2.1. Mean flow

Consider the near-wall compressible laminar flow on a swept cylinder of radius ${r^\mathrm{\ast }}$ and infinite length in the spanwise direction z (figure 1). In this case, the flow field does not depend on z. Gas is perfect with constant specific heat ratio $\gamma $ and Prandtl number Pr. The dimensionless viscosity $\mu = {\mu ^\mathrm{\ast }}/\mu _e^\mathrm{\ast }$ is approximated by the power law $\mu (T) = {T^m}$ or the Sutherland law $\mu = {T^{3/2}}(1 + S)/(T + S),\;S = 110.4K/T_e^\ast $.

In the attachment line vicinity, the near-wall flow can be described by the compressible Hiemenz solution (Theofilis et al. Reference Theofilis, Fedorov and Collis2006). For small values of ${\varDelta ^\mathrm{\ast }}/{r^\mathrm{\ast }}$, the cylinder-surface curvature can be neglected. Then, the coordinate system $(x,y,z) = ({x^\mathrm{\ast }},{y^\mathrm{\ast }},{z^\mathrm{\ast }})/{\varDelta ^\mathrm{\ast }}$ is locally Cartesian. The velocity components $({u^\mathrm{\ast }},{v^\mathrm{\ast }},{w^\mathrm{\ast }})$, temperature ${T^\mathrm{\ast }}$ and pressure ${P^\mathrm{\ast }}$ are scaled using $W_e^\mathrm{\ast }$, $T_e^\mathrm{\ast }$ and $\rho _e^\mathrm{\ast }W_e^{\mathrm{\ast }2}$, where the flow parameters at the upper boundary-layer edge are calculated using the analytical relations derived by Rechotko & Beckwith (Reference Rechotko and Beckwith1958).

In the attachment line vicinity,

(2.1a–c)\begin{equation}(u,v,w) = (\varepsilon xU(y),\varepsilon V(y),W(y)),\quad T = T(y),\quad P = {(\gamma M_e^2)^{ - 1}} - {\varepsilon ^2}{x^2}/2,\end{equation}

where $\varepsilon = {R^{ - 1}}$ is assumed to be small. The mean-flow profiles $U(y),V(y),W(y),T(y)$ are governed by the ordinary differential equation (ODE) system

(2.2) \begin{equation}\left. {\begin{gathered} {({U^2} + VU^{\prime})/T = 1 + \mu^{\prime}T^{\prime}U^{\prime} + \mu U^{\prime\prime}}\\ {VW^{\prime}/T = \mu^{\prime}T^{\prime}W^{\prime} + \mu W^{\prime\prime}}\\ {U - VT^{\prime}/T + V^{\prime} = 0}\\ {(\mu^{\prime}{{T^{\prime}}^2} + \mu T^{\prime\prime})/Pr - T^{\prime}V/T + M_e^2(\gamma - 1)\mu {{W^{\prime}}^2} = 0} \end{gathered}} \right\},\end{equation}

with the boundary conditions

(2.3) \begin{equation}\left. {\begin{gathered} {y = 0:\;U = V = W = 0}\\ {T = {T_w}\;\textrm{for}\;\textrm{isothermal}\;\textrm{wall},\;\textrm{or}\;T^{\prime} = 0\;\textrm{for}\;\textrm{thermally}\;\textrm{isolated}\;\textrm{wall}}\\ {y = \infty :\;U = W = T = 1} \end{gathered}} \right\}.\end{equation}

Note that for incompressible flows, the Hiemenz solution is an exact Navier–Stokes solution. For compressible flows, this is not the case owing to the error of the order of $O({\varepsilon ^2}M_e^2)$. The system (2.2) is integrated numerically using the fourth-order Runge–Kutta algorithm. More than 400 grid points equally spaced versus y are needed to compute the boundary-layer profiles with six significant digits.

