2.1. Computational model and set-up

The compression ramp flow is used to model the STBLI and the geometry and size are shown in figure 2. The 2-D auxiliary computational domain framed by a grey dotted line is used by an auxiliary flow simulation for providing the laminar inlet boundary condition and the initial value for the 3-D principal computational domain covered by the computational grid. The STBLI in the corner is the area of interest in the paper. The outlet extends a buffer area with a sparse grid to dissipate the disturbance reflected from the outlet. A Cartesian coordinate system is established with the origin set at the corner vertex, $x$ in the streamwise direction, $y$ in the wall-normal direction and $z$ in the spanwise direction.

Figure 2. Computational domain and boundary condition of ramp flow (unit: mm).

The governing equations are Navier–Stokes equations in the curvilinear coordinate system as

(2.1)\begin{equation} \frac{\partial \boldsymbol{U}}{\partial t} + \frac{\partial (\boldsymbol{F}_{\boldsymbol{i}}+\boldsymbol{F}_{\boldsymbol{\nu},\boldsymbol{i}})}{\partial \xi_i} = 0, \end{equation}

where $\boldsymbol {U}$ is the conserved variable, $\boldsymbol {F}_{\boldsymbol {i}}$ is the inviscid flux and $\boldsymbol {F}_{\boldsymbol {\nu },\boldsymbol {i}}$ is the viscous flux (see (Fu, Ma & Li Reference Fu, Ma and Li2010) for detailed expressions). Besides, the viscous stress is modelled by the Newtonian fluid, the viscosity of which is calculated by the Sutherland formula. The heat flux is modelled by Fourier's law and the thermal conductivity is calculated from the viscosity and Prandtl number of 0.7. As for the state equation, the ideal gas model is used with the specific ratio equal to 1.4. No turbulence model is used and the above equations are closed.

The main boundary conditions are described in figure 2 and listed in detail in table 1. The inlet is a laminar profile extracted from a 2-D simulation and the outlet is a supersonic outflow. The blowing and suction disturbance is introduced from $x=-440$ to $-410$ mm to force the boundary layer into the transition to generate turbulence. The specific form of the disturbance is the same as Pirozzoli et al. (Reference Pirozzoli, Grasso and Gatski2004) and cited as

(2.2)\begin{equation} v_{bs}(x,z,t)=AU_{\infty}f(x)g(z)h(t), \end{equation}

where in the present paper the intensity of the disturbance is set as $A=0.2$, and $f(x)$, $g(z)$ and $h(t)$ are defined as

(2.3)\begin{equation} \left.\begin{array}{@{}l@{}c@{}} \displaystyle f(x)=\dfrac{4}{\sqrt{27}} \sin{\theta_x}(1-\cos{\theta_x}),\quad \theta_x=2{\rm \pi}\dfrac{x-x_a}{x_b-x_a}, \\ \displaystyle g(z)=\sum_{n=1}^3 \dfrac{0.8^{n-1}}{\displaystyle \sum_{n=1}^{3}0.8^{n-1}}\sin{(2{\rm \pi} n(z/L_z+\phi_n))}, \\ \displaystyle h(t)=\sum_{m=1}^5 \dfrac{0.8^{m-1}}{\displaystyle \sum_{m=1}^{5}0.8^{m-1}}\sin{(2{\rm \pi} m(\beta t +\phi_m))}, \end{array}\right\} \end{equation}

where the basic disturbance frequency $\beta =0.32 U_{\infty }/{\delta _d}$ or $1.24 U_{\infty }/{\delta }$, where $\delta _d$ is the nominal thickness of the laminar boundary layer at the disturbance location while $\delta$ is the nominal thickness of the turbulent boundary layer at the reference location ($x=-60$ mm). The phase parameters, $\phi _n$ and $\phi _m$, are random numbers ranging from 0 to 1. According to the above form, the disturbance flow rate is zero on average. Ignoring the rarefied gas effect, no slip wall assumption is reasonable and applied to the wall condition of the velocity field. Considering the actual wall situations in flights, a cold wall condition is applied in the simulation. The isothermal bottom wall is used with $T_w/T_r=0.5$, where $T_r$ is the recovery temperature calculated by recovery coefficient $r=Pr^{1/3}$. The top boundary is non-reflective. The spanwise boundary condition is periodic.

