2.1. Compressible Rayleigh equation

We start from the compressible linearised Euler equations. As the focus lies on the inviscid acoustic modes in BLs of high Reynolds numbers, the effects of heat conduction on the disturbances are neglected (Rienstra & Hirschberg Reference Rienstra and Hirschberg2001; Delfs Reference Delfs2016). Under these assumptions, which are particularly valid for high frequency ranges (Aurégan et al. Reference Aurégan, Starobinski and Pagneux2001; Brambley Reference Brambley2011), the linearised Euler equations (LEEs) can be written in the form (Criminale, Jackson & Joslin Reference Criminale, Jackson and Joslin2018; Görtz Reference Görtz2025)

(2.1a) \begin{align} \frac {{\rm D}_0 \rho ^{\prime }}{{\rm D} t}+\rho _0 \boldsymbol{\nabla } \boldsymbol \cdot \boldsymbol{v}^{\prime }+\boldsymbol{v}^{\prime } \boldsymbol \cdot \boldsymbol{\nabla }\rho _0+\rho ^{\prime } \boldsymbol{\nabla } \boldsymbol \cdot \boldsymbol{v}_0 &=0 , \end{align}

(2.1b) \begin{align} \rho _0 \frac {{\rm D}_0 \boldsymbol{v}^{\prime }}{{\rm D} t}+\boldsymbol{\nabla }p^{\prime }+\rho _0 \boldsymbol{v}^{\prime } \boldsymbol \cdot \boldsymbol{\nabla }\boldsymbol{v}_0+\rho ^{\prime } \boldsymbol{v}_0 \boldsymbol \cdot \boldsymbol{\nabla } \boldsymbol{v}_0 &=0 , \end{align}

(2.1c) \begin{align} \frac {1}{c_0^{2}}\left (\frac {{\rm D}_0 p^{\prime }}{{\rm D} t}+\boldsymbol{v}^{\prime } \boldsymbol \cdot \boldsymbol{\nabla }p_0\right )+\rho _0 \boldsymbol{\nabla } \boldsymbol \cdot \boldsymbol{v}^{\prime }+\left (\rho _0 \frac {c^{\prime 2}}{c_0^{2}}+\rho ^{\prime }\right ) \boldsymbol{\nabla } \boldsymbol \cdot \boldsymbol{v}_0 &=0 , \end{align}

representing a set of partial differential equations (PDEs) for the small unsteady perturbations, denoted by prime, linearised around the mean flow state, indexed by 0. Here, $p$ denotes the pressure, $\rho$ the density and $\boldsymbol{v}=(u,v,w)^T$ the velocity vector, while $c$ is the speed of sound and $ {{\rm D}_0}/{{\rm D} t}:=({\partial }/{\partial t})+ (\boldsymbol{v}_0 \boldsymbol \cdot \boldsymbol{\nabla })$ the material derivative with respect to the mean flow.

In view of the BL problem, the base flow is assumed as a plane shear layer $\boldsymbol{v}_0=U_0(y) \cdot \boldsymbol{e}_x$ , depending only on the wall-normal direction $y$ , with $y=0$ denoting the wall surface. Furthermore, the base flow is considered to be non-isentropic, non-isothermal with a mean temperature gradient $T_0(y)$ normal to the wall and constant pressure $p_0$ . For ideal gas, the equation of state thus gives

(2.2) \begin{equation} \rho _0(y)=\frac {p_0}{RT_0(y)} \end{equation}

and the speed of sound can be expressed by

(2.3) \begin{equation} c_0^{2}(y)=\gamma R T_0(y), \end{equation}

with $R$ as the ideal gas constant and $\gamma$ the isentropic exponent.

The LEEs resulting for these BL assumptions are non-dimensionalised by using the BL thickness $\delta$ , the free stream velocity $U_{\infty }$ and the mean pressure $p_0$ scaled with $\gamma$ , giving as reference scales for the dynamic quantities

(2.4) \begin{equation} p_{\textit{ref}}=p_0 \gamma ,\qquad \rho _{\textit{ref}}=\frac {p_0\gamma M^2}{U_\infty ^2},\qquad T_{\textit{ref}}=T_\infty =\frac {U_\infty ^2}{\gamma R M^2}. \end{equation}

As a result, in the non-dimensionalised LEE system, the Mach number $M=U_{\infty }/c_{\infty }$ arises as dimensionless parameter, with $c_{\infty }$ referring to the speed of sound in the far field. Note that from now on all quantities are non-dimensionalised.

Using the homogeneity of the equations with respect to $x$ , $z$ and $t$ , the perturbations can be expressed by a normal mode approach according to

(2.5) \begin{align} q^\prime (x,y,z,t)=\hat {q}(y)e^{i(\alpha x+\beta z-\omega t)}, \quad \text{with} \; q^\prime \in \left \{u^\prime ,v^\prime , w^\prime ,p^\prime ,\rho ^\prime , T^\prime \right \} , \end{align}

where $\hat {q}(y)$ is the complex amplitude of the dimensionless perturbations. It is a function only of the wall-normal coordinate $y$ . In the context of temporal stability analysis as intended in this work, $\omega$ is assumed to be complex, i.e. $\omega \in \mathbb{C}$ , with its imaginary part $\omega _i$ representing the temporal growth rate. Due to the choice of signs in (2.5), $\omega _i\gt 0$ implies unstable exponential growth. The real part $\omega _r$ corresponds to the temporal frequency. Here, $\alpha , \beta \in \mathbb{R}$ describe the dimensionless streamwise and spanwise wavenumbers. Inserting (2.5) into the non-dimensionalised LEEs transforms the PDE system into a system of ODEs for the perturbation amplitudes. Decoupling this ODE system leads to a single second-order ODE for the pressure perturbation amplitude $\hat {p}(y)$ can be derived, the compressible Rayleigh equation (Criminale et al. Reference Criminale, Jackson and Joslin2018),

(2.6) \begin{equation} \frac {{\rm d}^2 \hat p }{{\rm d}y^2}+\left [\frac {2\alpha U_0^{\prime}(y)}{\omega -\alpha U_0(y)}+\frac {T_0^{\prime}(y)}{T_0(y)}\right ]\frac {{\rm d} \hat p }{{\rm d}y}+\left [\frac {M^2\left (\omega -\alpha U_0(y)\right )^2}{T_0(y)}-\alpha ^2-\beta ^2\right ]\hat p(y)=0 , \end{equation}

where from now on, the prime denotes the derivative with respect to $y$ . From $\hat {p} (y)$ , the perturbation amplitudes of velocity, density and temperature follow from the decoupled ODEs, reading

(2.7a) \begin{align} \hat {u}&=-\frac {U_0^{\prime}\,T_0}{M^{2}(\omega -U_0\alpha )^{2}}\,\frac {{\rm d}\hat p }{{\rm d}y}+\frac {\alpha T_0}{M^{2}(\omega -U_0\alpha )}\hat {p}, \end{align}

(2.7b) \begin{align} \hat {v}&=-\frac {i T_0}{M^{2}(\omega -U_0\alpha )}\,\frac {{\rm d} \hat p }{{\rm d}y}, \end{align}

(2.7c) \begin{align} \hat {w}&=\frac {\beta \, T_0}{M^{2}(\omega -U_0\alpha )}\,\hat p(y), \end{align}

(2.7d) \begin{align} \hat {\rho }&=\frac {1}{i(\omega -U_0\alpha )}\left [\frac {1}{T_0}\left (i\alpha \,\hat u + \frac {{\rm d}\hat v}{{\rm d}y} \right )+\hat v\frac {{\rm d}}{{\rm d}y}\left (\frac {1}{T_0}\right )\right ]\! , \end{align}

(2.7e) \begin{align} \hat {T}&=\gamma \, T_0 \hat p -T_0^2\hat \rho . \end{align}

2.2. Boundary conditions

A central part of the physics to be investigated subsequently lies in the problem-specific BCs for $\hat {p}$ . Regarding the behaviour in the far field ( $y\to \infty$ ), we require bounded perturbation amplitudes (Mack Reference Mack1984), as we focus on the stability behaviour of the BL itself without energy transport from the free stream into the BL. It should be noted that other authors, such as Brazier-Smith & Scott (Reference Brazier-Smith and Scott1984), Huerre & Monkewitz (Reference Huerre and Monkewitz1985), Crighton (Reference Crighton1989), Riedinger et al. (Reference Riedinger, Le Dizès and Meunier2010) and Brambley & Gabard (Reference Brambley and Gabard2014), suggest a different far-field condition grounded on causality arguments, which allows unbound behaviour in the far field for neutrally stable or damped modes ( $\omega _i\lt 0$ ). However, with respect to temporally unstable perturbations ( $\omega _i\gt 0$ ), there is agreement on the boundedness of the perturbations at infinity. In the following study, we will restrict our analysis to unstable disturbances due to their relevance for the BL transition.

