2.1. Base flow

The calculation of the base flow is the same as that in Song & Dong (Reference Song and Dong2023). Using the Mangler transformation and Dorodnitzyn–Howarth transformation

(2.7a,b)\begin{equation} {\rm d}\bar X = {{\bar r}^2}\,{\rm d} X,\quad {\rm d}\eta = \frac{{\rho _B}\bar r\,{\rm d} y}{{\sqrt {\bar X} }}, \end{equation}

where $\bar r=X \sin \theta$, we obtain the boundary-layer equations (BLEs)

(2.8) \begin{align} \left.\begin{gathered} \bar X \frac{{\partial U_B}}{{\partial \bar X }} + \frac{{\partial \bar V_B}}{{\partial \eta }} + \frac{1}{2}U_B = 0,\\ \bar X U_B\frac{{\partial U_B}}{{\partial \bar X }} + \bar V_B\frac{{\partial U_B}}{{\partial \eta }} - \frac{1}{3}{\bar \varOmega ^2}{\left( {\bar W_B + 1} \right)^2}{\left( {\frac{{3\bar X }}{{{{\sin }^2}\theta }}} \right)^{{2}/{3}}} = \frac{\partial }{{\partial \eta }}\left( {\frac{\mu_B }{T_B} \frac{{\partial U_B}}{{\partial \eta }}} \right),\\ \bar X U_B\frac{{\partial \bar W_B}}{{\partial \bar X }} + \bar V_B \frac{{\partial \bar W_B}}{{\partial \eta }} + \frac{2}{3}U_B \left( {\bar W_B + 1} \right) = \frac{\partial }{{\partial \eta }}\left( {\frac{\mu_B }{T_B}\frac{{\partial \bar W_B}}{{\partial \eta }}} \right),\\ \bar X U_B\frac{{\partial T_B}}{{\partial \bar X }} + \bar V_B\frac{{\partial T_B}}{{\partial \eta }} = \left({\gamma - 1} \right){M^2}\frac{\mu_B }{T_B}\left[ {{{\left( {\frac{{\partial U_B}}{{\partial \eta }}} \right)}^2} + {{\left( {\frac{{3\bar X }}{{{{\sin }^2}\theta }}} \right)}^{{2}/{3}}}{\bar \varOmega ^2}{{\left( {\frac{{\partial \bar W_B}}{{\partial \eta }}} \right)}^2}} \right] \\ +\, \frac{1}{{\Pr }}\frac{\partial }{{\partial \eta }}\left( {\frac{\mu_B }{T_B}\frac{{\partial T_B}}{{\partial \eta }}} \right), \end{gathered}\right\} \end{align}

with ${\bar V_B} = {{\bar r}}^{-1}{{\rho _B}\sqrt {\bar X}} ( {{V_B} + {X}^{-1}{y}{U_B}} ) + \bar X{U_B}{{ \eta }}_{{ \bar X}}$ and ${{\bar W}_B} = {{{W_B}}/{\bar \varOmega }}$. Note that, for a supersonic flow past a slender cone, a shock wave forms from the cone tip, after which the potential flow shows a conic-flow feature, where all the physical quantities stay almost unchanged along each ray from the cone tip. This was confirmed by performing full N-S calculations, as shown in appendix B of Song & Dong (Reference Song and Dong2023). Thus, the pressure gradient along the cone surface can be taken to be zero, which leads to $\rho _B T_B =1$. The boundary conditions read

(2.9)\begin{equation} \left.\begin{gathered} (U_B, \bar V_B, \bar W_B, T_B)=(0,0,0,T_w) \quad \text{at}\ \eta =0, \\ (\rho_B, U_B, \bar W_B, T_B)\rightarrow(1,1,-1,1) \quad \text{as}\ \eta \to \infty, \end{gathered}\right\} \end{equation}

where the wall is assumed to be isothermal.

The system (2.8) is parabolic with respect to $\bar X$, which requires an initial condition at $\bar X=0$; this initial condition was illustrated in detail in Song & Dong (Reference Song and Dong2023). The third-order backward finite differential scheme is employed to march from $\bar X=0$; at each $\bar X$ location, the flow field is governed by a group of ordinary differential equations, which is solved by the Chebyshev collocation method.

2.2. Perturbations

Substituting the decomposition (2.6) into the N-S equations (2.4), and subtracting the base flow out, we obtain the nonlinear equations governing the disturbance

(2.10)\begin{align} & {\boldsymbol{ G }}\frac{{\partial \tilde \phi }}{{\partial t}} + {\boldsymbol{A}} \frac{{\partial \tilde \phi }}{{\partial x}} + {\boldsymbol{B}} \frac{{\partial \tilde \phi }}{{\partial y}} + {\boldsymbol{C}} \frac{{\partial \tilde \phi }}{{\partial \varphi }} + {\boldsymbol{D}}\tilde \phi + {{\boldsymbol{V}}_{xx}} \frac{{{\partial ^2}\tilde \phi }}{{\partial {x^2}}} + {{\boldsymbol{V}}_{yy}} \frac{{{\partial ^2}\tilde \phi }}{{\partial {y^2}}} \nonumber\\ &\quad +{{\boldsymbol{V}}_{\varphi \varphi }}\frac{{{\partial ^2}\tilde \phi }}{{\partial {\varphi ^2}}} + {{\boldsymbol{V}}_{xy}}\frac{{{\partial ^2}\tilde \phi }}{{\partial x\partial y}} + {{\boldsymbol{V}}_{x\varphi }}\frac{{{\partial ^2}\tilde \phi }}{{\partial x\partial \varphi }} + {{\boldsymbol{V}}_{y\varphi }}\frac{{{\partial ^2}\tilde \phi }}{{\partial y\partial \varphi }} = \boldsymbol{ F}, \end{align}

where the coefficient matrices $\boldsymbol {G}$, $\boldsymbol {A}$, $\boldsymbol {B}$, $\boldsymbol {C}$, $\boldsymbol {D}$, ${\boldsymbol {V}}_{xx}$, ${\boldsymbol {V}}_{yy}$, ${\boldsymbol {V}}_{\varphi \varphi }$, ${\boldsymbol {V}}_{xy}$, ${\boldsymbol {V}}_{y \varphi }$, ${\boldsymbol {V}}_{x \varphi }$ and the nonlinear forcing $\boldsymbol {F}$ can be found in Appendix A.

2.2.1. Linear stability theory (LST)

Under the parallel-flow assumption, the perturbation $\tilde \phi$ at each $x$ is expressed in terms of a travelling-wave form

(2.11)\begin{equation} \tilde \phi= {\varepsilon} \hat \phi \left( y \right){\exp[{{\rm i} \left( {\alpha x + n \varphi - \omega t} \right)}]}+ \text{c.c.}, \end{equation}

where $\alpha$, $n$ and $\omega$ represent the streamwise wavenumber, circumferential wavenumber and frequency, $\textrm {c.c.}$ represents the complex conjugation, $\textrm {i}\equiv \sqrt {-1}$ and $\hat {\phi }$ denotes the normalised shape function. In what follows, the shape functions are normalised by the maximum of the streamwise velocity perturbation, such that ${\max }_{y} | {\hat u} | = 1$. Here, $\varepsilon \ll 1$ measures the amplitude of the perturbation. We are interested in the spatial mode for which only $\alpha = \alpha _ {r}+ \textrm {i}\alpha _ {i}$ is complex with $-\alpha _i$ representing its growth rate. Substituting (2.11) into (2.10) with the non-parallel terms and $O(\varepsilon ^2)$ terms neglected, we arrive at the compressible Orr–Sommerfeld (O-S) equation

(2.12)\begin{equation} {\boldsymbol{\tilde B}}\frac{{\partial \hat \phi }}{{\partial y}} + {{\boldsymbol{V}}_{yy}} \frac{{{\partial ^2}\hat \phi }}{{\partial {y^2}}} + {\boldsymbol{\tilde D}}\hat \phi = 0, \end{equation}

where

(2.13)\begin{equation} \left.\begin{gathered} {\boldsymbol{\tilde B}} = {\boldsymbol{B}} + {\rm{i}}\alpha {{\boldsymbol{V}}_{xy}} + {\rm{i}}n{{\boldsymbol{V}}_{y\varphi }},\\ {\boldsymbol{\tilde D}} ={-} {\rm{i}}\omega {\boldsymbol{G}} + {\rm{i}}\alpha {\boldsymbol{A}} + {\rm{i}}n{\boldsymbol{C}} + {\boldsymbol{D}} - {\alpha ^2}{{\boldsymbol{V}}_{xx}} - {n^2}{{\boldsymbol{V}}_{\varphi \varphi }} - \alpha n{{\boldsymbol{V}}_{x\varphi }} . \end{gathered}\right\} \end{equation}

The perturbation field is subject to the no-slip, isothermal boundary conditions $\hat {u}(0)=\hat {v}(0)=\hat {w}(0)=\hat {T}(0)=0$ at the wall and the attenuation conditions $\hat {\phi } \to 0$ as $y \to \infty$. Malik's scheme (Malik Reference Malik1990) is employed to solve the linear system (2.12).

