2.1. Tensorial notation and the invariant form

The equations in this work are presented in their invariant form, making them valid with independence of the coordinate system, and featuring the metric tensor $g_{ij}$. This allows us to keep track of not only the variable value changes, but also of the modification of the space itself owing to such basis changes. It is defined as

(2.1)\begin{equation} g_{ij} = \sum_{k=1}^{3} \frac{\partial \mathscr{X}^k}{\partial x^i} \frac{\partial \mathscr{X}^k}{\partial x^j}, \end{equation}

where $\mathscr {X}^i$ corresponds to the Cartesian coordinates, and $x^i$ to the actual coordinate system to be employed. For a Cartesian coordinate system, $g_{ij}$ is simply equal to the Dirac delta function $\delta _{ij}$. The contravariant metric tensor ($g^{ij}$) is simply the inverse of the covariant ($g_{ij}$). Moreover, the expressions presented feature velocities ($u^i$) in the non-Cartesian reference frame ($x^i$ rather than $\mathscr {X}^i$). Velocities in the Cartesian reference frame ($\mathscr {U}^i$) can be obtained through

(2.2)\begin{equation} \mathscr{U}^i = \sqrt{g_{ii}}\,u^i , \quad \forall i\in[1,2,3] . \end{equation}

The superscript/subscript notation refers to contravariant/covariant vectorial variables. A vectorial variable is contravariant $q^i$ when its components vary with the inverse transformation with respect to the basis change, that is, they ‘contra-vary’. On the other hand, it is covariant $q_i$ if its components vary with the same transformation, that is, they ‘co-vary’. A spatial covariant derivative is expressed with a comma followed by an index corresponding to the spatial direction with respect to which one is deriving: $q^i_{,j}$. The comma derivative notation is restricted to spatial derivatives. For the sake of notational simplicity, the spatial subindices are strictly kept as $i$, $j$, $k$ and $l$ throughout the text. Therefore, variable subindices containing commas followed by other symbols are not to be regarded as derivatives. The evaluation of covariant derivatives must be done also taking into consideration the curving of the space itself:

(2.3)\begin{equation} q^j_{,k} = \frac{\partial q^j}{\partial x^k} + \varGamma^j_{ik} q^i , \end{equation}

which features the Christoffel symbol of the second kind

(2.4)\begin{equation} \varGamma^j_{ik} = \sum_{l=1}^{3} \frac{1}{2 g^{jl}} \left(\frac{\partial g_{li}}{\partial x^k} + \frac{\partial g_{lk}}{\partial x^i} - \frac{\partial g_{ik}}{\partial x^l}\right) . \end{equation}

A short introduction to tensorial algebra can be found in Pinna & Groot (Reference Pinna and Groot2014) and Miró Miró (Reference Miró Miró2020). For more details, one should refer to the work of Brillouin (Reference Brillouin1964) and Aris (Reference Aris1962).

2.2. Flow equations

A dilute mixture of gases in CNE can be modelled with a separate mass-conservation equation for each species (2.5a), three momentum equations (2.5b) and an energy equation (2.5c):

(2.5a)\begin{gather} \frac{\partial \rho_s}{\partial t} + (u^j \rho_s)_{,j} = - J_{s,j}^j + \dot{\omega}_s, \quad \forall s \in \mathcal{S}, \end{gather}

(2.5b)\begin{gather}\rho \frac{\partial u^i}{\partial t} + \rho u^j u^i_{,j} = -g^{ij} p_{,j} + \mathbb{T}^{ij}_{,j}, \quad \forall i \in [1,2,3] , \end{gather}

(2.5c)\begin{gather}\rho \frac{\partial h}{\partial t} + \rho u^j h_{,j} = \frac{\partial p}{\partial t} + u^j p_{,j} + (\kappa^{Fr} \,g^{ij} T_{,i})_{,j} - \mathcal{J}^j_{,j} + g_{ik} \mathbb{T}^{kj} u^i_{,j} , \end{gather}

where $t$ is time, $\rho _s$ is the partial density of each species $s$, $\rho$ is the density of the mixture, $\mathcal {S}$ is the set of all species, $p$ is the mixture pressure, $\mathbb {T}^{ij}$ is the viscous stress tensor, $h$ is the mixture enthalpy, $\kappa ^{Fr}$ is the frozen thermal conductivity, $J_s$ is the species mass diffusion flux, $\omega _s$ is the species mass production rate and $\mathcal {J}^j$ is the total-energy diffusion flux. The full nomenclature used in this article is listed in the supplementary material available at https://doi.org/10.1017/jfm.2020.804.