A stricter requirement, also commonly used in stability theory, is the constraint of vanishing amplitudes in the far field, which gives us the boundary condition

(2.8) \begin{equation} \lim _{y\to \infty }\hat {p }(y)=0 . \end{equation}

It should be mentioned that for our model equations, boundedness of the far-field amplitudes can only be realised by vanishing amplitudes in the case of $\omega _i\neq 0$ , as we will see later in § 3.2. This implies that the same eigenvalues of the stability problem would also arise under the less strict far-field condition requiring bounded perturbations.

To describe the effect of acoustic metasurfaces like a porous wall on the BL disturbances, we introduce the complex acoustic wall impedance $Z$ , which links the pressure perturbation to the wall-normal velocity perturbation at the wall (Hubbard Reference Hubbard1991; Malmuth et al. Reference Malmuth, Fedorov, Shalaev, Cole, Hites, Williams and Khokhlov1998), which can be expressed in the frequency–wavenumber domain as

(2.9) \begin{equation} \hat {p}(y; \alpha , \beta , \omega )=-Z\,\hat {v}(y; \alpha , \beta , \omega ) \quad \text{at }\, y=0 . \end{equation}

The latter form results from the impedance wall condition given in the Laplace domain for vanishing mean wall flow, as found, for example, in Malmuth et al. (Reference Malmuth, Fedorov, Shalaev, Cole, Hites, Williams and Khokhlov1998) and Rienstra (Reference Rienstra2006). This is Fourier transformed under the assumptions of a straight wall with homogeneous impedance $Z$ , yielding (2.9). The wall impedance $Z$ itself is a function of the disturbance frequency and wavenumber, as well as on the wall properties and characteristics of the mean flow, like the Mach number (Malmuth et al. Reference Malmuth, Fedorov, Shalaev, Cole, Hites, Williams and Khokhlov1998; Zhao et al. Reference Zhao, Wen, Zhou, Tu and Lei2022b ). From (2.9), we derive a second BC for the pressure amplitude $\hat {p}$ by expressing $\hat {v}$ by $\hat {p}$ via (2.7b ), which yields

(2.10) \begin{equation} \left .\frac {{\rm d}\hat {p}}{{\rm d}y}\right |_{y=0}=-i\omega \,Y\frac {M^2}{T_w}\hat {p}(0), \, \end{equation}

where $Y=Z^{-1}$ is the acoustic admittance. Note that the limit $Z\to \infty$ or $Y=0$ corresponds to the case of a rigid wall (Campos & Serrão Reference Campos and Serrão1998).

Both BCs (2.8) and (2.10) together with the CRE (2.6) form the boundary value problem for the acoustic disturbances in the considered BL flow, providing an eigenvalue problem for the normal mode parameters $(\alpha , \beta , \omega )$ , ensuring compatibility of the CRE solutions with the BCs.

3.1. Velocity and temperature profile

To model the mean velocity in the BL flow, we adopt an exponential shear profile

(3.1) \begin{align} U_0(y)=1-e^{-y} , \end{align}

whereby (3.1) is the dimensionless form scaled with $U_{\infty }$ and $\delta$ . The advantage of describing the velocity profile explicitly as an exponential function is that it allows us to derive an analytical solution for the underlying CRE, which in turn enables the BL stability problem to be recast as an algebraic eigenvalue equation. Building on this equation, together with an algebraic solution algorithm, we are able to carry out comprehensive eigenvalue analyses over a wide parameter range without the occurrence of spurious modes. It should be noted, however, that the exponential profile (3.1) does not correspond to a solution of the compressible BL equations, but rather represents a simplifying model assumption. Due to this simplification, the results obtained in this work may deviate quantitatively in their absolute values from those obtained for a more realistic BL profile. Nevertheless, the observed qualitative behaviour is expected to be retained even under other, more common profiles, as explained at the beginning of 5.

A brief remark is additionally required concerning the definition of the boundary-layer thickness $\delta$ : from (3.1), it follows that at the height of $y=1$ , corresponding to a wall distance of $\delta$ , we have $U_0(1)=1-\mathrm{exp}(-1)\approx 0.63$ , i.e. $63\,\%$ of the free stream velocity $U_\infty$ . Thus, the BL thickness $\delta$ in this work does not represent the $\delta _{99}$ thickness.

Regarding the temperature profile $T_0(y)$ , we formulate a quadratic relationship to the velocity $U_0(y)$ . This goes back to the temperature relation derived by Crocco’s method (Crocco Reference Crocco1948; Van Driest Reference Van Driest1952a , Reference Van Driestb ) for a flat plate BL flow without pressure gradient, which can be written, in case of Prandtl number ${\textit{Pr}}=1$ , as

(3.2) \begin{equation} T_0(y)=T_w+(T_{\textit{a}w}-T_w)U_0(y)+(1-T_{\textit{a}w})U_0(y)^2 . \end{equation}

The latter form is also known as the Crocco temperature–velocity relation. Here, $T_w$ describes the wall temperature and $T_{\textit{a}w}$ the adiabatic wall temperature, which refers to the wall temperature in the case of an insulated wall. With regards to the Crocco relation (3.2), we note that its derivation assumes that the mean velocity profile satisfies the compressible Blasius boundary layer equations (Van Driest Reference Van Driest1952b ), which is not the case for the exponential profile adopted in the present study. Nevertheless, the temperature profile obtained from (3.2) when the exponential velocity profile is inserted reproduces the principal qualitative characteristics of the Crocco temperature profile associated with the Blasius solution sufficiently well. We therefore regard the exponential velocity profile, together with the resulting temperature profile, as an adequate model to represent the key physical features of the BL. This choice should be understood in view of the purpose of the present work to formulate a model problem that allows the fundamental effects to be investigated in a qualitative manner. In the non-dimensional relation (3.2), both $T_w$ and $T_{\textit{a}w}$ are dimensionless parameters, scaled by the far-field temperature $T_{\infty }$ . For ${\textit{Pr}}=1$ , the adiabatic wall temperature is determined by

(3.3) \begin{equation} T_{\textit{a}w}=1+\frac {\gamma -1}{2}M^2 , \end{equation}

where $\gamma$ is the specific heat capacity ratio, with $\gamma \approx 1.4$ for air.

In the context of this work, we propose a modified quadratic temperature–velocity relation

(3.4) \begin{equation} T_0(y)=T_w\,\left (1-\tilde T\,U_0(y)\right )^2 \quad \text{with}\quad \tilde {T}=1-\sqrt {\frac {1}{T_w}} , \end{equation}

which allows the derivation of an analytical solution of the CRE, as discussed in the next section.

