1. Introduction and problem formulation

Several recent studies have shown experimentally (Rasheed et al. Reference Rasheed, Hornung, Fedorov and Malmuth2002), theoretically (Fedorov et al. Reference Fedorov, Malmuth, Rasheed and Hornung2001) and numerically (De Tullio & Sandham Reference De Tullio and Sandham2010) that porous coatings with a regular microstructure can stabilize second-mode disturbances in hypersonic boundary layers. Hence, these coatings may constitute an attractive passive laminar flow control technique for hypersonic boundary layers (Fedorov Reference Fedorov2011).

Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) investigated the stabilization properties of fibrous absorbent materials and presented for the first time a quantitative comparison between experimental and theoretical results regarding the effect of ultrasonically absorptive coatings (UACs) used for hypersonic laminar flow control. Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) measured spectra of natural and artificially excited disturbances in a hypersonic boundary layer on a $7^{\circ }$ half-angle sharp cone, which were used to compute experimental amplification curves and which were compared with results from linear stability theory (LST).

As will be shown in § 2, the numerical results for the porous-wall case in Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) were erroneous due to incorrect boundary conditions. The corrected LST results are presented in § 3, where we discuss the stability properties of a hypersonic boundary layer on a solid and on a porous wall around a sharp cone with zero angle of attack. The mean-flow parameters are given in table 1, where $M_{e}$ , $T_{e}^{\ast }$ and $\mathit{Re}_{1e}^{\ast }$ are the Mach number, temperature and unit Reynolds number at the boundary-layer edge respectively. Here, $\mathit{Re}_{1e}^{\ast }$ is defined as $\mathit{Re}_{1e}^{\ast }:=\sqrt{U_{e}^{\ast }/{\it\nu}_{e}^{\ast }}$ , where $U_{e}^{\ast }$ and ${\it\nu}_{e}^{\ast }$ are the velocity and kinematic viscosity at the boundary-layer edge respectively. Additionally, table 1 gives the chosen numerical values for the heat capacity ratio ${\it\gamma}$ , the Prandtl number $\mathit{Pr}$ , the gas constant $\mathscr{R}^{\ast }$ and the non-dimensional wall temperature $T_{w}:=T_{w}^{\ast }/T_{e}^{\ast }$ . In table 1 as well as hereafter, $(\cdot )^{\ast }$ , $(\cdot )_{e}$ and $(\cdot )_{w}$ represent dimensional quantities, variables at the boundary-layer edge and variables at the wall respectively.

Table 1.

Mean-flow parameters for the studied hypersonic boundary layers.

In our analysis we will use the same problem formulation and numerical framework for the LST computations as discussed by Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) with the exception of the corrected boundary conditions presented in § 2. We denote the non-dimensional mean-flow variables, i.e. pressure, temperature, density, streamwise and wall-normal velocity, as $P:=P^{\ast }/({\it\rho}_{e}^{\ast }U_{e}^{\ast 2})$ , $T:=T^{\ast }/T_{e}^{\ast }$ , ${\it\rho}:={\it\rho}^{\ast }/{\it\rho}_{e}^{\ast }$ , $U:=U^{\ast }/U_{e}^{\ast }$ and $V:=V^{\ast }/U_{e}^{\ast }$ respectively. The coordinate system $(x^{\ast },y^{\ast },z^{\ast })$ , used in the computations and representing the streamwise, wall-normal and spanwise direction, is non-dimensionalized with the boundary-layer length scale $l^{\ast }:=\sqrt{{\it\nu}_{e}^{\ast }x^{\ast }/U_{e}^{\ast }}$ , where $x^{\ast }$ is the distance measured from the tip of the cone along the cone surface. The reference time is given by $l^{\ast }/U_{e}^{\ast }$ and consequentially the non-dimensional time is $t:=t^{\ast }U_{e}^{\ast }/l^{\ast }$ and the non-dimensional circular frequency can be defined as ${\it\omega}:={\it\omega}^{\ast }l^{\ast }/U_{e}^{\ast }=2{\rm\pi}f^{\ast }l^{\ast }/U_{e}^{\ast }$ , where $f^{\ast }$ is the dimensional frequency. As the numerical framework is the same as in Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003, Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2015), the boundary-layer equations as well as the parallel and non-parallel linear stability equations used for the computations are given in Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003, Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2015) and are not repeated in this paper. In conclusion, this implies that the normal-mode ansatz of equation (4.9) of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003), where $u$ , $v$ , $w$ , $p$ and ${\it\theta}$ denote the disturbance amplitudes of the velocity components, of the pressure and of the temperature respectively, is used in this study. The computations in the remainder of the analysis are all performed in the framework of spatial linear stability analysis, hence we have ${\it\omega}\in \mathbb{R}$ for the circular frequency, ${\it\beta}:={\it\beta}^{\ast }l^{\ast }\in \mathbb{R}$ for the transverse wavenumber and ${\it\alpha}:={\it\alpha}^{\ast }l^{\ast }\in \mathbb{C}$ for the eigenvalue of the LST problem. The streamwise wavenumber can, hence, immediately be defined as ${\it\alpha}_{r}=\text{Re}({\it\alpha})$ .

