\unicode[STIX]{x2202}z$ a change in the order of integration is performed. At supersonic speeds this step is justified by the assumption that shock waves, if present, are of infinitesimal strength and therefore the flow remains smooth and isentropic. It follows that

(2.32) $$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D706}_{z}}\int _{0}^{\unicode[STIX]{x1D706}_{z}}(\unicode[STIX]{x1D70C}_{tot}v_{tot})|_{y\rightarrow \infty }\,\text{d}z=-\frac{k_{x}}{\unicode[STIX]{x1D706}_{z}}\left[\int _{0}^{\unicode[STIX]{x1D706}_{z}}\int _{0}^{\infty }\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}_{tot}u_{tot})}{\unicode[STIX]{x2202}\bar{x}}\,\text{d}y\,\text{d}z+\int _{0}^{\unicode[STIX]{x1D706}_{z}}\int _{0}^{\infty }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{tot}}{\unicode[STIX]{x2202}\bar{t}}\,\text{d}y\,\text{d}z\right].\end{eqnarray}$$

The first term on the right-hand side of (2.32) represents the derivative with respect to $\bar{x}$ of the spanwise-averaged boundary-layer displacement thickness, defined as

(2.33) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D6FF}}(\bar{x},\bar{t})=\frac{1}{\unicode[STIX]{x1D706}_{z}}\int _{0}^{\unicode[STIX]{x1D706}_{z}}\int _{0}^{\infty }\left[1-(\unicode[STIX]{x1D70C}_{tot}u_{tot})\right]\,\text{d}y\,\text{d}z.\end{eqnarray}$$

The second term on the right-hand side of (2.32) is due to compressibility effects and is not present in Ricco et al. (Reference Ricco, Luo and Wu2011). It can be written as the derivative with respect to $\bar{t}$ of a spanwise-averaged boundary-layer thickness $\bar{\unicode[STIX]{x1D6FF}}^{c}$ defined as

(2.34) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D6FF}}^{c}(\bar{x},\bar{t})=\frac{1}{\unicode[STIX]{x1D706}_{z}}\int _{0}^{\unicode[STIX]{x1D706}_{z}}\int _{0}^{\infty }(1-\unicode[STIX]{x1D70C}_{tot})\,\text{d}y\,\text{d}z.\end{eqnarray}$$

Matching the left-hand side of (2.32) with the outer flow gives

(2.35) $$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D706}_{z}}\int _{0}^{\unicode[STIX]{x1D706}_{z}}(\unicode[STIX]{x1D70C}_{tot}v_{tot})|_{y\rightarrow \infty }\,\text{d}z=\frac{1}{\unicode[STIX]{x1D706}_{z}}\int _{0}^{\unicode[STIX]{x1D706}_{z}}(\unicode[STIX]{x1D70C}_{out}v_{out})|_{\bar{y}\rightarrow 0}\,\text{d}z=\unicode[STIX]{x1D716}\bar{v}_{0},\end{eqnarray}$$

where the terms $O(\unicode[STIX]{x1D716}^{2})$ have been neglected. Therefore, equation (2.32) leads to

(2.36) $$\begin{eqnarray}\bar{v}_{0}=\frac{k_{x}}{\unicode[STIX]{x1D716}}\left(\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D6FF}}^{c}}{\unicode[STIX]{x2202}\bar{t}}\right)\quad \text{as}~\bar{y}\rightarrow 0,\end{eqnarray}$$

where the compressibility effects appear in the definition of the displacement thickness $\bar{\unicode[STIX]{x1D6FF}}$ and in the additional term $\bar{\unicode[STIX]{x1D6FF}}^{c}$ . Equation (2.36) is used as a boundary condition on the system (2.29).

The proof employed by Ricco et al. (Reference Ricco, Luo and Wu2011) to show the irrotationality of the two-dimensional flow holds for the compressible case as long as the flow remains isentropic (i.e. shock waves are absent or, if present, their effect is negligible). Therefore, the potential $\bar{\unicode[STIX]{x1D719}}_{0}$ is introduced such that $\unicode[STIX]{x1D735}_{\bar{x}\bar{y}}\bar{\unicode[STIX]{x1D719}}_{0}=\{\bar{u}_{0},\bar{v}_{0}\}$ , where $\unicode[STIX]{x1D735}_{\bar{x}\bar{y}}$ denotes the gradient operator in the $\bar{x}{-}\bar{y}$ plane. By rewriting (2.29) in terms of $\bar{\unicode[STIX]{x1D719}}_{0}$ and eliminating the pressure from the continuity equation with the aid of the momentum equations, a single equation for the potential is derived,

(2.37) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\bar{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\bar{y}^{2}}+(1-M_{\infty }^{2})\frac{\unicode[STIX]{x2202}^{2}\bar{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\bar{x}^{2}}-2M_{\infty }^{2}\frac{\unicode[STIX]{x2202}^{2}\bar{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\bar{x}\unicode[STIX]{x2202}\bar{t}}-M_{\infty }^{2}\frac{\unicode[STIX]{x2202}^{2}\bar{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\bar{t}^{2}}=0,\quad M_{\infty }\neq 1,\end{eqnarray}$$

which is a wave (i.e. hyperbolic) equation, indicating the acoustic nature of the perturbation. It is interesting that streaks emit sound waves spontaneously during their nonlinear evolution. Equation (2.37) is of the same form as the linearized perturbation velocity potential equation governing the flow over a thin airfoil performing small unsteady (periodic) oscillations in the transverse direction (refer to Landahl Reference Landahl1989, equation (1.7)). The small thickness of the airfoil is represented here by the boundary-layer displacement thickness, the equivalent body being semi-infinite rather than finite and closed (i.e. with null thickness at both ends of the body).

The problem of the flow over a thin oscillating airfoil has been widely studied in aeroelasticity because of the loads and vibrations occurring on the wing. Thanks to the linearization, two separated cases are distinguished: the thickness problem (symmetrical) and the lifting problem (anti-symmetrical). Aeroelasticians are usually interested in the latter because of the contribution to the lift experienced by the wing (Dowell Reference Dowell2014). Here it suffices to state that the analogy between the boundary-layer displacement effect and the thin airfoil theory only concerns the thickness problem, as circulatory flow is absent. This analogy was first suggested by Lighthill (Reference Lighthill1958) for incompressible flows and its use in the study of steady flows outside boundary layers has been well established (Van Dyke Reference Van Dyke1975). However, to the best of our knowledge, this analogy has never been considered for unsteady boundary layers.

By employing Fourier decomposition in time,

(2.38) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D719}}_{0}(\bar{x},\bar{y},\bar{t})=\mathop{\sum }_{m}\hat{\unicode[STIX]{x1D719}}_{m}(\bar{x},\bar{y})\text{e}^{\text{i}m\bar{t}},\end{eqnarray}$$

equation (2.37) is recast into a generalized Helmholtz differential equation (i.e. telegraph equation) for the Fourier coefficients $\hat{\unicode[STIX]{x1D719}}_{m}(\bar{x},\bar{y})$ :

(2.39) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\hat{\unicode[STIX]{x1D719}}_{m}}{\unicode[STIX]{x2202}\bar{y}^{2}}+(1-M_{\infty }^{2})\frac{\unicode[STIX]{x2202}^{2}\hat{\unicode[STIX]{x1D719}}_{m}}{\unicode[STIX]{x2202}\bar{x}^{2}}-2\text{i}mM_{\infty }^{2}\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D719}}_{m}}{\unicode[STIX]{x2202}\bar{x}}+m^{2}M_{\infty }^{2}\hat{\unicode[STIX]{x1D719}}_{m}=0.\end{eqnarray}$$

Performing the change $\hat{\unicode[STIX]{x1D719}}_{m}\rightarrow \hat{\unicode[STIX]{x1D6F7}}_{m}$ , where

(2.40) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D719}}_{m}(\bar{x},\bar{y})=\hat{\unicode[STIX]{x1D6F7}}_{m}(\bar{x},\bar{y})\text{e}^{-b\bar{x}/2}\quad \text{with}~b\equiv \frac{2\text{i}mM_{\infty }^{2}}{M_{\infty }^{2}-1},\quad M_{\infty }\neq 1,\end{eqnarray}$$

the telegraph equation (2.39) is reduced to either the Helmholtz equation if $M_{\infty }<1$ or the Klein–Gordon equation if $M_{\infty }>1$ :

(2.41) $$\begin{eqnarray}|1-M_{\infty }^{2}|\frac{\unicode[STIX]{x2202}^{2}\hat{\unicode[STIX]{x1D6F7}}_{m}}{\unicode[STIX]{x2202}\bar{x}^{2}}\pm \frac{\unicode[STIX]{x2202}^{2}\hat{\unicode[STIX]{x1D6F7}}_{m}}{\unicode[STIX]{x2202}\bar{y}^{2}}+\frac{m^{2}M_{\infty }^{2}}{|1-M_{\infty }^{2}|}\hat{\unicode[STIX]{x1D6F7}}_{m}=0,\end{eqnarray}$$

where the $+$ or $-$ sign correspond to the subsonic or supersonic case, respectively. The appropriate boundary conditions for (2.41) are:

(2.42a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F7}}_{m}}{\unicode[STIX]{x2202}\bar{y}}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{k_{x}}{\unicode[STIX]{x1D716}}(\hat{\unicode[STIX]{x1D6FF}}_{m}^{\prime }+\text{i}m\hat{\unicode[STIX]{x1D6FF}}_{m}^{c})\text{e}^{b\bar{x}/2},\quad & \bar{y}=0,\bar{x}\geqslant 0,\\ 0,\quad & \bar{y}=0,\bar{x}<0;\end{array}\right. & \displaystyle\end{eqnarray}$$

(2.42b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F7}}_{m}}{\unicode[STIX]{x2202}\bar{x}},\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F7}}_{m}}{\unicode[STIX]{x2202}\bar{y}}~\text{finite,}\quad \bar{x}^{2}+\bar{y}^{2}\rightarrow \infty , & \displaystyle\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D6FF}}_{m}(\bar{x})$ and $\hat{\unicode[STIX]{x1D6FF}}_{m}^{c}(\bar{x})$ are the Fourier coefficients of $\bar{\unicode[STIX]{x1D6FF}}(\bar{x},\bar{t})$ , and $\bar{\unicode[STIX]{x1D6FF}}^{c}(\bar{x},\bar{t})$ and the prime represents differentiation with respect to $\bar{x}$ . Equation (2.42a ) corresponds to the tangency (i.e. no penetration) condition imposed on a thin airfoil, according to which the velocity component normal to the body is fixed by the airfoil motion. This analogy has given rise to the well-known interpretation of the boundary-layer displacement effect as a surface distribution of sources (Lighthill Reference Lighthill1958). The far-field boundary condition (2.42b ) requires that the displacement-induced velocity field remains finite as the distance from the plate increases.