2.2.2. Parabolised stability equations (PSE)

The PSE approach (Bertolotti, Herbert & Spalart Reference Bertolotti, Herbert and Spalart1992; Chang & Malik Reference Chang and Malik1994) is considered as a more accurate means because it allows the streamwise variation of the perturbation profiles and takes into account the non-parallelism of the base flow. The only approximation is that the $\partial _{xx}$ terms are neglected. Expressing $\tilde \phi$ and $\boldsymbol {F}$ in terms of the Fourier series with respect to $\varphi$ and $t$, we obtain

(2.14)\begin{equation} \left.\begin{gathered} \tilde \phi (x,y,\varphi,t) = \sum_{{\mathcal {M}} ={-}M_e}^{{M_e}} {\sum_{{\mathcal {N}} ={-} {N_e}}^{{N_e}} {\overset{\smile}{\phi}}_{{\mathcal {M}},{\mathcal {N}}}}(x,y) \exp [\text{i} \left({\mathcal {N}} {n_0} \varphi - {\mathcal {M}} {\omega_0} t\right)], \\ {\boldsymbol{F}} (x,y,\varphi,t) = \sum_{{\mathcal {M}} ={-}M_e}^{{M_e}} {\sum_{{\mathcal {N}} ={-} {N_e}}^{{N_e}} {{{{\boldsymbol{\tilde F}}}_{{\mathcal {M}},{\mathcal {N}}} (x,y) }{\exp [ {\text{i}\left( {{\mathcal {N}}{n_0}\varphi - {\mathcal {M}}{\omega _0}t} \right)} ] }} } , \end{gathered}\right\} \end{equation}

where $M_e$ and $N_e$ denote the order of the Fourier-series truncation. In this paper, we choose $M_e =8$ and $N_e =16$, which has been confirmed to be sufficient via resolution studies (see Appendix C). Considering that the perturbations are propagating with two length scales, a fast one with an oscillatory manner and a slow one related to the non-parallelism, we express the perturbation profile ${\overset {\smile }{\phi }}$ as a Wentzel–Kramers–Brillouin form

(2.15)\begin{equation} {{\overset{\smile}{\phi}}_{{\mathcal {M}},{\mathcal {N}}}} (x,y) = {{\check \phi }_{{\mathcal {M}},{\mathcal {N}}}}\left( {x,y} \right) {\exp\left({{\rm i} \int_{{x_0}}^x {{\alpha _{{\mathcal {M}},{\mathcal {N}}}} (\bar x) \,{\rm d}\bar x} }\right)}, \end{equation}

where each Fourier component is denoted as $({{\mathcal {M}},{\mathcal {N}}})$, $\omega _{0}$ and $n_{0}$ are the fundamental frequency and circumferential wavenumber, respectively, and $\alpha _{{\mathcal {M}},{\mathcal {N}}}$ represents the complex streamwise wavenumber of (${{\mathcal {M}},{\mathcal {N}}}$). The shape function $\check {\phi }_{{{\mathcal {M}},{\mathcal {N}}}}$ varies slowly with $x$. Here, $x_0$ is a reference streamwise position as defined in (2.2).

Neglecting the $\partial _{xx} \check {\phi }_{{\mathcal {M}},{\mathcal {N}}}$ terms, system (2.10) is parabolised to

(2.16)\begin{equation} {{{\boldsymbol{\tilde A}}}_{{\mathcal {M}},{\mathcal {N}}}} \frac{{\partial {{\check \phi }_{{\mathcal {M}},{\mathcal {N}}}}}}{{\partial x}} + {{{\boldsymbol{\tilde B}}}_{{\mathcal {M}},{\mathcal {N}}}} \frac{{\partial {{\check \phi }_{{\mathcal {M}},{\mathcal {N}}}}}}{{\partial y}} +{{\boldsymbol{V}}_{yy}} \frac{{{\partial ^2}{{\check \phi }_{{\mathcal {M}},{\mathcal {N}}}}}}{{\partial {y^2}}} + {{{\boldsymbol{\tilde D}}}_{{\mathcal {M}},{\mathcal {N}}}}{{\check \phi }_{{\mathcal {M}},{\mathcal {N}}}} = {{{{{\boldsymbol{\check F}}}_{{\mathcal {M}},{\mathcal {N}}}}}}, \end{equation}

where the matrices ${{{\boldsymbol {\tilde A}}}_{{\mathcal {M}},{\mathcal {N}}}}$, ${{{\boldsymbol {\tilde B}}}_{{\mathcal {M}},{\mathcal {N}}}}$ and ${{{\boldsymbol {\tilde D}}}_{{\mathcal {M}},{\mathcal {N}}}}$ are given by

(2.17) \begin{equation} \left.\begin{gathered} {{{\boldsymbol{\tilde A}}}_{{\mathcal {M}},{\mathcal {N}}}} = {\boldsymbol{A}} + 2{\rm{i}}{\alpha _{{\mathcal {M}},{\mathcal {N}}}}{{\boldsymbol{V}}_{xx}} + {\rm{i}}{\mathcal {N}}{n_0}{{\boldsymbol{V}}_{x\varphi }},\\ {{{\boldsymbol{\tilde B}}}_{{\mathcal {M}},{\mathcal {N}}}} = {\boldsymbol{B}} + {\rm{i}}{\alpha _{{\mathcal {M}},{\mathcal {N}}}}{{\boldsymbol{V}}_{xy}} + {\rm{i}}{\mathcal {N}}{n_0}{{\boldsymbol{V}}_{y\varphi }},\\ {{{\boldsymbol{\tilde D}}}_{{\mathcal {{\mathcal {M}}}},{\mathcal {N}}}} ={-} {\rm{i}}M{\omega _0} {\boldsymbol{G}} + {\rm{i}}{\alpha _{{\mathcal {M}},{\mathcal {N}}}}{\boldsymbol{A}} + {\rm{i}}{\mathcal {N}}{n_0}{\boldsymbol{C}} + {\boldsymbol{D}}- {{\mathcal {N}}^2}n_0^2{{\boldsymbol{V}}_{\varphi \varphi }}\\ \quad\qquad - \left( {\alpha _{{\mathcal {M}},{\mathcal {{\mathcal {N}}}}}^2 - {\rm{i}}\frac{{d{\alpha _{{\mathcal {M}},{\mathcal {N}}}}}}{{{\rm d}x}}} \right){{\boldsymbol{V}}_{xx}} - {\mathcal {N}}{\alpha _{{\mathcal {M}},{\mathcal {N}}}}{n_0}{{\boldsymbol{V}}_{x\varphi }}, \\ {{{{{\boldsymbol{\check F}}}_{{\mathcal {M}},{\mathcal {N}}}}}}= {{{{{\boldsymbol{\tilde F}}}_{{\mathcal {M}},{\mathcal {N}}}}}}{{{\exp({-\text{i}\int_{{x_0}}^x {{\alpha _{{\mathcal {M}},{\mathcal {N}}}}\,{\rm d}\bar x} })}}}. \end{gathered}\right\} \end{equation}

Such a system can be solved numerically using a marching scheme, for which the flow quantities only depend on their upstream information. Thus, as initial perturbations, a pair of oblique normal modes with the same frequency but the opposite circumferential wavenumbers is introduced at $X=1$ (or $x=x_0=R$)