Table 1. Key parameters of boundary condition and boundary layer at the reference position ($x=-60$ mm).

To describe the incoming turbulent boundary layer quantitatively, some characteristic thicknesses at the reference location are also listed in the second line of table 1. Here $\delta$ is the nominal thickness, $\theta$ is the momentum thickness and $\delta _\nu$ is the viscous scale on the wall. Based on these thicknesses, the corresponding Reynolds number at the reference location can be obtained and are $Re_\delta =\rho _\infty U_\infty \delta / \mu _\infty =8.46\times 10^4$, $Re_\theta =\rho _\infty U_\infty \theta / \mu _\infty =4652$ and $Re_\tau =\rho _wu_\tau \delta /\mu _w=\delta /\delta _\nu =700$. Note that $\delta$ and $\delta _\nu$ will be used to non-dimensionalize length quantities later.

In the present DNS, a finite difference method is used on the structured grid. The inviscid flux terms in (2.1) are split by the Steger–Warming flux vector splitting method and then solved by a hybrid scheme that is composed of a seventh-order low dissipation upwind scheme in the smooth region, a seventh-order WENO scheme in the non-smooth region and a fifth-order WENO scheme in the extremely non-smooth region. The smoothness of the flow field is given by a modified Jameson shock sensor (Jameson, Schmidt & Turkel Reference Jameson, Schmidt and Turkel1981). The viscous terms are solved by a sixth-order central difference scheme. To ensure the numerical accuracy at the boundary, virtual grid points are extrapolated from inner grid points. Time advance is solved by the third-order Runge–Kutta scheme and 153 000 time steps or $92 \delta /U_\infty$ are calculated during the sampling stage with sampling every 40 steps. The statistical results of the skin friction, the wall heat flux and the incoming turbulent boundary layer profiles all indicate that the sampling time is statistically convergent. The computational grid is an important factor of the DNS and will be introduced specially in the next subsection. The above calculation is performed on the open source code OpenCFD-SC, which has been widely used and validated by the development team in DNS of compressible turbulence (Li et al. Reference Li, Fu, Ma and Liang2010; Liang et al. Reference Liang, Li, Fu and Ma2012; Tong et al. Reference Tong, Li, Yu and Li2018; Tong, Duan & Li Reference Tong, Duan and Li2021; Tong et al. Reference Tong, Yuan, Lai, Duan, Sun and Dong2022b).

2.2. Computational grid

The DNS grid is required to solve turbulence at all scales and, hence, the computational grid needs to be designed and examined carefully. As shown in figure 2, the present grid in the streamwise direction is refined near the corner. The streamwise grid in the upstream turbulent boundary layer ($x<-60$ mm) is refined by a three order function from inlet grid scale to corner grid scale and the streamwise grid in the corner area ($x>-60$ mm) is kept uniform, as indicated in grey in the figure. The near-wall grid is stretched with the geometric form of $y_j=\Delta y_1(\alpha ^{j-1}-1)/(\alpha -1)$, where $\Delta y_1$ is the first grid height near the wall and $\alpha$ is the stretching rate and set to 1.0293 here. In the spanwise direction the grid distributes uniformly. The total grid number is $N_x\times N_y \times N_z = 2660 \times 204 \times 300 = 1.63\times 10^8$. Dimensionless by the viscous scale $\delta _\nu$, the first layer grid height is $y^+=0.33$ at the reference station and is $y^+=0.59$ at its maximum point on the ramp. The near-wall first layer grid scale is also on the same order with the turbulence dissipation scale considering the Kolmogorov dissipation scale, $\eta /\Delta y_1=1.04$, which is estimated by the method of Zhang et al. (Reference Zhang, Ciu, Xu and Huang2017). The streamwise grid scale is $\Delta x^+=6.6$ in the interaction area, $\Delta x^+=6.6\unicode{x2013}17.5$ in the upstream turbulent boundary layer and in the spanwise grid scale is $\Delta z^+=6.6$, which are dimensionless by the reference viscous scale. The spanwise scale of the computational domain should solve the largest vortices, which is proved by the spanwise two-point correlation. The results indicate that with the distance increasing, the correlations decay rapidly to near 0 both at the inlet and the corner station, as shown in figure 3. In the near-wall region ($y^+=6.6$) the correlation shows small fluctuations due to the streak-like coherent structures. The above scale analysis preliminarily suggests that the present grid can be applied to solve the near-wall turbulence.