Comparison of the simplified temperature relation (3.4) with the full Crocco relation (3.2) shows that the degree of qualitative agreement depends on the wall temperature $T_w$ relative to $T_{\textit{a}w}$ . This can be seen in figure 1, which compares the simplified temperature relation (solid line) with the full Crocco relation (dashed line) plotted over the wall-normal coordinate $y$ , both being evaluated using the exponential velocity profile. The two temperature profiles are shown for different wall temperature–Mach number configurations: (a) the insulated-wall case, $T_w = T_{\textit{a}w}$ , assumed throughout this study, shown for three Mach numbers $M=0.8, 2.5, 3$ ; and, for comparison, (b) a cooled-wall case with $T_w = 0.5\,T_{\textit{a}w}$ at $M=3$ . The figure reveals that, for the adiabatic wall cases ( $T_w = T_{\textit{a}w}$ ), both temperature profiles agree sufficiently well for the purposes of the model problem examined in the present work. This, however, does not hold for the cooled-wall case, where a local maximum appears in the full Crocco profile that is not reproduced by the simplified profile. It can therefore be concluded that the simplified temperature relation captures the qualitative shape of the full Crocco relation sufficiently well only for wall-temperature ratios $T_w/T_{\textit{a}w}$ for which the full Crocco profile does not exhibit a maximum within the BL. From this requirement, limits on permissible wall temperatures can be derived. For the Crocco relation (3.2), the maximum occurs at

(3.5) \begin{equation} U_{0,{\textit{max}}}=\frac {T_{\textit{a}w}-T_w}{2(T_{\textit{a}w}-1)} . \end{equation}

To ensure that this maximum does not occur within the BL region where $U_0\in (0,1)$ applies, we require $ U_{0,{\textit{max}}} \geqslant 1$ or $ U_{0,{\textit{max}}}\leqslant 0$ , which together with (3.5) results in the wall temperature limits

(3.6) \begin{equation} T_w \leqslant 2-T_{\textit{a}w} \quad \! \! \text{for} \; M\lt \sqrt {5} \quad \text{or} \quad T_w \geqslant T_{\textit{a}w} . \end{equation}

The restriction $M\lt \sqrt {5}$ for the lower limit results from the requirement of non-negative temperatures, $T_w \gt 0$ , as becomes evident when substituting the relation for $T_{\textit{a}w}$ with $\gamma =1.4$ into (3.6). Accordingly, (3.6) represent those ranges for $T_w$ for which the Crocco relation (3.2) does not show a local maximum in the BL domain, allowing our simplified relation (3.4) to be used as an approximation.

Figure 1.

Comparison of the full Crocco temperature relation (3.2) (dashed line) with the modified relation (3.4) (solid line) for adiabatic wall BLs of different Mach numbers and for a cooled-wall BL.

Figure 1. Long description

The line graph compares the full Crocco temperature relation (3.2) with the modified relation (3.4) for adiabatic wall boundary layers of different Mach numbers and for a cooled-wall boundary layer. The x-axis represents the temperature ratio (T), ranging from 1 to 3, while the y-axis represents the distance (y), ranging from 0 to 6. Four data lines are plotted: one for M equals 0.8, T_w equals T_aw; one for M equals 2.5, T_w equals T_aw; one for M equals 3.0, T_w equals T_aw; and one for M equals 3.0, T_w equals 0.5 T_aw. Each line shows how the temperature ratio changes with distance for different Mach numbers and wall temperatures. The dashed lines represent the full Crocco temperature relation, while the solid lines represent the modified relation. All values are approximated.

Further mathematical aspects concerning the simplification of the temperature profile and its influence on the solution of the CRE, introduced in the following section, are discussed in Appendices A and C.

3.2. Analytical solution and eigenvalue problem

With the chosen models for the velocity profile (3.1) and the temperature profile (3.4), we are able to derive an analytical solution for the CRE, as presented in the following. By incorporating this solution into the two BCs (2.8) and (2.10), we achieve to break down the differential boundary value problem into an algebraic eigenvalue equation for $(\alpha , \beta , \omega )$ , where $M$ , $T_w$ and $Y$ appear as model parameters.

For deriving the solution to the CRE, we recall that it follows from an inviscid consideration and that it therefore contains singularities, which determine the character of the solution. We find that with the exponential velocity profile (3.1) and the chosen quadratic temperature profile (3.4), the CRE (2.6) has four regular singularities. This allows to transform the CRE to the general Heun equation (GHE) (Ronveaux & Arscott Reference Ronveaux and Arscott1995), as explained in Appendix A. As a result, the general solution of the CRE with the BL profiles can be written in terms of the general Heun function (GHF)

(3.7) \begin{align} \hat p(z) &= (z-a)^{r_{3,1}}\big ( C_1z^{r_{1,1}} \cdot \mathrm{HeunG}\left (a,q,\alpha _H,\beta _H,\gamma ,\delta _H;z\right )\nonumber\\ &\quad + C_2 z^{-r_{1,1}} \cdot \mathrm{HeunG}\left (a,q_1,\alpha _{H,1},\beta _{H,1},\gamma _1,\delta _{H};z\right )\big ), \end{align}

where the parameters $a,q,\alpha _H,\beta _H,\gamma ,\delta _H$ of $\text{HeunG}(;z)$ are given as functions of the CRE parameters, see Appendix A, (A6), while $q_1,\alpha _{H,1},\beta _{H,1},\gamma _1,$ of the second solution branch follow from the previous parameters via the relations (A8). Note that the index $H$ serves to distinguish the commonly used Heun parameters from the wavenumbers $\alpha , \beta$ and the BL thickness $\delta$ . Here, $z$ is the transform of the coordinate $y$ , given by

(3.8) \begin{equation} z=\frac {\alpha }{\alpha -\omega }e^{-y} . \end{equation}

The roots $r_{1,1}$ and $r_{3,1}$ stem from the Frobenius theory and are determined for our equation by (A4a ) and (A4c ).

For the CRE solution (3.7), we now require compatibility with the BCs (2.8) and (2.10). Starting with the far-field BC (2.8), we first examine the behaviour of the solution for $y\to \infty$ , which is equivalent to $z=0$ . Using the relation $\text{HeunG}(;0)=1$ (Olver et al. Reference Olver, Lozier, Boisvert and Clark2010), we find that only the second solution branch of (3.7) with constant $C_2$ fulfils the far-field condition (2.8), as shown in Appendix B. In contrast, the branch with $C_1$ grows unrestrictedly for $y\to \infty$ in case of $\omega _i\neq 0$ , meaning that we must require $C_1=0$ to fulfil the far-field BC.

The remaining solution

(3.9) \begin{equation} \hat p(z) = C_2 \cdot z^{-r_{1,1}}(z-a)^{r_{3,1}} \mathrm{HeunG}\left (a,q_1,\alpha _{H,1},\beta _{H,1},\gamma _1,\delta _{H};z\right ) \end{equation}

must further satisfy the impedance wall BC (2.10). Inserting (3.9) into (2.10) provides us with an algebraic eigenvalue equation for the normal mode perturbations $(\alpha , \beta , \omega )$ in the form

(3.10) \begin{align} 0&= Y \cdot \mathrm{HeunG}\left (a,q_1,\alpha _{H,1},\beta _{H,1},\gamma _1,\delta _{H};z_w\right )\nonumber\\ &\quad +iz_w \frac {T_w}{M^2 \omega }\left (\mathrm{HeunG}^{\prime}(\ldots ; z_w)+\left (\frac {r_{3,1}}{z_w-a}-\frac {r_{1,1}}{z_w}\right ) \cdot \mathrm{HeunG}(\ldots ; z_w)\right ), \end{align}

where $z_w={\alpha }/{\alpha -\omega }$ corresponds, according to (3.8), to the wall surface. Recall that the Heun function in (3.10) originates from the second solution branch (3.9), with its coefficients given by (A8). Here, $\text{HeunG}^{\prime}(;z)$ indicates the derivation with respect to the argument $z$ . By determining the eigenvalues $(\alpha , \beta , \omega )$ of (3.10) for given wall and flow parameters $(Y, M, T_w)$ , we get normal mode perturbations with amplitude $\hat {p}$ of the form (3.9) which satisfy both the BCs (2.8) and (2.10), thus solving our BL stability problem. Therefore, the eigenvalue (3.10) represents the key equation of our inviscid linear stability analyses, which we solve for the complex eigenvalues $\omega (\alpha , \beta , Y, M, T_w)$ of the temporal stability problem.