(2.18)\begin{equation} \tilde \phi (x_{0},y,\varphi ,t) = {\varepsilon_{1,1}} \hat\phi_{1,1}(y)E_{1,1}+ {\varepsilon_{1,-1}} \hat\phi_{1,-1}(y)E_{1,-1}+ \text{c.c.}, \end{equation}

where $E_{{\mathcal {M}},{\mathcal {N}}}=\exp [\textrm {i} ({\mathcal {N}}n_0 \varphi -{\mathcal {M}}\omega _0 t)]$, and $\varepsilon _{{{\mathcal {M}},{\mathcal {N}}}}$ measures the amplitude of each mode. The shape functions of the unstable modes $\hat {\phi }_{{{\mathcal {M}},{\mathcal {N}}}}$ are obtained by solving numerically the compressible O-S equation (2.12). The nonlinear PSE calculation marches downstream until its blowup, indicating that the transition onset will not be far away (Dong, Zhang & Zhou Reference Dong, Zhang and Zhou2008; Mayer et al. Reference Mayer, Fasel, Choudhari and Chang2014). The wall-normal boundary conditions for the system (2.16) are

(2.19)\begin{equation} \left.\begin{gathered} ({{\check u}_{{{\mathcal {M}},{\mathcal {N}}}}},{{\check v}_{{{\mathcal {M}},{\mathcal {N}}}}}, {{\check w}_{{{\mathcal {M}},{\mathcal {N}}}}},{{\check T}_{{{\mathcal {M}},{\mathcal {N}}}}}) = (0,0,0,0) \quad \text{at}\ y = 0,\\ ({{\check \rho}_{{{\mathcal {M}},{\mathcal {N}}}}},{{\check u}_{{{\mathcal {M}},{\mathcal {N}}}}}, {{\check v}_{{{\mathcal {M}},{\mathcal {N}}}}},{{\check w}_{{{\mathcal {M}},{\mathcal {N}}}}}, {{\check T}_{{{\mathcal {M}},{\mathcal {N}}}}}) \to (0,0,0,0) \quad \text{as}\ y \to \infty. \end{gathered}\right\} \end{equation}

Because the complex streamwise wavenumber $\alpha _{{{\mathcal {M}},{\mathcal {N}}}}$ and the nonlinear terms ${\boldsymbol {{\check F}}}_{{{\mathcal {M}},{\mathcal {N}}}}$ are unknown, an iterative procedure is used to solve the flow quantities at each streamwise position. More details about the iterative procedures, the discretisation scheme and code validation can be found in our previous works (Zhao et al. Reference Zhao, Zhang, Liu and Luo2016; Song et al. Reference Song, Dong and Zhao2022).

Additionally, if $\check {\boldsymbol {F}}_{{{\mathcal {M}},{\mathcal {N}}}}$ is set to be zero, then (2.16) is recast to the linear PSE (LPSE), which can be used to track the evolution of each linear mode separately. To distinguish it, the PSE approach with $\check {\boldsymbol {F}}_{{{\mathcal {M}},{\mathcal {N}}}}$ retained is referred to as the nonlinear PSE (NPSE) in this paper.

2.3. Secondary instability analysis (SIA)

When the amplitude of the dominant perturbation has accumulated to a finite level, the SI may appear. In this subsection, two types of SIs are considered. The first one is based on a wavy profile driven by a saturated travelling mode, while the second one is based on a streaky profile driven by finite-amplitude longitudinal streaks.

2.3.1. The SIA for a wavy base flow

For simplicity, we assume that the perturbation field is dominated by only one travelling mode (whose frequency, streamwise wavenumber and circumferential wavenumber are denoted by $\omega _0$, $\alpha _{1,1}$ and $n_0$, respectively) and its harmonics. When the dominant mode reaches the nonlinear saturation state, its growth rate becomes almost zero, and so $\alpha _{1,1}$ to leading order is almost real. The SIA is performed in the wave-propagating direction, whose oblique angle to the $x$ axis is $\varTheta =\tan ^{-1}[\beta _{1,1}/{{\alpha _{r,(1,1)}} }]$ with $\beta _{1,1}=n_0/r_0$. The composite wavenumber in the wave direction is given by $\breve \alpha = \sqrt { {\alpha ^{2}_{r,(1,1)}}+ \beta _{1,1}^{2} }$, which is also real. The base flow for the SIA is a superposition of the steady base flow and the saturated wavy travelling waves and is expressed in terms of a Fourier series. Thus, the base flow $\breve {\varPhi }_{B} \equiv [\breve \rho _B,\breve U_B,\breve V_B,\breve W_B,\breve T_B]$ in a moving frame reads

(2.20)\begin{equation} {\breve {\varPhi}} _{B} (\tilde x, y) = [ \rho_{B}, \tilde U-c, 0, \tilde W, T_{B}](y) + \sum_{ { m} ={-} M_{W}}^{M_{W}} {{{\overset{\smile}{\phi}} }_{m,m}}\left( y \right)\exp({\text{i} m \breve \alpha \tilde x}) +\cdots, \end{equation}

where $c= \omega _{0}/{\breve \alpha }$, $\tilde x = x \cos \varTheta + z \sin \varTheta - ct$ is the Galilean transformed coordinate along the wave direction, $\tilde U= U_{B} \cos \varTheta +W_{B} \sin \varTheta$, $\tilde W=-U_{B} \sin \varTheta +W_{B} \cos \varTheta$ and $M_W$ denotes the order of the Fourier-series truncation. The laminar base flow $\varPhi _{B}$ develops with a length scale much greater than the wavelength of the fundamental mode $2 {\rm \pi}/ {\breve \alpha }$, and so the non-parallelism of $\varPhi _B$ in the local region is neglected in the present analysis. Thus, $\boldsymbol {\breve \varPhi }_B$ is periodic in $\tilde x$. We introduce the coordinate $\tilde z=-x \sin \varTheta + z \cos \varTheta$ to denote the direction perpendicular to the wave direction. Here, $\breve U_{B}$ and $\breve W_{B}$ are the velocities along the $\tilde x$- and $\tilde z$-directions, respectively.

According to Floquet theory, the periodic base flow supports the instability modes which can be expressed as

(2.21)\begin{equation} \left.\begin{gathered} {{\tilde \phi }(x,y,z,t)} = \varepsilon {\breve \phi _W}\left( {\tilde x,y} \right){\exp{[\text{i} \gamma_{W}( \tilde x+ct)+\text{i}\breve \beta \tilde z+{\text{i}{\sigma _d} \breve \alpha \tilde x}]}} +\text{c.c.}, \\ {\breve \phi _W}\left( {\tilde x,y} \right) = \sum_{k ={-} {N_W}}^{{N_W}} {{{\hat \phi }_{W,k}} \left( y \right)\exp \left( { \text{i} k \breve \alpha \tilde x} \right)}, \end{gathered}\right\} \end{equation}

where $\gamma _W$ is the complex wavenumber in the $\tilde x$ direction with its imaginary part representing the opposite of the growth rate, $\breve \beta$ is the wavenumber in the $\tilde z$ direction, $\sigma _{d}$ is the detuning parameter and $N_W$ is the order of the Fourier-series truncation. For the fundamental resonance, we take $\sigma _d=0$. Here, $\hat {\phi }_{W,0}$ denotes the streak component, and $\hat {\phi }_{W,k}$ represents the travelling mode for a non-zero $k$ value. Substituting (2.20) and (2.21) into (2.10) with $O(\varepsilon ^{2})$ terms neglected forms an eigenvalue problem