Figure 3. Spanwise two-point correlations of the fluctuating velocity at (a) $x=-60$ mm and (b) $x=0$ mm.

Furthermore, turbulent energy spectra also show a good resolution of small turbulence scales. As shown in figure 4, the near-wall turbulent energy spectra at the inlet and the corner both present three characteristic regions: the inertial region with a peak, the inertial sub-region showing typical $-$5/3 power law and the dissipation region where the energy decays faster with increasing wavenumber. In the incoming turbulent boundary layer (figure 4a), with the distance from the wall increasing, the peak wavenumber decreases, which indicates the energetic scale becomes larger, or quantitatively from $L_z/15$ near the wall to $L_z/3$ far from the wall where $L_z$ is the spanwise length of the computational domain. However, around the corner (figure 4b), the spectra almost remain the same at a different distance from the wall, which indicates that the turbulence is more uniform in the corner region. The above results are consistent with the existing knowledge and prove that the present grid covers the small turbulent scale and is capable of solving the turbulence.

Figure 4. Near-wall turbulent energy spectra at the inlet and the corner at (a) $x=-60$ mm and (b) $x=0$ mm.

The most convincing and direct method to prove the DNS grid resolution is the grid convergence study; see table 2 for the five different grid set-ups used in this work. Case M3 is the grid introduced at the beginning of the subsection. Here $N_x$, $N_y$, $N_z$ are grid numbers in the streamwise, wall-normal and spanwise directions and the total grid number of these cases is from $1.25\times 10^8$–$2.44\times 10^8$. In cases M1–M4, only the wall-normal grid is changed by changing the first grid height and maintaining the stretching rate, with $y^+$ ranging from 1.31–0.16 at the reference station and $y^+$ ranging from 2.08–0.30 at the maximum point that is in the interaction area. The wall-normal grid scale at the edge of the boundary layer ranges from 18.0–16.6 at the reference station, which is similar to previous literature on STBLIs (for example, $\Delta y^+_e$ is 15.0 in Zhu et al. (Reference Zhu, Yu, Tong and Li2017) and 23.0 in Tong et al. Reference Tong, Li, Yu and Li2018). The wall-normal grid scale in the shear layer above the separation bubble is normalized by local viscous scale, that is, $\Delta y^*_{sl} = \Delta y / (\sqrt {{\nu _{sl}}/{(\partial u / \partial y)_{sl}}})$, where $\nu _{sl}$ and $(\partial u / \partial y)_{sl}$ mean the local values on the zero velocity line in the shear layer. Here $\Delta y^*_{sl}$ at $x=0$ mm are all less than 1 for the five cases. The wall-normal grid Reynolds number is also given, which is defined as $Re_{\Delta _n}=\rho _w U_\infty \Delta _n / \mu _w$, where $\Delta _n$ is equal to $\Delta y_1$ in the paper. Previous studies of aerodynamic heating over blunt bodies show that the calculation reaches grid convergent generally at $Re_{\Delta _n}$ of around 10 and, for low-dissipation numerical schemes, the value can be relaxed to 40 (Ma & Liu Reference Ma and Liu2023). Here $\Delta x^+$ ranges from 6.6 to 17.5 for all cases, which is reasonable according to Georgiadis, Rizzetta & Fureby (Reference Georgiadis, Rizzetta and Fureby2010) who suggest $\Delta x^+$ should be 10–20 for DNS. As for case M5, the spanwise grid is refined based on case M3, with $z^+$ from 6.6 to 4.4 while the same grid is used for the other two directions. The final results of the grid convergence test are shown in figure 5. The results show that for the incoming turbulent boundary layer, $St$ is basically unchanged with the increase of grid number. For the peak location, the peak $St$ increases with the increase of the grid number until the grid size reaches that of case M3, which indicates the grid convergence is basically reached for case M3.

Table 2. Grid set-up for the grid convergence test.

Figure 5. The Stanton number at the reference position and at the maximum position for the varying grid sizes.