Alternatively one may substitute the mass-conservation equation of one of the species with the mixture continuity equation:

(2.6)\begin{equation} \frac{\partial \rho}{\partial t} + (u^j \rho)_{,j} = 0 . \end{equation}

In order to have a well-conditioned system of equations, it is preferable to enforce (2.6) instead of the mass conservation (2.5a) of the bath species (largest mass fraction), as proposed by Stuckert (Reference Stuckert1991).

The viscous stress tensor is defined as

(2.7)\begin{equation} \mathbb{T}^{ij} = \lambda g^{ij} u^k_{,k} + \mu \left(g^{jk} u^i_{,k} + g^{ik} u^j_{,k}\right) , \end{equation}

where $\mu$ and $\lambda$ are the first and second dynamic viscosity coefficients. A Maxwellian reactive regime is assumed, justifying the absence of a reactive pressure term in (2.5b) (see Giovangigli Reference Giovangigli1999). Body forces are also neglected, owing to the high speeds under investigation.

The energy diffusion flux is defined as

(2.8)\begin{equation} \mathcal{J}^j = \sum_{s\in\mathcal{S}} h_s J^j_s , \end{equation}

where $h_s$ is the species enthalpy.

A TPG, in general, requires the same equation set as a gas in CNE, yet neglects the species source term ($\dot {\omega }\approx 0$). However, when there is no surface injection of dissimilar gases, one can reduce the equation set to simply (2.6), (2.5b) and (2.5c).

The same applies to a CPG, the difference between a TPG with constant composition and a CPG is the assumption made on the vibrational and electronic energy modes. These are neglected in CPG, rendering the internal energy and the enthalpy a linear function of temperature; see appendix A.6.

2.3. Closure of the equation system: thermophysical gas properties

A modelling of the various thermodynamic, transport and chemical properties is needed to provide the system closure. All flow assumptions employ the same state-of-the-art models. The transport properties are obtained deploying Chapman & Enskog's model (see appendix A.1). The diffusion fluxes, when necessary, are obtained from solving the Stefan–Maxwell equation system (appendix A.3), and the necessary collisional cross-sections are approximated from polynomial-bilogarithmic fits to state-of-the-art data (see appendix A.5). All flow assumptions feature a single temperature to describe the thermodynamic state of the gas, with the thermal properties obtained with the RRHO model (see appendix A.6). The chemical source terms are obtained using the law of mass action (see appendix A.7), with the reaction-rate constants collected by Mortensen & Zhong (Reference Mortensen and Zhong2016).

In addition, another less-accurate (see the quantitative analysis in Miró Miró et al. Reference Miró Miró, Beyak, Pinna and Reed2019a) combination of transport and diffusion models is compared with the previous: the BEW transport model (appendix A.2) and the constant Schmidt number ${Sc}=0.5$ (§ 4). This corresponds to the modelling framework adopted by Mortensen & Zhong (Reference Mortensen and Zhong2016). The two sets of models are compared in order to establish whether or not the large differences in the predicted stability characteristics stemming from inaccurate transport modelling seen in Miró Miró et al. (Reference Miró Miró, Beyak, Pinna and Reed2019a) also manifest themselves in more complex thermophysical scenarios.