(2.22)\begin{equation} [ {{{\boldsymbol{M}_0}} + \left( {\text{i}{\gamma _W}} \right) { {\boldsymbol{M}_1}} + {{\left( {\text{i}{\gamma _W}} \right)}^2} { {\boldsymbol{M}_2}} }]{ \breve \phi _W} \left( {\tilde x,y} \right) = 0,\end{equation}

where

(2.23) \begin{equation} \left.\begin{gathered} {{\boldsymbol{M}_0}} = ( { {\boldsymbol{A}} + {\rm{i}} \breve \beta { {{\boldsymbol{V}}_{\tilde x\tilde z}}}}) \frac{\partial }{{\partial \tilde x}} + ( {{\boldsymbol{B}} + {\rm{i}} \breve \beta { {\boldsymbol{V}_{y\tilde z}}}})\frac{\partial }{{\partial y}} + [ { {\boldsymbol{D}} + {\rm{i}} \breve \beta {\boldsymbol{\tilde C}} + {{( {{\rm{i}} \breve \beta } )}^2}{ { {\boldsymbol{V}_{\tilde z\tilde z}}}}} ] \\ +\,{{\boldsymbol{V}_{xx}}}\frac{{{\partial ^2}}}{{\partial {{\tilde x}^2}}} + { {\boldsymbol{V}_{yy}}} \frac{{{\partial ^2}}}{{\partial {y^2}}} + { {\boldsymbol{V}_{xy}}} \frac{{{\partial ^2}}}{{\partial x\partial y}}, \\ { {\boldsymbol{M}_1}} = ( { {\boldsymbol{A}} + \text{i} \breve \beta { {\boldsymbol{V}_{\tilde x\tilde z}}}} ) + 2 { {\boldsymbol{V}_{xx}}} \frac{\partial }{{\partial \tilde x}} + {{\boldsymbol{V}_{xy}}} \frac{\partial }{{\partial y}} + c{\boldsymbol{G}} , \\ { {\boldsymbol{M}_2}} = { {\boldsymbol{V}_{xx}}}, \end{gathered}\right\} \end{equation}

with ${\boldsymbol {\tilde C}} = {r}{\boldsymbol {C}}$, ${{ \boldsymbol {V}_{\tilde x \tilde z}}} = {r}{{ \boldsymbol {V}_{ x \varphi }}}$, ${{ \boldsymbol {V}_{y \tilde z}}}= {r}{{ \boldsymbol {V}_{ y \varphi }}}$ and ${{ \boldsymbol {V}_{\tilde z \tilde z}} }= {r}^{2} {\boldsymbol {V_{ \varphi \varphi }}}$. The wall-normal boundary conditions read

(2.24)\begin{equation} \left.\begin{gathered} {\breve u}_{W}= {\breve v}_{W}={\breve w}_{W}={\breve T}_{W}=0 \quad \text{at}\ y=0, \\ ({\breve \rho}_{W},{\breve u}_{W}, {\breve v}_{W},{\breve w}_{W},{\breve T}_{W}) \to 0 \quad \text{as}\ y \to \infty. \end{gathered}\right\} \end{equation}

In the $\tilde x$-direction, the periodic boundary condition is employed. System (2.22) is discretised by using the fourth-order finite difference scheme with five points in the $y$-direction and the spectral method in the $\tilde x$-direction. A total of $J_W$ grid points are allocated in the wall-normal direction. The discretisation leads to a high-dimensional (the dimension of the coefficient matrix is $5{J_W}N_{W} \times 5{J_W}N_{W}$) eigenvalue problem with $\gamma _w$ being the eigenvalue, which is solved using the Rayleigh quotient iteration method (see the Appendix of Han et al. (Reference Han, Yuan, Dong and Fan2021)). In our calculations, we select $M_{W}=3$, $N_{W}=6$ and $J_W=581$. In general, only one unstable discrete mode exists for a given $\breve \beta$, and thus, we simply take the initial guess for the iteration to be $\gamma _w=(({(\bar \varOmega X \cos \varTheta + \sin \varTheta )}/{(-\bar \varOmega X \sin \varTheta + \cos \varTheta )}) \breve \beta,-0.01)$, which is based on the approximation that the streak direction is roughly along the potential-flow direction, and the convergence of the iteration is usually reached in 6 to 8 steps.

Such an analysis has also been used in the study of the SI of 2-D Mack second modes in a hypersonic boundary layer (Chen et al. Reference Chen, Zhu and Lee2017; Xu et al. Reference Xu, Liu, Mughal, Yu and Bai2020), and our code validation and resolution test are provided in Appendix D.

2.3.2. The SIA for a streaky base flow

Now we consider a base flow perturbed by longitudinal streaks, and the streak direction is not necessarily along the streamwise direction. We denote the angle between the streak direction and the streamwise direction by $\theta _s$. For convenience, we introduce a coordinate transformation

(2.25) \begin{equation} \left.\begin{array}{l@{}} {x_s} = x, \\ {z_s} = z - x\tan {\theta _s} , \\ \end{array}\right\} \end{equation}

which yields

(2.26) \begin{equation} \left.\begin{array}{l@{}} \dfrac{\partial }{{\partial x}} = \dfrac{\partial }{{\partial {x_s}}} - \tan \theta_s \dfrac{\partial }{{\partial {z_s}}}, \\ \dfrac{\partial }{{\partial z}} = \dfrac{\partial }{{\partial {z_s}}} . \\ \end{array}\right\} \end{equation}

The streaky base flow obtained by NPSE can be expressed as

(2.27) \begin{equation} \breve \varPhi_{B} (x,y,z) = \varPhi_{B} (y) +\sum_{m ={-} M_{S}}^{M_{S}} {{{\overset{\smile}{\phi}}_{0,m}}\left( y \right){\exp({\text{i} ( \alpha_{0,m} x +\beta_{0,m} z )}}}) +\cdots , \end{equation}

where $\beta _{0,m} =m n_{0}/r_{0}$ and $M_S$ is the Fourier-series truncation. The SIA is performed when the streak mode reaches a finite amplitude, for which the locally parallel assumption is also approximately valid. Therefore, the perturbation field $\tilde \phi$ can be expressed as

(2.28)\begin{equation} \left.\begin{gathered} \tilde \phi (x,y,z,t) = \varepsilon \breve{\phi}_{S}(y,z_s) \exp[{\text{ i} (\alpha_{s} x_{s}+ \sigma_{d} \beta_{0,1} z_s - \omega t)} ]+\text{c.c.}, \\ {\breve \phi _S}\left( {y,z_s} \right) = \sum_{k ={-} {N_S}}^{{N_S}} {{{\hat \phi }_{S,k}} \left( y \right)\exp \left( { \text{i} \beta_{0,k} z_s} \right)}, \end{gathered}\right\} \end{equation}

where $\sigma _d=0$ is a detuning parameter and $N_S$ is the Fourier-series truncation. For the spatial mode, the frequency $\omega$ is real and $\alpha _s$ is complex, whose real and imaginary parts represent the wavenumber and the opposite of the growth rate, respectively.

Substituting (2.28), (2.27) and (2.26) into (2.10) and collecting the $O(\varepsilon )$ terms, we obtain

(2.29)\begin{equation} [ {{ {{\boldsymbol{D}}_s}} + \text{i}{\alpha _s} {{ \boldsymbol{A}_s}} + {{\left( {\text{i}\alpha_s } \right)}^{2}} {\boldsymbol{V}_{xx}}} ] \breve \phi_S \left( {y,{z_s}} \right) = 0, \end{equation}

where

(2.30)\begin{equation} \left.\begin{gathered} {{{\boldsymbol{D}_s}}} = \boldsymbol{D} + \boldsymbol{B}\frac{\partial }{{\partial y}} + {{\boldsymbol{C}_s}}\frac{\partial }{{\partial {z_s}}} + \boldsymbol{D} \frac{{{\partial ^2}}}{{\partial {y^2}}} + {\boldsymbol{V}_{zz,s}} \frac{{{\partial ^2}}}{{\partial z_s^2}} + { \boldsymbol{V}_{yz,s}} \frac{{{\partial ^2}}}{{\partial y\partial {z_s}}} - \text{i} \omega {\boldsymbol{ G}}, \\ {\boldsymbol{A}_s} = \boldsymbol{A} + {\boldsymbol{V}_{xy}}\frac{{{\partial ^2}}}{{\partial y}} + {\boldsymbol{V}_{xz,s}}\frac{{{\partial ^2}}}{{\partial {z_s}}}, \\ \end{gathered}\right\} \end{equation}

with ${\boldsymbol {C}_s} = {r}\boldsymbol {C} + h \boldsymbol {A}$, ${\boldsymbol {V}_{zz,s}} = r^2 {\boldsymbol {V}_{\varphi \varphi }}+ {r}h{\boldsymbol {V}_{x\varphi }} + {h^2}{\boldsymbol {V}_{xx}}$, ${\boldsymbol {V}_{yz,s}} = r {\boldsymbol {V}_{y\varphi }} + h{\boldsymbol {V}_{xy}}$, ${\boldsymbol {V}_{xz,s}} = r {\boldsymbol {V}_{x\varphi }} + 2h{\boldsymbol {V}_{xx}}$ and $h=-\tan \theta _s$. The periodic boundary condition is employed in the $z_s$ direction, and the same boundary conditions in the wall-normal direction as in (2.24) are employed. The numerical scheme is the same as that in § 2.3.1, and we choose $(M_S,N_S,J_S)=(8,16,581)$. The Arnoldi method is used to search the initial eigenvalues because a multiplicity of unstable discrete modes may exist, and the Rayleigh quotient iteration method is employed to confirm the accuracy of each eigenvalue solution. This method has been used to analyse the SI of the Görtler vortices over a concave wall in our previous work (Song, Zhao & Huang Reference Song, Zhao and Huang2020).