2. Corrected porous-wall boundary condition

In the spatial linear stability calculations performed by Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) the acoustic properties of a thin coating, made of a fibrous absorbent material (felt-metal), were described by the semi-empirical model of Allard & Champoux (Reference Allard and Champoux1992) in conjunction with laboratory measurements of the felt-metal characteristics. Unfortunately, the equations describing the complex dynamic density $\widetilde{{\it\rho}}:={\it\rho}^{\ast }({\it\omega}^{\ast })/{\it\rho}_{w}^{\ast }$ , the complex dynamic compressibility $\widetilde{C}:={\it\gamma}P_{w}^{\ast }/K^{\ast }({\it\omega}^{\ast })$ , where $K^{\ast }({\it\omega}^{\ast })$ is the complex dynamic bulk modulus, and the propagation constant ${\it\Lambda}:={\it\Lambda}^{\ast }\,\cdot \,l^{\ast }$ were erroneous. The equations describing $\widetilde{{\it\rho}}$ and $\widetilde{C}$ , i.e. equations (4.34)–(4.36) of the study by Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003), were chosen identical to those of the work by Allard & Champoux (Reference Allard and Champoux1992) (equations (13) and (30) in conjunction with (46)), thereby neglecting the fact that the time dependences in the two studies are opposite. In the original work by Allard & Champoux (Reference Allard and Champoux1992) the harmonic time dependence was of the form $\text{exp}(\text{i}{\it\omega}t)$ ; however, Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) used a time dependence of the form $\text{exp}(-\text{i}{\it\omega}t)$ in the normal-mode ansatz in equation (4.9). This change in time dependence requires a change in the characteristic impedance from $Z_{0}:=Z_{0}^{\ast }/({\it\rho}_{e}^{\ast }U_{e}^{\ast })$ to $Z_{0}^{\dagger }$ , where $\dagger$ represents the complex conjugate. Similarly, it can be shown that the complex dynamic density $\widetilde{{\it\rho}}$ , the complex dynamic compressibility $\widetilde{C}$ , as well as the propagation constant ${\it\Lambda}$ and the wall admittance $A_{y}$ , are related via complex conjugation in both representations (Allard & Atalla Reference Allard and Atalla2009). This is a direct consequence of the fact that a real-valued pressure disturbance should yield a real-valued velocity disturbance, cf. Landau, Lifshitz & Pitaevskii (Reference Landau, Lifshitz and Pitaevskii1984) and Johnson, Koplik & Dashen (Reference Johnson, Koplik and Dashen1987).

Equations (4.34)–(4.36) in Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) should consequently be replaced by

(2.1a,b ) $$\begin{eqnarray}\widetilde{{\it\rho}}=a_{\infty }\left[1+\frac{g({\it\lambda}_{1})}{{\it\lambda}_{1}}\right],\quad \widetilde{C}={\it\gamma}-\frac{{\it\gamma}-1}{1+g({\it\lambda}_{2})/{\it\lambda}_{2}},\end{eqnarray}$$

(2.2) $$\begin{eqnarray}g({\it\lambda})=\sqrt{1+\frac{4a_{\infty }{\it\mu}_{w}^{\ast }{\it\lambda}}{{\it\sigma}^{\ast }{\it\phi}\,r_{p}^{\ast 2}}},\end{eqnarray}$$

