2. Problem and methods

2.1. Problem and coordinate definition

The problem is defined in figure 3. We consider a swept flow over a spanwise-infinite plate. Two sets of coordinates, based on the chordwise geometry $(x,y,z)$ and (local) potential streamline direction$(x_s,y,z_s)$ will be used for the subsequent analysis. Velocity components $(u,v,w)$ and $(u_s,v,w_s)$ are thus defined accordingly. The ambient pressure is $p^*_\infty =80$ bar (we use superscript $*$ to denote dimensional quantities and subscript $\infty$ for ambient values). The effective swept angle is $\phi ^*_\infty =45^\circ$. The distribution of the pressure coefficient $c_p(x)$ is matched to the experimental measurements (Lohse, Barth & Nitsche Reference Lohse, Barth and Nitsche2016; Barth, Hein & Rosemann Reference Barth, Hein and Rosemann2018). The flow is subject to the following dimensionless numbers:

(2.1a–d)\begin{equation} {{Re}}=\frac{\rho_{\infty}^{*}u_{\infty}^{*}l_0^{*}}{\mu_{\infty}^{*}},\quad {{Ma}}=\frac{u_{\infty}^{*}}{a^*_\infty},\quad Pr=\frac{\mu_{\infty}^{*}C_{p\infty}^{*}}{\kappa^*_\infty},\quad Ec=\frac{u_{\infty}^{*2}}{C_{p\infty}^{*}T^*_\infty}. \end{equation}

The Prandtl and Eckert number (Pr and Ec) are not independent and can be calculated from the Mach and Reynolds numbers (${{Ma}}$ and ${{Re}}$). To allow for a comparable discussion, ${{Ma}}=0.2$ and ${{Re}}=1.4687\times 10^5$ are prescribed as in the reference case (Dörr & Kloker Reference Dörr and Kloker2017). We consider the following cases listed in table 1, traversing different thermodynamic regimes by specifying the free stream and wall temperatures ($T^*_\infty$ and $T_w^*$). In each regime, the following wall-to-free stream ratios are studied: $T_w/T_\infty =1.0667$, 1.0333, 0.9688 and 0.9375, accounting for both heating and cooling walls. An overview of the thermodynamic regimes is displayed in the density-temperature diagram in figure 4. The pink areas correspond to thermodynamic states covered by the flows in table 1. The parameters remain close to the isobar of 80, indicating that the pressure gradient is not large enough to change the thermodynamic properties significantly. On the other hand, we employ the NIST Refprop database (Lemmon, Huber & Mclinden Reference Lemmon, Huber and Mclinden2013) to generate look-up tables $\chi (\rho, T)$, accounting for the highly non-ideal equation of states and transport properties. Here, $\chi$ collectively symbolises the required quantities (e.g. pressure, internal energy, viscosity, thermal conductivity). The dashed squares in figure 4 that enclose different regimes stand for the range of look-up tables employed for each regime. It has been verified that the tables are good enough, whose size does not influence the results.

Figure 3. Problem and coordinate definition.

Figure 4. An overview of the thermodynamic regimes in density–temperature ($\rho$–$T$) diagram.

Table 1. Case definition and parameters. All three cases have the same dimensionless numbers ${{Ma}}=0.2$, ${{Re}}=1.4687\times 10^5$, pressure coefficient $C_p(x)$ and sweep angles $\phi (x)$.

2.2. The laminar baseflow

In the redesigned DLR (German Aerospace Center)-experimental configuration, the baseflow is not self-similar. We solve the ‘parabolised’ Navier–Stokes equations (PNS) for the laminar baseflow. In steady boundary-layer flows without separation, the streamwise viscous gradient is much smaller than the other component in wall-normal direction. The PNS are thus derived from the complete Navier–Stokes equations by dropping the streamwise gradient in the viscous terms, reading (in 2-D form)

(2.2)\begin{gather} \boldsymbol{\mathsf{A}}\frac{\partial\boldsymbol{q}}{\partial x}+\boldsymbol{\mathsf{B}}\frac{\partial\boldsymbol{q}}{\partial y}={\text{RHS}}; \end{gather}

(2.3)\begin{gather} \boldsymbol{\mathsf{A}}=\left(\begin{array}{@{}ccccc@{}} u & \rho & 0 & 0 & 0\\ 0 & \rho u &-\dfrac{1}{Re}\dfrac{\partial\mu}{\partial y} & 0 & 0\\ 0 &-\dfrac{1}{Re}\dfrac{\partial\lambda}{\partial y} & \rho u & 0 & 0\\ 0 & 0 & 0 & \rho u & 0\\ \rho u\dfrac{\partial e}{\partial\rho} & p & 0 & 0 & \rho u\dfrac{\partial e}{\partial T} \end{array}\right), \end{gather}