In the supersonic case it is convenient to solve for $\hat{\unicode[STIX]{x1D719}}_{m}$ directly. The fluid ahead of the body remains undisturbed and so the Laplace transform in the $\bar{x}$ direction can be employed,

(2.43) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D719}}_{m}(s,\bar{y})=\int _{0}^{\infty }\hat{\unicode[STIX]{x1D719}}_{m}(\bar{x},\bar{y})\text{e}^{-s\bar{x}}\,\text{d}\bar{x}.\end{eqnarray}$$

The telegraph equation (2.39) and the boundary condition (2.42a ) for $\bar{x}\geqslant 0$ become:

(2.44a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\tilde{\unicode[STIX]{x1D719}}_{m}}{\unicode[STIX]{x2202}\bar{y}^{2}}=c^{2}\tilde{\unicode[STIX]{x1D719}}_{m},\quad \frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D719}}_{m}}{\unicode[STIX]{x2202}\bar{y}}(s,0)=\tilde{v}_{m}(s,0),\end{eqnarray}$$

where $c^{2}=s^{2}(M_{\infty }^{2}-1)+mM_{\infty }^{2}(2\text{i}s-m)$ and $\tilde{v}_{m}(s,0)$ indicates the Laplace transform of $\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D719}}_{m}/\unicode[STIX]{x2202}\bar{y}$ at $\bar{y}=0$ . The solution to (2.44) is

(2.45) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D719}}_{m}(s,\bar{y})=-\frac{\tilde{v}_{m}(s,0)}{c}\text{e}^{-c\bar{y}},\quad \bar{y}\geqslant 0.\end{eqnarray}$$

Inverting (2.45) and using the convolution theorem, one finds

(2.46) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D719}}_{m}(\bar{x},\bar{y};M_{\infty }>1) & = & \displaystyle -\frac{k_{x}}{\unicode[STIX]{x1D716}\sqrt{M_{\infty }^{2}-1}}\int _{0}^{\bar{x}-\bar{y}\sqrt{M_{\infty }^{2}-1}}\!\left[\hat{\unicode[STIX]{x1D6FF}}_{m}^{\prime }(\bar{x}_{0})+\text{i}m\hat{\unicode[STIX]{x1D6FF}}_{m}^{c}(\bar{x}_{0})\right]\text{e}^{-(\text{i}mM_{\infty }^{2}/(M_{\infty }^{2}-1))(\bar{x}-\bar{x}_{0})}\nonumber\\ \displaystyle & & \displaystyle \times \,J_{0}\left[\frac{mM_{\infty }}{M_{\infty }^{2}-1}\sqrt{\left(\bar{x}-\bar{x}_{0}\right)^{2}-\left(M_{\infty }^{2}-1\right)\bar{y}^{2}}\right]\,\text{d}\bar{x}_{0},\end{eqnarray}$$

where $J_{0}$ is the zeroth-order Bessel function of the first kind (Abramowitz & Stegun Reference Abramowitz and Stegun1964). The procedure to derive (2.46) closely follows the theory proposed by Stewartson (Reference Stewartson1950) for harmonically oscillating thin airfoils in supersonic flows.

In the subsonic case, the disturbance is felt in all directions and so the use of the Laplace transform in $\bar{x}$ is not appropriate. The Fourier transform is instead employed and the analytic continuation of the Neumann boundary condition (2.42a ) to the complex plane $\bar{\unicode[STIX]{x1D709}}=\bar{x}+\text{i}\bar{y}$ is considered. The solution reads

(2.47) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D719}}_{m}(\bar{x},\bar{y};M_{\infty }<1) & = & \displaystyle -\frac{\text{i}k_{x}}{2\unicode[STIX]{x1D716}\sqrt{1-M_{\infty }^{2}}}\int _{\bar{x}-\text{i}\bar{y}\sqrt{1-M_{\infty }^{2}}}^{\bar{x}+\text{i}\bar{y}\sqrt{1-M_{\infty }^{2}}}\left[\hat{\unicode[STIX]{x1D6FF}}_{m}^{\prime }(\bar{\unicode[STIX]{x1D709}})+\text{i}m\hat{\unicode[STIX]{x1D6FF}}_{m}^{c}(\bar{\unicode[STIX]{x1D709}})\right]\text{e}^{(\text{i}mM_{\infty }^{2}/(1-M_{\infty }^{2}))(\bar{x}-\bar{\unicode[STIX]{x1D709}})}\nonumber\\ \displaystyle & & \displaystyle \times \,J_{0}\left[\frac{mM_{\infty }}{1-M_{\infty }^{2}}\sqrt{\left(\bar{x}-\bar{\unicode[STIX]{x1D709}}\right)^{2}+\left(1-M_{\infty }^{2}\right)\bar{y}^{2}}\right]\,\text{d}\bar{\unicode[STIX]{x1D709}}.\end{eqnarray}$$

The solution can also be expressed in terms of the Green’s function by modelling the boundary-layer thickness as a distribution of pulsating sources as suggested by Lighthill’s theory. The free-space Green’s function associated with the Helmholtz equation (2.41) is given by Dragos (Reference Dragos2004) on p. 33 for a single source located in the origin. Considering a line distribution of these sources at the point $(\bar{x}_{0},0)$ , where $\bar{x}_{0}$ spans the $\bar{x}$ axis, and using the method of images to include the boundary $\bar{y}=0$ , we obtain

(2.48) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D719}}_{m}(\bar{x},\bar{y};M_{\infty }<1) & = & \displaystyle \frac{\text{i}k_{x}}{2\unicode[STIX]{x1D716}\sqrt{1-M_{\infty }^{2}}}\int _{0}^{\infty }\left[\hat{\unicode[STIX]{x1D6FF}}_{m}^{\prime }(\bar{x}_{0})+\text{i}m\hat{\unicode[STIX]{x1D6FF}}_{m}^{c}(\bar{x}_{0})\right]\text{e}^{(\text{i}mM_{\infty }^{2}/(1-M_{\infty }^{2}))\left(\bar{x}-\bar{x}_{0}\right)}\nonumber\\ \displaystyle & & \displaystyle \times \,H_{0}^{(2)}\left[\frac{mM_{\infty }}{1-M_{\infty }^{2}}\sqrt{\left(\bar{x}-\bar{x}_{0}\right)^{2}+\left(1-M_{\infty }^{2}\right)\bar{y}^{2}}\right]\,\text{d}\bar{x}_{0},\end{eqnarray}$$

where $H_{0}^{(2)}$ is the zeroth-order Hankel function of the second kind (Abramowitz & Stegun Reference Abramowitz and Stegun1964). The solution with $H_{0}^{(2)}$ has been chosen instead of that with $H_{0}^{(1)}$ to ensure outgoing waves radiating from the source. As $M_{\infty }\rightarrow 0$ , equation (2.47) matches the solution obtained in the incompressible case (Ricco et al. Reference Ricco, Luo and Wu2011):

(2.49) $$\begin{eqnarray}\displaystyle \lim _{M_{\infty }\rightarrow 0}\hat{\unicode[STIX]{x1D719}}_{m}(\bar{x},\bar{y};M_{\infty }<1) & = & \displaystyle -\frac{\text{i}k_{x}}{2\unicode[STIX]{x1D716}}\int _{\bar{x}-\text{i}\bar{y}}^{\bar{x}+\text{i}\bar{y}}\hat{\unicode[STIX]{x1D6FF}}_{m}^{\prime }(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}=-\frac{\text{i}k_{x}}{2\unicode[STIX]{x1D716}}[\hat{\unicode[STIX]{x1D6FF}}_{m}(\bar{x}+\text{i}\bar{y})-\hat{\unicode[STIX]{x1D6FF}}_{m}(\bar{x}-\text{i}\bar{y})]\nonumber\\ \displaystyle & = & \displaystyle \frac{k_{x}}{\unicode[STIX]{x1D716}}\operatorname{Im}[\hat{\unicode[STIX]{x1D6FF}}_{m}(\bar{x}+\text{i}\bar{y})],\end{eqnarray}$$

where $\operatorname{Im}$ indicates the imaginary part and use has been made of the property of holomorphic functions, $\hat{\unicode[STIX]{x1D6FF}}_{m}(\bar{\unicode[STIX]{x1D709}}^{\star })=$ $\hat{\unicode[STIX]{x1D6FF}}_{m}(\bar{\unicode[STIX]{x1D709}})^{\star }$ . The limit $M_{\infty }\rightarrow \infty$ is not considered because for very large hypersonic Mach numbers the Blasius boundary-layer assumption of negligible wall-normal pressure gradient is invalid (refer to Anderson Reference Anderson2006, p. 275).