3.3. Porous wall model

For the wall admittance $Y$ in the eigenvalue equation (3.10), we introduce a porous wall model following Fedorov et al. (Reference Fedorov, Malmuth, Rasheed and Hornung2001), representing a regular microstructured porous surface with cylindrical pores. The same model was previously employed in our work on isothermal boundary layers (De Broeck et al. Reference De Broeck, Görtz, Alter, Hennings de Lara and Oberlack2025), thereby facilitating a direct comparison with the effects observed in the isothermal case. The proposed solution framework, however, is not restricted to this choice and can be extended to alternative impedance models. In particular, given the exponential mean-flow profile considered here, which is commonly associated with suction effects, the use of impedance models that explicitly account for suction represents a natural extension of the present analysis.

Within the chosen porous wall model of Fedorov et al. (Reference Fedorov, Malmuth, Rasheed and Hornung2001), the wall admittance is described as

(3.11) \begin{equation} Y=-\frac {\phi }{Z_0}\mathrm{tanh}\left (\varLambda h\right ) , \end{equation}

where $h$ denotes the porous layer thickness and $\phi$ the porosity. The latter is defined by the pore radius $r$ relative to the pore spacing $s$ in the form $\phi =\pi (r/s )^2$ , which implies $\phi \in [0,\pi /4 )$ due to $r\lt s/2$ . The quantities are non-dimensionalised with the BL thickness $\delta$ . For vanishing porosity $\phi =0$ or vanishing porous layer thickness $h=0$ , (3.11) returns the admittance $Y=0$ for rigid (i.e. non-porous) walls. The complex quantities $Z_0$ and $\varLambda$ in (3.11) represent the characteristic impedance and propagation constant, which depend on the perturbation frequency $\omega$ , the wall temperature $T_w$ , and the mean density $\rho _w$ and viscosity $\mu _w$ at the wall as well as on the characteristic flow quantities $M$ , $ Re$ , ${\textit{Pr}}$ , $\gamma$ . Following from the works of Daniels (Reference Daniels1950), Benade (Reference Benade1968) and Stinson & Champoux (Reference Stinson and Champoux1992), both $Z_0$ and $\varLambda$ can be described in terms of Bessel functions of the first kind. It applies that ${\textit{Re}}(\varLambda )\lt 0$ . For the detailed relations for $Z_0$ and $\varLambda$ in the same nomenclature used here, the reader is refered to Fedorov et al. (Reference Fedorov, Malmuth, Rasheed and Hornung2001) or De Broeck et al. (Reference De Broeck, Görtz, Alter, Hennings de Lara and Oberlack2025).

To apply the porous wall model to inviscid investigations, as in this work, we consider the model equations in the limit of high Reynolds number $ Re \to \infty$ . As derived and validated by De Broeck et al. (Reference De Broeck, Görtz, Alter, Hennings de Lara and Oberlack2025), for this case where $ Re \to \infty$ , $Z_0$ and $\varLambda$ can be expressed by

(3.12) \begin{equation} Z_0 = \frac {1}{M\sqrt {T_w}}, \qquad \varLambda =\mathrm{sign}(\omega _i)\frac {i\omega M}{\sqrt {T_w}} , \end{equation}

where $\operatorname {\textit{Re}}(\varLambda )\lt 0$ is ensured by $\mathrm{sign}(\omega _i)$ . Equation (3.12) together with (3.11) thus serve as a porous wall model for the inviscid limit. It should be emphasised that the inviscid relations (3.12) are reached continuously from the viscous model relations as the Reynolds number increases. Thus, the inviscid limit is inherently included in Fedorov’s viscous formulation. Based on this inviscid model, implemented in the eigenvalue equation (3.10), we aim to investigate the effects of porous walls on the temporal stability behaviour.

5.1. Damping and destabilising effect

To examine the influence of the porous wall parameters $h$ and $\phi$ on the stability behaviour, figure 2 shows the maximum growth rate $\omega _{i,{\textit{max}}}$ as a function of the porous layer thickness $h$ for several porosity values $\phi$ . Here, $\omega _{i,{\textit{max}}}$ corresponds to the most unstable mode occurring across a broad wavenumber range $(\alpha , \beta )$ under given flow and wall parameters, i.e.

(5.3) \begin{equation} \omega _{i,{\textit{max}}}(h,\phi , M, T_w) = \max \limits _{\alpha ,\,\beta } \omega _i(\alpha , \beta , h, \phi , M, T_w). \end{equation}

A distinction is made between (a) cases leading to damping (i.e. where the growth rate $\omega _{i,{\textit{max}}}$ of the most unstable mode decreases for $h\gt 0$ compared with the rigid (i.e. non-porous) wall case $h=0$ ), shown in figures 2(a), and (b) cases leading to destabilisation (i.e. where $\omega _{i,{\textit{max}}}$ increases for $h\gt 0$ relative to $h=0$ ), shown in figure 2(b).

Figure 2.

Maximum growth rate $\omega _{i,{\textit{max}}}= \max _{\alpha ,\,\beta } \omega _i$ versus porous layer thickness $h$ for different porosities $\phi$ . (a) Damping at small porosities $\phi$ , shown for $M=4.2$ . (b) Destabilisation arising at larger porosities $\phi$ , shown here for $M = 4.2$ and $M=0.8$ . Right panel shows a magnified view of the smaller $h$ -range, which clearly reveals the onset of inviscid instability at low Mach number, e.g. $M=0.8$ , above $h_{{\textit{crit}}}$ .

Figure 2. Long description

The image contains two line graphs. The first graph, labeled (a), shows the damping at small porosities for Mach numbers 4.2 with porosities 0.0003 and 0.0001. The x-axis represents the porous layer thickness (h) ranging from 0 to 1000, and the y-axis represents the maximum growth rate (ωi,max) ranging from 0 to 5 times 10^-4. The second graph, labeled (b), illustrates destabilization at larger porosities for Mach numbers 4.2 and 0.8 with porosities 0.1 and 0.05. The x-axis represents the porous layer thickness (h) ranging from 0 to 20, and the y-axis represents the maximum growth rate (ωi,max) ranging from 0 to 0.03. The right panel of the second graph provides a magnified view of the smaller h-range, revealing the onset of inviscid instability at low Mach number, e.g., 0.8, above h_crit.

5.1.2. Destabilisation effect in high-Mach-number BLs

The trend that an increase in porosity $\phi$ leads to stronger damping of the most unstable growth rate, however, only holds within the range of small $\phi$ -values. At higher porosities, the trend reverses, with destabilisation occurring relative to the rigid-wall case. This destabilisation effect can be seen in figure 2(b), which presents the most unstable growth rate $\omega _{i,{\textit{max}}}$ over the porous layer thickness $h$ for the two porosity values $\phi =0.05$ and $0.1$ . The results are shown for both a supersonic Mach number ( $M=4.2$ ) and a subsonic Mach number ( $M=0.8$ ).

For the higher Mach number $M=4.2$ , for which inviscid instability is already present under the rigid-wall condition, the introduction of porous layers ( $h\gt 0$ ) leads to pronounced destabilisation, with growth rates significantly exceeding those of the rigid-wall case. This destabilising effect is stronger for higher porosity, as comparison of the curves for $\phi = 0.05$ and $0.1$ reveals. Even for small values of $h$ , a significant increase in $\omega _{i,{\textit{max}}}$ is achieved compared with the rigid-wall instability at $h=0$ , indicating that destabilisation occurs even for thin porous coatings. The maximum amplification occurs at a specific layer thickness $h$ , which is found at $h \approx 2.8$ for $\phi =0.1$ and at $h\approx 3$ for $\phi =0.05$ under the present flow conditions. For larger $h$ beyond the maximum amplification, the growth rate decreases again in both cases, while remaining significantly higher than in the rigid-wall case even for thick porous layers. The kink observed in the curve for $M=4.2$ with $\phi =0.1$ results from a shift in the dominant mode as $h$ increases, as will be shown in the following sections.