(2.4)\begin{gather}\boldsymbol{\mathsf{B}}=\left(\begin{array}{@{}ccccc@{}} v & 0 & \rho & 0 & 0\\ 0 & b_{2,2} & 0 & 0 & 0\\ \dfrac{\partial p}{\partial\rho} & 0 & b_{3,3} & 0 &\dfrac{\partial p}{\partial T}\\ 0 & 0 & 0 & b_{4,4} & 0\\ \rho v\dfrac{\partial e}{\partial\rho} &-\dfrac{\mu}{Re}\dfrac{\partial u}{\partial y} &p-\dfrac{2\mu+\lambda}{Re}\dfrac{\partial v}{\partial y} &-\dfrac{\mu}{Re}\dfrac{\partial w}{\partial y} & b_{5,5} \end{array}\right), \end{gather}

(2.5)\begin{gather} \left.\begin{gathered} b_{2,2} = \rho v-\frac{1}{Re}\frac{\partial\mu}{\partial y}-\frac{\mu}{Re}D_{y}\\ b_{3,3} = \rho v-\frac{1}{Re}\frac{\partial\left(2\mu+\lambda\right)}{\partial y}-\frac{2\mu+\lambda}{Re}D_{y}\\ b_{4,4} = \rho v-\frac{1}{Re}\frac{\partial\mu}{\partial y}-\frac{\mu}{Re}D_{y}\\ b_{5,5} = \rho v\frac{\partial e}{\partial T}-\frac{1}{RePrEc}\frac{\partial\kappa}{\partial y}- \frac{\kappa}{RePrEc}D_{y} \end{gathered}\right\}. \end{gather}

The equations are linearised by ‘lagging’ coefficients $\boldsymbol{\mathsf{A}}$ and $\boldsymbol{\mathsf{B}}$ relative to the solution vector $\boldsymbol {q}=(\rho,u,v,w,T)^{\textrm {T}}$ in an iteration procedure, i.e.  subiterations are performed to update $\boldsymbol{\mathsf{A}}$ and $\boldsymbol{\mathsf{B}}$ from $\boldsymbol {q}$ at each station of the streamwise marching to get the correct, fully nonlinear values. Note that the term ‘parabolised’ is bearably a misnomer, while the equations are a hybridised set of hyperbolic–parabolic equations (Tannehill, Anderson & Pletcher Reference Tannehill, Anderson and Pletcher1997). To ensure a strictly parabolic manner, the streamwise pressure gradient is formulated as a forcing term on the right-hand side of the PNS: ${\textrm {right-hand side}}=(0,-\textrm {d}p/{\textrm {d} x},0,0,0)^\textrm {T}$. The boundary conditions are

(2.6a)$$\begin{gather} y=\infty:\ \frac{\partial u}{\partial y}=\frac{\partial w}{\partial y}=0, \quad v=v\left(\textrm{continuity eq.}\right),\quad \rho=\rho_{e}\left(x\right),\quad T=T_{e}\left(x\right); \end{gather}$$

(2.6b)$$\begin{gather}y=0:\ u=v=w=0,\quad \rho=\rho\left(\frac{\partial p}{\partial y}=0\right),\quad T=T_{w}. \end{gather}$$

We employ subscript $e$ for boundary layer edge values. Here $\rho _{e}(x)$ and $T_{e}(x)$ are potential-flow values given by the isentropic relations

(2.7a,b)\begin{equation} S\left(\rho_{e}\left(x\right),p\left(x\right)\right)=S\left(\rho_{\infty},p_{\infty}\right), \quad S\left(T_{e}\left(x\right),p\left(x\right)\right)=S\left(T_{\infty},p_{\infty}\right), \end{equation}

where $S$ stands for entropy, and $p(x)$ is the measured pressure with $\textrm {d}p/{\textrm {d} x}<0$. The PNS are integrated downstream using an implicit Euler scheme, starting from an initial profile at $x=x_0$. In this study, we prescribe the streamwise and spanwise velocities $u(x_0,y)$, $w(x_0,y)$ using the Falkner–Skan–Cooke (FSC) solution and $v(x_0,y)=0$. The thermodynamic variables ($\rho, T$) are given either as the potential-flow values (applying isentropic relations) or extrapolated from existing downstream data. The validation of the PNS results with different initial profiles is provided in Appendix A, ensuring that the influence of the initial profiles is insignificant.

2.3. Linear stability theory

We consider linear modal instability of the laminar baseflow obtained in § 2.2. Since both steady and unsteady CF modes are subject to modal instabilities, they shall serve as fundamental mechanisms leading to transition. The algebraic instability (Levin & Henningson Reference Levin and Henningson2003) under these conditions is thus not considered. On the other hand, the non-ideal effects of entirely bypass mechanisms (e.g. with massive free stream turbulence) remain of interest for further investigations. The non-ideal framework was derived based on the state postulate that a simple compressible system is defined by two independent thermodynamics properties. The perturbation vector is defined as $\boldsymbol {q}=(\rho ^{\prime },u^{\prime },v^{\prime },w^{\prime },T^{\prime })^{\textrm {T}}$. Therefore, perturbations in the other transport and thermodynamic properties (e.g. $e^{\prime }$, $\kappa ^{\prime }$) are formulated as a function of $(\rho ^{\prime },T^{\prime })$ through 2-D Taylor expansion, see Ren et al. (Reference Ren, Marxen and Pecnik2019b). The stability equations are obtained from the Navier–Stokes equations by subtracting the governing equations of the unperturbed baseflow, formulated as

