Acoustic receptivity of high-speed boundary layers on a flat plate at angles of attack

Abstract

Direct numerical simulation and theoretical analysis of acoustic receptivity are performed for the boundary layer on a flat plate in Mach 6 flow at various angles of attack (AoA). Slow or fast acoustic wave passes through: a bow shock at AoA $=-5^{\circ }$, a weak shock induced by the viscous–inviscid interaction at AoA $=0^{\circ }$ or an expansion fan emanating from the plate leading edge at AoA $=5^{\circ }$. The study is focused on cases where the integral amplification of unstable mode S (or Mack second mode) is sufficiently large $(N\approx 8.4)$ to be relevant to transition in low-disturbance environments. It is shown that excitation of dominant modes F and S occurs in a small vicinity of the plate leading edge. The initial disturbance propagates further downstream in accord with the two-mode approximation model accounting for the mean-flow nonparallel effects and the intermodal exchange mechanism. This computationally economical model can be useful for predictions of the second mode dominated transition onset using the physics-based amplitude method.

Graphic abstract

Appendix

Herein we evaluate the forced response of the boundary layer to mode S using the two-mode approximation. Consider the system of equations (4.2) in the region $x{>}x_{n1}$

$$\begin{aligned} \frac{dC_{k}}{dx} = \sum _{j=1}^ {2} {C_{j}W_{kj}e^{i (S_{j}-S_{k} )}} ,\quad k=1,2, \end{aligned}$$

(A1)

with the initial conditions $C_{k} (x_{n1} )=C_{k}^{(0)}$. Hereafter the amplitude coefficient $C_ 1$ corresponds to mode F and $C_ 2$ to mode S. The substitution $C_{k} (x )={\tilde{C}}_{k} (x{){\mathrm {exp}}}\left( \int \nolimits _{x_{n1}}^x {W_{kk}dx} \right)$ gives

$$\begin{aligned}&\frac{d{\tilde{C}}_{k}}{dx} = {\tilde{C}}_{j}W_{kj}e^{i ({\tilde{S}}_{j}-{\tilde{S}}_{k} )},\quad {\tilde{C}}_{k} (x_{n} )=C_{k}^{(0)} , \nonumber \\&{\tilde{S}}_{k} = \int \limits _{x_{n1}}^x {\tilde{\alpha }_{k}dx} , \quad \tilde{\alpha }_{k} = \alpha _{k}-iW_{kk}. \end{aligned}$$

(A2)

Since the wavelength $\lambda _{k}$ is of the order of the boundary layer thickness, the wave number is large: $\tilde{\alpha }_{k}{=2}\pi /\lambda _{k} = O (L^{{*}} /\lambda _{k}^{{*}} )=O (\varepsilon ^{-1})$, where $\varepsilon ={\mathrm {{R}{e}}}_{e}^{{-1/2}}$ is a small parameter. After rescaling: $\tilde{\alpha }_{k} = \varepsilon ^{-1}{\bar{\alpha }}_{k}$, where $\bar{\alpha }_{k} = O$(1), the system (A2) reads

$$\begin{aligned} \frac{d{\tilde{C}}_{k}}{dx}= & {} {\tilde{C}}_{j}W_{kj}e^{\varepsilon ^-1i ({\bar{S}}_{j}-{\bar{S}}_{k} )}, {\tilde{C}}_{k} (x_{n1} )=C_{k}^{(0)} , \nonumber \\ {\bar{S}}_{k}= & {} \int \limits _{x_{n1}}^x {{\bar{\alpha }}_{k}dx} . \end{aligned}$$

(A3)

Integrating by parts, we get

$$\begin{aligned} {\tilde{C}}_{k} (x )=C_{k}^{(0)} +O (\varepsilon {)+}{\tilde{C}}_{j} (x )\left[ \varepsilon \frac{W_{kj} (x )}{i (\bar{\alpha }_{j}-{\bar{\alpha }}_{k} )}+O (\varepsilon ^ 2 ) \right] e^{\varepsilon ^{-1i} ({\bar{S}}_{j}-{\bar{S}}_{k} )}. \end{aligned}$$

(A4)

Substituting the amplitude coefficients (A4) into (4.1) and dropping the exponentially small terms proportional to $e^{\varepsilon ^-1i{\bar{S}}_ 1}$, we obtain

$$\begin{aligned} \varvec{\varPsi } (x,y,t )=C_ {2}^{(0)} e^{i\varepsilon ^-1{\bar{S}}_ {2}}\left\{ \left[ {(1+}O (\varepsilon ) \right] \varvec{\xi }_{2}+\varepsilon \frac{W_{{12}}}{i (\bar{\alpha }_{2}-{\bar{\alpha }}_ {1} )}\left[ {(1+}O (\varepsilon ) \right] \varvec{\xi }_{1} \right\} . \end{aligned}$$

(A5)

Projection of (A5) on BES leads us to the wrong conclusion that the disturbance field consists of the two modes: mode S of the amplitude ${\mathrm {abs}(C_ 2^{(0)} e^{i\varepsilon ^-1{\bar{S}}_ 2}} )$ corresponding to the first term, and mode F of the amplitude ${\mathrm {abs}}\left[ \varepsilon C_ {2}^{(0)} e^{i\varepsilon ^{-1}{\bar{S}}_ 1}\frac{W_{{12}}}{i (\bar{\alpha }_ {2}-{\bar{\alpha }}_{1} )} \right]$ corresponding to the second term. In fact, the both terms in (A5) refer to mode S having the eikonal ${\bar{S}}_ 2$. The second term in (A5) is just a correction of the mode S eigenfunction due to the nonparallel effect. It is associated with a forced response of the boundary layer to mode S.