Figure 3.

Maximum growth rate $\omega _{i,{\textit{max}}}= \max _{\alpha ,\,\beta } \omega _i$ shown as contour lines in the $h$ – $\phi$ plane: (a) for M = 4.2; (b) for M = 0.8. The right-hand panel in (a) shows a magnified view of the left-hand panel in (a), focusing on the same $h$ – $\phi$  range as in (b).

Figure 3. Long description

The image contains two contour plots labeled (a) and (b). Plot (a) shows the maximum growth rate for M equals 4.2, with a magnified view of a smaller range on the right. Plot (b) shows the maximum growth rate for M equals 0.8. Both plots have contour lines representing different growth rates, with the right-hand graph in each pair providing a closer look at a specific smaller range of values. The x-axis represents the variable h, and the y-axis represents the variable phi. The color bar on the right of each plot indicates the maximum growth rate values, ranging from 0.015 to 0.040 in plot (a) and from 0.001 to 0.010 in plot (b).

The effects of the porous-wall parameters $\phi$ and $h$ on the stability of the Mach $4.2$ BL are examined more comprehensively in figure 3(a), showing the maximum temporal growth rate $\omega _{i,{\textit{max}}}$ over a continuous variation of both wall parameters $\phi$ and $h$ for $M=4.2$ . Regarding the $\phi$ -range plotted in figure 3(a), it should be noted that we did not resolve the region of very small porosities, where damping occurs, in detail due to its small scale; instead, the focus lies on the destabilising regime. Regarding the upper bound of $\phi$ , we recall that, due to its definition, the porosity is limited to $\phi \lt \pi /4 \approx 0.74$ and results are shown up to this limit. It should be noted, however, that large porosity values affect the mean flow such that the results at high porosities should be interpreted with care.

The results shown in figure 3(a) confirm the observation from figure 2(b) that an increase in porosity enhances the destabilisation. This amplifying effect of $\phi$ applies for all $h$ . In contrast, the porous layer thickness $h$ exhibits a maximum destabilisation at a specific value, as seen earlier in figure 2(b). This optimal thickness $h$ appears nearly independent of $\phi$ , exhibiting only a slight shift towards smaller values as $\phi$ increases (from $h\approx 3$ at $\phi =0.05$ to $h\approx 2.2$ at $\phi =0.7$ ). It can thus be stated that destabilisation is maximised for large porosity $\phi$ and a specific layer thickness $h$ . The latter depends greatly on the Mach number, as evidenced by the comparison of the curves for $M=0.8$ and $4.2$ .

5.1.4. Transition point from damping to destabilisation

Figure 4.

Maximum growth rate $\omega _{i,{\textit{max}}}= \max _{\alpha ,\,\beta } \omega _i$ versus porosity $\phi$ for different $M{-}h$ combinations. The plot focuses on small $\phi$ , capturing the transition from damping to destabilisation.

Figure 4. Long description

The line graph presents the maximum growth rate versus porosity for different combinations, focusing on small values and capturing the transition from damping to destabilization. The x-axis represents porosity, ranging from 0 to approximately 0.004. The y-axis represents the maximum growth rate, ranging from 0 to approximately 4 times 10 to the power of −3. Three data lines are shown: one for M equals 4.2 and h equals 10, another for M equals 4.2 and h equals 500, and the third for M equals 0.8 and h equals 500. The black line representing M equals 4.2 and h equals 10 shows a significant increase in the maximum growth rate as porosity increases. The brown line representing M equals 4.2 and h equals 500 shows a moderate increase. The orange line representing M equals 0.8 and h equals 500 shows a slight increase. All values are approximated.

To investigate the transition between damping for very low porosity and destabilisation for higher porosity, figure 4 displays the maximum growth rate $\omega _{i,{\textit{max}}}$ as a function of the porosity $\phi$ for three different $M$ – $h$ combinations, focusing on the small $\phi$ regime.

The results for $M=4.2$ confirm the impression of the dual role of $\phi$ . It is found that damping of the growth rates is only achieved by porosities below a certain critical value $\phi _{{\textit{crit}}}$ . In this damping regime ( $\phi \lt \phi _{{\textit{crit}}}$ ), the reduction of the maximum growth rate becomes more pronounced with rising $\phi$ . However, this trend holds only up to the critical porosity $\phi _{{\textit{crit}}}$ . Once $\phi$ exceeds this threshold, destabilisation sets in, with the growth rate increasing with $\phi$ . Comparing the two Mach $4.2$ curves shows that increasing $h$ from $10$ to $500$ enhances the reduction of the growth rate in the damping regime (cf. figure 2 a), while reducing the amplification in the destabilising regime (cf. figure 2 b).

Considering the curve for $M=0.8$ , we likewise observe that the newly emerging inviscid instability only occurs for sufficiently large porosity. It can thus be concluded that, at lower Mach numbers, the induction of inviscid instability modes requires both a minimum layer thickness $h_{{\textit{crit}}}$ and a minimum porosity $\phi _{{\textit{crit}}}$ . As $\phi$ increases beyond $\phi _{{\textit{crit}}}$ , the growth rates increase. It should be explicitly emphasised that the critical value $\phi _{{\textit{crit}}}$ identified in figure 4 is not a sharply defined boundary, but rather represents the qualitative phenomenon of a minimum required porosity. Weakly unstable eigenvalues with smaller positive growth rates may arise even for $\phi$ slightly below the value indicated in figure 4. However, this does not alter the qualitative conclusion that a minimum porosity is required to induce inviscid instability at low Mach numbers.

What stands out in the comparison of the three curves in figure 4 is that, although these cases differ substantially in either Mach number or layer thickness, the onset of the destabilisation lies within a similar range of $\phi$ . We see that the critical value $\phi _{{\textit{crit}}}$ shifts only slightly towards smaller porosities $\phi$ when the Mach number $M$ is increased (cf. curves for $M=0.8$ and $M=4.2$ at $h=500$ ), or when the layer thickness is reduced from $h=500$ to $h=10$ (cf. curves for $M=4.2$ ). This is consistent with the observation that in both cases, i.e. increasing $M$ or decreasing $h$ to $10$ , the destabilising effect becomes more pronounced.

It is worth noting that the effects observed in figure 4 could likely also be identified in the stability maps shown in figures 3(a) and 3(b), if the region of very small $\phi$ values had been explicitly resolved there.

5.2. Mechanisms of damping and destabilisation

To understand the mechanisms behind the damping and the destabilising effect of porous walls on the various unstable modes, we no longer limit our investigations to the most unstable mode, but consider the full eigenvalue spectrum in the following. Figures 5 and 6 show the full eigenvalue spectrum of the Mach $4.2$ BL with adiabatic wall temperature for different porous-wall configurations. In particular, figure 5 presents the rigid wall eigenvalue spectrum as a reference and compares it with the spectra for different porosities $\phi$ at $h=10$ , including both small and larger values of $\phi$ to shed light on the damping effect and the transition to destabilisation. Figure 6, in contrast, shows the eigenvalue spectra for different layer thicknesses $h$ under sufficiently high porosity $\phi =0.1$ to elucidate the mechanisms responsible for destabilisation at different $h$ . In all the cases, both the growth rates $\omega _i$ (left column) and the corresponding streamwise phase velocities $c_x=\omega _r/\alpha$ (right column) of the eigenvalues are shown, each plotted against the streamwise wavenumber $\alpha$ . Since only unstable eigenvalues are considered, the phase velocity $c_x$ lies within the range of the mean flow velocity, $c_x \in [0,1]$ . This results from the existence of a critical layer associated with instabilities, where the phase velocity $c_x$ of the disturbances equals the local mean flow velocity $U_0$ , allowing for optimal energy exchange between the disturbance and the flow (Maslowe Reference Maslowe1986). Only 2-D eigenvalues ( $\beta =0$ ) are considered in this section, as they already reveal all the decisive phenomena. An extension of the investigations to 3-D modes will be provided in the following section.