The pressure is obtained from the $y$ -momentum equation (2.29) as

(2.50) $$\begin{eqnarray}\bar{p}_{0}=-\!\left(\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\bar{x}}\right),\end{eqnarray}$$

or, in Fourier space,

(2.51) $$\begin{eqnarray}\hat{p}_{m}=-\!\left(\text{i}m\hat{\unicode[STIX]{x1D719}}_{m}+\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D719}}_{m}}{\unicode[STIX]{x2202}\bar{x}}\right).\end{eqnarray}$$

From substitution of the supersonic and subsonic expressions for the potential, equations (2.46) and (2.48), into (2.51), it follows that

(2.52) $$\begin{eqnarray}\displaystyle & & \displaystyle \hat{p}_{m}(\bar{x},\bar{y};M_{\infty }>1)=\frac{k_{x}}{\unicode[STIX]{x1D716}\sqrt{M_{\infty }^{2}-1}}\left\{\int _{0}^{\bar{x}-\bar{y}\sqrt{M_{\infty }^{2}-1}}f(\bar{x_{0}},\bar{x},\bar{y})\,\text{d}\bar{x}_{0}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \left.\vphantom{\int _{0}^{\bar{x}-\bar{y}\sqrt{M_{\infty }^{2}-1}}}-\,\left[\hat{\unicode[STIX]{x1D6FF}}_{m}^{\prime }(\bar{x}-\bar{y}\sqrt{M_{\infty }^{2}-1})+\text{i}m\hat{\unicode[STIX]{x1D6FF}}_{m}^{c}(\bar{x}-\bar{y}\sqrt{M_{\infty }^{2}-1})\right]\text{e}^{-(\text{i}mM_{\infty }^{2}/\sqrt{M_{\infty }^{2}-1})\bar{y}}\right\}\qquad\end{eqnarray}$$

in the supersonic case, where

(2.53) $$\begin{eqnarray}\displaystyle f & = & \displaystyle \left[\hat{\unicode[STIX]{x1D6FF}}_{m}^{\prime }(\bar{x}_{0})+\text{i}m\hat{\unicode[STIX]{x1D6FF}}_{m}^{c}(\bar{x}_{0})\right]\text{e}^{-(\text{i}mM_{\infty }^{2}/(M_{\infty }^{2}-1))\left(\bar{x}-\bar{x}_{0}\right)}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{\frac{\text{i}m}{M_{\infty }^{2}-1}J_{0}\left[\frac{mM_{\infty }}{M_{\infty }^{2}-1}\sqrt{\left(\bar{x}-\bar{x}_{0}\right)^{2}-\left(M_{\infty }^{2}-1\right)\bar{y}^{2}}\right]\vphantom{\frac{mM_{\infty }}{\sqrt{(\bar{x}-\bar{x}_{0})^{2}-(M_{\infty }^{2}-1)\bar{y}^{2}}}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\frac{mM_{\infty }(\bar{x}-\bar{x}_{0})}{(M_{\infty }^{2}-1)\sqrt{(\bar{x}-\bar{x}_{0})^{2}-(M_{\infty }^{2}-1)\bar{y}^{2}}}J_{1}\left[\frac{mM_{\infty }}{M_{\infty }^{2}-1}\sqrt{\left(\bar{x}-\bar{x}_{0}\right)^{2}-\left(M_{\infty }^{2}-1\right)\bar{y}^{2}}\right]\right\}\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and $J_{1}$ is the first-order Bessel function of the first kind (Abramowitz & Stegun Reference Abramowitz and Stegun1964), and in the subsonic case

(2.54) $$\begin{eqnarray}\displaystyle \hat{p}_{m}(\bar{x},\bar{y};M_{\infty }<1) & = & \displaystyle \frac{\text{i}k_{x}}{2\unicode[STIX]{x1D716}\sqrt{1-M_{\infty }^{2}}}\int _{0}^{\infty }g(\bar{x_{0}},\bar{x},\bar{y})\,\text{d}\bar{x}_{0},\end{eqnarray}$$

where

(2.55) $$\begin{eqnarray}\displaystyle g & = & \displaystyle \left[\hat{\unicode[STIX]{x1D6FF}}_{m}^{\prime }(\bar{x}_{0})+\text{i}m\hat{\unicode[STIX]{x1D6FF}}_{m}^{c}(\bar{x}_{0})\right]\text{e}^{(\text{i}mM_{\infty }^{2}/(1-M_{\infty }^{2}))(\bar{x}-\bar{x}_{0})}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{\frac{\text{i}m}{1-M_{\infty }^{2}}H_{0}^{(2)}\left[\frac{mM_{\infty }}{1-M_{\infty }^{2}}\sqrt{\left(\bar{x}-\bar{x}_{0}\right)^{2}+\left(1-M_{\infty }^{2}\right)\bar{y}^{2}}\right]\right.\nonumber\\ \displaystyle & & \displaystyle -\,\frac{mM_{\infty }\left(\bar{x}-\bar{x}_{0}\right)}{(1-M_{\infty }^{2})\sqrt{(\bar{x}-\bar{x}_{0})^{2}+(1-M_{\infty }^{2})\bar{y}^{2}}}\nonumber\\ \displaystyle & & \displaystyle \left.\times \,H_{1}^{(2)}\left[\frac{mM_{\infty }}{1-M_{\infty }^{2}}\sqrt{(\bar{x}-\bar{x}_{0})^{2}+(1-M_{\infty }^{2})\bar{y}^{2}}\right]\right\},\end{eqnarray}$$

and $H_{1}^{(2)}$ is the first-order Hankel function of the second kind (Abramowitz & Stegun Reference Abramowitz and Stegun1964). The temperature is found from the energy equation (2.30) as

(2.56) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D70F}}_{0}=(\unicode[STIX]{x1D6FE}-1)M_{\infty }^{2}\bar{p}_{0}.\end{eqnarray}$$

2.3.2 Viscous three-dimensional flow

By substituting the expansion (2.28) into the Navier–Stokes equations and subtracting the equations (2.29) and (2.30) for the displacement-induced disturbance, the leading-order three-dimensional part of the perturbation is found to satisfy the equations:

(2.57) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}v_{0}}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}z}=0,\end{eqnarray}$$

(2.58) $$\begin{eqnarray}\left\{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x1D716}}{k_{x}}\left[(\bar{v}_{0}+v_{0})\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}+w_{0}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right]\right\}\left\{\begin{array}{@{}c@{}}u_{0}\\ v_{0}\\ w_{0}\\ \unicode[STIX]{x1D70F}_{0}\end{array}\right\}=-\frac{\unicode[STIX]{x1D716}}{k_{x}}\left\{\begin{array}{@{}c@{}}0\\ \unicode[STIX]{x2202}p_{1}/\unicode[STIX]{x2202}y\\ \unicode[STIX]{x2202}p_{1}/\unicode[STIX]{x2202}z\\ 0\end{array}\right\}+\unicode[STIX]{x1D705}^{2}\unicode[STIX]{x1D6FB}_{yz}^{2}\left\{\begin{array}{@{}c@{}}u_{0}\\ v_{0}\\ w_{0}\\ Pr^{-1}\unicode[STIX]{x1D70F}_{0}\end{array}\right\},\end{eqnarray}$$

where $p_{0}=0$ without loss of generality and $\unicode[STIX]{x1D735}_{yz}$ is the gradient operator in the $y{-}z$ plane.

The three-dimensional part of the perturbation in (2.57) and (2.58) does not directly depend on the slow variable $\bar{y}$ at the leading order, but the momentum equations (2.58) include the inviscid wall-normal velocity component $\bar{v}_{0}$ , which depends on $\bar{y}$ . Therefore, the dependence of the leading-order three-dimensional disturbance upon $\bar{y}$ is parametric. The aim of the following analysis is to decouple (2.58) from $\bar{v}_{0}$ to obtain a system of equations in the only unknowns $\left\{u_{0},v_{0},w_{0},\unicode[STIX]{x1D70F}_{0},p_{1}\right\}$ . This is accomplished by introducing the Prandtl transformation, ${\hat{y}}=y-\operatorname{Re}[\unicode[STIX]{x1D6E5}(\bar{\unicode[STIX]{x1D709}},\bar{t})]$ , where $\operatorname{Re}$ denotes the real part and $\unicode[STIX]{x1D6E5}$ is suitably chosen to eliminate the coupling between $\left\{u_{0},v_{0},w_{0},\unicode[STIX]{x1D70F}_{0},p_{1}\right\}$ and $\bar{v}_{0}$ from (2.58). The new variable ${\hat{y}}$ is written in this form for consistency with the previous analyses of Wundrow & Goldstein (Reference Wundrow and Goldstein2001) and Ricco et al. (Reference Ricco, Luo and Wu2011), although it will turn out that only the dependence of $\unicode[STIX]{x1D6E5}$ on the real variables $\bar{x}$ and $\bar{t}$ is relevant since we are interested in the matching at $\bar{y}=0$ . Written in terms of ${\hat{y}}$ , equations (2.57) and (2.58) become

(2.59) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}v_{0}}{\unicode[STIX]{x2202}{\hat{y}}}+\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}z}=0, & \displaystyle\end{eqnarray}$$

(2.60) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}\left\{\begin{array}{@{}c@{}}u_{0}\\ v_{0}\\ w_{0}\\ \unicode[STIX]{x1D70F}_{0}\end{array}\right\}=-\frac{\unicode[STIX]{x1D716}}{k_{x}}\left\{\begin{array}{@{}c@{}}0\\ \unicode[STIX]{x2202}p_{1}/\unicode[STIX]{x2202}{\hat{y}}\\ \unicode[STIX]{x2202}p_{1}/\unicode[STIX]{x2202}z\\ 0\end{array}\right\}+\unicode[STIX]{x1D705}^{2}\unicode[STIX]{x1D6FB}_{{\hat{y}}z}^{2}\left\{\begin{array}{@{}c@{}}u_{0}\\ v_{0}\\ w_{0}\\ Pr^{-1}\unicode[STIX]{x1D70F}_{0}\end{array}\right\}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D735}_{{\hat{y}}z}$ is the gradient operator in the ${\hat{y}}{-}z$ plane and ${\mathcal{N}}$ is the differential operator:

(2.61) $$\begin{eqnarray}{\mathcal{N}}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x1D716}}{k_{x}}\left(v_{0}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}{\hat{y}}}+w_{0}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right)-\left(\operatorname{Re}\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}}{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D709}}}\right]-\frac{\unicode[STIX]{x1D716}}{k_{x}}\bar{v}_{0}\right)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}{\hat{y}}}.\end{eqnarray}$$

The dependence on $\bar{v}_{0}$ is removed by choosing $\unicode[STIX]{x1D6E5}$ to satisfy (Ricco et al. Reference Ricco, Luo and Wu2011)