(2.8)\begin{align} &\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{t}}}\frac{\partial\boldsymbol{q}}{\partial t}+\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{x}}}\frac{\partial\boldsymbol{q}}{\partial x}+\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{y}}}\frac{\partial\boldsymbol{q}}{\partial y}+\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{z}}}\frac{\partial\boldsymbol{q}}{\partial z}+\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{q}}}\boldsymbol{q} \nonumber\\ &\quad +\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{xx}}}\frac{\partial^{2}\boldsymbol{q}}{\partial x^{2}}+\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{xy}}}\frac{\partial^{2}\boldsymbol{q}}{\partial x\partial y}+\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{xz}}}\frac{\partial^{2}\boldsymbol{q}}{\partial x\partial z}+\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{yy}}}\frac{\partial^{2}\boldsymbol{q}}{\partial y^{2}}+\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{yz}}}\frac{\partial^{2}\boldsymbol{q}}{\partial y\partial z}+\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{zz}}}\frac{\partial^{2}\boldsymbol{q}}{\partial z^{2}}=0. \end{align}

The definition of the matrices in (2.8) is provided in Appendix B. Under normal-mode form, $\boldsymbol {q}(x,y,z,t)=\hat {\boldsymbol {q}}(y)\exp (\textrm {i}\alpha x+\textrm {i}\beta z-\textrm {i}\omega t)+\textrm {c.c.}$ In this work, we seek the solution in the spatial mode, where the spanwise wavenumber $\beta$ and angular frequency $\omega$ are given as real input and $\alpha$ is the complex eigenvalue to be determined by the following resulting nonlinear eigenvalue problem:

(2.9) \begin{align} &(-{\rm i}\omega\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{t}}}+\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{y}}} \boldsymbol{\mathsf{D}}+{\rm i}\beta\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{z}}}+\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{q}}}+ \boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{yy}}}\boldsymbol{\mathsf{D}}^{2}+{\rm i}\beta\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{yz}}} \boldsymbol{\mathsf{D}}-\beta^{2}\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{zz}}})\hat{\boldsymbol{q}}\nonumber\\ &\quad =\alpha\left(\beta\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{xz}}}-{\rm i}\boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{xy}}} \boldsymbol{\mathsf{D}}-{\rm i}\boldsymbol{\mathsf{L}}_{\boldsymbol{\mathsf{x}}}\right)\hat{\boldsymbol{q}}+\alpha^{2} \boldsymbol{\mathsf{V}}_{\boldsymbol{\mathsf{xx}}}\hat{\boldsymbol{q}}. \end{align}

Here $\boldsymbol{\mathsf{D}}$ stands for the differential matrix (using the Chebyshev spectral method) with $\boldsymbol{\mathsf{D}}\hat {\boldsymbol {q}}=\textrm {d}\hat {\boldsymbol {q}}/{\textrm {d} y}$. The perturbations are subject to Dirichlet boundary conditions ($u^{\prime }=v^{\prime }=w^{\prime }=T^{\prime }=0$) at the wall and in the free stream. Note that, compared with the conventional linear stability theory (LST) framework, the current method has been verified to reproduce the classic results in the ideal-gas regime, see Appendix C.

2.4. Disturbance pattern evolution according to controlled disturbance sources

Previous research has revealed the receptivity of disturbance modes in dependence of spanwise wavenumber, see, e.g. Stemmer, Kloker & Wagner (Reference Stemmer, Kloker and Wagner2000). It is found that the receptivity coefficients are of the same order of magnitude. Compared with the exponential growth that follows, this paves the way for constructing the disturbance pattern evolution assuming a constant, or incorporating known results of receptivity coefficients. As an example, we consider a harmonic point disturbance

(2.10)\begin{equation} f_{v}\left(x,z,t\right)=A(14r^{5}-45r^{4}+50r^{3}-20r^{2}+1)\sin(\omega t+\varphi_0), \end{equation}

with

(2.11)\begin{equation} r^{2}=\frac{\left(x-x_{c}\right)^{2}+\left(z-z_{c}\right)^{2}}{R^{2}},\quad 0\leqslant r\leqslant1. \end{equation}

The function (2.10) satisfies that $\iint f_{v}(x,z,t)\,\textrm {d}x\,\textrm {d}z=0$. Therefore, at any time step, no net flow flux is introduced and the generation of sound is minimised. By choosing the frequency $\omega$, point source centre $(x_c,z_c)$ and the radius $R$, we will be able to mimic various excitations. Figure 5 provides such an example that we locate the source at the domain centre. Accordingly, the Fourier amplitudes are symmetric with respect to wavenumber $\beta =0$. To connect to the LST results, we collect the Fourier amplitudes

(2.12a,b)\begin{equation} \hat{f}_{v,c}\left(\beta\right)=\hat{f}_{v}\left(x_{c},\beta,t_{0}\right)\quad\textrm{or}\quad \hat{f}_{v,a}\left(\beta\right)=\int\hat{f}_{v}\left(x,\beta,t_{0}\right)\textrm{d}x. \end{equation}

The use of either the centre line of integration along $x$ in (2.12a,b) does not cause a qualitative difference. The physical results are obtained as