Given that the paper focuses on the near-wall aerodynamic heating, the wall-normal grid convergence is investigated in detail and the results of the corresponding cases M1–M4 are shown in figure 6. According to the skin friction results (figure 6a), it can be found that with the refinement of the near-wall grid, the flow transition is slightly advanced. For all cases in the current test, the results of the incoming turbulent boundary layer are almost the same. With the refinement of the near-wall grid, the separation point is basically unchanged but the reattachment point is advanced and then the downstream friction coefficient increases until case M3 reaches the grid convergence of the flow field. As for heat transfer (figure 6b), the results of the incoming turbulent boundary layer are also basically consistent for all cases, which indicates that incoming flow has been well solved. With the near-wall grid refinement, the downstream heat transfer coefficient increases and M3 also achieves grid convergence of the heat transfer simulation. Moreover, as expected, the pressure results are not sensitive to changes of the wall-normal grid as shown in figure 6(c).

Figure 6. (a) Skin friction coefficients, (b) the Stanton number and (c) wall pressure ratio distribution along the streamwise direction for the cases of varying wall-normal grid sizes.

The spanwise grid convergence details are shown in figure 7. For heat transfer, as shown in figure 7(a), with the refinement of the spanwise grid, the $St$ before the peak point and the peak $St$ remain unchanged but the $St$ after the peak point increases slightly, showing that grid convergence is essentially achieved for the present study. For the pressure ratio, as shown in figure 7(b), the results are also almost identical between the cases of different spanwise grids.

Figure 7. (a) The Stanton number and (b) wall pressure ratio distribution along streamwise direction for the cases of varying spanwise grid sizes.

Furthermore, the grid convergence of some representative higher-order statistics are examined carefully as shown by the root mean square (r.m.s.) of streamwise turbulent velocity in figure 8 and the wall-normal turbulent scalar flux in figure 9. In the undisturbed boundary layer ($x=-60$ mm) there is no significant difference in the $u_{rms}$ profiles among the five different grids listed in table 2. However, as the boundary layer is interacted by the shock, the required grid resolution increases. In the near-wall region the r.m.s. of the streamwise turbulent velocity for cases M1 and M2 is slightly higher until case M3 achieves grid convergence in the separation bubble region ($x=0$ mm) and downstream boundary layer ($x=60$ mm), as shown in figure 8(b,c). The turbulent scalar flux profiles show similar grid convergence as shown in figure 9. Compared with the wall gradients, the turbulence statistics profiles are less sensitive to the grid and show better grid convergence.

Figure 8. The r.m.s. of streamwise turbulent velocity profiles at the (a) upstream boundary layer ($x=-60$ mm), (b) separation bubble region ($x=0$ mm) and (c) downstream boundary layer ($x=60$ mm) for the cases of varying grid sizes.

Figure 9. The wall-normal turbulent scalar flux profiles at the (a) upstream boundary layer ($x=-60$ mm), (b) separation bubble region ($x=0$ mm) and (c) downstream boundary layer ($x=60$ mm) for the cases of varying grid sizes.

In summary, by careful analysis and validation, case M3 basically achieved grid convergence and can solve the small-scale turbulence near the wall. The results of case M3 are used to analyse the aerodynamic heating mechanisms in this work.

2.3. Validation of the incoming turbulent boundary layer

The turbulent boundary layer is generated by the forced transition with spatial development. Hence, in this section the development degree and the calculation accuracy are carefully checked for the skin friction distribution, velocity and temperature profile at the reference station $x_{ref}=-60$ mm. Note that the results in this paper are mostly average so the average symbol is omitted without confusion for the reason of simplification. The averaging operations in the present study are conducted in both time and spanwise direction.

The skin friction $C_f$ describes the boundary layer development from the laminar flow to turbulence as shown in figure 10, which also provides the $C_f$ prediction in laminar compressible boundary layers of Eckert (Reference Eckert1955) and the prediction in turbulent compressible boundary layers of White & Majdalani (Reference White and Majdalani2006), which is also given as

(2.4a,b)\begin{equation} C_{f}=0.455 \left/\, \left[S \ln \left(\frac{0.06}{S} Re_x \frac{\mu_e}{\mu_w} \sqrt{\frac{T_e}{T_w}} \right)\right]^2\right., \quad S=\frac{(T_r/T_e-1)^{{1}/{2}}}{\sin^{{-}1}{A}+\sin^{{-}1}{B}}, \end{equation}

where

(2.5)\begin{equation} \left.\begin{array}{@{}c@{}} \displaystyle A=\dfrac{2a^2-b}{(b^2+4a^2)^{{1}/{2}}},\quad B=\dfrac{b}{(b^2+4a^2)^{{1}/{2}}} ;\\ \displaystyle a=\left(r\dfrac{\gamma-1}{2}Ma_e^2 \dfrac{T_e}{T_w}\right)^{{1}/{2}},\quad b=\dfrac{T_r}{T_w}-1. \end{array}\right\} \end{equation}