(2.62) $$\begin{eqnarray}\operatorname{Re}\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}}{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D709}}}\right]=\frac{\unicode[STIX]{x1D716}}{k_{x}}\bar{v}_{0},\end{eqnarray}$$

which, in view of (2.36), becomes

(2.63) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}}{\unicode[STIX]{x2202}\bar{x}}=\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D6FF}}^{c}}{\unicode[STIX]{x2202}\bar{t}}\quad \text{at}~\bar{y}=0.\end{eqnarray}$$

Decomposing $\unicode[STIX]{x1D6E5}$ into Fourier series as

(2.64) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}\left(\bar{x},\bar{t}\right)=\mathop{\sum }_{m}\hat{\unicode[STIX]{x1D6E5}}_{m}(\bar{x})\text{e}^{-\text{i}m(\bar{x}-\bar{t})}\quad \text{at}~\bar{y}=0,\end{eqnarray}$$

and using the time Fourier series of $\bar{\unicode[STIX]{x1D6FF}}$ and $\bar{\unicode[STIX]{x1D6FF}}^{c}$ , we obtain the expression for $\hat{\unicode[STIX]{x1D6E5}}_{m}$ ,

(2.65) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6E5}}_{m}(\bar{x})=\hat{\unicode[STIX]{x1D6FF}}_{m}(\bar{x})\text{e}^{\text{i}m\bar{x}}+\text{i}m\int _{0}^{\bar{x}}\left[\hat{\unicode[STIX]{x1D6FF}}_{m}(\check{x})-\hat{\unicode[STIX]{x1D6FF}}_{m}^{c}(\check{x})\right]\text{e}^{\text{i}m\check{x}}\,\text{d}\check{x},\quad \text{at}~\bar{y}=0,\end{eqnarray}$$

where the condition $\unicode[STIX]{x1D6E5}(0,\bar{t})=0$ , $\forall \bar{t}>0$ has been employed.

Equation (2.60) shows that the streamwise velocity $u_{0}$ and the temperature $\unicode[STIX]{x1D70F}_{0}$ are decoupled from the transverse velocities $\{v_{0},w_{0}\}$ and from the pressure $p_{1}$ . At leading order, only $\{v_{0},w_{0},p_{1}\}$ enter the matching with the boundary-layer solution, while the matching of $\{u_{0},\unicode[STIX]{x1D70F}_{0}\}$ needs to be considered when a solution with an order of accuracy higher than $O(R_{\unicode[STIX]{x1D706}}^{-1})$ is sought.

In view of the continuity equation (2.59), the streamfunction $\unicode[STIX]{x1D713}_{0}(\bar{x},{\hat{y}},z,\bar{t})$ is introduced such that $\unicode[STIX]{x1D735}_{{\hat{y}}z}\unicode[STIX]{x1D713}_{0}=\{-w_{0},v_{0}\}$ . In terms of the streamfunction the transverse momentum equations are recast into a transport equation for the longitudinal vorticity $\unicode[STIX]{x1D6FB}_{{\hat{y}}z}^{2}\unicode[STIX]{x1D713}_{0}$ ,

(2.66) $$\begin{eqnarray}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{t}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x1D716}}{k_{x}}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{0}}{\unicode[STIX]{x2202}z}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}{\hat{y}}}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{0}}{\unicode[STIX]{x2202}{\hat{y}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\hat{z}}\right)\right]\unicode[STIX]{x1D6FB}_{{\hat{y}}z}^{2}\unicode[STIX]{x1D713}_{0}=\unicode[STIX]{x1D705}^{2}\unicode[STIX]{x1D6FB}_{{\hat{y}}z}^{4}\unicode[STIX]{x1D713}_{0},\end{eqnarray}$$

along with the Poisson equation for the pressure,

(2.67) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{{\hat{y}}z}^{2}p_{1}=2\left[\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}_{0}}{\unicode[STIX]{x2202}{\hat{y}}^{2}}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}_{0}}{\unicode[STIX]{x2202}z^{2}}-\left(\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}_{0}}{\unicode[STIX]{x2202}{\hat{y}}\unicode[STIX]{x2202}z}\right)^{2}\right].\end{eqnarray}$$

In the general case of a full spectrum of free-stream vortical disturbances, the streamfunction is decomposed as:

(2.68) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{0}\left(\bar{x},{\hat{y}},z,\bar{t}\right)=\mathop{\sum }_{m,n,j}\hat{\unicode[STIX]{x1D713}}_{m,n}^{(j)}(\bar{x})\text{e}^{\text{i}(m\bar{t}+jk_{y}{\hat{y}}+nk_{z}z)}.\end{eqnarray}$$

By substituting (2.68) into (2.66) and expanding the resulting equation for a pair of oblique forcing modes, the nonlinear terms cancel out and (2.66) reduces to

(2.69) $$\begin{eqnarray}\left(\text{i}m+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{x}}\right)\hat{\unicode[STIX]{x1D713}}_{m,n}^{(j)}=-\unicode[STIX]{x1D705}^{2}\left(j^{2}k_{y}^{2}+n^{2}k_{z}^{2}\right)\hat{\unicode[STIX]{x1D713}}_{m,n}^{(j)},\end{eqnarray}$$

with $\{m,j,n\}=\{1,-1,\pm 1\}$ . The solution is:

(2.70) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D713}}_{1,\pm 1}^{(-1)}=\mp \text{i}c_{\infty }\text{e}^{-\text{i}\bar{x}-\unicode[STIX]{x1D705}^{2}(k_{y}^{2}+k_{z}^{2})\bar{x}}+\text{c.c.},\end{eqnarray}$$

where $c_{\infty }=1/k_{y}$ is chosen to normalize the amplitude of the free-stream spanwise velocity to unity. From (2.68) and (2.70) it follows that

(2.71) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{0}\left(\bar{x},{\hat{y}},z,\bar{t}\right)=-2\text{i}c_{\infty }\cos (\bar{x}-\bar{t}+k_{y}{\hat{y}})\text{e}^{-\unicode[STIX]{x1D705}^{2}(k_{y}^{2}+k_{z}^{2})\bar{x}+\text{i}k_{z}z}+\text{c.c.}.\end{eqnarray}$$

The transverse velocities are obtained as

(2.72) $$\begin{eqnarray}\left\{v_{0},w_{0}\right\}=2c_{\infty }\left\{k_{z}\cos (\bar{x}-\bar{t}+k_{y}{\hat{y}}),-\text{i}k_{y}\sin (\bar{x}-\bar{t}+k_{y}{\hat{y}})\right\}\text{e}^{-\unicode[STIX]{x1D705}^{2}(k_{y}^{2}+k_{z}^{2})\bar{x}+\text{i}k_{z}z}+\text{c.c.}.\end{eqnarray}$$

By inserting (2.71) into (2.67), the pressure is found to be

(2.73) $$\begin{eqnarray}p_{1}=2k_{y}^{2}c_{\infty }^{2}\text{e}^{-2\unicode[STIX]{x1D705}^{2}(k_{y}^{2}+k_{z}^{2})\bar{x}+2\text{i}k_{z}z}-2k_{z}^{2}c_{\infty }^{2}\text{e}^{-2\unicode[STIX]{x1D705}^{2}(k_{y}^{2}+k_{z}^{2})\bar{x}}\cos (2\bar{x}-2\bar{t}+2k_{y}{\hat{y}})+\text{c.c.}.\end{eqnarray}$$

At $\bar{y}=0$ , the expression ${\hat{y}}=y-\unicode[STIX]{x1D6E5}(\bar{x},\bar{t})$ is substituted into (2.71)–(2.73), and the time-dependent terms are expanded into Fourier series,

(2.74a,b ) $$\begin{eqnarray}\text{e}^{\text{i}\bar{t}+\text{i}k_{y}\unicode[STIX]{x1D6E5}(\bar{x},\bar{t})}=\mathop{\sum }_{m}\hat{\unicode[STIX]{x1D712}}_{m}(\bar{x})\text{e}^{\text{i}m\bar{t}},\quad \text{e}^{2\text{i}\bar{t}+2\text{i}k_{y}\unicode[STIX]{x1D6E5}(\bar{x},\bar{t})}=\mathop{\sum }_{m}\hat{\unicode[STIX]{x03C0}}_{m}(\bar{x})\text{e}^{\text{i}m\bar{t}}.\end{eqnarray}$$

Rewriting (2.72) and (2.73) with the aid of (2.74), the outer-flow solution for $y=O(1)$ is obtained as

(2.75) $$\begin{eqnarray}\left\{v_{0},w_{0},p_{1}\right\}=\mathop{\sum }_{m,n}\left\{v_{m,n}^{\dagger },w_{m,n}^{\dagger },p_{m,n}^{\dagger }\right\}\text{e}^{\text{i}m\bar{t}+\text{i}nk_{z}z},\end{eqnarray}$$

with

(2.76a ) $$\begin{eqnarray}\displaystyle & \displaystyle v_{m,\pm 1}^{\dagger }=k_{z}c_{\infty }\text{e}^{-\unicode[STIX]{x1D705}^{2}(k_{y}^{2}+k_{z}^{2})\bar{x}}\left[\hat{\unicode[STIX]{x1D712}}_{m}\text{e}^{-\text{i}(\bar{x}+k_{y}y)}+\hat{\unicode[STIX]{x1D712}}_{-m}^{\star }\text{e}^{\text{i}(\bar{x}+k_{y}y)}\right], & \displaystyle\end{eqnarray}$$

(2.76b ) $$\begin{eqnarray}\displaystyle & \displaystyle w_{m,\pm 1}^{\dagger }=\pm k_{y}c_{\infty }\text{e}^{-\unicode[STIX]{x1D705}^{2}(k_{y}^{2}+k_{z}^{2})\bar{x}}\left[\hat{\unicode[STIX]{x1D712}}_{m}\text{e}^{-\text{i}(\bar{x}+k_{y}y)}-\hat{\unicode[STIX]{x1D712}}_{-m}^{\star }\text{e}^{\text{i}(\bar{x}+k_{y}y)}\right], & \displaystyle\end{eqnarray}$$

