Stabilization of a hypersonic boundary layer using an ultrasonically absorptive coating – CORRIGENDUM

1. Error in the complex representation of the porous-coating model

The authors are grateful to Mr R. C. Tritarelli and Professor S. K. Lele for highlighting a significant error in their analysis of the stabilization effect of felt-metal coatings on hypersonic boundary layers (Fedorov et al. 2003).

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 black lines: porous wall with the corrected boundary condition; solid red lines: erroneous porous-wall result as presented in Fedorov et al. (2003). Correction of figure 13 of Fedorov et al. (2003) as presented in Tritarelli, Lele & Fedorov (2015) in combination with the original erroneous results.

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 black lines: porous wall with the corrected boundary condition; solid red lines: erroneous porous-wall result as presented in Fedorov et al. (2003). Correction of figure 14 of Fedorov et al. (2003) as presented in Tritarelli et al. (2015) in combination with the original erroneous results.

Figure 3. Growth rate of two-dimensional disturbances as a function of frequency $f^{\ast }$ at various $x^{\ast }$ , $T_{w}^{\ast }=2T_{e}^{\ast }$ , $Re_{1e}^{\ast }=10^{7}~\text{m}^{-1}$ ; 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 black lines: porous wall with the corrected boundary condition; solid red lines: erroneous porous-wall result as presented in Fedorov et al. (2003). Correction of figure 17 of Fedorov et al. (2003) as presented in Tritarelli et al. (2015) in combination with the original erroneous results.

As discussed in Tritarelli et al. (2015), the complex representation of the porous-coating model used in Fedorov et al. (2003) was in discordance with the time convention used in the normal-mode ansatz of the linear stability analysis. The numerical results from Fedorov et al. (2003) are hereby negated.

For the sake of completeness, the corrected equations, which were already presented and discussed in Tritarelli et al. (2015), are repeated in this corrigendum. For a detailed discussion we refer to Tritarelli et al. (2015).

The equations describing the felt-metal porous-coating model in Fedorov et al. (2003), i.e. equations (4.34)–(4.36), have to be substituted by

(1.1a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{{\it\rho}}=a_{\infty }\left[1+\frac{g({\it\lambda}_{1})}{{\it\lambda}_{1}}\right],\quad \widetilde{C}={\it\gamma}-\frac{{\it\gamma}-1}{1+g({\it\lambda}_{2})/{\it\lambda}_{2}}, & \displaystyle\end{eqnarray}$$

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

(1.3a,b ) $$\begin{eqnarray}\displaystyle \displaystyle {\it\lambda}_{1}=-\text{i}a_{\infty }{\it\rho}_{w}^{\ast }{\it\omega}^{\ast }/({\it\phi}{\it\sigma}^{\ast }),\quad {\it\lambda}_{2}=4\text{Pr}{\it\lambda}_{1}. & & \displaystyle\end{eqnarray}$$

Additionally, the boundary conditions need to be replaced in order to be consistent with the time convention used. This implies that (4.31)–(4.33) should be reformulated as follows:

(1.4a,b ) $$\begin{eqnarray}\displaystyle \displaystyle y=0:\quad u=w={\it\theta}=0,\quad v=A_{y}\,p, & & \displaystyle\end{eqnarray}$$

(1.5) $$\begin{eqnarray}\displaystyle A_{y}=-\frac{{\it\phi}}{Z_{0}}\tanh ({\rm\Lambda}h), & & \displaystyle\end{eqnarray}$$

(1.6a,b ) $$\begin{eqnarray}\displaystyle \displaystyle Z_{0}=\frac{\sqrt{\widetilde{{\it\rho}}/\widetilde{C}}}{M_{e}\sqrt{T_{w}}},\quad {\rm\Lambda}=-\frac{\text{i}{\it\omega}M_{e}}{\sqrt{T_{w}}}\sqrt{\widetilde{{\it\rho}}\,\widetilde{C}}. & & \displaystyle\end{eqnarray}$$

The corrected version of the figures of Fedorov et al. (2003), which is presented in Tritarelli et al. (2015), is partially reprinted in figures 1–3 in combination with the original erroneous results in order to show the impact of the corrected boundary conditions. Figures 1–3 present the corrected versions of figures 13, 14 and 17 of the original paper. For the correction of figures 15 and 16 we refer to Tritarelli et al. (2015). The physical interpretations of the new results regarding the effect of felt-metal coatings on hypersonic boundary layers are discussed in Tritarelli et al. (2015). In addition to the erroneous complex representation of the porous-coating model, several typographical errors existed in the original work, which are partially corrected in appendix A.