(2.3a,b ) $$\begin{eqnarray}{\it\lambda}_{1}=-\text{i}a_{\infty }{\it\rho}_{w}^{\ast }{\it\omega}^{\ast }/({\it\phi}{\it\sigma}^{\ast }),\quad {\it\lambda}_{2}=4\mathit{Pr}{\it\lambda}_{1},\end{eqnarray}$$

where $a_{\infty }$ , ${\it\phi}$ , ${\it\sigma}^{\ast }$ , $r_{p}^{\ast }$ and ${\it\mu}_{w}^{\ast }$ represent the tortuosity of the porous medium, the porosity, the flow resistivity, the characteristic viscous length scale inside the porous coating and the dynamic viscosity at the wall respectively. It should be noted that the tortuosity represents the high-frequency asymptotic limit of the dynamic density, i.e. $a_{\infty }=\text{lim}_{{\it\omega}^{\ast }\rightarrow \infty }\,\widetilde{{\it\rho}}({\it\omega}^{\ast })$ . The definition of the flow resistivity can be found in equation (4.37) of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003), where furthermore the viscous length scale $r_{p}^{\ast }$ was correctly defined in equation (4.40) as

(2.4) $$\begin{eqnarray}r_{p}^{\ast }=\frac{{\rm\pi}d^{\ast }}{2(1-{\it\phi})(2-{\it\phi})},\end{eqnarray}$$

with $d^{\ast }$ referring to the diameter of the stainless steel fibres of the felt-metal coating. It should be highlighted that equation (4.39) for $r_{p}^{\ast }$ of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) was erroneous, see Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2015). The definition of the porosity ${\it\phi}$ is given in (A 7).

Equations (2.1)–(2.3) correspond to the porous-media model derived by Allard & Champoux (Reference Allard and Champoux1992), which can be considered to be a special case (applicable to fibrous absorbent materials only) of the model by Champoux & Allard (Reference Champoux and Allard1991). The model by Allard & Champoux (Reference Allard and Champoux1992) was derived for a material with non-touching fibres. Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) applied this model to a UAC with touching fibres and therefore decided to evaluate $r_{p}^{\ast }$ differently from Allard & Champoux (Reference Allard and Champoux1992). In Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) the pore size $r_{p}^{\ast }$ was treated as a hydraulic radius, which was defined as the radius of the equivalent pore. It was derived based on a statistical analysis (see appendix A).

Compared with the original work, changing the sign of ${\it\lambda}_{1}$ in equation (4.36) of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) results in (2.3); this makes the model compatible with the time dependence $\text{exp}(-\text{i}{\it\omega}t)$ and thereby yields the correct values for $\widetilde{{\it\rho}}$ and $\widetilde{C}$ .

The correct boundary conditions for the linear stability computations, replacing equations (4.31)–(4.33) of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003), can now be written in non-dimensional form as

(2.5a,b ) $$\begin{eqnarray}y=0:\quad u=w={\it\theta}=0,\quad v=A_{y}\,p,\end{eqnarray}$$

(2.6) $$\begin{eqnarray}A_{y}=-\frac{{\it\phi}}{Z_{0}}\tanh \!({\it\Lambda}\,h),\end{eqnarray}$$

(2.7a,b ) $$\begin{eqnarray}Z_{0}=\frac{\sqrt{\widetilde{{\it\rho}}/\widetilde{C}}}{M_{e}\sqrt{T_{w}}},\quad {\it\Lambda}=-\frac{\text{i}{\it\omega}M_{e}}{\sqrt{T_{w}}}\sqrt{\widetilde{{\it\rho}}\,\widetilde{C}},\end{eqnarray}$$

where $h:=h^{\ast }/l^{\ast }$ is the non-dimensional porous-layer thickness and where (2.7) is obtained after the non-dimensionalization of the characteristic impedance $Z_{0}^{\ast }$ and the propagation constant ${\it\Lambda}^{\ast }$ , given by $Z_{0}^{\ast }=\sqrt{{\it\rho}^{\ast }({\it\omega}^{\ast })K^{\ast }({\it\omega}^{\ast })}$ and ${\it\Lambda}^{\ast }=-\text{i}{\it\omega}^{\ast }\sqrt{{\it\rho}^{\ast }({\it\omega}^{\ast })/K^{\ast }({\it\omega}^{\ast })}$ . In the definition of the admittance $A_{y}$ in (2.6), the minus sign is required because an excess pressure induces a flow into the wall (in the $-y$ direction). In Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) the minus sign due to the direction of propagation was incorporated into the definition of $Z_{0}$ ; however, in the present work $\text{Re}(Z_{0})>0$ is preferred. Additionally, it should be highlighted that the sign of the propagation constant ${\it\Lambda}$ was changed in comparison to Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003), in order to make the boundary condition compatible with the normal-mode ansatz.