3.1. Base flow

In the following calculations, the computational parameters are listed in table 1. Note that the values of the oncoming Mach number and temperature represent those at the outer reach of the boundary layer, and from conic-flow theory (Sims Reference Sims1964), we can estimate that the oncoming Mach number and temperature before the shock are $M_\infty =3.214$ and $T_\infty =48.28$ K, respectively. A detailed verification can be seen in Song & Dong (Reference Song and Dong2023). The apex angle of the cone follows the models in the numerical studies (Zhong & Ma Reference Zhong and Ma2006; Balakumar Reference Balakumar2009; Hader & Fasel Reference Hader and Fasel2021) and the experimental studies (Schuele et al. Reference Schuele, Corke and Matlis2013; Craig & Saric Reference Craig and Saric2016), and the Mach number and the temperature of the oncoming flow are chosen from the studies of the Magnus effect in Sturek et al. (Reference Sturek, Dwyer, Kayser, Nietubicz, Reklis and Opalka1978) and Klatt et al. (Reference Klatt, Hruschka and Leopold2012). The base flow is obtained by solving numerically the BLEs (2.8) with boundary conditions (2.9) in a 2-D domain $\bar {X}\in [0,0.1]$ and $\eta \in [0,15]$, with $401$ and $121$ grid points allocated in the $\bar {X}$- and $\eta$-directions, respectively. The numerical method was also employed and confirmed to be sufficiently accurate in the study of the linear instability of a rotating-cone boundary layer in Song & Dong (Reference Song and Dong2023).

In this paper, we focus on a series of case studies with different $\varOmega$ values, as shown in table 2, and the $U_B$- and $W_B$-profiles at $X=1$ are shown in figure 4(a,b), respectively. In the boundary layer, as $\varOmega$ increases, both $U_{B}$ and $|W_B|$ increase monotonically, leading to higher wall shears. The streamwise distributions of the wall shears of $U_{B}$ and $|W_{B}|$ are shown in figure 4(c,d), respectively. For $\varOmega =0$, the self-similar solution determines that the boundary-layer thickness grows like $X^{1/2}$, and hence the wall shear $U_{B}|_{y=0}$ decays like $X^{-1/2}$, as confirmed by the black circles. Such a scaling is not strictly satisfied as $\varOmega$ increases, and an appreciable discrepancy is observed when $\varOmega =0.0071$ for case VI. According to (b), $W_{B}$ is approximately proportional to $X$, and the boundary-layer thickness almost grows like $X^{1/2}$. Therefore, the wall shear of the spanwise velocity $-W_{B}|_{y=0}$ grows like $X^{1/2}$; such a scaling law is approximately correct even when $\varOmega =0.0071$ for case VI. The velocity profiles in the potential-flow direction $U_p$ and the cross-flow direction $U_c$ for different cases at $X=1$ are plotted in figure 5, where $U_p$ and $U_c$ are defined as

(3.1a–c)\begin{align} U_p = U_B \cos \varPhi_e + W_B \sin \varPhi_e, \quad U_c ={-}U_B \sin \varPhi_e + W_B \cos \varPhi_e, \quad \varPhi_e \equiv \tan ^{{-}1} \bar \varOmega. \end{align}

As $\varOmega$ increases, $U_p$ increases monotonically since $W_B$ induced by the rotation increases. The value of $U_c$ is zero at $y = 0$ and $y \to \infty$, and an inflectional point appears for $\varOmega \ne 0$, which is likely to support the inviscid cross-flow instability.

Figure 4. Base flow for different cases. (a,b) Are the profiles of $U_B$ and $-W_B$ at $X=1.0$, respectively. (c,d) Are the streamwise distributions of the wall shears of the streamwise and spanwise velocities, respectively. The circles in (c) denote the scaling of $X^{-1/2}$, and the triangles in (d) denote the scaling of $X^{1/2}$.

Figure 5. Wall-normal profiles of (a) $U_p$ and (b) $U_c$ at $X = 1$ for different cases.

3.2. Linear instability analysis

The linear instability analysis can be performed using either the LST, for which the local growth rate is predicted based on the base flow at each streamwise location, or the LPSE, for which the amplitude evolution is predicted by a marching scheme.

The LST analysis for a rotating cone at this Mach number in Song & Dong (Reference Song and Dong2023) reveals that, when the rotation speed is small, only one unstable zone exists, but for a sufficiently high rotation rate, an additional unstable zone near the zero-frequency band appears. To be distinguished, the unstable modes in the two zones are referred to as the type-I and type-II instabilities, respectively, following Song & Dong (Reference Song and Dong2023). Note that this notation is different from that in Garrett et al. (Reference Garrett, Hussain and Stephen2009), where the two instability types are referred to as the inviscid CFM and the viscous Coriolis mode. The type-I instability may be a MM or a quasi-stationary CFM, while the type-II instability may be a CM at a moderate $R$ or a CFM at a large $R$. For cases V and VI, both type-I CFM and type-II CM exist. As confirmed by figure 20 in Song & Dong (Reference Song and Dong2023), the CFM appears as a pair of co-rotating vortices near the critical line, while the CM appears as a pair of counter-rotating vortices in the near-wall region, agreeing with the experimental observations in Kobayashi et al. (Reference Kobayashi, Kohama and Kurosawa1983) and Kobayashi (Reference Kobayashi1994). The CM shows a much greater growth rate than that of the CFM. In figure 6, we show the contours of the growth rates for the type-I instability modes for different cases. For $\varOmega =0$, shown in (a), the unstable zone is symmetric about the $n=0$ line, and the most unstable mode appears as a pair of oblique waves. This mode is the Mack first mode, abbreviated as MM. When $\varOmega$ is slightly increased, as shown in (b), the unstable zone leans towards the positive-$n$ direction, also showing an MM-instability nature. For $\varOmega =0.0024$ and 0.0036 (cases III and IV), the right-branch unstable zone crosses the zero-frequency line, indicating the appearance of the stationary mode. Previous analysis (Song & Dong Reference Song and Dong2023) confirmed that the quasi-stationary mode is driven by the cross-flow effect, and so it is referred to as the type-I CFM, while the majority of the travelling mode is still the MM instability. As $\varOmega$ is further increased such that $\bar \varOmega$ reaches 0.75 and 1.01, the unstable zone of the type-I CFM enlarges. Simultaneously, as shown in figure 7, the type-II CM instability appears in the vicinity of the zero-frequency line, whose growth-rate peak (0.017) exceeds that of the MM instability (0.009) for case VI when $\bar \varOmega$ reaches 1.01.

Figure 6. Growth-rate contours of type-I instability in the $\omega$–$n$ plane for different cases at $X=1$. The black contour in each panel represents the neutral curve where $\alpha _i=0$.

Figure 7. Growth-rate contours of the type-I and type-II instabilities in the $\omega$–$n$ plane for cases V (a) and VI (b) at $X=1$.

A clearer observation by showing the dependence of the growth rate on $\omega$ for different $n$ values for representative cases is shown in figure 8. For a moderate rotation rate (case IV), shown in (a), only the type-I instability appears. The curve for $n=25$ shows a single-peak feature, representing the MM instability, whereas for $n=55$, the growth rate shows two mild maxima near $\omega =0$, corresponding to an MM and a CFM, as marked in the plot. For a strong rotating case (case VI), shown in (b), the CFM already appears when $n=25$, and becomes the dominant mode for type-I instability when $n=55$. Additionally, the type-II CM instability appears in the low-frequency band when $n=55$ with a much greater growth rate.

Figure 8. Dependence on $\omega$ of the growth rate $-\alpha _i$ for different $n$ values for cases IV (a) and VI (b) at $X=1$.