3.1. Base-flow problem

The analysis presented in this work solves the laminar base-flow problem by decoupling the inviscid and the viscous flow regions. First the oblique shock-jump relations, followed by the one-dimensional Euler equations are solved in the inviscid region. Both the shock-jump relations and the Euler equations vary depending on the flow assumption employed (see § 7 in Miró Miró Reference Miró Miró2020). The inviscid wall values are subsequently imposed as a boundary condition at the BL edge, followed by the resolution of the viscous BL region. This effectively imposes a zeroth-order coupling of the inviscid and viscous solutions (see Brazier, Aupoix & Cousteix Reference Brazier, Aupoix and Cousteix1991). The viscous problem is solved after simplifying the flow equations presented in § 2.2 with the steady BL assumptions (see White Reference White1991):

(3.2a–c)\begin{equation} \frac{\partial }{\partial t} = 0 , \quad \frac{\partial^{2} }{\partial x^{2}}, \ \frac{\partial }{\partial z} \approx 0 , \quad \frac{\partial }{\partial x} \ll \frac{\partial }{\partial y} . \end{equation}

The resulting BL equations are solved after imposing the appropriate wall boundary conditions. The no-slip condition requires null streamwise and spanwise velocities ($u_w = w_w = 0$, where the $w$ subindex denotes the wall value). In the absence of gas–surface reactions, and the associated outgassing, the wall-normal velocity $v_w$, can be considered zero:

(3.3)\begin{equation} v_w = 0 . \end{equation}

The absence of surface chemical reactions implies that there is a non-catalytic wall, which leads to Neumann conditions on the species mass fractions:

(3.4)\begin{equation} \left.g^{ij}Y_{s,j}\right|_w = 0 , \quad \begin{array}{l} \forall s\in\mathcal{S}\\ i=2 \end{array} . \end{equation}

Similarly the wall temperature can either be assumed constant or given by a certain surface thermal distribution:

(3.5a)\begin{gather} T_w = \text{cst}. , \end{gather}

(3.5b)\begin{gather}T_w = T_w(x) . \end{gather}

Alternatively, to account for the radiation of energy from the surface, one may also consider a radiative-equilibrium boundary condition:

(3.6)\begin{equation} g^{ij} \kappa^{Froz} T_{,j} - \dot{\omega}_{rad} =0 , \quad i=2 , \end{equation}

where $\dot {\omega }_{rad}$ corresponds to the surface's radiative heat flux:

(3.7)\begin{equation} \dot{\omega}_{rad} = \sigma^\text{SB} \epsilon_{C} T^4 , \end{equation}

where $\sigma ^\text {SB}$ is the Stefan–Boltzmann constant and $\epsilon _{C}$ is the emissivity of graphite $({\approx } 0.9)$. As the pressure is constant in the wall-normal direction, owing to the BL assumptions, at the wall it is equal to that at the edge, and thus no wall boundary condition is needed.

The accurate modelling of an ablating wall requires other wall boundary conditions. As a consequence of the outgassing of ablation subproducts, the surface injection velocity cannot be zero as in (3.3). One must therefore enforce

(3.8)\begin{equation} \rho_w v_w = \dot{m}_w . \end{equation}

where the injected mass flux ($\dot {m}_w$) can be obtained from a gas–surface interaction model, such as that laid out in appendix A.8. Similarly, individual mass-conservation equations must be imposed on each species, rather than the non-catalytic condition in (3.4):

(3.9)\begin{equation} \rho_s u^i + J_s^i - \dot{m}_{sw} = 0 , \quad \begin{array}{l}\forall s\in\mathcal{S} \\ i=2 \end{array} , \end{equation}

or, alternatively, if the composition of the injected gas is known, one may impose

(3.10)\begin{equation} Y_{sw} = Y_{sw}(x) , \quad \forall s\in\mathcal{S} . \end{equation}

A surface energy balance is also necessary, instead of the isothermal wall condition (3.5):

(3.11)\begin{equation} g^{ij} \kappa^{Froz} T_{,j} - \dot{\omega}_{rad} - \sum_{s\in\mathcal{S}} h_{s} \dot{m}_{sw} =0 , \quad i=2 , \end{equation}

Equation (3.11) is the result of simplifying (21) in Mortensen & Zhong (Reference Mortensen and Zhong2016) by summing all the species conservation equations (equation (23) in Mortensen & Zhong Reference Mortensen and Zhong2016) multiplied by the species enthalpy $h_s$.

The resolution of the base-flow problem is carried out with the DEKAF flow solver (see Groot et al. Reference Groot, Miró Miró, Beyak, Moyes, Pinna and Reed2018, Miró Miró et al. Reference Miró Miró, Beyak, Mullen, Pinna and Reed2018 or § 7 of Miró Miró Reference Miró Miró2020). It employs a pseudo-spectral collocation method in the wall-normal direction, and a second-order finite-difference method in the marching direction.