The numerical parameters needed for the evaluation of the boundary conditions and the porous-coating model are given in table 2, with the exception of the dynamic viscosity ${\it\mu}_{w}^{\ast }$ , which is given by Sutherland’s law as

(2.8) $$\begin{eqnarray}{\it\mu}_{w}^{\ast }=1.458\times 10^{-6}\,\frac{T_{w}^{\ast 3/2}}{T_{w}^{\ast }+C^{\ast }},\end{eqnarray}$$

with $C^{\ast }=110.4~\text{K}$ in accordance with the non-dimensional form of the Sutherland law, which was used in the boundary-layer and stability equations, i.e.

(2.9) $$\begin{eqnarray}{\it\mu}:={\it\mu}^{\ast }/{\it\mu}_{e}^{\ast }=\frac{1+S}{T+S}\,T^{3/2}\end{eqnarray}$$

with $S=110.4/T_{e}^{\ast }$ .

Table 2.

Numerical parameters describing the felt-metal coating.

It should be mentioned that, although equations (4.34)–(4.36) in Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) provided an incorrect representation of the porous coating, the identification of the felt-metal flow resistivity ${\it\sigma}^{\ast }$ via the method of acoustic standing waves provided the correct results. Despite the fact that the predicted imaginary parts of $Z_{0}$ , of $A_{y}$ and of the reflection coefficient for plane acoustic waves of normal incidence $R_{ref}$ , which can be expressed as

(2.10) $$\begin{eqnarray}R_{ref}:=\frac{M_{e}\sqrt{T_{w}}+A_{y}}{M_{e}\sqrt{T_{w}}-A_{y}},\end{eqnarray}$$

have the wrong sign in Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003), the results in figure 12 remain correct as the normal-incidence energy absorption coefficient $K_{a}:=1-|R_{ref}|^{2}=1-|R_{ref}^{\dagger }|^{2}$ is taken as the basis for the identification of ${\it\sigma}^{\ast }$ , which therefore is not affected by the erroneous time dependence. The experimentally obtained value of ${\it\sigma}^{\ast }$ in Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) can consequently be used in the present work in order to complete the semi-empirical model presented in equations (2.1)–(2.4).

3. Stability calculations and comparison with experiment

3.1. Linear stability calculations

In this section the corrected version of the erroneous linear stability results of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) is presented. Figures 1, 2 and 4 present the corrected results of figures 13, 14 and 17 of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) respectively. Figure 1(a) depicts the growth rate of two-dimensional disturbances ${\it\sigma}_{{\it\alpha}}^{\ast }(x^{\ast },f^{\ast },{\it\beta}^{\ast }=0)$ in a boundary layer corresponding to the mean-flow parameters given in table 1 at a position $x^{\ast }$ equivalent to a Reynolds number $R:=U_{e}^{\ast }\,l^{\ast }/{\it\nu}_{e}^{\ast }=\sqrt{U_{e}^{\ast }\,x^{\ast }/{\it\nu}_{e}^{\ast }}$ of $R=1761.2$ . The growth rate ${\it\sigma}_{{\it\alpha}}^{\ast }(x^{\ast },f^{\ast },{\it\beta}^{\ast })$ is defined as ${\it\sigma}_{{\it\alpha}}^{\ast }:=-\text{Im}({\it\alpha}^{\ast })$ for the case of locally parallel LST. For the definition of ${\it\sigma}_{{\it\alpha}}^{\ast }$ applicable in the case of the non-parallel LST, we refer to Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003). Additionally, figure 1 presents the maximum growth rate of three-dimensional disturbances, i.e. $\textstyle {\it\sigma}_{{\it\alpha},max}^{\ast }:=\max _{{\it\beta}^{\ast }}[{\it\sigma}_{{\it\alpha}}^{\ast }(x^{\ast },f^{\ast },{\it\beta}^{\ast })]$ , and the corresponding wave angle ${\it\psi}^{\ast }$ . Figure 2 presents similar results for a different position in the boundary layer. Figures 1 and 2 correspond to the experimental conditions of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003). Whereas these conditions represent a nearly adiabatic wall temperature, figure 4 represents a scenario with a cooled wall.