(2.76c ) $$\begin{eqnarray}\displaystyle & \displaystyle p_{0,\pm 2}^{\dagger }=2k_{y}^{2}c_{\infty }^{2}\text{e}^{-2\unicode[STIX]{x1D705}^{2}(k_{y}^{2}+k_{z}^{2})\bar{x}}, & \displaystyle\end{eqnarray}$$

(2.76d ) $$\begin{eqnarray}\displaystyle & \displaystyle p_{m,0}^{\dagger }=-2k_{z}^{2}c_{\infty }^{2}\text{e}^{-2\unicode[STIX]{x1D705}^{2}(k_{y}^{2}+k_{z}^{2})\bar{x}}\left[\hat{\unicode[STIX]{x03C0}}_{m}\text{e}^{-2\text{i}(\bar{x}+k_{y}y)}+\hat{\unicode[STIX]{x03C0}}_{-m}^{\star }\text{e}^{2\text{i}(\bar{x}+k_{y}y)}\right]. & \displaystyle\end{eqnarray}$$

The other components $v_{m,n}^{\dagger },w_{m,n}^{\dagger }$ with $n\neq \pm 1$ and $p_{m,n}^{\dagger }$ with $n\neq 0,\pm 2$ are null. Although the upstream flow is forced only by a pair of oblique modes with opposite spanwise wavenumbers, further downstream the disturbance is composed of all the temporal harmonics. These are generated by nonlinear interactions in the boundary layer and transmitted to the outer flow via the displacement effect. Although the Fourier components (2.76) are of the same form as those obtained in the incompressible case (refer to Ricco et al. Reference Ricco, Luo and Wu2011, equation (2.28)), the coefficients $\hat{\unicode[STIX]{x1D712}}_{m}$ and $\hat{\unicode[STIX]{x03C0}}_{m}$ defined in (2.74) now include the compressible effects, the function $\unicode[STIX]{x1D6E5}(\bar{x},\bar{t})$ being related to $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6FF}^{c}$ through (2.65).

2.4 Initial and boundary conditions

As the boundary-region equations (2.23)–(2.27) constitute a parabolic system in the streamwise direction, initial conditions are needed for $\bar{x}\ll 1$ . Since the velocity and temperature fluctuations are of small amplitude near the leading edge, their governing equations become linear. This was elucidated by Ricco et al. (Reference Ricco, Luo and Wu2011), who showed that the full nonlinear regime develops gradually from the initial linear stage in the upstream region corresponding to $R_{\unicode[STIX]{x1D706}}^{-1}\ll \bar{x}\ll 1$ . The initial conditions for the forced modes $(m,n)=(1,\pm 1)$ are thus the same as those in Ricco & Wu (Reference Ricco and Wu2007).

The upstream conditions are found by first seeking the power series solution for $\unicode[STIX]{x1D702}=O(1)$ and $\bar{x}\ll 1$ ,

(2.77) $$\begin{eqnarray}\{\bar{u},\bar{v},\bar{w},\bar{\unicode[STIX]{x1D70F}},\bar{p}\}=\mathop{\sum }_{n=0}^{\infty }(2\bar{x})^{n/2}\left\{2\bar{x}U_{n}(\unicode[STIX]{x1D702}),V_{n}(\unicode[STIX]{x1D702}),W_{n}(\unicode[STIX]{x1D702}),2\bar{x}T_{n}(\unicode[STIX]{x1D702}),(2\bar{x})^{-1/2}P_{n}(\unicode[STIX]{x1D702})\right\},\end{eqnarray}$$

and by constructing a composite solution that is valid for $\bar{x}\ll 1$ and $\forall \unicode[STIX]{x1D702}$ :

(2.78) $$\begin{eqnarray}\{\hat{u} ,\hat{v},{\hat{w}},\hat{\unicode[STIX]{x1D70F}},\hat{p}\}_{-1,\pm 1}\rightarrow \pm \frac{\text{i}\unicode[STIX]{x1D705}^{2}}{k_{z}}u_{z,w,\pm }\left\{U_{in},V_{in},\mp \text{i}W_{in},T_{in},P_{in}\right\},\end{eqnarray}$$

where $\{U_{in},V_{in},W_{in},T_{in},P_{in}\}$ are equal to the right-hand sides of equations (4.12)–(4.16) in Ricco & Wu (Reference Ricco and Wu2007). The term $u_{z,w,\pm }$ represents the spanwise slip velocity at the surface of the plate (refer to Leib et al. Reference Leib, Wundrow and Goldstein1999, equation (3.14)). In the case of a pair of oblique modes, it is given by

(2.79) $$\begin{eqnarray}u_{z,w,\pm }=\mp 1\pm \frac{\text{i}k_{z}}{\sqrt{k_{x}^{2}+k_{z}^{2}}}\hat{u} _{y,\pm }^{\infty }.\end{eqnarray}$$

For all the other harmonics generated by nonlinear effects, null initial velocity, temperature and pressure profiles are imposed.

The velocity fluctuations are required to vanish at the wall (no-slip condition). Two types of different thermal boundary conditions at the wall may be imposed: the Dirichlet boundary condition, $\bar{\unicode[STIX]{x1D70F}}(0)=0$ , which was also employed by Ricco & Wu (Reference Ricco and Wu2007), and the Neumann boundary condition, $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}|_{\unicode[STIX]{x1D702}=0}=0$ . The condition $\bar{\unicode[STIX]{x1D70F}}(0)=0$ is used herein as Ricco et al. (Reference Ricco, Tran and Ye2009) showed that there is no substantial difference in the development of the Klebanoff modes.

As $\unicode[STIX]{x1D702}\rightarrow \infty$ , the boundary-region solution must match to the limit $\bar{y}\rightarrow 0$ of the outer flow. On taking into account the decomposition (2.16) and (2.17) within the boundary layer, the matching with the outer solution (2.75) requires that

(2.80) $$\begin{eqnarray}\left\{\hat{u} _{m,n},\hat{v}_{m,n},{\hat{w}}_{m,n},\hat{p}_{m,n},\hat{\unicode[STIX]{x1D70F}}_{m,n}\right\}\rightarrow \left\{0,\frac{\unicode[STIX]{x1D705}}{\sqrt{2\bar{x}}}v_{m,n}^{\dagger },\unicode[STIX]{x1D705}^{2}w_{m,n}^{\dagger },\frac{\unicode[STIX]{x1D716}}{k_{x}}p_{m,n}^{\dagger },0\right\}\quad \text{as}~\unicode[STIX]{x1D702}\rightarrow \infty\end{eqnarray}$$

for $\bar{x}=O(1)$ , where the condition on $\hat{v}_{m,n}$ holds only for $n\neq 0$ . In the spanwise-averaged equations ( $n=0$ ) the pressure only appears in the $y$ -momentum equation. The velocities and the temperature are calculated by solving the continuity, $x$ - and $z$ -momentum and the energy equations. The condition on $\hat{v}_{m,0}$ as $\unicode[STIX]{x1D702}\rightarrow \infty$ is not needed because the order of the system has decreased. As an a posteriori check, the free-stream value of $\hat{v}_{m,0}$ is obtained from the large- $\unicode[STIX]{x1D702}$ limit of the continuity equation,

(2.81) $$\begin{eqnarray}\displaystyle \hat{v}_{m,0} & = & \displaystyle \frac{k_{x}}{\unicode[STIX]{x1D716}}\frac{\unicode[STIX]{x1D705}}{\sqrt{2\bar{x}}}\left(\frac{\text{d}\hat{\unicode[STIX]{x1D6FF}}_{m}}{\text{d}\bar{x}}-\frac{\text{d}\unicode[STIX]{x1D6FF}_{bl}}{\text{d}\bar{x}}+\text{i}m\hat{\unicode[STIX]{x1D6FF}}_{m}^{c}\right)\end{eqnarray}$$

(2.82) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\int _{0}^{\infty }\left(\frac{\unicode[STIX]{x2202}\hat{u} _{m,0}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\hat{u} _{m,0}}{2\bar{x}}\right)\,\text{d}\unicode[STIX]{x1D702}+\int _{0}^{\infty }\frac{F^{\prime }}{T}\left(\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}_{m,0}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\hat{\unicode[STIX]{x1D70F}}_{m,0}}{2\bar{x}}\right)\,\text{d}\unicode[STIX]{x1D702}+\int _{0}^{\infty }\frac{\text{i}m}{T}\hat{\unicode[STIX]{x1D70F}}_{m,0}\text{d}\unicode[STIX]{x1D702}\nonumber\\ \displaystyle & & \displaystyle -\,r_{t}\int _{0}^{\infty }T\left(\frac{\unicode[STIX]{x2202}\widehat{\bar{\unicode[STIX]{x1D70C}}\bar{u}}_{m,0}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\widehat{\bar{\unicode[STIX]{x1D70C}}\bar{u}}_{m,0}}{2\bar{x}}\right)\,\text{d}\unicode[STIX]{x1D702}+r_{t}\int _{0}^{\infty }\frac{\text{i}m}{T}\widehat{\bar{\unicode[STIX]{x1D70C}}\bar{\unicode[STIX]{x1D70F}}}_{m,0}\,\text{d}\unicode[STIX]{x1D702}\nonumber\\ \displaystyle & & \displaystyle +\,r_{t}\int _{0}^{\infty }F^{\prime }\left(\frac{\unicode[STIX]{x2202}\widehat{\bar{\unicode[STIX]{x1D70C}}\bar{\unicode[STIX]{x1D70F}}}_{m,0}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\widehat{\bar{\unicode[STIX]{x1D70C}}\bar{\unicode[STIX]{x1D70F}}}_{m,0}}{2\bar{x}}\right)\,\text{d}\unicode[STIX]{x1D702}\quad \text{as}~\unicode[STIX]{x1D702}\rightarrow \infty ,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{bl}$ is the compressible Blasius displacement thickness,