Figure 5.

Growth rate $\omega _i$ (left column) and streamwise phase velocity $c_x$ (right column) of the 2-D modes versus streamwise wavenumber $\alpha$ in the $M=4.2$ BL: (a) for a rigid wall, and (b–d) for a porous wall with $h=10$ and (b) $\phi =0.002$ , (c) $\phi =0.1$ , (d) $\phi =0.3$ . $\blacklozenge$ , most unstable perturbation. The horizontal dashed line in the panels on the right indicates $c_x=1-1/M$ .

Figure 5. Long description

The image contains four sets of graphs, each with two subplots. The left subplot in each set shows the growth rate (omega_i) versus the streamwise wavenumber (alpha), while the right subplot shows the streamwise phase velocity (c_x) versus the streamwise wavenumber (alpha). The graphs are labeled (a) for a rigid wall, and (b), (c), and (d) for a porous wall with different conditions. Each set of graphs includes multiple lines representing different modes, with the most unstable perturbation highlighted. The horizontal dashed line in the right subplots indicates a specific phase velocity value. The graphs illustrate how the growth rate and phase velocity vary with the streamwise wavenumber under different conditions, providing insights into the stability of the boundary layer.

Figure 6.

Growth rate $\omega _i$ (left column) and streamwise phase velocity $c_x$ (right column) of the 2-D modes versus streamwise wavenumber $\alpha$ in the $M=4.2$ BL: (a–d) for a porous wall with $\phi =0.1$ and (a) $h=0.5$ , (b) $h=3$ , (c) $h=13$ , (d) $h=700$ . $\blacklozenge$ , most unstable perturbation. The horizontal dashed line in the panels on the right indicates $c_x=1-1/M$ .

Figure 6. Long description

The image contains four sets of graphs, each with two subplots. The left subplots show the growth rate (omega_i) versus the streamwise wavenumber (alpha) for different values of h (0.5, 3, 13, and 700). The right subplots display the streamwise phase velocity (c_x) versus the streamwise wavenumber (alpha) for the same values of h. Each set of graphs represents the behavior of 2-D modes in boundary layers with varying porous wall conditions. The horizontal dashed line in the right subplots indicates a specific phase velocity value. The graphs illustrate how the growth rate and phase velocity change with different streamwise wavenumbers and porous wall values, highlighting the most unstable perturbation in each scenario.

5.2.1. Characterisation of modes from their eigenfunctions

Figure 7.

Pressure eigenfunction $\hat {p}$ for 2-D eigenvalue solutions $(\alpha , \omega )$ under different wall configurations: real part (black) and imaginary part (orange) of $\hat {p}$ plotted separately; far-field solution $\hat {p}_{\textit{ff}}$ shown as real part (black dashed) and imaginary part (black dotted). (a) Rigid wall: most unstable mode occurring at $\alpha \approx 1.96$ , $\omega \approx 1.72+0.0005i$ . Porous wall: (b) $h=10, \phi =0.0002$ : most unstable mode occurring at $\alpha \approx 0.18$ , $\omega \approx 0.08 + 0.0018i$ ; (c) $h=10, \phi =0.3$ : most unstable mode occurring at $\alpha \approx 0.33$ , $\omega \approx 0.26 + 0.0277i$ ; (d) $h=10, \phi =0.3$ : 2nd most unstable mode occurring at $\alpha \approx 0.58$ , $\omega \approx 0.39 + 0.0229i$ ; (e) $h=0.5, \phi =0.1$ : most unstable mode occurring at $\alpha \approx 0.97$ , $\omega \approx 0.75 + 0.0135i$ ; (f) $h=3, \phi =0.1$ : most unstable mode occurring at $\alpha \approx 0.41$ , $\omega \approx 0.29 + 0.0286i$ ; (g) $h=13, \phi =0.1$ : most unstable mode occurring at $\alpha \approx 0.25$ , $\omega \approx 0.19 + 0.0171i$ ; (h) $h=700, \phi =0.1$ : most unstable mode occurring at $\alpha \approx 0.12$ , $\omega \approx 0.10 + 0.0060i$ .

Figure 7. Long description

The image contains eight subplots labeled (a) through (h), each showing pressure eigenfunctions for 2-D eigenvalue solutions under different wall configurations. Each subplot displays the real part of the eigenfunction in black and the imaginary part in orange. The far-field solution is shown as the real part in black dashed lines and the imaginary part in black dotted lines. Subplot (a) represents a rigid wall with the most unstable mode. Subplots (b) through (h) represent porous walls with varying parameters and different unstable modes. Each subplot includes specific values for alpha and omega, indicating the characteristics of the unstable modes. The graphs illustrate how different wall configurations affect the stability of the system, highlighting the complex interactions between the real and imaginary parts of the pressure eigenfunctions.

The type of the instability modes shown in figures 5 and 6 is inferred in from their corresponding eigenfunction. Accordingly, figure 7 shows, for selected modes, the corresponding pressure-disturbance eigenfunctions $\hat {p}(y)$ . As described by Mack (Reference Mack1984), the type of each mode follows from the number of zero crossings of the eigenfunction. In this context, only those zeros are relevant that do not originate from the oscillatory behaviour of the far-field solution. For this reason, in addition to the eigenfunction $\hat {p}$ , figure 7 also shows the corresponding far-field solution of each mode, denoted by $\hat {p}_{\textit{ff}}$ .

The pressure eigenfunctions $\hat {p}(y)$ are given by (3.9) together with (3.8), resulting from the analytical solution of the CRE for the chosen BL profiles, derived in Appendix A. The far-field solution for the pressure perturbations, $\hat {p}_{\textit{ff}}$ , follows from (B2), with $C_1=0$ , as imposed by the far-field condition. With $r_{1,1}$ being complex, as described in Appendix B, the far field solution can be written in the form

(5.4) \begin{equation} \hat p_{\textit{ff}}(y) =C_{2}e^{\sqrt {\theta } y /\big(\tilde {T}-1 \big)}=C_{2}e^{\big (\big (\sqrt {\theta }\big )_r+i\big (\sqrt {\theta }\big )_i\big )y/\big(\tilde {T}-1 \big)} , \end{equation}

thereby revealing its oscillatory nature.

5.3. Destabilisation under variation of the layer thickness h

We now want to examine the influence of $h$ in more detail by extending the investigations to a continuous variation of $h$ and by additionally considering 3-D modes (i.e. $\beta \geqslant 0$ ). For this, figure 8 shows the most unstable eigenvalue of the full 3-D eigenvalue spectrum plotted continuously over $h$ , again for $M=4.2$ , $T_w=T_{\textit{a}w}$ and $\phi =0.1$ . In comparison to figure 2(b), we analyse not only the growth rate $\omega _{i,{\textit{max}}}$ , but also the corresponding wavenumber $\alpha$ , streamwise phase velocity $c_x$ and propagation angle $\gamma =\mathrm{arctan}(\beta /\alpha )$ of the most unstable mode.

Figure 8.

Key characteristics of the most unstable mode (maximised over $\alpha$ – $\beta$ space) versus layer thickness $h$ , at $M=4.2$ and $\phi =0.1$ : (a) maximum growth rate $\omega _{i,{\textit{max}}}$ and corresponding phase velocity $c_x$ , (b) wavenumber $\alpha$ and propagation direction $\gamma$ .

Figure 8. Long description

The image contains two line graphs that illustrate key characteristics of the most unstable mode versus layer thickness. The first graph (a) on the left shows the maximum growth rate (omega_i,max) on the left y-axis and the corresponding phase velocity (c_x) on the right y-axis, both plotted against the layer thickness (h) on the x-axis. The second graph (b) on the right displays the wavenumber (alpha) on the left y-axis and the propagation direction (gamma) on the right y-axis, also plotted against the layer thickness (h) on the x-axis. The graphs highlight specific points such as h_s, h_max, and h3D, indicating critical values or transitions in the data. The trends and relationships between these variables are visually represented to show how the characteristics of the most unstable mode change with varying layer thickness.