(2.13)\begin{equation} \boldsymbol{q}\left(x,y,z,t\right)=\hat{\boldsymbol{q}}\left(y\right)\underset{\beta}{\sum} [\hat{f}_{v}\left(\beta\right)\exp\left({\rm i}\alpha_{r}\left(\beta\right)x+{\rm i}\beta z-{\rm i}\omega t+N\left(\beta,x\right)\right)]+{\rm c.c.} \end{equation}

in which the $N$-factor is collected by integrating the growth rate from $x_c$ of the point source

(2.14)\begin{equation} N\left(\beta,x\right)={-}\int_{x_{c}}^{x}\alpha_{i}\left(\beta\right)\textrm{d} x. \end{equation}

Figure 5. An illustration of the point-source type wall perturbation. (a) A 3-D overview of a snapshot ($t=t_0$); (b) A 2-D view at the centre line ($x=x_c, t=t_0$); (c) Fourier amplitudes $\hat {f}_{v,c}(\beta )$ and $\hat {f}_{v,a}(\beta )$ of the point source, both are normalised with $\beta =0$.

To remove the influence of persistent damped modes (throughout the considered domain), their $N$ factors are specified as $-\infty$. Equation (2.13) gives a fast route revealing the flow scenario in the linear regime once the linear stability results are obtained.

3. Supercritical and subcritical regimes: the non-ideal CF modes

Serving as a reference for the investigation of non-ideality, we start our analysis from the ideal regime. Figure 6 shows stability diagrams in the $x$–$\beta$ and $\omega$–$\beta$ frames, accounting for both steady and unsteady/travelling CF modes. Given the modal growth rate, unsteady modes (with positive $\beta$) usually dominate, while steady modes have been deemed more important at low free stream turbulence levels due to surface roughness. Here we place our focus on the influence of the temperature ratio. The isothermal case (${T_{w}/T_{\infty }=1}$) corresponds to the experimental set-up, whose stability has been validated against published results in Appendix C. As is indicated in the stability diagram, wall-heating destabilises the CF mode and vice versa. We explain the influence of the temperature ratio through the baseflows presented in figure 7. The definition of the boundary layer parameters are

(3.1a,b)$$\begin{gather} \delta_{1,s}=\int_{y=0}^{\infty}\left(1-\dfrac{\rho u_{s}}{\rho_{e}u_{s,e}}\right){{\rm d} y},\quad \delta_{2,s}=\int_{y=0}^{\infty}\dfrac{\rho u_{s}}{\rho_{e}u_{s,e}}\left(1-\dfrac{\rho u_{s}}{\rho_{e}u_{s,e}}\right){{\rm d}y}. \end{gather}$$

(3.2a,b)$$\begin{gather}\phi_{e}=\arctan\left(\frac{w_{e}}{u_{e}}\right),\quad H_{12,s}=\frac{\delta_{1,s}}{\delta_{2,s}}. \end{gather}$$

Note that the compressible definition of the displacement thickness $\delta _{1,s}$ and momentum thickness $\delta _{2,s}$ has been used to take into account the variation of flow density. The shape factor $H_{12,s}(x)$ would also not be constant for self-similar flow (with $H_{12,x}(x) = \textrm {const}.$). The underlying influence of temperature is two-fold. First, an increase in the displacement thickness ($\delta _{1,s}$) is caused by wall heating (and vice versa). Profiles of $u$, $w$, and accordingly $u_s$, become less full, as seen in the baseflow profiles at $x=1$ in figure 7(b). Second, a notable increase of $\max _y(-w_s)$ is observed, leading to the larger magnitude of $w_s$. Both aspects point to the destabilisation of CF modes. The shape of $u_s(y)$ stands for the bulk of the baseflow, upon which $w_s(y)$ determines the degree of its inflection-point influence. Both profiles characterise the CF instability. We revisit the mathematical connection which is established by returning to the governing equations. Near the wall the momentum equations in $x$ and $z$ reduce to the boundary-layer equations,

(3.3a)$$\begin{gather} \frac{\partial^{2}u}{\partial y^{2}} = \frac{Re}{\mu}\frac{{\rm d} p}{{\rm d} x}+ \frac{Re}{\mu}\rho v\frac{\partial u}{\partial y}+\frac{Re}{\mu}\rho u\frac{\partial u}{\partial x}- \frac{1}{\mu}\frac{\partial\mu}{\partial y}\frac{\partial u}{\partial y}, \end{gather}$$

(3.3b)$$\begin{gather}\frac{\partial^{2}w}{\partial y^{2}} = \frac{Re}{\mu}\rho u\frac{\partial w}{\partial x}+ \frac{Re}{\mu}\rho v\frac{\partial w}{\partial y}-\frac{1}{\mu}\frac{\partial\mu}{\partial y}\frac{\partial w}{\partial y}. \end{gather}$$

On the wall,

(3.4a)$$\begin{gather} \left.\frac{\partial^{2}u}{\partial y^{2}}\right|_{w}=\frac{Re}{\mu_{w}}\frac{{\rm d} p}{{\rm d} x}- \left.\left(\frac{1}{\mu}\frac{\partial\mu}{\partial y}\frac{\partial u}{\partial y}\right)\right|_{w}, \end{gather}$$