The accumulated amplitude of the instability mode at each frequency and circumferential wavenumber can be measured by an $N$ factor, which can be predicted by the LST-predicted growth rate via

(3.2)\begin{equation} N(x) =\exp\left[\int_{{x_0}}^x { - {\alpha _i} (\tilde x) \,{\rm d}\tilde x}\right], \end{equation}

or by the LPSE-calculated amplitude via

(3.3)\begin{equation} N(x) = \ln [ {{{{A_u}\left( { x} \right)}/{{A_u}\left( {{x_0}} \right)}}} ], \end{equation}

where $A_{u}(x)= {\max }_{y} | { {{\overset {\smile }{u}}} (x,y)} + \textrm {c.c.} |$ denotes the amplitude of the streamwise velocity disturbance. For the type-I instability, the travelling MM is always more unstable than the quasi-stationary CFM, and the most unstable MMs appear around $n=20$. However, the frequency of the most unstable mode varies with the angular rotation rate $\varOmega$. In figure 9(a), we compare the streamwise evolution of the $N$ factors of the type-I MM for different $\varOmega$ values. The greatest amplitude appears for case IV with $\varOmega =0.0036$, further increase or decrease of the rotation speed leads to a reduction of the accumulated amplitude. (b) Plots $N$-factor accumulation of the stationary type-I CFM for cases III to VI, because for smaller $\varOmega$ values the CFM are stable. The accumulated $N$ factor of the CFM is much smaller than that of the MM for each case, indicating that the CFM is not the dominant perturbation to trigger transition. Figure 9(c) shows the $N$-evolution for the type-II CM instability for cases V and VI. In the interval $X\in [1,2]$, the type-II CM instability can be amplified approximately twenty times, remarkably greater than that for the type-I instability. The CM instability is more unstable for a higher rotation speed. Additionally, for both the type-I and type-II instabilities, the LST predictions agree well with the LPSE predictions, indicating that the non-parallel effect is rather weak.

Figure 9. The comparison of the $N$-factor obtained by LST (solid lines) and LPSE (symbols) for the most unstable type-I MM (a), the most unstable stationary type-I CFM (b) and the most unstable stationary type-II CM (c).

In figure 10(a), we summarise the dependence of the $N$ factor at $X=2$ on $\varOmega$ for the travelling MM. It is clearly seen that the MM amplification reaches its peak at $\varOmega \approx 0.0036$, corresponding to case IV. As shown in (b), the frequency of the most amplified MM decreases with the increase of $\varOmega$, but approaches a constant 0.06 when $\varOmega$ is greater than 0.0036. The circumferential wavenumber stays unchanged for all rotation rates considered. Therefore, from the traditional $e^N$ transition prediction method, we may conclude that for the travelling-mode-induced transition, the earliest transition occurs when $\varOmega \approx 0.0036$. However, things are not that simple. In the following, we will show that the nonlinear effect will lead to a different dependence of the transition onset on $\varOmega$. In (c), we plot the dependence on $\varOmega$ of the $N$-factor accumulation and the circumferential wavenumber of the stationary CFM. Note that the CFM is stable for $\varOmega \le 0.0007$. Similar to the results in (a), the $N$ factor of the stationary CFM reaches its maximum at $\varOmega \approx 0.0036$, but its value is much smaller than that of the MM. The most unstable circumferential wavenumber decreases with $\varOmega$. In (d), we plot the dependence of the $N$ factor for the CM on $\varOmega$. Because the CM is unstable only for cases V and VI, the plot starts from $\varOmega =0.0053$. The accumulated $N$ factor increases with $\varOmega$, and its value is remarkably greater than that for the MM.

Figure 10. (a) Dependence on $\varOmega$ of the $N$ factors at $X=2$, $N(X=2)$, for the type-I MMs; (b) dependence on $\varOmega$ of the most unstable frequency $\omega _{max}$ and circumferential wavenumber $n_{max}$ for the type-I MMs; (c,d) dependences of $N(X=2)$ and $n_{max}$ on $\varOmega$ for the stationary CFMs and CMs, respectively.

4.1. Parameters for the NPSE calculations

Based on the base flow obtained from § 3.1, we perform the NPSE calculations starting from a streamwise position $X=1$, which march downstream until their blowup. The streamwise grid spacing is chosen to be $dX=0.0043$ (or ${\textrm {d}x}=5.07$). In the wall-normal direction, we choose $y \in [0,2500]$ (such a large $y$-domain is employed to ensure that the attenuation conditions are satisfied in the far field) and 581 non-uniform grid points that are clustered in the near-wall region are employed, for which the coordinate $y_j$ at the $j^{th}$ point is

(4.1)\begin{equation} y_j=\frac{ j /J}{1+(y_{J}/\delta_{99}-2)(1-j/J)} y_J, \quad j \in [0,J], \end{equation}

where $\delta _{99}=8.7$ denotes the nominal boundary-layer thickness at $x=x_0$ and $J=580$. Such an allocation ensures that half of the total grid points are allocated to the boundary layer, which has also been used in our previous work (Zhao, Dong & Yang Reference Zhao, Dong and Yang2019; Song et al. Reference Song, Dong and Zhao2022).

From § 3, we know that, for cases I and II, there is only one linear instability mode, the travelling MM; for cases III and IV, the type-I MM and the type-I CFM coexist; for cases V and VI, there are three instability modes, namely, type-I MM, type-I CFM and type-II CM. The type-I CFM is always less unstable than the MM, but the type-II CM, if exists, is much more unstable. In this paper, the NPSE calculations start from a location where the linear instability modes have already been excited, and so a careful selection of the initial perturbations is required. First, we decide to exclude the type-I CFM as the initial perturbation because of its small growth rate. Second, the type-II CM shows a quasi-stationary nature, which is more likely to be excited by either surface imperfections, such as roughness, or the nonlinear interactions between travelling modes. In this paper, we assume the cone to be smooth without any surface imperfection, and so the CM instability is not introduced in the first place. However, for high rotation rates, the nonlinear interaction of the travelling modes can support its initial amplitude, and its high growth rate for strong rotation cases ensures its dominant role in the downstream locations.

For each of the six rotation rates as studied in § 3.1, we choose the most linearly amplified type-I travelling MM as one of the two initial perturbations, whose frequency and circumference wavenumber can be found in figure 10(a,b), respectively. They are denoted by the fundamental frequency $\omega _0$ and the fundamental wavenumber $n_0$, respectively. In the following, each Fourier component with a frequency ${\mathcal {M}} \omega _0$ and a circumference spanwise wavenumber ${\mathcal {N}} n_0$ is denoted by (${\mathcal {M}},{\mathcal {N}}$) for convenience, and so the aforementioned introduced mode is denoted by ($1,1$). An MM with the same frequency but the opposite circumferential wavenumber, denoted by ($1,-1$), is also introduced. The detailed parameters for the two introduced modes are listed in table 3, and the shape functions of $\hat u_{1,\pm 1}$ and $\hat w_{1,\pm 1}$ are shown in figure 11. Note that the calculations are performed in a moving frame, and are not affected by the rotation. In the NPSE calculations, the nonlinear interaction between the introduced MMs leads to generation of the MFD ($0,0$), the streak mode ($0,2$), the high-order harmonics ($2,0$) and so on. Sketches of the wall pressure in the $t$–$\varphi$ plane for representative Fourier components, $(1,1), (1,-1), (2,0)$ and ($0,2$), are shown in figure 12. When the amplitudes of the perturbations have accumulated to sufficiently high levels, the mean flow would undergo such a rapid distortion due to the strong Reynolds stress that the parabolic assumption ceases to be valid. This leads to a blowup of the NPSE calculation. As confirmed by a number of previous direct numerical simulation (DNS) studies (Dong et al. Reference Dong, Zhang and Zhou2008; Zhang & Zhou Reference Zhang and Zhou2008; Mayer et al. Reference Mayer, Fasel, Choudhari and Chang2014), such a blowup phenomenon indicates the onset of laminar–turbulent transition, and the NPSE calculations upstream of the blowup point agree well with the DNS results.

Table 3. Parameters of the introduced MMs for the NPSE calculations.

Figure 11. Eigenfunctions of the introduced MMs obtained by LST at $X=1$: (a) $|\hat {u}_{1,1}|$; (b) $|\hat {u}_{1,-1}|$; (c) $|\hat {w}_{1,1}|$; (d) $|\hat {w}_{1,-1}|$.

Figure 12. Sketches of the wall pressure in the $t$–$\varphi$ plane for representative Fourier components: (a) (1,1); (b) ($1,-1$); (c) ($2,0$); (d) ($0,2$).