(2.83) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{bl}=\int _{0}^{\infty }\left[1-\frac{F^{\prime }(\unicode[STIX]{x1D702})}{T(\unicode[STIX]{x1D702})}\right]\,\text{d}y=\sqrt{\frac{2\bar{x}}{k_{x}R_{\unicode[STIX]{x1D706}}}}(\unicode[STIX]{x1D6FE}_{c}+\unicode[STIX]{x1D6FD}_{c}),\end{eqnarray}$$

and $\unicode[STIX]{x1D6FE}_{c}=\unicode[STIX]{x1D702}_{c}-\unicode[STIX]{x1D702}$ as $\unicode[STIX]{x1D702}\rightarrow \infty$ . In the limit $M_{\infty }\rightarrow 0$ the result of Ricco et al. (Reference Ricco, Luo and Wu2011) is recovered:

(2.84) $$\begin{eqnarray}\hat{v}_{m,0}=\frac{k_{x}}{\unicode[STIX]{x1D716}}\frac{\unicode[STIX]{x1D705}}{\sqrt{2\bar{x}}}\left(\frac{\text{d}\hat{\unicode[STIX]{x1D6FF}}_{m}}{\text{d}\bar{x}}-\frac{\text{d}\unicode[STIX]{x1D6FF}_{bl}}{\text{d}\bar{x}}\right)=-\int _{0}^{\infty }\left(\frac{\unicode[STIX]{x2202}\hat{u} _{m,0}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\hat{u} _{m,0}}{2\bar{x}}\right)\,\text{d}\unicode[STIX]{x1D702}\quad \text{as}~\unicode[STIX]{x1D702}\rightarrow \infty ,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{bl}=\int _{0}^{\infty }[1-F^{\prime }(\unicode[STIX]{x1D702})]\,\text{d}y=\unicode[STIX]{x1D6FD}\sqrt{2\bar{x}/k_{x}R_{\unicode[STIX]{x1D706}}}$ is the incompressible Blasius displacement thickness and $\unicode[STIX]{x1D6FD}=1.2168$ . In the compressible case the additional term related to $\bar{\unicode[STIX]{x1D6FF}}^{c}$ in (2.81) is present or absent depending on the motion being unsteady or steady, while in the incompressible case the transpiration velocity is not affected by unsteady effects.

5.1 Choice of parameters

5.1.1 Turbomachinery applications

With a reference to typical experimental works on turbomachinery applications (e.g. Camci & Arts Reference Camci and Arts1990) the following parameters are adopted: unit Reynolds number $R_{1\infty }^{\ast }=$ $U_{\infty }^{\ast }/\unicode[STIX]{x1D708}_{\infty }^{\ast }=8\times 10^{6}~\text{ m}^{-1}$ , chord length $\ell ^{\ast }=8~\text{ cm}$ , free-stream temperature $T_{\infty }^{\ast }=500~\text{ K}$ and kinematic viscosity of air $\unicode[STIX]{x1D708}_{\infty }^{\ast }=3.9\times 10^{-5}~\text{ m}^{2}~\text{s}^{-1}$ , free-stream Mach number $M_{\infty }=0.69$ and turbulence level $Tu=0.8-5.2\,\%$ . Although we have chosen our flow parameters to be as close as possible to the experimental ones, important extra factors are present in real experimental and technological flow systems, such as acoustic free-stream forcing, surface curvature, pressure gradient and wall cooling. These can be taken into account by suitable extensions of the present framework. For example, the boundary-region equation approach has been extended to study the generation and development of Görtler vortices in the incompressible boundary layer over a concave wall (Wu, Zhao & Luo Reference Wu, Zhao and Luo2011). With further progress, precise quantitative comparisons with experiments and applications to practical situations would be possible. Presently, the results are of qualitative value as far as their relevance to turbomachinery is concerned. The adiabatic wall temperature is calculated using the following relation valid for a perfect gas,

(5.1) $$\begin{eqnarray}T_{ad}^{\ast }=T_{\infty }^{\ast }\left(1+C_{t}\frac{\unicode[STIX]{x1D6FE}-1}{2}M_{\infty }^{2}\right),\end{eqnarray}$$

where $C_{t}=\sqrt{Pr}$ is the recovery factor for a laminar boundary layer over a flat plate. The resulting adiabatic wall temperature, $T_{ad}^{\ast }=540.5~\text{ K}$ , is in the range of typical turbine rotor applications. The ratio of the wall temperature to the adiabatic wall temperature $T_{w}^{\ast }/T_{ad}^{\ast }=0.7$ is chosen to mimic realistic aero-engine conditions (Zhang & He Reference Zhang and He2014), where blade cooling is most often applied to avoid excessive surface temperature. The resulting non-dimensional wall temperature is $T_{w}=0.75$ .

As the relevant length scales and spectra of the free-stream turbulence are not documented in the experiments, we choose to assume that the ratio of the streamwise integral scale to the chord is equal to $1.5$ . It follows that $\unicode[STIX]{x1D706}_{x}^{\ast }=0.12~\text{ m}$ and the frequency $f^{\ast }=U_{\infty }^{\ast }/\unicode[STIX]{x1D706}_{x}^{\ast }=2.5~\text{ kHz}$ . At downstream locations where $x^{\ast }=O(\unicode[STIX]{x1D706}^{\ast }R_{\unicode[STIX]{x1D706}})$ , the boundary-layer thickness is comparable with the spanwise length scale $\unicode[STIX]{x1D706}^{\ast }$ . The laminar boundary-layer thickness is proportional to $\sqrt{\unicode[STIX]{x1D708}_{\infty }^{\ast }x^{\ast }/U_{\infty }^{\ast }}$ , and it is estimated that transition occurs at $x_{T}^{\ast }/\ell ^{\ast }=0.67$ (i.e. $x_{T}^{\ast }=5.36~\text{ cm}$ ), which corresponds to a critical Reynolds number of $5\times 10^{5}$ (Schlichting & Gersten Reference Schlichting and Gersten2000). It follows that the boundary-layer thickness $\unicode[STIX]{x1D6FF}_{99}^{\ast }$ , defined as the wall-normal location where $U=0.99$ , is $0.41~\text{ mm}$ , and the displacement thickness is $\unicode[STIX]{x1D6FF}^{\ast }=0.14~\text{ mm}$ . The latter value is taken as the reference length scale $\unicode[STIX]{x1D706}^{\ast }$ . The Reynolds number $R_{\unicode[STIX]{x1D706}}$ and the scaled streamwise wavenumber $\unicode[STIX]{x1D705}$ are worked out by use of (2.5) and (2.7). The free-stream disturbance intensity is defined as $Tu=2\unicode[STIX]{x1D716}\sqrt{(\hat{u} _{x,+}^{\infty })^{2}+(\hat{u} _{x,-}^{\infty })^{2}}$ . For our choice of $\hat{u} _{x,\pm }^{\infty }=1$ , $Tu=2\sqrt{2}\unicode[STIX]{x1D716}$ . Given $\unicode[STIX]{x1D716}$ and $R_{\unicode[STIX]{x1D706}}$ , the turbulence Reynolds number $r_{t}$ is calculated from (2.8). The influence of nonlinear effects is investigated by varying the turbulence level (i.e.  $\unicode[STIX]{x1D716}$ ) with all the other parameters kept constant, as shown in table 1(a). Two different values of $\unicode[STIX]{x1D706}_{x}^{\ast }/\ell ^{\ast }=0.77$ , $3.1$ are considered in order to study the effect of the streamwise wavenumber (refer to table 1(b)). The relation between $\unicode[STIX]{x1D706}_{z}^{\ast }$ , shown in table 1, and $\unicode[STIX]{x1D706}^{\ast }$ is $\unicode[STIX]{x1D706}_{z}^{\ast }=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}^{\ast }$ .

Table 1. Choice of the parameters for the turbomachinery, flight and wind tunnel cases.

5.2 Evolution of nonlinear compressible streaks

As anticipated on the basis of the linear analysis by Ricco & Wu (Reference Ricco and Wu2007), free-stream disturbances of the hydrodynamic kind (i.e. convected gusts) generate thermal fluctuations inside the boundary layer due to the velocity–temperature coupling. Just as the temperature profile affects the stability of a two-dimensional flow (Lees Reference Lees1947; Mack Reference Mack1975), thermal streaks are bound to influence the secondary instability of the streaky boundary layer. We will therefore focus on the fluctuations of the streamwise velocity and temperature inside the boundary layer.

The overall intensity of the streak signature, which comprises all the harmonics in the nonlinear regime, is measured by the root-mean-square (r.m.s. hereinafter) of the fluctuating quantity, defined as

(5.2) $$\begin{eqnarray}q_{rms}\equiv r_{t}\sqrt{\mathop{\sum }_{m=-\bar{N}_{t}}^{\bar{N}_{t}}\mathop{\sum }_{n=-\bar{N}_{z}}^{\bar{N}_{z}}|\hat{q}_{m,n}|^{2}},\quad m\neq 0,\end{eqnarray}$$

where $q$ may stand for the streamwise velocity or the temperature and $\bar{N}_{t}=(N_{t}-1)/2$ , $\bar{N}_{z}=(N_{z}-1)/2$ . The downstream development of the maximum $q_{rms}$ along $\unicode[STIX]{x1D702}$ ,

(5.3) $$\begin{eqnarray}q_{rms,max}(\bar{x})=\max _{\unicode[STIX]{x1D702}}q_{rms}(\bar{x},\unicode[STIX]{x1D702}),\end{eqnarray}$$

is shown in figures 5 (turbomachinery case), 6 (flight case) and 7 (wind tunnel case), where the nonlinear solutions are compared with the corresponding linearized approximations for different values of $Tu$ . In the linear case, the peak of the r.m.s. is given by

(5.4) $$\begin{eqnarray}q_{rms,max}=\unicode[STIX]{x1D716}\sqrt{2}(k_{z}/k_{x})\sqrt{|u_{z,w,+}|^{2}+|u_{z,w,-}|^{2}}\max _{\unicode[STIX]{x1D702}}|\bar{q}_{l}(\bar{x},\unicode[STIX]{x1D702})|,\end{eqnarray}$$

where $\bar{q}_{l}$ represents the solution provided by Ricco & Wu (Reference Ricco and Wu2007). Since $|u_{z,w,+}|=|u_{z,w,-}|=|u_{z,w}|$ , (5.4) simplifies to:

(5.5) $$\begin{eqnarray}q_{rms,max}=2\unicode[STIX]{x1D716}(k_{z}/k_{x})|u_{z,w}|\max _{\unicode[STIX]{x1D702}}|\bar{q}_{l}(\bar{x},\unicode[STIX]{x1D702})|.\end{eqnarray}$$

Sufficiently upstream, the linear and nonlinear solutions overlap as the influence of nonlinearity is still weak. Due to the continued amplification of the disturbance, the streak signature becomes stronger and the linear and nonlinear solutions start diverging for moderate $r_{t}$ . In the case with the lowest $r_{t}$ for all the three scenarios considered, the linear and nonlinear curves are almost indistinguishable. In the cases with higher $r_{t}$ , a stabilizing effect of nonlinearity is observed on the streamwise velocity and temperature and, as the turbulence intensity increases, the attenuation of the disturbances is enhanced. The higher the turbulence level, the slower and the weaker the disturbance growth becomes, and the farther upstream the nonlinear effects start asserting their influence. The stabilizing effect of nonlinearity was already observed by Ricco et al. (Reference Ricco, Luo and Wu2011) in the incompressible regime for high turbulence levels. Here, it is shown that nonlinear effects play the same role on thermal (temperature) streaks.