(3.4b)$$\begin{gather}\left.\frac{\partial^{2}w}{\partial y^{2}}\right|_{w}={-}\left.\left(\frac{1}{\mu} \frac{\partial\mu}{\partial y}\frac{\partial w}{\partial y}\right)\right|_{w}. \end{gather}$$

For flows with fluids in the ideal regime subject to favourable gradient $\textrm {d}p/{\textrm {d} x}<0$, the inflection point (IP) is inside the wall, and the flow cannot separate since both velocity gradients ($\partial u/\partial y|_w$ and $\partial w/\partial y|_w$) are positive on the wall. In (3.4), the wall temperature leads to the following relation:

(3.5) \begin{equation} \left.\frac{\partial \mu}{\partial y}\right|_w \left\{ \begin{array}{@{}lll} >0, & \textrm{wall cooling}\ \left(\left.\dfrac{\partial T}{\partial y}\right|_w>0\right), & \textrm{IP moves towards the wall;} \\ <0, & \textrm{wall heating}\ \left(\left.\dfrac{\partial T}{\partial y}\right|_w<0\right), & \textrm{IP moves away from the wall.} \end{array} \right. \end{equation}

Accordingly, the velocity profiles get fuller (cooling wall) or less full (heating wall), leading to smaller/larger growth rates of the CF modes. Note that the wall distance of the inherent IP in the true CF profile $w_s(y)$ reduces with cooling and increases with heating. (Cooling acts like suction on $w_s(y)$, cf. Messing & Kloker (Reference Messing and Kloker2010), and the closer the IP to the wall is the smaller is the inviscid instability caused by $w_s(y)$.) The relation (3.5) will be shown later to exert a critical influence on the non-ideal regime.

Figure 6. Stability diagram in the ideal regime ($T_\infty ^{*}=700$ K) with different temperature ratios ($T_{w}/T_{\infty }$). (a) Steady CF modes with $N$-factor lines of 1, 3, 5, 7, 9; (b) steady and unsteady modes at $x=1$.

Figure 7. Baseflows in the ideal regime. (a) Boundary layer parameters as functions of $x$; (b) baseflow profiles at $x=1$.

On the other hand, the magnitude of $w_s$ is mathematically explained from the momentum equation along $z_{s}$. The bulk cross-pressure equation (without considering viscous forces), a balance between centripetal and centrifugal (volume) forces on a curved streamline, reads

(3.6)\begin{equation} \frac{{\rm d}p}{{\rm d}z_{s}}={-}\frac{\rho}{R}u_{s}^{2}. \end{equation}

Here $R$ is the local radius of the curved streamline. Across the boundary layer the pressure gradient keeps constant, and since $u_{s}$ decreases towards the wall, $R$ must follow at constant density, deflecting the flow in the direction of the curvature centre thus giving rise to $w_{s}$. With the density varying in the layer, the density itself has a share of the balance. For example, the density decreases from its potential value to the wall with wall heating. Accordingly, $R$ will be smaller towards the wall leading to a more curved streamline as demonstrated in figure 8. This results in a larger/smaller amplitude of $w_s$ being generated from $u_{s,e}$ as seen in figure 7(b).

Figure 8. Streamlines in the ideal regime. The same colour style as in figure 7 is used to denote temperature ratios.

The above analysis of the ideal regimes serves as an important foundation and reference when evaluating the non-ideal effects. We compare the stability diagrams between the subcritical and supercritical regimes in figure 9. The coordinates are mirrored in $x$ and $\omega$, respectively, to compare the two non-ideal regimes better. Here, $N$-factors in the ideal regime (figure 6a) are included in the $x$–$\beta$ diagrams ($\omega =0$) as white dashed lines. These diagrams in figure 9 indicate that non-ideal CF modes, both steady and unsteady, follow such behaviour.

1. (i) The subcritical regime. An inversed influence of the temperature ratio is seen, i.e. wall cooling destabilises the CF mode.

2. (ii) The supercritical regime. The CF mode receives a similar but slightly enhanced influence of the temperature ratio as in the ideal regime, leading to a smaller/larger $N$-factor than the ideal regime subject to wall cooling/heating of the same ratio.

Figure 9. The LST of the subcritical and supercritical regimes. (a) Stability diagram for steady CF modes. The white lines are contours of $N$-factors ($1,3,\ldots,9$) for the non-ideal (solid lines) and ideal (dashed lines) regimes; (b) steady and unsteady modes at $x=1$.

Note that non-ideal effects are not present for isothermal cases since thermodynamic and transport properties reduce to constant functions of $y$. To understand the above stability results, we present the baseflow in figure 10. In either case, the flow temperature remains above or below the pseudocritical point. We focus on the profiles of $u_s$ and $w_s$. Contrary to the ideal regimes, the $u_s$ profiles of the subcritical cases get fuller with wall heating. In the supercritical regime, $u_s$ is not much influenced by non-ideality. In terms of $w_s$, a similar trend as in the ideal regime is held in both subcritical and supercritical regimes, i.e. heating increases the CF components. The difference is that non-ideality enhances the influence of temperature ratios on $w_s$ in supercritical regimes while weakens it in subcritical regimes. This is in accordance with the density profiles shown in figure 10(b). The density gradient $\partial \rho /\partial y$ near the edge of the boundary layer gets smaller in the subcritical regime compared with the ideal-gas reference. According to the momentum balance relation (3.6), the influence of wall heating/cooling on $w_{s}$ therefore gets weakened.