Most prominently, figure 8 reveals that the kink in the $\omega _{i,{\textit{max}}}$ – $h$ curve, as observed in figure 2(b), coincides with jumps in the curves for $\alpha$ , $\gamma$ and $c_x$ , all occurring at the same value $\tilde {h}$ , which in the present case ( $M=4.2, \phi =0.1$ ) is $\tilde {h} \approx 12.9$ . This simultaneous occurrence of discontinuities in the various characteristics of the most unstable mode indicates that, at this thickness $\tilde {h}$ , another mode becomes dominant. This is consistent with the observation in figure 6 that for $h=13$ , the highest growth rates are attained by the second-fastest branch, originating at larger $\alpha$ , rather than by the fastest branch observed at smaller layer thicknesses. In line with this interpretation, the jumps in the curves for $\alpha$ and $c_x$ in figure 8 shift towards higher wavenumbers and higher phase velocities, indicating that for layer thicknesses $h\gt \tilde {h}$ , a slower mode (relative to the free stream velocity $U_\infty =1$ ) becomes dominant.

Furthermore, the $\alpha$ -curve (figure 8 b) reveals that increasing $h$ causes the most unstable mode to occur at smaller wavenumbers $\alpha$ , consistent with the earlier observation that thicker porous layers increasingly promote the destabilisation of longer-wavelength structures.

The influence of $h$ on the character of the destabilisation is further evident in the propagation direction of the most unstable mode, described by the angle $\gamma$ , $\gamma =\mathrm{arctan}(\beta /\alpha )$ , between the propagation direction and the mean flow direction in the $x$ – $z$ plane. The curve of $\gamma$ over $h$ (figure 8 b) shows that for small layer thicknesses $h$ , the most unstable mode propagates in streamwise direction ( $\gamma \approx 0$ ), meaning that the dominant instability is a 2-D mode. This is in line with Fedorov et al. (Reference Fedorov, Brès, Inkman and Colonius2011), who observed that porous walls with shallow pores and high porosity give rise to a new instability that attains its maximum for 2-D waves, as is typical for the acoustic Mack modes. For layer thicknesses above a certain value $h_{{3\text{-}D}}$ , the most unstable growth occurs at $\gamma \gt 0$ , indicating that for larger layer thicknesses $h$ , the dominant mode acquires 3-D character.

Regarding the transition point $h_{{3\text{-}D}}$ from 2-D to 3-D behaviour, two conclusions can be drawn. First, $h_{{3\text{-}D}}\lt \tilde {h}$ , indicating that the onset of 3-D behaviour is not associated with a switch of the dominant mode to another solution branch $\tilde {h}$ . Second, and more importantly, the transition to 3-D behaviour occurs at larger $h$ than the transition of the dominant mode from second- to first-mode type: while in the present case, the dominant mode can already be interpreted as a first mode at $h=3$ (see § 5.2.5), 3-D behaviour emerges only for $h_{{3\text{-}D}} \approx 5.5$ . This implies the existence of a range of $h$ in which the most unstable mode is a 2-D first mode, in contrast to the commonly known dominance of first modes in the 3-D direction.

In the 3-D regime, $\gamma$ increases with $h$ , implying that increasing the layer thickness causes the unstable mode to become more oblique to the main flow direction. Only the jump at $\tilde {h}$ , associated with the shift of the dominant solution branch, leads to a decrease in $\gamma$ . Overall, in the present case, propagation angles of up to $60^\circ$ are observed for the dominant mode within the considered $h$ -range. Note that owing to the symmetry (5.2) of the governing equations with respect to the reflection of $\beta$ , oblique modes always occur pairwise with reflected propagation angle, i.e. $(\gamma , -\gamma )$ .

The curve of the phase velocity $c_x$ (figure 8 a) demonstrates that the layer thickness $h$ also strongly influences whether the most unstable mode exhibits subsonic or supersonic character relative to the free stream. While, for the Mach $4.2$ BL considered here, the dominant mode propagates subsonically in the rigid-wall case ( $c_x \gt 1 - 1/M$ ), introducing a porous layer $h\gt 0$ leads to a steep increase in phase velocity relative to the free stream velocity $U_\infty =1$ . As a result, for $h\gt h_s$ , supersonic velocities $c_x \lt 1 - 1/M$ are attained, such that the most unstable mode exhibits supersonic character over a broad $h$ -range. Immediately after the shift to a slower mode at $\tilde {h}$ , the dominant mode is initially subsonic (cf. figure 6 c), but becomes supersonic again for larger $h$ , with phase velocities close to the sonic limit $1-1/M$ .

Taken together, the layer thickness $h$ has a pronounced impact on both the magnitude of the growth rate and the character of the dominant mode, in terms of its subsonic or supersonic and 2-D or 3-D behaviour.

A comparison of figures 6 and 8 with the results of De Broeck et al. (Reference De Broeck, Görtz, Alter, Hennings de Lara and Oberlack2025) for the isothermal BL shows that the effects observed here for the non-isothermal BL are qualitatively consistent with those previously found for the isothermal model. In particular, it was also found by De Broeck et al. (Reference De Broeck, Görtz, Alter, Hennings de Lara and Oberlack2025) that thin layers lead to second-mode destabilisation at large wavenumbers $\alpha$ , while thicker layers cause first-mode destabilisation at small $\alpha$ . The main difference to the results presented here is that, in the isothermal case, the growth rates are significantly higher for rigid walls, but lower for large layer thicknesses $h$ . As a result, the isothermal case allowed for damping to be achieved by employing sufficiently large layer thicknesses $h$ . This difference can be attributed to the fact that, under the isothermal assumption, the wall temperature $T_w=1$ is significantly lower than the adiabatic wall temperature $T_{\textit{a}w}\approx 4.53$ for $M=4.2$ . Consequently, the isothermal consideration may be compared, in a broader sense, to the case of wall cooling, which is known to damp first modes while amplifying second modes (Lees & Lin Reference Lees and Lin1946; Mack Reference Mack1984). Therefore, it is intuitive that in the isothermal case, the rigid-wall second mode exhibits higher growth rates, whereas the first mode, dominant at large h, shows lower growth rates than in the adiabatic wall case considered here. However, it should be emphasised that the constant temperature profile $T(y)=1$ of the isothermal assumption is a simplified model and does not correspond to the non-isothermal temperature profile of BLs with cooled wall.

6.1. Influence of $M$ and $h$ on the growth rate

Figure 9 shows the maximum growth rate $\omega _{i,{\textit{max}}}$ as a function of $M$ and $h$ . The values are obtained over a broad wavenumber range $(\alpha ,\beta )$ , and are presented for $\phi =0.1$ and an adiabatic wall temperature $T_{\textit{a}w}$ .

Figure 9.

Maximum growth rate $\omega _{i,{\textit{max}}}= \max _{\alpha ,\,\beta } \omega _i$ plotted over the $M$ – $h$ plane for $\phi =0.1$ . Black dashed line, onset of inviscid instability; grey solid line, Mach number of maximum amplification for each $h$ .

Figure 9. Long description

The line graph presents the maximum growth rate plotted over the plane for various parameters. The x-axis represents the parameter M, ranging from 0 to 4, and the y-axis represents the parameter h, ranging from 0 to 12. The graph includes a dashed line indicating the onset of inviscid instability and a grey line representing the Mach number of maximum amplification for each parameter. The color gradient on the right side of the graph indicates the values of omega i max, ranging from 0.005 to 0.025. The graph shows multiple contour lines with varying colors, representing different levels of omega i max. The dashed line starts at the bottom left and curves upwards, while the grey line starts at the top left and curves downwards. The contour lines form closed loops and vary in thickness and color intensity, indicating different levels of maximum growth rate. All values are approximated.