In addition, the laminar flow past a swept cylinder is calculated using the in-house N-S solver. The 2-D computational domain is limited by: the detached bow shock with the Rankine–Hugoniot conditions; the body surface with no-slip and isothermal boundary conditions; the symmetry line; and the chordwise edge, which corresponds to the azimuthal angle ${\rm \pi}/2$, with the soft boundary condition – the linear extrapolation of dependant variables.

The problem is solved numerically using the method of bow-shock isolation (Egorov Reference Egorov1992). For approximation of convective and diffusion components of the flow vector in semi-integer nodes on the elementary cell edges, the second-order central difference scheme on a nine-point “box” stencil is used. This scheme does not belong to the family of monotone difference schemes and, therefore, cannot be used for solving problems with discontinuities. However, in the case where dependant variables are smooth and the physical dissipation is present, central difference schemes allow us to obtain more accurate solutions compared with monotone schemes on the same computational grids. To solve the nonlinear difference equations approximating the differential equations, the modified Newton–Raphson method is used. The system of linear algebraic equations obtained at each iteration step, is solved using the direct method of LU-decomposition with preliminary renumbering of unknowns by the nested dissection method.

Computations are performed on a non-uniform computational grid containing 201 nodes along the cylinder surface and 301 nodes in the wall normal direction. Near the shock wave and the body surface there are thin layers, which contain, after clustering, approximately 10 % and 55 % of the total number of nodes, respectively.

Figure 2 compares the theoretical and N-S mean-flow solutions for the case C3373a specified in table 2 of Appendix A. The profiles $U(y),W(y),T(y)$ and their derivatives agree very well. The discrepancy in $V(y)$ is not appreciable, because the vertical velocity $\varepsilon V(y)$ is not involved into the leading-order approximation of the eigenvalue problem for disturbances (see § 2.2).

Figure 2.

The mean flow profiles in case C3373a: lines, compressible Hiemenz solution; symbols, our N-S solution.

This result disagrees with similar comparisons reported by Xi et al. (Reference Xi, Ren and Fu2021). Namely, the N-S mean-flow profiles differ significantly from the boundary layer approximation. As shown in Appendix B, the N-S mean-flow solutions of Xi et al. (Reference Xi, Ren and Fu2021) are not correct. This led to the wrong conclusion: “the traditional boundary layer model fails to take the influence of inviscid flow into consideration this influence sometimes may change the physics of flow instability significantly…”

2.2. Stability analysis

The disturbance vector is expressed as

(2.4)\begin{equation}\boldsymbol{q}(x,y,z,t) \equiv {\left( {u,\frac{{\partial u}}{{\partial y}},v,p,\theta ,\frac{{\partial \theta }}{{\partial y}},w,\frac{{\partial w}}{{\partial y}}} \right)^T} = \boldsymbol{F}(x,y)\textrm{exp[i}(\beta z - \omega t)\textrm{]},\end{equation}

where $t = {t^\mathrm{\ast }}W_e^\mathrm{\ast }/{\varDelta ^\mathrm{\ast }}$ is time and $\theta $ is non-dimensional temperature. Because the attachment-line flow is convectively unstable, we consider the spatial stability problem, where $\omega $ is real and $\beta $ is a complex eigenvalue with $\sigma ={-} {\beta _i}$ representing the growth rate of a wave propagating along the attachment line.

For quasi-2-D disturbances, the vector-function $\boldsymbol{F}$ is expanded as (Theofilis et al. Reference Theofilis, Fedorov and Collis2006)

(2.5)\begin{equation}\boldsymbol{F} = {\boldsymbol{Z}_0}(y;{x_1},{z_1}) + \varepsilon {\boldsymbol{Z}_1}(y;{x_1},{z_1}) + \cdots ,\end{equation}

where ${x_1} = \varepsilon x$ and ${z_1} = \varepsilon z$ are slow variables. In accord with the Gaster (Reference Gaster1974) approach (see also Nayfeh Reference Nayfeh1980) the zero-order term is expressed as ${\boldsymbol{Z}_0} = C({x_1},{z_1})\boldsymbol{\xi }(y,{x_1})$, where $\boldsymbol{\xi }$ is solution of the problem