Figure 10. Baseflow in the subcritical and supercritical regimes. (a) Amplitude of the CF component $\max _y(-w_s)$ versus $x$; (b,c) baseflow profiles at $x=1$. In each panel, dashed lines indicate the ideal regime and line colours stand for different temperature ratios.

Mathematically, the behaviour of $u_s$ is explained by the momentum equation at the wall (3.4) that holds also for non-ideal cases. In particular, the viscosity gradient at the wall

(3.7)\begin{equation} \left.\frac{\partial\mu}{\partial y}\right|_{w}=\underset{\textrm{term }T}{\underbrace{\left(\frac{\partial\mu}{\partial T}\frac{\partial T}{\partial y}\right)_{w}}}+\underset{\textrm{term }\rho}{\underbrace{\left(\frac{\partial\mu}{\partial\rho}\frac{\partial\rho}{\partial y}\right)_{w}}}, \end{equation}

determines the shape of the $u_s$ profiles: positive leads to fuller profiles (and vice versa). The role of non-ideality thus sets in through the considerable modulation of the viscosity gradient (3.7) enumerated in figure 11. A first knowledge provided in figure 11(a) indicates that in the supercritical regimes (fluids are gas-like), the viscosity is dominated by term $T$ while term $\rho$ marginally reduces the amplitude no matter whether with wall cooling or heating. Moreover, the value of $\partial \mu /\partial y|_{w}$ remains similar to the ideal-gas regime, which explains that the $u_s(y)$ profiles are not much influenced by non-ideality in the supercritical regime. However, in the subcritical case, term $\rho$ becomes important and dominates significantly. The transport property tables in figure 11(b) provide a global view of the $T$ and $\rho$ gradients of viscosity (${\partial \mu ^*}/{\partial T^*}$ and ${\partial \mu ^*}/{\partial \rho ^*}$). Both terms broadly stay positive throughout different thermodynamics regimes. A notable difference is found in the subcritical regime where ${\partial \mu ^*}/{\partial T^*}$ turns negative. In this case, term $T$ and term $\rho$ (much larger) may thus join forces. By scrutinising each term in figure 11(a), ${\partial \rho }/{\partial y}$ and ${\partial T}/{\partial y}$ are in the same order of magnitude. The crucial factor driving the subcritical regime far from the ideal gas lies in the term ${\partial \mu }/{\partial \rho }$, which gets much larger than ${\partial \mu }/{\partial T}$, leading to a significant dominance of term $\rho$ in (3.7). Considering that ${\partial T}/{\partial y}$ and ${\partial \rho }/{\partial y}$ have differing signs, the influence of wall heating/cooling, therefore, becomes reversed in the subcritical regime. The above analysis is summarised in figure 11(c), explaining how non-ideality and wall-heating/cooling regulate the $u_s$ profiles.

Figure 11. (a) The $y$-gradient of the viscosity and its compositing terms (term $T$ and term $\rho$ in (3.7)); (b) property tables for the viscosity gradient ${\partial \mu ^*}/{\partial T^*}|_\rho$ and ${\partial \mu ^*}/{\partial \rho ^*}|_T$; (c) summary of $y$-gradient of viscosity for the subcritical and supercritical regimes. Here, $\bigstar$ and $\bigstar \bigstar$ symbolically stand for the degree of dominance.

4. The transcritical regime: a changeover of the leading instability mechanism by wall cooling

Now we focus on the transcritical regime with the stability diagrams shown in figure 12. Compared with the ideal regime, wall heating serves to stabilise the CF mode just like in the subcritical regime. Noteworthy is the cooled case: besides the CF mode, an intensely unstable region appears for higher frequencies when the temperature ratio is sufficiently low. We note that the maximum growth rate is one order of magnitude larger than the one of the CF modes, and its instability band in frequency $\omega$ and spanwise wavenumber $\beta$ is considerably larger. The analysis below shows that the tremendous growth rate is due to an inviscid Tollmien–Schlichting (TS) mode instability.

Figure 12. Stability diagram in the transcritical regimes with different temperature ratios ($T_{w}/T_{\infty }$). (a) Steady CF modes with $N$-factor lines of 1, 3, 5, 7, 9; (b) steady and unsteady modes at $x = 1$.

Based on the observation in figure 12(b), we present the eigenspectrum and eigenvectors at $x=1$ in figure 13 in a comparable manner; $(\beta,\omega ) = (80, 1.75)$ and $(111, 34)$, correspond to the most unstable CF and TS mode, respectively. The eigenvalues of the two cases are $(c,\alpha _i)=(-0.019, -2.644)$ and $(0.466, -23.656)$. Here $c=\omega /\alpha _r$ is the phase velocity. The eigenvector shows that the TS mode is more cramped near the wall while the CF mode largely takes up the space of the whole boundary layer. A noteworthy characteristic of the TS mode is that the density perturbation dominates strongly over the other perturbations.