3.2. Perturbation problem

The LST equations are retrieved from substituting (3.1) for all flow variables in the flow equations in § 2.2, and then subtracting the base-flow equation. The result is a generalised eigenvalue problem that is solved to obtain the complex streamwise wavenumber ($\alpha$) for real values of the perturbation frequency ($\omega$) and spanwise wavenumber ($\beta$). For an introduction to LST, see the work of Arnal (Reference Arnal1993) or Mack (Reference Mack1984). The natural logarithm of the amplitude of perturbations for fixed $\omega$ and $\beta$ can be obtained by integrating the minus imaginary part of the solution of the generalised eigenvalue problem. This is known as the $N$ factor (see van Ingen Reference van Ingen1956 and Smith & Gamberoni Reference Smith and Gamberoni1956):

(3.12)\begin{equation} N = -\int_{x_{NP}}^x \alpha_\Im\left(\omega,\beta,\check{x}\right) \text{d} \check{x} , \end{equation}

where $x_{NP}$ denotes the streamwise location of the neutral point.

The boundary conditions imposed on the perturbation-amplitude quantities must retain the mathematical homogeneity of the resulting LST generalised eigenvalue problem. This implies that all perturbation boundary conditions must be an exclusive function of perturbation amplitudes, without forcing terms that would otherwise change the mathematical nature of the problem. The perturbation boundary conditions differ slightly from those on the base-flow quantities. For impenetrable walls, or for instances of ideal mass injection (without perturbations) one may enforce Dirichlet conditions on all three components of the velocity perturbation amplitude:

(3.13)\begin{equation} \tilde{u}^j_w = 0 , \quad \forall j\in [1,2,3] . \end{equation}

For isothermal walls, it is also reasonable to impose a Dirichlet condition on the temperature perturbation amplitude:

(3.14)\begin{equation} \tilde{T}_w = 0 . \end{equation}

When assuming radiative equilibrium, one may either impose (3.14) or (3.6) after applying the corresponding LST ansatz (3.1). It is unclear whether one should follow the former or the latter approach. Given the high frequency of second-mode waves, and the high thermal inertia of the graphite wall, temperature instabilities cannot adapt fast enough to satisfy the radiative condition (3.6). The near-zero reaction time with which radiation occurs, however, suggests that it should, in general, be accounted for in a perturbation energy-balance condition. Both of these modelling possibilities are explored.

The pressure perturbation, however, is not zero. The LST wall-normal momentum equation (2.5b) with $i=2$ after substituting (3.1) and operating, is commonly imposed at the wall as a compatibility condition (see Mack Reference Mack1984). For flow assumptions with the species concentrations as state quantities (CNE), one can either apply the LST ansatz on (3.4), or employ separate species-wall-normal momentum equations for each individual species:

(3.15)\begin{equation} \rho_s \frac{\partial u^i}{\partial t} + \rho_s u^j u^i_{,j} = -g^{ij} p_{s,j} + \mathbb{T}^{ij}_{,j} , \quad \begin{array}{l} \forall s\in\mathcal{S} \\ i=2 \end{array} , \end{equation}

where $p_s$ is the species partial pressure. Miró Miró & Pinna (Reference Miró Miró and Pinna2017) observed that the use of the species momentum compatibility condition instead of the non-catalytic improved the matrices’ condition number by four orders of magnitude, whilst not modifying the stability characteristics.

Note that the contributions of species interdiffusion and momentum exchange to the momentum balance have been neglected in (3.15). This may seem like an overly restrictive simplification, but this restriction is already implicit to the usage of mixture momentum equations (2.5b) to describe the motion of all species. There is therefore no loss in generality in enforcing (3.15).

Alternatively, for ablating surfaces, one may simply apply the LST ansatz (3.1) on (3.8), (3.9), or (3.11). However, as pointed out for the radiative-equilibrium boundary condition, it is unclear whether the thermal inertia of the wall allows for rapidly varying second-mode perturbations to adapt to such local changes.

Regarding the free-stream perturbation boundary conditions, two different methodologies are investigated in this work. They are compared in order to identify the isolated effect of the shock wave on the instabilities.