(2.6) \begin{equation}\left. {\begin{gathered} {\dfrac{{\partial \boldsymbol{\xi }}}{{\partial y}} = \boldsymbol{A\xi }}\\ {{\xi_1} = {\xi_3} = {\xi_5} = {\xi_7} = 0,\quad y = 0}\\ {{\xi_1} = {\xi_3} = {\xi_5} = {\xi_7} = 0,\quad y = \infty } \end{gathered}} \right\},\end{equation}

which delivers the eigenvalue $\beta = {\beta _0}(\omega ,R)$. Here $\boldsymbol{A}$ is an 8 × 8 matrix of the linear stability problem. Its explicit form can be found in Mack (Reference Mack1979), Nayfeh (Reference Nayfeh1980), Tumin (Reference Tumin2007) and other papers. Note that some elements of this matrix include $\varepsilon $ and therefore (2.5) is not a self-consistent asymptotic expansion. Nevertheless, Tumin (Reference Tumin2006) showed that this inconsistency does not lead to an erroneous asymptotic behaviour of Tollmien–Schlichting waves. Numerous papers demonstrated robustness of this approach to receptivity and stability problems for weakly non-parallel compressible boundary-layer flows (for example, see the survey of Tumin Reference Tumin2011).

For the compressible Hiemenz flow, ${\beta _0}$ does not depend on ${x_1}$ while the eigenvector $\boldsymbol{\xi }$ can be expressed in the form

(2.7)\begin{equation}\boldsymbol{\xi }({x_1},y) = {({x_1}{\xi _{01}}(y),{x_1}{\xi _{02}}(y),{\xi _{03}}(y),{\xi _{04}}(y),{\xi _{05}}(y),{\xi _{06}}(y),{\xi _{07}}(y),{\xi _{08}}(y))^T}.\end{equation}

In the first-order approximation, we obtain the inhomogeneous problem

(2.8) \begin{equation}\left. {\begin{gathered} {\dfrac{{\partial {\boldsymbol{Z}_1}}}{{\partial y}} = \boldsymbol{A}{\boldsymbol{Z}_1} + {\boldsymbol{G}_x}\dfrac{{\partial {\boldsymbol{Z}_0}}}{{\partial {x_1}}} + {\boldsymbol{G}_z}\dfrac{{\partial {\boldsymbol{Z}_0}}}{{\partial {z_1}}} + \boldsymbol{H}{\boldsymbol{Z}_0}}\\ {{Z_{11}} = {Z_{13}} = {Z_{15}} = {Z_{17}} = 0,\quad y = 0}\\ {{Z_{11}} = {Z_{13}} = {Z_{15}} = {Z_{17}} = 0,\quad y = \infty } \end{gathered}} \right\},\end{equation}

where ${\boldsymbol{G}_z} ={-} \textrm{i}\partial \boldsymbol{A}/\partial \beta $, ${\boldsymbol{G}_x} ={-} \textrm{i}\partial \boldsymbol{A}/\partial \alpha $ and the matrix $\boldsymbol{H}$ depends on the mean-flow profiles $U(y)$ and $V(y)$.

The problem (2.8) has a non-trivial solution if the inhomogeneous part is orthogonal to the solution $\boldsymbol{\zeta }$ of the adjoint problem. This leads to the equation for the amplitude function $C({x_1},{z_1})$