Figure 13. (a,b) Eigenspectrum and (c,d) eigenvectors of the transcritical regime with wall cooling ($T_w/T_\infty =0.9375$) at $x=1$: (a,c) $\beta =80$, $\omega =1.75$ (CF mode); (b,d) $\beta =111$, $\omega =34$ (TS mode). Note the differing scalings for $\rho$ and $T$ between (c) and (d).

To understand the occurrence of the inviscid TS mode we look into the details of the cooling case by gradually reducing the temperature ratio. Correspondingly, the stability diagrams are summarised in figure 14. Despite the decreasing growth rates of the CF mode with the cooling wall, the TS mode rushes to dominate and reaches tremendous growth rates. For a view in the streamline-based coordinate system we present the $\alpha _{r,s}$–$\beta _{r,s}$ diagram in figure 14(b). The result shows that the inviscid TS mode reaches its maximum growth rate around $\beta _{r,s}=0$, implying its essential dependency on the $u_s$ profile. These appearances are in accordance with the typical nature of TS modes. To explore the mechanisms further, we show the pertinent baseflow parameters in figure 15. According to the cross-pressure equation (3.6), the amplitude of the CF $w_{s}$ decreases with cooling and thus the growth rate of the CF mode. More importantly, the $u_s(y)$-profile becomes inflectional due to the cooling wall, with the generalised IP moving away from the wall. Since the $w_{s}(y)$-profiles are also inflectional, this leads to a coexistence and competition of two inviscid mechanisms, CF and TS. Once the temperature ratio is low enough, the inviscid TS instability overtakes the CF instability. For an ideal gas, this situation only holds for the case of a strong adverse pressure gradient acting on the 3-D boundary layer, cf. Wassermann & Kloker (Reference Wassermann and Kloker2005).

Figure 14. Stability diagram of the transcritical regimes at $x=1$. (a) Cases with $T_{w}^{*}= 310, 308, 306,\ldots, 300\ \textrm {K}$; (b) cases with $T_{w}^{*}= 304$, 302 and 300 K (shown in the $\alpha _{r,s}$–$\beta _{r,s}$ coordinate). The blue dashed line gives the constant frequency $\omega =20$.

Figure 15. Baseflows in the transcritical regimes with wall cooling. (a) Amplitude of the CF component $\max _y(-w_s)$ versus $x$; (b,c) baseflow profiles at $x=1$. In each panel, line colours stand for different wall temperatures.

We further explain the occurrence of the generalised IP in figure 16. As can be seen, the viscosity gradient at the wall is immense ($O(10^2\sim 10^3)$), particularly for wall cooling compared with the other regimes discussed earlier (see figure 11a). Following the analysis of the wall momentum equation (3.4), the amplitude of $\partial \mu /\partial y|_w$ serves to move the IP into the wall (positive) or towards the edge of the boundary layer (negative). Thus the flow profiles become strongly inflectional with wall cooling. Unlike the other regimes, a prime reason causing large $\partial \mu /\partial y|_w$ lies in the fact that the density gradient $\partial \rho /\partial T|_p$ becomes considerably large (as shown in figure 16a,b) near the Widom line. Following the relation

(4.1)\begin{equation} \frac{\partial \rho}{\partial y}= \left.\frac{\partial \rho}{\partial T}\right|_p\frac{\partial T}{\partial y}, \end{equation}

the term $\rho$ in (3.7) thus erupts, turning into the most important feature of the transcritical regime.

Figure 16. (a) The $y$-gradient of viscosity in the transcritical regimes with wall cooling and heating; (b) density $\rho ^*$ and its gradient $\partial \rho ^*/\partial T^*|_p$ versus temperature at $p^*=80$ bar. The yellow shaded area stands for strong gradients of thermodynamic properties near the Widom line (pseudocritical point).

5. Scenarios of the flow instability – streamwise perturbation patterns

It has been clear that the consequence of wall heating/cooling on the flow instability has become intensified in the supercritical regimes but reversed in the subcritical regime. The most striking results are found in the transcritical case with wall cooling, where the inviscid TS mode coexists with and dominates the CF mode despite the acceleration of the flow. This changeover of the leading primary instability mechanism causes not only a much earlier transition but also a different disturbance structure of the flow. To reveal this feature in physical space, we study the spatiotemporal evolution of the perturbations in the transcritical regime.

The perturbation fields are reconstructed based on the results in the stability diagrams shown in figure 17. The diagrams provide a comprehensive comparison of the influence of non-ideality, frequency ($\omega = 3, 20$) and wall temperature ($T_w/T_\infty = 1.0667, 0.9375$) on the growth rate, wave angle and $N$-factors. For all cases, the growth rate for positive spanwise wavenumbers $\beta$ is larger than for negative ones. For example, for both wall heating and cooling, the ideal-gas case reaches an $N$-factor of 9 ($\beta >0$) and 5 ($\beta <0$) by the end of the domain considered ($x=5.6$). A stabilisation by the non-ideality is seen for the CF mode ($\omega =3$), which is more significant in the cooling condition. With high frequency $\omega =20$, as predicted in figure 12(b), only positive $\beta$ are unstable, indicating a deterioration of the CF mode. The maximum growth rate is reduced by non-ideality in the heating case while the instability remains for larger $x$ leading to a slightly higher $N$-factor. The inviscid TS mode shown in figure 17(b) presents a way larger parameter range of instability. With a tremendous growth rate, an $N$-factor of 60 is reached around $x=4.2$. Of course, the strong growth rate will cause a much earlier transition.