Figure 1.

(a) Growth rate of two-dimensional disturbances versus frequency at $x^{\ast }=200.1$ mm ( $R=1761.2$ ); locally parallel approximation. (b) Maximum growth rate of three-dimensional disturbances versus frequency at $x^{\ast }=200.1$ mm ( $R=1761.2$ ); locally parallel approximation. (c) The wave angle ${\it\psi}^{\ast }=\arctan \!({\it\beta}/{\it\alpha}_{r})$ of the most unstable three-dimensional waves versus frequency at $x^{\ast }=200.1$ mm ( $R=1761.2$ ); locally parallel approximation. Dashed lines: solid wall; solid lines: porous wall with the corrected boundary condition. Correction of figure 13 of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003).

Figure 2.

(a) Growth rate of two-dimensional disturbances versus frequency at $x^{\ast }=283.2$  mm ( $R=2095$ ); locally parallel approximation. (b) Maximum growth rate of three-dimensional disturbances versus frequency at $x^{\ast }=283.2$ mm ( $R=2095$ ); locally parallel approximation. Dashed lines: solid wall; solid lines: porous wall with the corrected boundary condition. Correction of figure 14 of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003).

The stabilization of the second-mode disturbances due to the porous coating is also observed with the corrected boundary conditions in figures 1, 2 and 4, similarly to the original results. Furthermore, it can be seen that the porous medium shifts the maximum growth rate associated with second-mode disturbances to lower frequencies. This observation is in accordance with earlier studies of the stabilization effect of porous coatings with regular microstructure, i.e. circular cylindrical holes, performed by Fedorov et al. (Reference Fedorov, Malmuth, Rasheed and Hornung2001), whose results remain unaffected by the present analysis. The shift of the second-mode instability to lower frequencies therefore seems to be independent of the particular porous coating under investigation. This conclusion is in disagreement with the originally presented results, which indicated that the felt-metal coating shifts the second-mode instability to higher frequencies. This difference can be observed best when figure 17 of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) is compared with the present figure 4.

As a direct consequence of the observed frequency shift, the changeover from the oblique first-mode instability to the two-dimensional second-mode instability takes place at lower frequencies, as can be observed by comparing the results of figure 13(c) of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) with figure 1(c). Apart from the shift of the frequency, a considerably increased value of the maximum second-mode growth rate is observed with the corrected boundary condition, indicating a strong dependence of the stabilization on the phase of the wall admittance and thereby indicating that the norm of the reflection coefficient of the wall is not a good criterion for the design of porous coatings.

The corrected linear stability results help to explain the presence of the low-frequency lobe in the disturbance spectrum for natural disturbances measured on the porous surface (see figure 4(b) of Fedorov et al. Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003). This lobe in the frequency band from $100$ to $200~\text{kHz}$ was originally attributed to the destabilization of the first mode. As a consequence of the aforementioned corrected results, it can be concluded that the presence of this lobe is associated with two independent trends: along with the increase of the first-mode growth rate there is a shift of the second-mode instability to lower frequencies, as can be seen in figure 2(a). The importance of this shift is emphasized if the ratio $A_{f}(\text{S5})/A_{f}(\text{S1})$ is analysed. This corresponds to the ratio of the disturbance amplitude at location S5 ( $x^{\ast }=289.4$  mm) and location S1 ( $x^{\ast }=189.1$  mm), near the leading edge of the porous surface ( $x^{\ast }=187$  mm), where the experimental mass-flow disturbance amplitude $A_{f}(x^{\ast },{\it\Theta}^{\ast },f^{\ast })$ for natural disturbances and its corresponding phase ${\it\Phi}_{f}^{\ast }$ are computed via the discrete Fourier transform, i.e.