The first consists of ignoring the shock position, and extending the wall-normal domain far beyond it. All perturbation amplitudes must damp in the farfield, and therefore one can impose Dirichlet conditions. However, in order to account for the finite size of the domain, one of the perturbation amplitudes is often liberated with an additional compatibility condition in the freestream. In this work, the wall-normal momentum equation is employed to account for $\tilde {p}$, which are liberated in the CPG, and TPG solvers. Similarly, in the CNE solvers, it is $\tilde {v}$ that is liberated in the farfield, using the continuity equation (2.6) as a compatibility condition. This standard approach is hereinafter referred to as the ‘freestream Dirichlet’, and was also taken by Malik (Reference Malik1989b), Hudson et al. (Reference Hudson, Chokani and Candler1997), Mortensen & Zhong (Reference Mortensen and Zhong2016) and many others.

The alternative approach is to truncate the wall-normal domain at the shock location, and to impose the Rankine–Hugoniot shock-jump relations, after substituting the LST ansatz (3.1) on both the postshock variables and the shock height. The preshock region is considered to be unperturbed. Such a treatment of the farfield boundary was previously explored by Esfahanian (Reference Esfahanian1991), Stuckert & Reed (Reference Stuckert and Reed1994) or Pinna & Rambaud (Reference Pinna and Rambaud2013), among others. The full mathematical development with the nomenclature of this article, detailed in the article's supplementary material, can be found in § 6 of Miró Miró (Reference Miró Miró2020).

The LST computations are performed with the VESTA toolkit (see Pinna Reference Pinna2013), exploiting the capabilities of the ADIT (see Pinna & Groot Reference Pinna and Groot2014, Pinna et al. Reference Pinna, Miró Miró, Zanus, Padilla Montero and Demange2019 or § 8 in Miró Miró Reference Miró Miró2020).

4.1. Flow assumptions

The simplest flow assumption is CPG, where air is modelled as a homogeneous mixture with a constant heat capacity. TPG assumptions add to it by relaxing the constant-heat-capacity constraint, and allowing for a nonlinear thermal law based on assuming that molecules behave like rigid rotors and harmonic oscillators (RRHOs); see appendix A.6. Depending on the list of species conforming the non-reacting mixture, TPG assumptions can incrementally include different physical phenomena. TPG1 (one-species mixture) assumes a constant mixture composition with 23.65 % $\textrm {O}_2$ and 76.35 % $\textrm {N}_{2}$ in mass. This implies that species diffusion is completely neglected. However, TPG11 models the test gas as a mixture with variable concentrations of five air species (N, O, NO, $\textrm {N}_{2}$ and $\textrm {O}_{2}$), together with six ablation subproducts ($\textrm {CO}_{2}$, $\textrm {C}_{3}$, $\textrm {C}_{2}$, C, CO and CN). The comparison of the predictions made with CPG and TPG1 allows to isolate the contribution of internal-energy-mode excitation to instability growth. Similarly, the comparison of the TPG11 with the TPG1 assumption displays the contribution of the diffusion of the ablation subproducts.

CNE assumptions additionally allow for the various species composing the mixture to react between them. The actual chemical reactions accounted for are subordinate to the choice of species forming the mixture. CNE5 assumes the gas to be composed of the five species in dissociating/recombining air (N, O, NO, $\textrm {O}_{2}$ and $\textrm {N}_{2}$). CNE11 also includes six ablation subproducts ($\textrm {CO}_{2}$, $\textrm {C}_{3}$, $\textrm {C}_{2}$, C, CO and CN), thus allowing for the dissociation/recombination of carbon species, as well as the gas–surface reactions occurring with the graphite wall. The comparison between the CNE5 and the TPG1 predictions allows to conclude on the effect molecular dissociation of air species ($\textrm {N}_{2}$ and $\textrm {O}_{2}$) has on second-mode development. Similarly, the comparison of CNE11 and CNE5 isolates the reactions occurring between the carbon and air species, noted that one subtracts the contribution of the interdiffusion of carbon species, already isolated by comparing the TPG11 and TPG1 assumptions.