In the prediction, the effective development origin of the turbulent boundary layer is hard to determine due to the uncertainty of the forced transition and, hence, it is simply set at the leading edge of the plate in the paper to provide a reference. In the inlet of the computational domain, $C_f$ is around the laminar value. At $x>-400$ mm, $C_f$ increases quickly, which indicates that the transition occurs while around $x=-250$ mm, $C_f$ reaches a maximum value that indicates the transition is near completion. After $x=-150$ mm, $C_f$ decreases slowly and steadily as predicted by the theory, which indicates the turbulence can be considered fully developed, that is to say, the turbulence is fully developed at the corner inlet ($x=-60$ mm).

Figure 10. Turbulence development validation by comparing the skin friction coefficient with the values in laminar and turbulent boundary layer.

The turbulent boundary layer profiles at the corner inlet are further validated and the sources of reference data are listed in table 3. Duan et al. (Reference Duan, Beekman and Martin2010) and Chu, Zhuang & Lu (Reference Chu, Zhuang and Lu2013) provided DNS results of hypersonic turbulent boundary layers for similar conditions to this paper. The DNS results of Huang et al. (Reference Huang, Lian and Choudhari2022) and Zhang, Duan & Choudhari (Reference Zhang, Duan and Choudhari2018) are also cited for later comparison and discussion. Spalart (Reference Spalart1988) provided classical DNS results of incompressible turbulent boundary layers while Sillero, Jimenez & Moser (Reference Sillero, Jimenez and Moser2013) provided recent DNS data of higher Reynolds numbers.

Table 3. Reference data with key incoming conditions for the hypersonic turbulent boundary layer validation.

The mean velocity and temperature results at the reference station are compared with the previous DNS results and classic theories as shown in figure 11. The velocity profile in wall coordinates is shown in figure 11(a) after van Driest transformation defined as

(2.6)\begin{equation} U_{vd}^+=\int_0^{U^+} \sqrt{\rho/\rho_w} \,{\rm d}U^+. \end{equation}

The transformation collapses the profiles well to the classic incompressible wall law, showing linear law at $y^+<4$ and logarithmic law at $y^+=30\unicode{x2013}200$. Compared with the DNS results of Duan et al. (Reference Duan, Beekman and Martin2010), there is no significant difference overall except for the wake layer. The slight deviation of up to 3 % can be attributed to the difference of the Reynolds number and the turbulence generation method (Wenzel et al. Reference Wenzel, Selent, Kloker and Rist2018). The accuracy of the velocity profile is considered enough for the current research.

Figure 11. (a) Mean streamwise velocity profiles and (b) temperature–velocity relation of the turbulent boundary layer at the reference station.

The temperature–velocity relation profile is also a key result of the compressible turbulent boundary layer, which is shown in figure 11(b). Compared with the canonical Walz relation (Walz & Oser Reference Walz and Oser1969),

(2.7a,b)\begin{equation} \frac{T}{T_e}=\frac{T_w}{T_e}+\frac{T_r-T_w}{T_e}\frac{U}{U_e}+\frac{T_e-T_r}{T_e} \left(\frac{U}{U_e}\right)^2, \quad \frac{T_r}{T_e}=1+r\frac{\gamma-1}{2}Ma_e^2 , \end{equation}

the present DNS also shows good consistency and indicates that the temperature presents a good quadratic relation with the velocity. The slight deviation in the middle part is because the cold wall deviates from an assumption of the Walz relation, i.e. turbulent Prandtl number $Pr_t=1$. Zhang et al. (Reference Zhang, Bi, Hussain and She2014) considered the wall heat flux effect and proposed a more generalized relation that just replaces $r$ in (2.7) with

(2.8)\begin{equation} r_g = \frac{2(T_w/T_e-1)}{(\gamma-1)Ma_e^2}+4Pr\frac{(T_r/T_e-T_w/T_e)}{(\gamma-1)Ma_e^2}\frac{St}{C_f}. \end{equation}

The result of the Zhang relation shows good agreement with the present DNS. Compared with previous DNS results, there are some minor deviations mainly near the wall and the present DNS profile is closer to that of Chu et al. (Reference Chu, Zhuang and Lu2013), which is because the wall temperatures are slightly different for these cases and the wall temperature of the present DNS is closer to that of Chu et al. (Reference Chu, Zhuang and Lu2013). On the whole, the temperature results of the present DNS is accurate and reliable.