(3.1) $$\begin{eqnarray}A_{f}(x^{\ast },{\it\Theta}^{\ast },f^{\ast })\,\text{exp}(\text{i}{\it\Phi}_{f}^{\ast }(x^{\ast },{\it\Theta}^{\ast },f^{\ast })):=\frac{2}{N}\mathop{\sum }_{j=1}^{N}{\dot{m}}_{max}^{\prime }(x^{\ast },{\it\Theta}^{\ast },t_{j}^{\ast })\,\text{exp}(-2{\rm\pi}\text{i}\,f^{\ast }\,t_{j}^{\ast }).\end{eqnarray}$$

Here, $N$ denotes the sample count, ${\it\Theta}^{\ast }$ is the circumferential angle on the cone and ${\dot{m}}_{max}^{\prime }$ is the maximum of the non-dimensional mass-flow disturbance ${\dot{m}}^{\prime }$ , which is defined as ${\dot{m}}^{\prime }:={\it\rho}^{\prime }U+{\it\rho}U^{\prime }$ , where ${\it\rho}^{\prime }$ and $U^{\prime }$ are the experimentally measured density disturbance and velocity disturbance respectively. In the experiments the maximum of the mass-flow disturbance is computed over the coordinate normal to the axis of the cone. Analogously to $A_{f}$ and ${\it\Phi}_{f}^{\ast }$ for natural disturbances, the amplitude $A(x^{\ast },{\it\Theta}^{\ast })$ and the phase ${\it\Phi}^{\ast }(x^{\ast },{\it\Theta}^{\ast })$ can be defined for artificially excited disturbances of a given frequency $f^{\ast }$ . The disturbance amplitudes plotted in figure 4 of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) were measured at a circumferential angle of ${\it\Theta}=0^{\circ }$ .

The ratio $A_{f}(\text{S5})/A_{f}(\text{S1})$ , which corresponds to the growth of the instability on the porous surface, has a peak between $150$ and $250~\text{kHz}$ and can most likely be associated with a second mode, see figure 2(a). However, the presence of the lobe cannot be attributed definitely to one or the other mode, as the frequency ranges of both modes are overlapping for the porous-wall case, which is also apparent from the computational results, e.g. figure 2(a). This conclusion is drastically different from the conclusion made by Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003), who believed that the low-frequency lobe is due entirely to first-mode instabilities.

Figure 3.

Amplification of the two-dimensional component of the artificially excited wavepacket of frequency $f^{\ast }=280$  kHz, i.e. $\text{SA}^{\ast }(x^{\ast },{\it\beta}_{exp}^{\ast }=0)$ , versus the theoretical amplification curve $A_{num}$ ; (a) locally parallel theory versus experiment, (b) non-parallel theory versus experiment. Theoretical result for the solid-wall case (dashed lines), theoretical result for the porous-wall case with the corrected boundary condition (dot-dashed lines), theoretical result for the porous-wall case as presented in Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) (solid lines), experiment on the solid wall (dashed lines with ▵ symbol), experiment on the porous wall (solid lines with ○ symbol). Correction of figures 15 and 16 of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003).

Figure 4.

Growth rate of two-dimensional disturbances as a function of frequency $f^{\ast }$ at various $x^{\ast }$ , $T_{w}^{\ast }=2~T_{e}^{\ast }$ , $Re_{1e}^{\ast }=10^{7}~\text{m}^{-1}$ ; locally parallel approximation; (i)  $x^{\ast }=206.35$  mm, (ii)  $x^{\ast }=298.72$  mm, (iii)  $x^{\ast }=406.05$  mm, (iv) $x^{\ast }=515.79$  mm. Dashed lines: solid wall; solid lines: porous wall with the corrected boundary condition. Correction of figure 17 of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003).