The large number of species that are necessary to accurately model the ablation of graphite often places a computational burden in investigating such scenarios. It is therefore interesting to explore species-list-reduction techniques. To that end, two additional mixtures are employed, featuring five air species (N, O, NO, $\textrm {O}_{2}$ and $\textrm {N}_{2}$), and a single non-reacting carbon species ($\textrm {CO}_{2}$): TPG6 (where the chemical activity is neglected) and CNE6 (allowing for dissociation and exchange reactions between the air species). The comparison of the 6-species assumptions (TPG6 and CNE6) with their 11-species counterparts (TPG11 and CNE11) allows to conclude on the appropriateness of such a simplification in chemically frozen (TPG) or non-equilibrium scenarios (CNE).

Note that none of these flow assumptions include charged or electron species. Namely ionisation effects on stability are neglected (see Miró Miró et al. Reference Miró Miró, Beyak, Mullen, Pinna and Reed2018). TNE is also neglected in the present work, in order to restrict the coexisting physical phenomena and facilitate the subsequent analysis.

In this work, the flow assumptions are imposed consistently on both the base-flow ($\bar {q}$) and perturbation quantities ($q'$). This is done in order to simplify the (already large) test matrix. However, additional insight can be obtained from deploying distinct assumptions on the base-flow and perturbation quantities, as done, for instance, by Bitter & Shepherd (Reference Bitter and Shepherd2015), Miró Miró et al. (Reference Miró Miró, Beyak, Mullen, Pinna and Reed2018) or Miró Miró (Reference Miró Miró2020, § 10.4).

4.2. Base-flow surface boundary conditions

Four combinations of the base-flow wall conditions reviewed in § 3.1 are hereinafter presented, named and subsequently employed in the numerical investigation:

(i) noBw: no blowing. Assumes an impenetrable (3.3), isothermal (3.5a) and non-catalytic (3.4) wall.

(ii) SSBw: self-similar blowing. Assumes a blowing wall with a mass injection such that the self-similar blowing parameter $\bar {f}_w$ is constant:

(4.1)\begin{equation} \bar{f}_w = -\frac{\bar{v}_w \, \bar{\rho}_w \, \sqrt{2\xi}}{\rho_eu_e\mu_e} , \end{equation}

where $q_e$ are the BL edge quantities, $q_w$ are the wall quantities and $\xi$ is the marching variable (see Groot et al. Reference Groot, Miró Miró, Beyak, Moyes, Pinna and Reed2018 or White Reference White1991),

(4.2)\begin{equation} \mathrm{d} \xi = \rho_e u_e \mu_e \, \mathrm{d} x . \end{equation}

The wall is assumed isothermal (3.5a), and either non-catalytic (3.4) for mixtures without carbon species or, for mixtures with carbon species, with an imposed mass-fraction profile (3.10) coming from the solution to the Abl case.

(iii) AblBwCstT: prescribed ablation blowing with constant temperature. The mass flux obtained from the Abl case is imposed at the wall (3.8). The treatment of the wall temperature and mass fractions is the same as in the SSBw case.

(iv) Abl: full ablation model. Uses the gas–surface interaction model to obtain the species mass fractions (3.9), the total mass flux (3.8) and the temperature (3.11) at the wall. That is, for the CNE11 flow assumption. For all other flow assumptions, the mass-injection and temperature profiles obtained with CNE11 are imposed at the wall. For these flow assumptions, and similarly to the SSBw case, the wall is assumed either non-catalytic (3.4), for mixtures without carbon species, or with an imposed mass-fraction profile (3.10) coming from the solution to the Abl case, for mixtures with carbon species.

The comparison between the noBw, the SSBw and the AblBwCstT boundary conditions provides insight into how the modification of the laminar flow field owing to different mass-injection profiles ultimately affects the propagation of instabilities. The Abl condition considers a varying wall-temperature profile, whereas the AblBwCstT condition assumes it constant. The comparison of their predictions thus allows conclusions to be drawn on the isolated effect of the wall-temperature distribution on perturbation growth.