The r.m.s. profile of the fluctuating velocity provides more detail on the turbulent boundary layer as shown in figure 12. Good qualitative agreement is achieved compared with the previous DNS of Duan et al. (Reference Duan, Beekman and Martin2010) and Chu et al. (Reference Chu, Zhuang and Lu2013) both in global coordinates and wall coordinates. The present DNS is also close to the results of Huang et al. (Reference Huang, Lian and Choudhari2022) for similar $Ma$ and $Re_\tau$, except the peak of Huang et al. (Reference Huang, Lian and Choudhari2022) is sharper near the wall. The deviation can be attributed to the wall temperature difference, which can be inferred from the comparison with DNS data for a lower wall temperature (Zhang et al. Reference Zhang, Duan and Choudhari2018). The profiles present a peak near the wall in the buffer region of the near-wall turbulence. At the edge of the boundary layer, the present DNS fluctuation is slightly higher than the reference data, which might be explained by the difference of the Reynolds number as listed in table 3. The relative deviation at the edge of the boundary layer is 5 $\%$ compared with data of Huang et al. (Reference Huang, Lian and Choudhari2022) for similar $Re_\tau$, which is acceptable for the present study. The r.m.s. profile of the fluctuating velocity after Morkovin transformation also confirms the Morkovin's assumption, as shown in figure 12(a), that predicts that the profile will collapse to the incompressible boundary layer profile after the transformation. It can be seen from the results that in $y/\delta <0.4$ the Morkovin's assumption is well validated while in $y/\delta >0.4$ the DNS compressible results are all a little bit larger compared with classical data from Spalart (Reference Spalart1988), which is also observed by Pirozzoli et al. (Reference Pirozzoli, Grasso and Gatski2004). Considering recent data for higher Reynolds number from Sillero et al. (Reference Sillero, Jimenez and Moser2013), better consistency can be observed between compressible and incompressible boundary layers. On the whole, the r.m.s. profile of velocity fluctuation shows reasonable agreement with the previous turbulent boundary layer theories and results.

Figure 12. Root mean square of the fluctuating streamwise velocity of the turbulent boundary layer at reference station in (a) global coordinate and (b) wall coordinate.

In summary, the compressible turbulent boundary layer at the corner inlet is fully developed and the present DNS results of the incoming turbulence including the flow field, temperature field and fluctuations are accurate and reliable enough for the current study.

2.4. The flow results in the interaction region

The STBLI results are validated from the flow field including the wave structure, pressure distribution and interaction length in this subsection and the flow field is also analysed preliminarily as the starting point of the aerodynamic heat analysis.

The shock wave structure of the present DNS is given in schlieren visualization in figure 13 with the previous experiment record of Li (Reference Li2007), and also given in velocity field in figure 14. The shock structure in the compression ramp flow is qualitatively consistent with the experimental schlieren image as shown in figure 13 for similar $Ma$ and the results both show that the hypersonic shock wave is deeply immersed in the boundary layer and obviously curved due to the interaction with the near-wall flow at the bottom. More details of the flow field are described in figure 14. The shock wave agrees well with the inviscid shock wave theory when far away from the corner. As for the turbulent boundary layer, it is compressed after passing through the shock wave and separates slightly at the bottom around the corner. The recirculation bubble lifts the shear layer leading to the enhancement of the momentum exchange between the bottom of the boundary layer and the mainstream so that the low velocity fluid can obtain momentum as the pressure rises. Since the separation area only occupies a very small part of the boundary layer, there is no clear reattachment shock wave in the flow field. Quantitatively, the boundary layer separates at $x_S=-8.66$ mm and reattaches at $x_R=2.25$ mm on the ramp. Above all, the shock wave structure of the present DNS is qualitatively consistent with the previous study and agrees well with the inviscid theory for the mainstream away from the wall with the near-wall boundary layer in a state of small separation.