Figure 5. Evolution of the maximum r.m.s. of the streamwise velocity (a) and the temperature (b) for different values of the turbulence Reynolds number: $r_{t}=2.7$ (solid lines), $r_{t}=7.9$ (dashed lines) at $M_{\infty }=0.69$ (refer to table 1(a)). Thick lines: nonlinear solutions, thin lines: linearized solution.

Figure 6. Evolution of the maximum r.m.s. of the streamwise velocity (a) and the temperature (b) for different values of the turbulence Reynolds number: $r_{t}=3.37$ (solid line), $r_{t}=6.25$ (dashed lines) at $M_{\infty }=3$ (refer to table 1(d)). Thick lines: nonlinear solutions, thin lines: linearized solutions.

Figure 7. Evolution of the maximum r.m.s. of the streamwise velocity (a) and temperature (b) for different values of the turbulence Reynolds number: $r_{t}=2.1$ (dashed-dotted lines), $r_{t}=4.4$ (dashed lines), $r_{t}=6$ (solid lines) at $M_{\infty }=3$ (refer to table 1(e)). Thick lines: nonlinear solutions, thin lines: linearized solutions.

For the turbomachinery case (figure 5), the attenuation of the streaks is more pronounced than in flight conditions as the turbulence levels are higher, and the deviation of the kinematic and thermal streaks from the linearized solution is of similar magnitude. The kinematic fluctuations are approximately one order of magnitude higher than the thermal streaks relatively to the correspondent free-stream mean velocity and temperature.

In flight conditions (figure 6), because of a higher Mach number, the temperature fluctuation acquires a large intensity, which is comparable with that of the velocity fluctuation. Although the influence of $Tu$ on the streamwise velocity is quite weak even in the highest turbulence intensity case, $r_{t}=6.25$ , appreciable attenuation of the nonlinear thermal streaks with respect to the linearized ones is observed. The stabilizing effect is more marked on the temperature than on the streamwise velocity. The nonlinear and linear r.m.s. of the temperature streaks diverge farther upstream than the r.m.s. of the streamwise velocity. Even for a low-disturbance environment such as free flight, it seems necessary to describe the formation and amplification of the streaks correctly by taking into account the nonlinear interactions inside the boundary layer.

In the wind tunnel case shown in figure 7, a peculiar behaviour of the nonlinear curve for the highest turbulence level case is observed. A sharp deviation of the nonlinear solution from the linear one occurs at $\bar{x}\approx 0.5$ , which corresponds to a physical downstream position of $x^{\ast }\approx 10~\text{ cm}$ . This phenomenon was not observed in the incompressible cases studied by Ricco et al. (Reference Ricco, Luo and Wu2011). The parameters used in this case are very similar to those of the flight case with highest $r_{t}$ (refer to tables 1(d) and 1(e)), except that $k_{x}$ is approximately half of that in the flight case. As a consequence, the effect of nonlinearity is stronger. The streamwise wavenumber of the wind tunnel case is comparable with that of the flight case with $M_{\infty }=6$ (refer to table 1(c)) but $r_{t}$ is smaller and $M_{\infty }$ is higher in the latter, thus resulting in weaker nonlinear effects. Additional calculations suggest that this abrupt change occurs when a sufficiently small streamwise wavenumber is employed, i.e. $k_{x}\leqslant 1.5\times 10^{-3}$ , together with a high turbulence Reynolds number, $r_{t}\geqslant 6$ , at supersonic speed. In the subsonic regime no such sharp deviation was observed, even at high $r_{t}$ .

The effect of the frequency is displayed in figure 8, where the downstream evolutions of $u_{rms,max}$ and $\unicode[STIX]{x1D70F}_{rms,max}$ are plotted for different values of $k_{x}$ in the turbomachinery case (refer to table 1(b)). The fluctuations of the streamwise velocity and temperature are higher for smaller $k_{x}$ , but the dependence of the stabilizing effect of nonlinearity on the frequency is very weak and it is the same for the streamwise velocity component and the temperature. For $k_{x}\geqslant 0.01$ the amplitude saturates at an almost constant value before decaying farther downstream.

Figure 8. Evolution of the maximum r.m.s. of the streamwise velocity (a) and the temperature (b) for different streamwise wavenumber: $k_{x}=0.0036$ (solid lines) and $k_{x}=0.0142$ (dashed lines) at $M_{\infty }=0.69$ (refer to table 1(b)). Thick lines: nonlinear solutions, thin lines: linearized solutions.

Figure 9 displays the signature of the streamwise velocity and temperature for different Mach numbers in the flight case (refer to table 1(c)). The variation of $M_{\infty }$ is only due to a variation of $U_{\infty }^{\ast }$ , while $T_{\infty }^{\ast }$ is constant. As the Mach number increases the r.m.s. of the streamwise velocity is attenuated while the temperature fluctuations are intensified. For $M_{\infty }=6$ , the latter acquire an intensity as large as 26 % of $T_{\infty }^{\ast }$ , whereas the velocity fluctuations merely reach 5 % of $U_{\infty }^{\ast }$ . The result suggests that thermal streaks are likely to be the primary cause of secondary instability in high speed flows. Figure 9 also shows that the attenuation effect on the velocity is stronger than the enhancement of the temperature. For example, tripling the Mach number from 2 to 6 results in a decrease of more than three times in the r.m.s. of the streamwise velocity and an increase of twice in the temperature signature at $x=600$ . Despite $r_{t}$ being the same, for higher $M_{\infty }$ the stabilizing effect of nonlinearity becomes less pronounced and this is more evident for the streamwise velocity r.m.s., whose nonlinear evolution is indistinguishable from the linearized curve in the highest $M_{\infty }$ case. This behaviour is attributed to the turbulence level being smaller for higher values of the free-stream velocity, although $r_{t}$ is the same in all the cases.

Figure 9. Evolution of the maximum r.m.s. of the streamwise velocity (a) and the temperature (b) for different free-stream Mach numbers: $M_{\infty }=2$ (dashed lines), $M_{\infty }=6$ (solid lines). The turbulence Reynolds number is $r_{t}=4.8$ (refer to table 1(c)). Thick lines: nonlinear solutions, thin lines: linearized solutions.

In figure 10 the results obtained in the wind tunnel case with $r_{t}=6$ for the hot-wall condition $T_{w}^{\ast }/T_{ad}^{\ast }=1.1$ are compared with those relative to a cold-wall condition $T_{w}^{\ast }/T_{ad}^{\ast }=0.8$ and an adiabatic temperature at the wall $T_{w}^{\ast }=T_{ad}^{\ast }$ . As the wall heat flux increases from negative (heating), to zero (adiabatic) and to positive (cooling), the signature of the streamwise velocity is enhanced while the temperature disturbance is attenuated. The position where the abrupt deviation occurs moves from $\bar{x}\approx 0.5$ in the hot-wall case to $\bar{x}\approx 0.7$ in the cold-wall case. Therefore the wall heat flux influences the position where the onset of the stabilizing effect due to nonlinearity occurs. This suggests that the employment of the adiabatic wall condition in wind tunnel experiments may lead to an inaccurate prediction of the transition location for high Mach number supersonic flight conditions, where wall cooling usually needs to be employed for thermal protection.

Figure 10. Evolution of the maximum r.m.s. of the streamwise velocity (a) and the temperature (b) for different wall temperature conditions: hot wall (h.w.), adiabatic wall (a.w.) and cold wall (c.w.) at $M_{\infty }=3$ .

5.3 Wall-normal profiles of the perturbation for flight condition

The wall-normal profiles of the streamwise velocity and the temperature are now examined for flight conditions with $M_{\infty }=3$ and $r_{t}=4.8$ , as this case features significant effects of nonlinearity and compressibility, whereas in the turbomachinery case compressible effects are weak. The parameters correspond to case (d) in table 1, except that $Tu=0.27\,\%$ and $r_{t}=4.8$ . The flow is symmetric with respect to the $z$ direction because the Fourier forcing modes have opposite spanwise wavenumbers but equal amplitude. Therefore, results will only be presented for modes with $n\geqslant 0$ , since modes $(m,n)$ and $(m,-n)$ have the same amplitude and shape.

Figure 11 shows the profiles of the spanwise-uniform time-averaged flow distortion $(0,0)$ , the forcing mode $(-1,1)$ , and the second and third harmonics with $|m|=|n|$ , i.e. $(-2,2)$ and $(-3,3)$ , at $\bar{x}=0.5$ and $1.2$ . As expected, the forcing mode has a higher amplitude than the other components. The mean-flow distortion makes a significant contribution to the overall flow. The magnitude of the higher harmonics decreases so quickly that the third harmonic becomes almost negligible. The profiles of the temperature perturbation are similar to those of the streamwise velocity and evolve in a similar manner, but the amplitude of the thermal fluctuations is slightly higher than those of the streamwise velocity (relatively to the free-stream values).