3.2. Comparison of numerical and experimental amplification curves

Figures 3(a) and (b) present the correction of figures 15 and 16 of the original paper and present a comparison between theoretical and experimental amplification curves. The experimental amplification curve represents the amplification of the two-dimensional component of the artificially excited wavepacket, denoted $\text{SA}^{\ast }$ $(x^{\ast },{\it\beta}_{exp}^{\ast }=0)$ , where ${\it\beta}_{exp}^{\ast }$ is the azimuthal wavenumber. The azimuthal wavenumber ${\it\beta}_{exp}^{\ast }$ used in the experiments ( $[{\it\beta}_{exp}^{\ast }]=\text{rad}~\text{deg}^{-1}$ ) should be discerned from the transverse wavenumber ${\it\beta}^{\ast }$ used in the linear stability calculations ( $[{\it\beta}^{\ast }]=\text{rad}~\text{m}^{-1}$ ). The amplitude $\text{SA}^{\ast }$ of the azimuthal wavenumber ${\it\beta}_{exp}^{\ast }$ component and its corresponding phase $\text{SF}^{\ast }$ can be determined via

(3.2) $$\begin{eqnarray}\text{SA}^{\ast }(x^{\ast },{\it\beta}_{exp}^{\ast })\text{exp}(\text{i}\text{SF}^{\ast }(x^{\ast },{\it\beta}_{exp}^{\ast })):=\int _{-{\it\Theta}_{0}^{\ast }}^{{\it\Theta}_{0}^{\ast }}A(x^{\ast },{\it\Theta}^{\ast })\text{exp}(\text{i}({\it\Phi}^{\ast }(x^{\ast },{\it\Theta}^{\ast })-{\it\beta}_{exp}^{\ast }{\it\Theta}^{\ast }))\text{d}{\it\Theta}^{\ast },\end{eqnarray}$$

where the limit of integration ${\it\Theta}_{0}^{\ast }$ was chosen in order to cover the entire wavepacket in the experiments, and where the amplitude $A(x^{\ast },{\it\Theta}^{\ast })$ was calculated for artificially exited disturbances with a frequency of $f^{\ast }=280~\text{kHz}$ . The experimentally obtained amplification curve is compared with the numerical amplification curve $A_{num}(x^{\ast })$ , defined as

(3.3) $$\begin{eqnarray}A_{num}(x^{\ast }):=\,A_{num,0}\text{exp}\left(\int _{x_{1}^{\ast }(S_{1})}^{x^{\ast }}{\it\sigma}_{{\it\alpha}}^{\ast }(\widetilde{x}^{\ast },f_{0}^{\ast },0)\text{d}\widetilde{x}^{\ast }\right),\end{eqnarray}$$

where $A_{num,0}$ is the amplitude at position S1, which is adjusted to match the experimental results. The local growth rate for two-dimensional disturbances, is determined by either the locally parallel LST or the non-parallel LST. For the results presented in figure 3, the local growth rate was computed for a fixed frequency of $f^{\ast }=f_{0}^{\ast }=280~\text{kHz}$ and for the mean-flow parameters given in table 1.

Comparing the experimental and numerical results in figure 3, we can see that with the corrected boundary condition, the theoretical results on the porous wall based on the parallel flow assumption are in better agreement with the experimental measurements than the originally published results (see figure 3(a)). In opposition to this improved agreement for the locally parallel theory, the results based on the weakly non-parallel theory show a considerable discrepancy compared with the experimental data for the porous-wall case, for which in Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003) a nearly perfect agreement was erroneously observed (see figure 3(b)). Compared with the locally parallel theory, the non-parallel theory predicts a smaller amplification for the solid wall and for the porous wall. This observation is in opposition to the results of Fedorov et al. (Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003), where the non-parallel theory predicted an increased amplification for the porous-wall case. The fact that for the porous-wall case the presumably more accurate non-parallel theory shows less satisfactory agreement with the experimental results needs to be analysed in the future. At this point, one needs to be reminded that the growth rate of the non-parallel theory is dependent on the distortion of the eigenfunction and therefore is affected by the choice of the disturbance norm and the location of its evaluation. In the present work, the disturbance measure was given by the mass-flow disturbance, which was evaluated at the wall-normal location of the maximum mass-flow disturbance. Additionally, it is reasonable to mention that the distributed roughness of the felt-metal surface could lead to additional growth of the boundary-layer disturbances. In the experiments, acoustic noise present in the hypersonic wind tunnel could interact with this roughness, which could lead to a downstream increase in the disturbance amplitude. The linear stability theory does not account for this effect, which could be present in the experiment. Therefore, it is natural to assume that the stability theory can overpredict the UAC performance. This is consistent with the results shown in figure 3(b).