(2.9)\begin{equation}\langle {\boldsymbol{G}_x}\boldsymbol{\xi },\boldsymbol{\zeta }\rangle \frac{{\partial C}}{{\partial {x_1}}} + \langle {\boldsymbol{G}_z}\boldsymbol{\xi },\boldsymbol{\zeta }\rangle \frac{{\partial C}}{{\partial {z_1}}} + C\left[ {\left\langle {{\boldsymbol{G}_x}\frac{{\partial \boldsymbol{\xi }}}{{\partial {x_1}}},\boldsymbol{\zeta }} \right\rangle + \langle \boldsymbol{H\xi },\boldsymbol{\zeta }\rangle } \right] = 0,\end{equation}

where the scalar product is defined as $\langle \boldsymbol{\xi },\boldsymbol{\zeta }\rangle = \int_0^\infty {(\sum\nolimits_{j,k = 1}^8 {{\xi _k}{\zeta _j}} )\,\textrm{d}y} $. Equation (2.9) can be written in a form similar to the incompressible case (Theofilis et al. Reference Theofilis, Fedorov, Obrist and Dallman2003)

(2.10)\begin{equation}{D_1}{x_1}\frac{{\partial C}}{{\partial {x_1}}} + {D_2}\frac{{\partial C}}{{\partial {z_1}}} + C{D_3} = 0,\end{equation}

where ${D_{1,2,3}}$ are constants. Equation (2.10) admits the family of solutions

(2.11)\begin{equation}{C_n} = Bx_1^n\textrm{exp(i}{\beta _{1n}}{z_1}\textrm{),}\quad {\beta _{1n}} = i({D_3} + n{D_1})/{D_2},\end{equation}

where $n = 0,1,2, \ldots $ and B is constant.

Thus, quasi-2-D modes can be expressed as

(2.12) \begin{equation}\left. {\begin{gathered} {{\boldsymbol{q}_n}(x,y,z,t) = Bx_1^n\boldsymbol{\xi }({x_1},y)[1 + O(\varepsilon )]\textrm{exp[}i({\beta_n}z - \omega t)\textrm{]}}\\ {{\beta_n} = {\beta_0} + \varepsilon {\beta_{1n}} + O({\varepsilon^2})} \end{gathered}} \right\}.\end{equation}

Here $n = 0,2, \ldots $ correspond to symmetric modes S ₁, S ₂,… and $n = 1,3, \ldots $ to antisymmetric modes A ₁, A ₂, … These modes are calculated using the following algorithm: (1) solve the zero-order problem (2.6) at ${x_1} = 0$, which is a standard stability problem for 2-D waves in a 2-D boundary layer with the mean-flow profiles $W(y)$ and $T(y)$; (2) solve the corresponding adjoint problem; (3) calculate the eigenvalues ${\beta _n}$ and the vector-functions ${\boldsymbol{q}_n}$ using (2.11) and (2.12).

For 3-D disturbances of the type

(2.13)\begin{equation}\boldsymbol{F} = [{\boldsymbol{Z}_0}(y;{x_1},{z_1}) + \varepsilon {\boldsymbol{Z}_1}(y;{x_1},{z_1}) + \cdots ]\textrm{exp(i}\alpha x\textrm{)},\end{equation}

the zeroth-order vector function is expressed as ${\boldsymbol{Z}_0} = C({z_1})\boldsymbol{\xi }({x_1},y)$, where $\boldsymbol{\xi }$ is a solution of the eigenvalue problem (2.6) with the complex eigenvalue $\beta = {\beta _0}(\omega ,\alpha )$. In the first-order approximation, ${\boldsymbol{Z}_1}$ is a solution of the problem (2.8), while the amplitude coefficient is governed by the equation

(2.14)\begin{equation}\langle {\boldsymbol{G}_z}\boldsymbol{\xi },\boldsymbol{\zeta }\rangle \frac{{\partial C}}{{\partial {z_1}}} + C\left[ {\left\langle {{\boldsymbol{G}_x}\frac{{\partial \boldsymbol{\xi }}}{{\partial {x_1}}},\boldsymbol{\zeta }} \right\rangle + \langle \boldsymbol{H\xi },\boldsymbol{\zeta }\rangle } \right] = 0.\end{equation}