It is important to note that, in order for the base-flow treatment to be valid, one must assume steady surface ablation. This implies assuming a flow regime where the ablation rate at the surface can be considered independent of time. Real ablating surfaces are obviously evolving over time. However, as long as the rate of recession is slow enough in comparison with the flow and instability time scale (see Schrooyen Reference Schrooyen2015) it is fair to assume the surface recession to be steady. One can consequently place the reference frame on the receding ablating surface, and use the theoretical treatment presented in § 3.1.

4.3. Perturbation surface boundary conditions

Regarding the surface boundary conditions on the perturbation quantities, the no-slip condition allows $\tilde {u}_w=\tilde {w}_w=0$ to be fixed. For all other variables, four different sets of boundary conditions are distinguished.

1. (i) HSBC: homogeneous stability boundary condition. One assumes that either there is no wall blowing, or the blowing is done such that there are no wall-normal velocity perturbations (3.13). This approach was followed by the majority of authors previously studying wall-blowing effects on stability, such as Wagnild et al. (Reference Wagnild, Candler, Leyva, Jewell and Hornung2010), Li et al. (Reference Li, Choudhari, Chang and White2013), Fedorov & Soudakov (Reference Fedorov and Soudakov2014), Malik (Reference Malik1989a), Johnson et al. (Reference Johnson, Gronvall and Candler2009) or Ghaffari et al. (Reference Ghaffari, Marxen, Iaccarino and Shaqfeh2010). Regarding the temperature perturbation, as laid out in § 3.2, the thermal inertia of the wall normally allows the assumption that it is homogeneous (3.14). Compatibility conditions are imposed to account for the fact that the pressure or density perturbations at the wall are not necessarily zero. For mono-species flow assumptions (CPG and TPG1) the mixture $y$-momentum equation (2.5b) is linearised and evaluated at the wall (see Mack Reference Mack1984). For multi-species assumptions, separate $y$-momentum equations for each species (3.15) are linearised and evaluated at the wall.

2. (ii) RESBC: radiative-equilibrium stability boundary condition. The assumptions are the same as with the HSBC, with the exception that the boundary condition on the temperature perturbation ($\tilde {T}_w$) is obtained from the linearisation of the surface radiative-equilibrium condition (3.6).

3. (iii) ATSBC: ablation isothermal stability boundary condition. This approach imposes the mixture surface mass balance (3.8) and species surface mass balance (3.9), linearised and evaluated at the wall, and with the source terms provided by (A 38) and (A 39). The ATSBC does, however, assume homogeneous temperature perturbations (3.14). The multi-species character of (3.9) obviously restricts the use of the ATSBC to 11-species mixtures.

4. (iv) ASBC: full ablation stability boundary condition. An additional equation is included with respect to ATSBC: the surface energy balance. Equation (3.11) is linearised and evaluated at the wall. Its usage implies that, unlike in the previous set of boundary conditions, the temperature perturbation is no longer homogeneous $\tilde {T}_w\neq 0$. Equations (3.8) and (3.9), linearised and evaluated at the wall, complete the set of boundary conditions. This implies that the wall-normal velocity and species partial-density perturbations are also inhomogeneous at the wall $\tilde {v}_w, \tilde {\rho }_{s\,w} \neq 0$.

Unlike for the base-flow boundary condition, where the Abl is undeniably more accurate than any of the others, it is unclear which of the four perturbation boundary conditions is more appropriate (see § 3.2). One may think that the RESBC is more physically accurate than the HSBC, because it includes a modelling of the surface perturbation energy balance. However, as commented in § 3.2, it is also fair to argue that, given the high frequency of second-mode waves and the high thermal inertia of the graphite wall, temperature instabilities cannot adapt fast enough to satisfy the RESBC. The near-zero reaction time with which radiation occurs, however, suggests that it should, in general, be accounted for in a perturbation energy-balance condition. It is therefore unclear whether it is preferable to employ the ASBC, the RESBC or the HSBC. Previous authors also did not conclude on which is more relevant. Mortensen & Zhong (Reference Mortensen and Zhong2016) compared both, and Johnson & Candler (Reference Johnson and Candler2005) employed the HSBC, despite having a base-flow solution with a temperature profile cooling in the streamwise direction owing to radiation and ablation. The four boundary treatments (HSBC, RESBC, ATSBC and ASBC) are thus explored in this work in order to gain insight into the dispersion of the predictions associated with the choice of one or another.