Figure 17. Stability diagram of the unsteady perturbations with $\omega =3$ and 20. Colours indicate the growth rate ($-\alpha _i$), white solid lines show $N$ factors and white dashed line present the wave angle ($\arctan {\beta /\alpha _r}$). Comparison of the transcritical regimes versus the ideal cases. (a) Wall heating ($T_w/T_\infty =1.0667$); (b) wall cooling ($T_w/T_\infty =0.9375$).

Using the technique introduced in § 2.4, the streamwise perturbation patterns are shown in figure 18 (see supplementary movie available at https://doi.org/10.1017/jfm.2022.845 for temporal evolutions). A point-source-type perturbation is introduced into the domain at $x=1$. The frequency of the source is $\omega =3$ and 20, respectively, and we present plots spanning four periods in the spanwise direction. The dashed line is the potential streamline, and the dash–dotted line shows the wall streamline. The colour map has been designed such that the structures are shown in black and white when the amplitudes have become large. Since in the stability framework all perturbations ($\rho ^\prime, u^\prime, v^\prime, w^\prime, T^\prime$ and secondary variables) have been considered as a whole and grow with the same growth rate, we choose to show the wall-normal gradient of the streamwise velocity perturbation at the wall $\partial u^{\prime }/\partial y|_{y\rightarrow 0}(x,z,t)$. This variable is related to the skin friction perturbation (that rises in an averaged manner in the nonlinear stages during flow transition).

Figure 18. Stability scenario shown with physical perturbation $\partial u^{\prime }/\partial y|_{y\rightarrow 0}(x,z,t)$. Movies showing the temporal evolution are available as supplemental materials. The dashed and dash–dotted lines correspond to the potential and wall streamlines. We present four transcritical cases in accordance with the results in figure 17. The colourmap displays perturbation whose amplitude is large than a threshold with black-and-white contours. The point sources (shown with blue circles) are located at $x=1$ and $z= 1.05$, 3.14, 5.24 and 7.33. An additional case for wall cooling and $\omega =20$ are shown with a reduced growth rate of $\alpha _i/20$.

As presented in figure 18, we consider the flow field until $x=5$. Note that figure 18(a,b,c) share the same colourmap. In the low-frequency regime ($\omega =3$, figure 18a,b), the CF modes dominate both wall heating and cooling. As seen in supplementary movies, the perturbations travel along the wall streamline consistent with the direction of the group velocity. The wave crests are about parallel to the wall streamline, and inside each wave packet train, the crests move in a positive spanwise direction because of the dominating CF modes travelling against the CF direction (downwards in figure 18a,b,c) that have higher amplitudes than those travelling in the CF direction. This travel is in a local manner since the amplitude decays at the sides of each train towards its neighbouring train. By comparing figure 18(a,b), we see that the amplitude of the cooling-wall case is smaller compared with the heating-wall case.

A noteworthy difference in the $\omega =20$ case for the heating wall (figure 18c) is that the perturbations grow only for positive $\beta$, and, of course, the wavelengths are shorter because of the higher frequency. The wave crests are now more oriented to the $x$-direction according to the pure existence of waves travelling downwards against the CF. Under such conditions, perturbations are not cancelled by the counterpart of negative $\beta$. Next, we focus on the TS-dominated flow with the cooling wall $\omega =20$ (figure 18d,e) in the transcritical regime. Due to the tremendous growth rates, we have indicated the $N$ factor of 9 that is immediately reached at $x=1.48$ (grown from $x=1$). The crests within the wave train are now perpendicular to the potential streamline and travel along it. This corresponds to the typical nature of incompressible-flow TS instability with the most amplified waves being 2-D and travelling along the streamline, as seen in figure 14(b), where the stability diagram is quasisymmetric with respect to $\beta _{r,s}=0$.

We make use of figure 19 to explain the observations in the perturbation patterns. The physical perturbations for various individual $\beta$ are plotted for two cases (heating wall with $\omega =3$ and cooling wall with $\omega =20$). The $N$ factors at $x=5$ are shown at the bottom of each panel. Since the final perturbation is a linear superposition of these structures, the wavenumber that reaches the largest $N$ factor will dominate the structures shown in figure 18. In the two cases, this corresponds to $\beta =63$ and 54, whose crests are parallel to the wall streamline or perpendicular to the potential streamline, respectively.

Figure 19. Display of physical perturbations $\partial u^{\prime }/\partial y|_{y\rightarrow 0}(x,z,t)$ for chosen spanwise wavenumber $\beta$. The maximum $N$ factors at $x=5$ are written at the bottom of each subpanel. (a) Wall heating and $\omega =3$, $x\in [2,3]$, $z\in [0,2.08]$; (b) wall cooling and $\omega =20$, $x\in [2,3]$, $z\in [0,2.08]$. Movies showing the temporal evolution are available as supplemental materials.