4.2. Cases I and II: OB regime

The OB regime usually appears in a quasi-2-D supersonic boundary layer, as for cases I and II in this paper. The solid curves in figure 13 show the evolution of the $\tilde u$-amplitude of representative Fourier components obtained by NPSE for these two cases. Note that the local Reynolds number $R_x$ and the local rotation rate $\bar \varOmega _x$ vary with the streamwise location $X$, and so they are also marked in the horizontal axis in each panel, where

(4.2a,b)\begin{equation} R_x=\frac{\rho^*_e U^*_e x^*}{\mu^*_e},\quad \bar \varOmega_x=\frac{ \varOmega^{*}x^{*} \sin \theta} {U_e^{*}}. \end{equation}

In case I, the amplitude evolution of the fundamental modes ($1,1$) and ($1,-1$) are identical, and so only the former is plotted. We also show the LPSE predictions by the circles for comparison. For each case, the evolution of the fundamental modes $(1,\pm 1)$ agrees with the LPSE prediction until $X\approx 1.8$, indicating the linear feature of the fundamental modes for the majority of the laminar phase. The growth of the streak mode ($0,2$) is mainly attributed to the direct interaction of ($1,1$) and ($1,-1$). It shows a super-exponential growth in the close neighbourhood of $X=1$, followed by a growth rate that is the sum of the fundamental modes $(1, \pm 1)$, as confirmed by the pink dashed lines; although its amplitude is preliminarily smaller than those of the fundamental modes, its high growth rate ensures its dominant role in the downstream region. The 2-D travelling component ($2,0$), also driven by the direct interaction of $(1,\pm 1)$, shows the same growth rate as that of ($0,2$) (see the green dashed lines), but its amplitude is not as large as the streak component ($0,2$) because of its weaker amplification initially. The MFD is also much weaker than the streak mode in the early laminar phase. For $\varOmega =0$ (case I), the extra amplification of the streak mode ($0,2$) in the early laminar phase has been explained by an asymptotic theory based on the weakly nonlinear analysis (WNA) in Song et al. (Reference Song, Dong and Zhao2022); the explanation for a non-zero $\varOmega$ (case II) is in principle the same, and can be found in Appendix B.

Figure 13. The $A_u$-evolution of the Fourier components obtained by NPSE for cases I (a) and II (b). The dashed lines denote the theoretical prediction of the growth rate by direct interaction of the introduced oblique modes.

For case I, in the late laminar phase, $X>1.88$, the harmonics $(2,0), (2,2), (2,4)$ and ($2,6$) undergo drastic amplification with almost the same growth rate. Because the dominant perturbations in this region are the streak mode ($0,2$) and the MFD ($0,0$), this phenomenon is a reminiscent of the SI of a streaky profile. In order to confirm the occurrence of the SI of a streaky base flow, we compare the perturbation profiles $|{\tilde u}_{2 \omega _{0}}|$ in the $y$–$\varphi$ plane at different $X$ locations for case I in figure 14, where

(4.3)\begin{equation} {{\tilde u}_{2{\omega _0}}}\left( {x,y,\varphi } \right) = \int_0^{2{\rm \pi} /{\omega _0}} {\tilde u\left( {x,y,\varphi ,t} \right)\exp \left( { - 2i{\omega _0}} \right){\rm d} t}, \end{equation}

with $\tilde u$ being obtained by NPSE. The dashed black lines denote the streaky base flow consisting of the laminar base flow, the MFD and the streak components including $(0,2), (0,4)$, etc. It can be seen that, as $X$ increases, the profile of the unsteady travelling perturbation becomes concentrated around the top of the streak ‘head’, and the perturbation is found to be symmetric about the centre of the streak, showing a varicose feature.

Figure 14. Contours of the perturbation profiles $|{\tilde u}_{2 \omega _{0}}|$ in the $y$–$\varphi$ plane for case I. The dashed lines denote the contours of $\breve \varPhi _{B}(y,\varphi )$.

Then, we perform the SIA (introduced in § 2.3.2) based on the streaky base flow obtained by NPSE, and the eigenvalues of the growth rates at $X=2$ for case I are plotted in figure 15(a). Four dominant unstable modes are found, including two sinuous modes (whose eigenfunctions are anti-symmetric about the centre line) and two varicose modes (whose eigenfunctions are symmetric about the centre line), and the varicose I mode is the most unstable one. The other eigen-solutions with quasi-neutral or stable features are not of our interest, and so are not labelled. Tracing the growth rates of the two varicose modes, as shown by the symbols in (b), we find that the growth rate of the varicose I agrees well with the NPSE results of ($2,0$) until $X=1.97$, confirming that the dominant mechanism of the growth of this component is the SI of the ($0,2$)-induced streaky base flow. However, the higher-order modes, $(2,2), (2,4)$ and ($2,6$) show greater growth rates than the SIA-predicted growth rate, because these modes are also affected by the nonlinear interaction of ($2,0$) with other components. The eigenfunctions of the two varicose modes are shown in figure 15(c,d), respectively, which can well predict the local peaks and the overall shape of the perturbation profile in figure 14(h).

Figure 15. (a) Eigenvalues obtained by SIA for $\omega =2\omega _{0}$ at $X=2$; (b) comparison of the growth rates between SIA and NPSE. (c,d) Eigenfunctions of the varicose I and varicose II modes obtained by SIA, respectively.

For case II, shown in figure 13(b), the nonlinear evolution is quite similar, but the only difference is that the growth rate of the MFD, which is twice the growth rate of ($1,1$), is not equal to that of the streak mode, which is the sum of the growth rates of ($1,1$) and ($1,-1$). Because for a rotating cone, where the growth rate of ($1,1$) is larger than that of ($1,-1$), the MFD grows with a faster rate than the streak mode ($0,2$) and the travelling mode ($2,0$). Thus, although the amplitude of the MFD is initially smaller than the streak mode, it overwhelms the latter in a downstream position, $X=1.96$. It is also predicted that, if $\varOmega$ is increased further, the streak-mode-dominated region would shrink and eventually disappear, and so the oblique breakdown regime would be replaced by another regime, as will be introduced in the next subsection.

4.3. Cases III and IV: generalised fundamental resonance (GFR) regime

When $\varOmega$ is increased to 0.0024 and 0.0036, or $\bar \varOmega$ reaches 0.34 and 0.51 (cases III and IV), the growth rates of the two introduced oblique modes are remarkably different, namely, ($1,1$) becomes more unstable but ($1,-1$) becomes almost neutral. This is because the base flow shows a strong asymmetric feature due to the rotation, for which the oblique-breakdown regime as in the quasi-symmetric cases, cases I and II, is not likely to occur. Figure 16(a,c) shows the amplitude evolution of the streamwise and spanwise velocity perturbations for each Fourier component for case III, respectively. The component ($1,1$) shows a linear amplification until around $X=1.80$, after which it saturates due to the nonlinear effect; the other introduced mode, ($1,-1$), shows a much weaker amplification due to its small growth rate before $X=1.80$, which is followed by a drastic amplification until reaching the same magnitude as that for ($1,1$). Additionally, the evolution of both ($1,1$) and ($1,-1$) before $X=1.8$ agrees well with the linear prediction (LPSE). Before $X=1.8$, the MFD grows with a doubled growth rate of that of ($1,1$), which ensures that the amplitude of the MFD reaches the same magnitude of ($1,1$) in a short distance, leading to the nonlinear saturation of ($1,1$). In this region, the growth rates of the streak mode ($0,2$) and the harmonic travelling mode ($2,0$) are the sum of the growth rates of ($1,1$) and ($1,-1$), and are smaller than that of the MFD. For case IV, as shown in (b,d), the scenario is the same, but the saturation position of the fundamental oblique mode $(1, 1)$ appears earlier. Comparing with cases I and II, we find that an increase of the angular rotation rate $\varOmega$ leads to a greater difference between the MFD and the streak mode. As a consequence, for a moderate $\varOmega$, the streak mode no longer becomes dominant before the MFD overwhelms ($1,1$), and so the OB regime does not appear in cases III and IV. After the saturation of the dominant component ($1,1$), the amplitude of the MFD ceases to grow, but the components $(0,2), (2,0)$ and ($1,-1$) undergo rapid amplifications with almost the same rate, which is attributed to the SI of the saturated ($1,1$) component, as will be proven in the following. Being different from the SI of the streaky profiles $\breve \varPhi _{B}(y,\varphi )$ discussed in § 4.2, the SI here is supported by a wavy base flow $\breve \varPhi _B(\tilde x,y)$.