Its solution is

(2.15)\begin{equation}C = B {\rm exp} \textrm{(i}{\beta _1}{z_1}\textrm{)},\quad {\beta _1} = {{i\left[ {\left\langle {{\boldsymbol{G}_x}\frac{{\partial \boldsymbol{\xi }}}{{\partial {x_1}}},\boldsymbol{\zeta }} \right\rangle + \langle \boldsymbol{H\xi },\boldsymbol{\zeta }\rangle } \right]} / {\langle {\boldsymbol{G}_z}\boldsymbol{\xi },\boldsymbol{\zeta }\rangle }}.\end{equation}

In summary, the 3-D mode is expressed as

(2.16) \begin{equation}\left. {\begin{gathered} {\boldsymbol{q}(x,y,z,t) = B\boldsymbol{\xi }({x_1},y)[1 + O(\varepsilon )]{\rm exp} [\textrm{i}(\alpha x + \beta z - \omega t)]}\\ {\beta = {\beta_0} + \varepsilon {\beta_1} + O({\varepsilon^2})} \end{gathered}} \right\},\end{equation}

Computations of quasi-2-D modes (2.12) and 3-D mode (2.16) are performed using an in-house linear stability solver. The fundamental solutions of the direct and adjoint eigenvalue problems are obtained by numerical integration of the equations from the upper boundary toward the wall with the known analytical asymptotic solutions outside the boundary layer using the Gramm–Schmidt orthonormalisation procedure and the fourth-order Runge–Kutta algorithm. Eigenfunctions of the direct and adjoint problems are expressed as a linear combination of the fundamental solutions, with coefficients being determined from the boundary conditions on the wall. To satisfy these conditions, the eigenvalues $\beta $ are determined using the Newton–Raphson iterations.

The asymptotic solution (2.12) is validated by comparison with solutions of the 2-D-PDE stability problem reported by Li & Choudhari (Reference Li and Choudhari2008) for the compressible Hiemenz flow at ${M_e} = 1.69$, $T_e^\mathrm{\ast } = 300\;\textrm{K}$, ${T_w} = {T_{ad}}$ (thermally isolated wall). Figure 3 shows that the theoretical growth rates $\sigma ={-} {\beta _i}(\omega )$ agree well with those predicted by the 2-D-PDE stability analysis for the first symmetric mode S ₁. Similar comparisons for subsonic attachment-line flows (${M_e} = 0.5$ and 0.9) were reported by Gennaro et al. (Reference Gennaro, Rodríguez, Medeiros and Theofilis2013).

Figure 3.

Distributions $\sigma ={-} {\beta _i}(\omega )$ of mode S ₁ for the compressible Hiemenz flow at ${M_e} = 1.69$, $T_e^\mathrm{\ast } = 300\;\textrm{K}$, ${T_w} = {T_{ad}}$ and ${x_1} = 0$: lines, asymptotic solution (2.12); symbols, 2-D-PDE solution of Li & Choudhari (Reference Li and Choudhari2008).

Because the compressible Hiemenz mean-flow profiles agree well with the corresponding N-S profiles (figure 2), it is expected that stability characteristics of these flows also agree well. This is confirmed by figure 4 showing the maximal growth rates ${\sigma _{max}}(R) = {\rm max}_\omega [ - {\beta _i}(\omega ,R)]$ of mode S ₁ predicted by the theory in the case C3373a. A slight downward shift of ${\sigma _{max}}$ in the case of N-S mean flow seems to arise from the curvature effect.

Figure 4.

Distribution of maximum growth rate ${\sigma _{max}}(R)$ for mode S ₁ in case C3373a.