The figure reveals that the Mach number at which the strongest amplification is attained depends on the porous-layer thickness $h$ . This dependence is illustrated by the curve $M_{\omega _{i,{\textit{max}}}}(h)$ (grey solid line), which, for each $h$ , denotes the Mach number at which $\omega _{i,{\textit{max}}}$ reaches its maximum over $M$ , i.e. the strongest amplification at a given $h$ . The monotonic decrease of the curve shows that the strongest amplification shifts to lower Mach numbers as $h$ increases. While for the rigid-wall case and small $h$ the strongest amplification lies beyond the Mach-number range shown, it occurs at progressively lower values of $M$ with increasing $h$ , but remains in the supersonic regime for all $h$ .

Over the entire $M$ – $h$ parameter space examined here, the global maximum of $\omega _{i,{\textit{max}}}$ occurs at $M=4.2$ and $h \approx 2.8$ , representing the most unstable configuration within this parameter space for $\phi =0.1$ and insulated-wall conditions considered here.

The critical porous-wall thickness $h_{{\textit{crit}}}(M)$ (dashed black line) for the onset of inviscid instabilities as a function of Mach number is included in figure 9, indicating the minimum layer thickness required for the occurrence of inviscid unstable modes at a given $M$ . The curve reveals that $h_{{\textit{crit}}}$ increases with decreasing $M$ . For a rigid wall, inviscid unstable modes arise only for $M \gtrsim 2.2$ , consistent with previous studies that report a similar lower bound for inviscid instabilities under insulated-wall conditions (Mack Reference Mack1984; Malmuth et al. Reference Malmuth, Fedorov, Shalaev, Cole, Hites, Williams and Khokhlov1998). Introducing a porous wall shifts the onset of inviscid instabilities to lower, and even subsonic, Mach numbers once the layer thickness exceeds a critical value $h_{{\textit{crit}}}(M)$ . The slight increase of $h_{{\textit{crit}}}$ with decreasing $M$ indicates that progressively thicker porous layers are required to induce inviscid instabilities at lower Mach numbers.

It should be noted that, close to the lower threshold $h_{{\textit{crit}}}$ , inviscid instabilities are confined to higher wavenumbers $\alpha$ . This suggests that extending the analysis to larger $\alpha$ (beyond the range $\alpha \le 6$ considered here) may lead to the appearance of unstable eigenvalues at slightly smaller layer thicknesses than identified in figure 9, thereby shifting the $h_{{\textit{crit}}}$ -curve slightly downwards. However, the overall trend of the $M$ -dependence of $h_{{\textit{crit}}}$ remains qualitatively unchanged and the eigenvalues at higher $\alpha$ exhibit only vary small growth rates, of the order of $10^{-5}$ in the present case. For $h$ - or $M$ -values well above the curve $h_{{\textit{crit}}}(M)$ , the maximum growth is shifted to lower $\alpha$ values, as observed in figure 2(b).

Taken together, within the $M$ – $h$ range considered here, both the lower Mach number limit for the onset of inviscid instabilities and the Mach number for the strongest amplification decrease with increasing porous layer thickness $h$ .

Figure 10.

Phase velocity of the fastest instability relative to the sonic line, i.e. $c^{\textit{rel}}_{x,{\textit{max}}}=\max _{\alpha ,\,\beta } (1-1/M-c_x)$ , plotted over the $M$ – $h$ plane for $\phi =0.1$ . Thus, $c^{\textit{rel}}_{x,{\textit{max}}}\gt 0$ implies supersonic propagation.

Figure 10. Long description

A line graph showing phase velocity of the fastest instability relative to the sonic line. The x-axis represents the variable M, ranging from 1 to 4. The y-axis represents the variable h, ranging from 0 to 12. The graph includes multiple lines representing different values of C rel x max, ranging from 0 to 0.6. Each line shows how the phase velocity changes with respect to M and h. The dashed line represents h crit (M). All values are approximated.

6.2. Influence of $M$ and $h$ on the phase velocity

We now consider the influence of $M$ and $h$ on the phase velocity of inviscid instabilities, with particular emphasis on the occurrence of supersonic instabilities, which are known to contribute to BL noise emission. This coupling between supersonic instabilities and acoustic radiation was demonstrated by Tam & Burton (Reference Tam and Burton1984a , Reference Tam and Burtonb ) for a two-dimensional shear flow and an asymmetric jet flow. Using the method of matched asymptotic expansions, they matched an instability, forming the inner solution of the flow, with an outer solution of the acoustic wave equation, allowing for far-field sound radiation. They showed that this matching of the inner instability solution and the outer wave-like solution is possible when the instabilities propagate with a phase velocity that is supersonic relative to the free stream velocity, giving rise to so-called Mach waves that radiate to the far field and thereby contribute to noise generation.

In view of this, we consider the unstable mode with the highest streamwise phase velocity $c_x$ relative to the free stream, identified over a broad $(\alpha ,\beta )$ -space to assess the presence of supersonic instabilities. Only modes with sufficiently large growth rates $\omega _i$ , exceeding $10^{-4}$ , are retained in this analysis. Accordingly, figure 10 shows the streamwise phase velocity of this fastest mode as contour lines in the $M$ – $h$ plane, again for $\phi = 0.1$ . The phase velocity is plotted relative to the sonic line $1 - 1/M$ , i.e.

(6.1) \begin{equation} c^{\textit{rel}}_{x,{\textit{max}}}=\max \limits _{\alpha ,\,\beta }\left (1-1/M-c_x\right ) . \end{equation}

Thus, positive values $c^{\textit{rel}}_{x,{\textit{max}}}\gt 0$ indicate supersonic propagation relative to the free stream.

Figure 10 shows that, at fixed $h$ , $c^{{\textit{rel}}}_{x,{\textit{max}}}$ increases with $M$ , a trend that is observed consistently across the entire $M$ – $h$ range considered. This implies that, with increasing Mach number, faster-propagating instabilities emerge among those with relevant growth rates. At fixed $M$ , increasing $h$ similarly leads to an increase in $c^{{\textit{rel}}}_{x,{\textit{max}}}$ in the range of smaller $h$ values, corresponding to the promotion of faster instabilities with increasing $h$ . For larger $h$ , however, the variation of $c^{{\textit{rel}}}_{x,{\textit{max}}}$ with $h$ becomes progressively weaker. These trends are also reflected in the sonic line $c^{\textit{rel}}_{x,{\textit{max}}}=0$ , which marks the transition to supersonic propagation and, since the fastest instability among those with growth rates of order $10^{-4}$ is considered here, the onset of relevant supersonic instabilities. It can be seen that, relative to the rigid-wall case ( $h=0$ ), increasing $h$ shifts the sonic line to lower Mach numbers, indicating that porous walls promote the occurrence of supersonic instabilities at significantly lower Mach numbers than in rigid-wall BLs, consistent with the observations in figure 5. For larger $h$ , the sonic line asymptotically approaches $M\approx 1.15$ , implying that supersonic instabilities occur only under supersonic flow conditions. This directly follows from the fact that, for $M\lt 1$ , we have $0 \gt 1 - 1/M$ . Consequently, for all unstable solutions characterised by a critical layer (i.e. $c_x \in [0,1]$ ), it holds that $c_x \gt 1 - 1/M$ if $M\lt 1$ , implying that all unstable modes are inherently subsonic relative to the free stream under subsonic flow conditions.

Regarding the quantitative interpretation of the results in figure 10, it should be recalled that only eigenvalues with sufficiently large growth rates of at least order $10^{-4}$ are considered. Including modes with smaller growth rates arising at higher wavenumbers $\alpha$ yields instabilities with larger relative phase velocities at given $M$ – $h$ combinations, as the phase velocity of the unstable branches increases with $\alpha$ relative to the free stream (see figures 5 and 6) . Consequently, the onset of supersonic instabilities is shifted to slightly smaller $M$ – $h$ combinations if such modes with marginal growth rates are taken into account.