Figure 16. The amplitude evolution of the Fourier components obtained by NPSE for cases III and IV. (a,c) Case III; (b,d) case IV. (a,b) $A_u$; (b,d) $A_w$. The vertical dashed line in each panel represents the position where ($1,1$) saturates.

Actually, the amplification of the streak mode ($0,2$) may be attributed to a few mechanisms, including (A) the nonlinear interaction of components ($1,1$) and ($1,-1$), (B) the linear growth of the type-I CFM, (C) the linear growth of the type-II CM and (D) the SI induced by the wavy base flow $\breve \varPhi _B(\tilde x,y)$. In order to prescribe the dominant mechanism leading to the streak amplification, we compare the theoretical predictions of the $A_u$ evolution of ($0,2$) based on these candidates with the NPSE calculations for case IV in figure 17. It is seen that neither of the mechanisms (B) and (C) can reproduce the streak amplification. In the region $X<1.78$, the NPSE result agrees with the WNA prediction as introduced in Appendix B, confirming that the growth of the streak mode in the early phase is driven by mechanism (A). This is also the reason why the streak mode ($0,2$) shows a greater amplitude than the MFD ($0,0$) in the close neighbourhood of $X=1$. Performing the SIA introduced in § 2.3.1 based on a wavy base flow consists of the laminar base flow, the MFD, the saturated fundamental mode (1,1) and its harmonics $(2,2), (3,3)$, etc., we calculate the amplitude growth of the SI mode for a skewed spanwise wavenumber $\breve \beta = -\sin \varTheta \alpha _{r,(0,2)} +\cos \varTheta 2 n_0 /r_0$, as shown by the crosses in figure 17, which is confirmed to agree well with the NPSE calculation for $X>1.78$. As shown in figure 18, the agreement of the eigenfunctions obtained by SIA and NPSE is also satisfactory, confirming that the late-stage growth of the streak mode is attributed to the mechanism (D).

Figure 17. Comparison of the $A_u$-evolution for ($0,2$) obtained by different approaches for case IV.

Figure 18. Comparison of the eigenfunction of ($0,2$) obtained by the NPSE and SIA at $X=1.9$: (a) $|\hat {u}_{0,2}|$; (b) $|\hat {v}_{0,2}|$; (c) $|\hat {w}_{0,2}|$; (d) $|\hat {T}_{0,2}|$.

Such an SI is a reminiscent of the fundamental resonance regime in hypersonic boundary layers, as observed in Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2015), Chen et al. (Reference Chen, Zhu and Lee2017) and Hader & Fasel (Reference Hader and Fasel2019). In the present study, the introduced fundamental mode ($1,1$) plays the same role as the 2-D Mack second mode in the 2-D boundary layers in the fundamental resonance regime. It dominates the perturbation field in the early laminar phase, and promotes the growth of the MFD due to its direct interaction with its complex conjugate. When the fundamental mode saturates, its SI encourages the rapid amplification of the streak mode, as well as other travelling waves. Therefore, this regime is referred to as the GFR regime. In the very late stage, as shown in figure 16(a,b), the streak mode becomes the dominant perturbation eventually, supporting the SI modes as for the OB regime. The blowup of the NPSE calculation appears when the mean flow undergoes sufficient distortion forced by the SI modes.

4.4. Cases V and VI: centrifugal-instability-induced transition (CIT) regime

Now we consider the nonlinear evolution of the perturbations for strong rotation rates. The solid curves in figure 19(a) display the $A_u$-evolution of the representative Fourier components obtained by NPSE for case V. It is seen that the introduced fundamental component ($1,1$) amplifies with its linear growth rate until $X \approx 1.8$, where it reaches the saturation state with an amplitude of approximately 0.1. The other introduced mode ($1,-1$) decays with its linear decay rate until $X\approx 1.67$, after which a sharp amplification appears. The streak component ($0,2$) also shows a growth rate that is the sum of ($1,1$) and ($1,-1$) before $X \approx 1.52$, confirmed by the pink dashed line in (a), after which a much greater growth rate is observed. Increasing $\varOmega$ to $0.0071$ or $\bar \varOmega$ reaches 1.01, as shown in figure 19(b), the linear growth regions of ($1,1$) and ($1,-1$) become shorter, and the steak component ($0,2$) shows a much greater growth rate than the sum of ($1,1$) and ($1,-1$) from a position rather close to $X=1$. This phenomenon is in contrast to the cases with smaller $\varOmega$ values.

Figure 19. The $A_u$-evolution of the Fourier components obtained by NPSE for cases V (a) and VI (b).

In order to explain the greater growth rate of the ($0,2$) component, we perform the linear stability analysis of the type-II CM instability by the LPSE approach for these two cases, and compare their accumulated amplitudes with the NPSE results, as shown by the pink circles in figure 19. For case V, the linear growth of the CM can well predict the evolution of ($0,2$) when $X>1.5$. For case VI, because the CM shows a much greater growth rate, the linear stability analysis yields a good prediction from almost $x=x_0$.

For case VI, in the late laminar phase, $X > 1.7$, the harmonics $(2,0), (2,2), (2,4)$ and ($2,6$) undergo drastic amplification with almost the same growth rate as shown in figure 19(b). Similar to case I, the dominant perturbations in this region are the streak mode ($0,2$) and the MFD ($0,0$), and the rapid growths of the harmonics are attributed to the SI of the streaky base flow. However, the difference is that the streak mode ($0,2$) is driven by the rapid growth of the CM, instead of the nonlinear interaction between ($1,1$) and ($1,-1$). In figure 20, we plot the perturbation profiles of $|{\tilde u}_{2 \omega _{0}}|$ in the $y$–$\varphi$ plane at different $X$ locations for case VI. The perturbations become localised downstream of $X\approx 1.7$, where the streak mode becomes the dominant perturbation.

Figure 20. Contours of the perturbation profiles $|{\tilde u}_{2 \omega _{0}}|$ in the $y$–$\varphi$ plane for case VI. The dashed lines denote the contours of $\breve \varPhi _{B}(y,\varphi )$.

Performing the SIA for a streaky base flow as introduced in § 2.3.2, we can obtain a set of unstable eigenvalue solutions for $\omega =2\omega _0$ at $X=1.73$, as shown in figure 21(a), where the most unstable two modes are marked mode I and mode II. Tracing the growth rates of the two modes as shown by the symbols in (b), the growth rates of the travelling components $(2,0), (2,2), (2,4)$ and ($2,6$) obtained by NPSE are compared with the SIA predictions, and favourable agreement is observed in the late phase ($X>1.71$), confirming that the rapid amplifications of the travelling modes in this region are attributed to the SI of the streaky base flow. The eigenfunctions of the two modes obtained by SIA are shown in 21(c,d), respectively. Both eigenfunctions can predict the overall shape of the NPSE perturbation profile in figure 20(g).

Figure 21. (a) Eigenvalues obtained by SIA at $X=1.73$ for $\omega =2\omega _{0}$; (b) comparison of the growth rates between SIA and NPSE. (c,d) Eigenfunctions of mode I and mode II obtained by SIA, respectively.

4.5. Summary

In summary, there exist three mechanisms leading to the rapid amplification of the streak component in a supersonic boundary layer over a rotating cone. For small $\varOmega$ values including $\varOmega =0$, the growth of the streak mode is driven by the direct interaction between the introduced fundamental modes ($1,1$) and ($1,-1$), which can be well explained by the asymptotic analysis based on the WNA. Such a nonlinear process belongs to the OB regime. Increasing $\varOmega$ to a moderate level, say $\varOmega =0.0036$, the accumulated amplitude of ($1,1$) is much greater than that of ($1,-1$), and the rapid growth of the streak mode is due to the SI of the saturated travelling mode ($1,1$). In the late laminar phase, the streak mode would become the dominant perturbation. Such a nonlinear process belongs to the GFR regime. Further increasing $\varOmega$ to 0.0071 or $\bar \varOmega = O(1)$, the linear growth rate of the centrifugal instability becomes greater than the amplification due to the aforementioned two regimes, which ensures that the stationary CM (that also shows a stationary streak nature) becomes the dominant perturbation in a short distance. Such a process belongs to the CIT regime.