Figure 11. Profiles of the streamwise velocity (a,b) and the temperature (c,d) of the mean-flow distortion and harmonics with $|m|=|n|=1,2,3$ at $\bar{x}=0.5$ (a,c) and $\bar{x}=1.2$ (b,d). The parameters correspond to the flight case with $M_{\infty }=3$ and $r_{t}=4.8$ .

The streamwise velocity and temperature of the seeded modes attain their respective maxima at $\unicode[STIX]{x1D702}=1.5$ and $\unicode[STIX]{x1D702}=2$ . The most pronounced peak of the second harmonic occurs at a larger wall-normal distance, $\unicode[STIX]{x1D702}=3$ . The mean-flow distortion of the streamwise velocity is positive near the wall and negative for $\unicode[STIX]{x1D702}>2$ , while the temperature profile $(0,0)$ is negative close to the plate (excluding a small positive region at $\unicode[STIX]{x1D702}<0.5$ ) and positive in the outer layer. The $(0,0)$ components of the streamwise velocity and temperature grow significantly with the downstream distance and their amplitudes become greater than that of the seeded modes in the outer portion of the boundary layer, $\unicode[STIX]{x1D702}>4$ . The difference between the maximum values of the linear and nonlinear profiles is larger for the temperature than for the streamwise velocity. This confirms the stronger effect of nonlinearity on temperature for high-speed flows, which was already observed in figure 6. At $\bar{x}=0.5$ the linear solution of $r_{t}|\hat{u} _{-1,1}|$ and $r_{t}|\hat{\unicode[STIX]{x1D70F}}_{-1,1}|$ are almost indistinguishable from their nonlinear counterparts. At $\bar{x}=1.2$ the peaks of $r_{t}|\hat{u} _{-1,1}|$ and $r_{t}|\hat{\unicode[STIX]{x1D70F}}_{-1,1}|$ in the nonlinear case have moved closer to the wall and have decreased in comparison to the linear case. After reaching their maxima, the nonlinear solutions decay more slowly and become larger than the linearized approximations in the outer region of the boundary layer. The mean-flow distortion acquires an amplitude comparable with that of the seeded modes. Therefore, the effect of nonlinearity is to move the location of the disturbance peaks nearer to the wall, to weaken the fluctuations in the core of the boundary layer while enhancing them close to the free stream, and most notably to generate significant mean-flow distortion.

As was pointed out by Ricco (Reference Ricco2006), the nonlinear interactions generate only Fourier modes with $m=n$ when the flow is forced by a single free-stream mode, while in the case of a pair of oblique free-stream modes additional components with $m\neq n$ are induced. The nonlinearly generated modes are those with $|m|+|n|$ equal to an even integer, i.e. they are arranged as a checkerboard in spectral space. Those generated at the second and third orders are displayed in figure 12 at the downstream locations $\bar{x}=0.5$ and $1.2$ . The amplitudes of the components $(-2,0)$ and $(0,2)$ are comparable with each other but approximately three times smaller than that of the mean-flow distortion $(0,0)$ . At $\bar{x}=0.5$ , the peak positions and the magnitudes of the modes $(-2,0)$ and $(0,2)$ are very similar (or almost identical for $\unicode[STIX]{x1D702}>2$ ), while at $\bar{x}=1.2$ the first peak of the mode $(-2,0)$ becomes much higher than that of the component $(0,2)$ and the second peak moves further from the wall. The third-order harmonics feature two or three humps, with the first or second peak being coincident with the valley of the second-order components. Profiles with three peaks were not observed in the incompressible case (Ricco et al. Reference Ricco, Luo and Wu2011) and they are more evident in the temperature streaks.

Figure 12. Profiles of the streamwise velocity (a,b) and the temperature (c,d) of harmonics with $m\neq n$ at $\bar{x}=0.5$ (a,c) and $\bar{x}=1.2$ (b,d). The parameters correspond to the flight case with $M_{\infty }=3$ and $r_{t}=4.8$ .

The profiles of the cross-flow velocity components and of the pressure are displayed in figure 13. No mean spanwise velocity component is generated because the disturbances are symmetric with respect to the plane $z=0$ . The profiles of $\hat{v}_{m,0}$ asymptotically approach a constant value (with respect to $\unicode[STIX]{x1D702}$ ) in the free stream, as the right-hand side of (2.82) is a function of $\bar{x}$ only. The amplitudes of the wall-normal and spanwise velocity components are much smaller than that of the streamwise velocity, as $\sqrt{k_{x}/R_{\unicode[STIX]{x1D706}}}=O(10^{-4})$ and $k_{x}/k_{z}=O(10^{-3})$ .

Figure 13. Profiles of the wall-normal velocity (a,b), spanwise velocity (c,d) and the pressure (e,f) of the mean-flow distortion and harmonics at $\bar{x}=1.2$ . The parameters correspond to the flight case with $M_{\infty }=3$ and $r_{t}=4.8$ .

The pressure fluctuations at the wall are analysed in order to show that these do not represent an aeroelasticity problem. The root-mean-square of the wall pressure is calculated and transformed in dimensional terms by multiplying it by $\unicode[STIX]{x1D70C}_{\infty }^{\ast }U_{\infty }^{\ast 2}$ , where $U_{\infty }^{\ast }=888~\text{ m}~\text{s}^{-1}$ (table 1(d)) and $\unicode[STIX]{x1D70C}_{\infty }^{\ast }=0.09~\text{kg}~\text{m}^{-3}$ at $20~\text{ km}$ altitude (Champion, Cole & Kantor Reference Champion, Cole and Kantor1985). The value obtained $p_{rms}^{\ast }=0.055~\text{ Pa}$ is compared to the pressure difference on the wing. The latter is derived from the lift coefficient $C_{l}$ , which is evaluated using the supersonic linearized theory (refer to Anderson Reference Anderson2007, equation (12.23)) and assuming an angle of attack equal to $10^{\circ }$ . The resulting pressure difference $\unicode[STIX]{x0394}p^{\ast }=4200~\text{ Pa}$ is five orders of magnitude higher than the pressure oscillations obtained in our calculations. A comparison is also performed with the wall pressure oscillations in a turbulent boundary layer for the same set of parameters. The experimental data provided by Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007), who measured the r.m.s. of the wall pressure at different friction Reynolds number $R_{\unicode[STIX]{x1D70F}}$ , are used to evaluate the $p_{rms}$ at the wall at the current friction Reynolds number $R_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}^{\ast }x^{\ast }/\unicode[STIX]{x1D708}_{\infty }^{\ast }$ , where $u_{\unicode[STIX]{x1D70F}}^{\ast }$ is the friction velocity and $x^{\ast }$ is the distance from the leading edge. From our calculations we obtain $u_{\unicode[STIX]{x1D70F}}^{\ast }=13~\text{ m}~\text{s}^{-1}$ and we choose $x^{\ast }=0.7~\text{ m}$ corresponding to $\bar{x}=1.2$ . It follows that $R_{\unicode[STIX]{x1D70F}}=3.4\times 10^{4}$ . The relation proposed by Farabee & Casarella (Reference Farabee and Casarella1991) $(p_{rms}^{+})^{2}=6.5+1.86\ln (R_{\unicode[STIX]{x1D70F}}/333)$ is employed to extrapolate the wall pressure r.m.s. $p_{rms}^{+}$ scaled with inner viscous units. The experiments of Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) refer to an incompressible boundary layer and therefore the effect of the Mach number on the wall pressure needs to be included in our calculations. As a first estimate, the effect of compressibility on the skin-friction coefficient $C_{f}$ is evaluated by means of figure 19.1 of Schlichting & Gersten (Reference Schlichting and Gersten2000) and the same ratio $C_{f}(M_{\infty }=3)/C_{f}(M_{\infty }=0)=0.6$ is assumed to be valid for the wall pressure. It follows that $p_{rms}^{\ast }=35~\text{Pa}$ and therefore the ratio between the wall pressure turbulent and pre-transitional fluctuations is approximately $6\times 10^{2}$ .

The total time-averaged ( $m=0$ ) streaks of the streamwise velocity and temperature, which will be referred to as $u_{str}$ and $\unicode[STIX]{x1D70F}_{str}$ , are defined as

(5.6) $$\begin{eqnarray}\left\{u_{str}(\bar{x},\unicode[STIX]{x1D702},z),\unicode[STIX]{x1D70F}_{str}(\bar{x},\unicode[STIX]{x1D702},z)\right\}=r_{t}\mathop{\sum }_{n=-\bar{N}_{z}}^{\bar{N}_{z}}\left\{\hat{u} _{0,n},\hat{\unicode[STIX]{x1D70F}}_{0,n}\right\}\text{e}^{\text{i}nk_{z}z},\end{eqnarray}$$

in which only modes with $n=0,2,4,\ldots$ provide a non-zero contribution, but the component $(0,4)$ is almost negligible. They represent a time-averaged spanwise modulation superimposed onto the Blasius boundary layer.

Figure 14 shows the contours of the time-averaged streamwise velocity and the temperature streaks plotted in the $\unicode[STIX]{x1D702}{-}z$ plane at different downstream locations. A positive value of the contours means that the mean flow is higher than the local Blasius solution, while a negative value means that it is lower. Therefore, near the wall the flow is accelerated and cooled, while close to the free stream it is decelerated and heated. The mean-flow distortion of the streamwise velocity has been interpreted as an increase of the mean wall shear stress and backward jets at the edge of the boundary layer, both of which have been observed in experiments (Ricco et al. Reference Ricco, Luo and Wu2011).

Figure 14. Contours of the time-averaged streamwise velocity streaks $u_{str}$ (a,b) and the temperature streaks $\unicode[STIX]{x1D70F}_{str}$ (c,d) in the $\unicode[STIX]{x1D702}{-}z$ plane, at different downstream locations: $\bar{x}=0.5$ (a,c), $\bar{x}=1.2$ (b,d). The parameters correspond to the flight case with $M_{\infty }=3$ and $r_{t}=4.8$ .