For small $\alpha :\alpha = a\varepsilon ,\;a = O(1)$, the factor $\textrm{exp(i}\alpha x\textrm{)}$ in (2.16) can be expanded in the Taylor series: $\textrm{exp(i}\alpha x\textrm{)} = \sum\nolimits_{n = 0}^\infty {{{(\textrm{i}a)}^n}x_1^n/n!} $, where $0! = 1$. Then the solution can be treated as a sum of symmetric $(n = 0,2, \ldots )$ and antisymmetric $(n = 1,3, \ldots )$ modes of the form (2.12). The dominant term of this expansion corresponds to $n = 0$. Consequently, in the attachment-line vicinity ${x_1} \ll 1$, the 3-D mode of (2.16) tends to the symmetric quasi-2-D mode S ₁ as $\alpha \to 0$. This trend is illustrated in figure 5 for a supersonic attachment-line flow at ${M_e} = 1.55$, $T_e^\mathrm{\ast } = 300\;\textrm{K}$, ${T_w} = {T_{ad}}$, and in figure 6 for a hypersonic attachment-line flow at ${M_e} = 5$, $T_e^\mathrm{\ast } = 300\;\textrm{K}$, ${T_w} = 0.7{T_e}$. In the first case, the most unstable are oblique waves having the wave-front angle $\psi = \textrm{ta}{\textrm{n}^{ - 1}}(\alpha /{\beta _r}) \approx 42^\circ $. In the second case, the most unstable are plane waves of $\alpha = 0$.

Figure 5.

Growth rate $\sigma (\alpha )$ at $R = 1000$, $\omega = 0.0568$ and ${x_1} = 0$. The compressible Hiemenz mean flow at ${M_e} = 1.55$, $T_e^\mathrm{\ast } = 300\;\textrm{K}$, ${T_w} = {T_{ad}}$: line, 3-D mode (2.16); symbols, quasi-2-D modes (2.12); $\psi = \textrm{ta}{\textrm{n}^{ - 1}}(\alpha /{\beta _r})$, wave front angle.

Figure 6.

Growth rate $\sigma (\alpha )$ at $R = 1000$, $\omega = 0.5479$ and ${x_1} = 0$. The compressible Hiemenz mean flow at ${M_e} = 5$, $T_e^\mathrm{\ast } = 300\;\textrm{K}$, ${T_w} = 0.7{T_e}$: line, 3-D mode (2.16); symbols, quasi-2-D modes (2.12).

This situation is similar to the case of boundary layer flow on a flat plate. For moderate supersonic speeds, the dominant attachment-line instability behaves as the TS-like first mode. It is natural to assume that for sufficiently large ${M_e}$, the dominant instability is associated with the Mack second mode of acoustic nature. Indeed, figure 7 shows that there are fast and slow modes of the discrete spectrum. Owing to their synchronisation in the frequency band between the two vertical dashed lines, the slow mode becomes more unstable while the fast mode becomes more stable. This behaviour is in full accordance with the flat-plate case (see, for example, Fedorov Reference Fedorov2011 and Fedorov & Tumin Reference Fedorov and Tumin2011). Therefore, in what follows, we use the terminology of Mack for the dominant attachment-line instabilities. Further analysis is focused on the close vicinity of the attachment line and all computations are performed at ${x_1} = 0$.

Figure 7.

Phase speeds ${c_r}(\omega ) = {(\omega /\beta )_r}$ and the spatial decrements ${\beta _i}(\omega )$ of fast and slow modes at $R = 4000$ and $\alpha = 0$ for the compressible Hiemenz mean flow at ${M_e} = 5$, $T_e^\mathrm{\ast } = 300\;\textrm{K}$, ${T_w} = 5.308{T_e} \approx {T_{ad}}$ and ${x_1} = 0$: horizontal dashed lines, phase speeds of 2-D fast and slow acoustic waves of zero angle of incidence $({c_r} = 1 \pm 1/{M_e})$ and vortical/entropy waves $({c_r} = 1)$.

It should also be noted that the foregoing results contradict the statement of Xi et al. (Reference Xi, Ren and Fu2021) that the attachment-line mode associated with the Mack second mode is absent if the basic flow is calculated with boundary layer assumptions.