Figure 13. Present numerical schlieren visualization of shock wave structure (a) compared with the experiment result of Li (Reference Li2007) (b).

Figure 14. Basic structures in the flow field of the STBLI generated from the compression ramp.

The pressure ratio and interaction length are another two key parameters to describe the STBLI, which are shown in figure 15(a). Generally, the pressure ratio is used to describe the shock wave strength and the pressure ratio of the present DNS agrees well with the prediction of the inviscid oblique shock wave theory with the pressure ratio of 13. Note that in figure 15(a), and later similar figures of the streamwise distribution, when $x \leq 0$, this is equal to the global horizontal coordinate and, when $x > 0$, this represents the local coordinate $x_r$ along the ramp wall as shown in the figure 15(a). Since there must be a subsonic region at the bottom of the boundary layer due to the no-slip wall condition, the high pressure propagates upstream and, hence, another important parameter in the STBLI is the upstream interaction length that is defined as the distance from the point where the pressure starts to rise to the starting point of the ideal shock wave (which here is the corner vertex). Souverein, Bakker & Dupont (Reference Souverein, Bakker and Dupont2013) made an intelligent theoretical analysis of the interaction length in the STBLI and proposed a simple prediction method that established the relation between the dimensionless interaction length $L^*$ and the dimensionless shock intensity $S_e^*$. Later, Jaunet, Debieve & Dupont (Reference Jaunet, Debieve and Dupont2014), Hong, Li & Yang (Reference Hong, Li and Yang2021), Zuo et al. (Reference Zuo, Wei, Hu and Pirozzoli2022), Volpiani et al. (Reference Volpiani, Bernardini and Larsson2020) have corrected the relation for non-adiabatic wall conditions and hypersonic conditions. This paper uses the non-adiabatic wall corrected $S_e^*$ proposed by Jaunet et al. (Reference Jaunet, Debieve and Dupont2014) and the result is shown in figure 15(b) with the previous results compiled by Souverein et al. (Reference Souverein, Bakker and Dupont2013) and the recent non-adiabatic cases of Jaunet et al. (Reference Jaunet, Debieve and Dupont2014), Spaid & Frishett (Reference Spaid and Frishett1972), Schreyer et al. (Reference Schreyer, Sahoo, Williams and Smits2018), Bhagwandin, Helm & Martin (Reference Bhagwandin, Helm and Martin2019), Volpiani et al. (Reference Volpiani, Bernardini and Larsson2020) and Zuo et al. (Reference Zuo, Wei, Hu and Pirozzoli2022). The Souverein's correlation, $L^*=1.3S_e^{*3}$, is also plotted in the same figure as a dotted line. The present DNS result is in the trend shown by all the previous data, especially by non-adiabatic ones, which indicates the present result is qualitatively reasonable. Compared with the Souverein's theory, the predicted interaction length is 18 mm while the present DNS results is 14 mm, with a relative deviation of 20 %. Considering that Souverein's fitting formula still contains some uncertainty and the fitting data have some scattering, this deviation can be considered reasonable. Besides, no plateau area is observed in the pressure distribution, indicating that the separation is in the incipient separation state. Above all, the calculated pressure ratio and interaction length in the STBLI are also generally consistent with previous theories.

Figure 15. (a) Mean wall pressure through interaction and (b) interaction length compared with the previous prediction (Souverein et al. Reference Souverein, Bakker and Dupont2013) and data (the grey symbols are from Souverein et al. Reference Souverein, Bakker and Dupont2013, for adiabatic cases, the red symbols represent hot wall conditions and the blue symbols represent cold wall conditions).

In conclusion, the above analysis proves that the DNS results are physically reasonable and reliable again, and indicates that the STBLI in this work is in the initial separation state.

3.1. Wall heat flux prediction

The wall heat flux in the turbulent boundary layer can be predicted accurately with the current theory (van Driest Reference van Driest1956; Qu et al. Reference Qu, Liu, Zeng and Liu2001), so the essential question of the wall heat flux prediction in the STBLI lies in how many times the heat flux increases across the shock.

As reviewed in the introduction, Q-P theory is a method to predict the wall heat flux ratio in STBLIs in which the basic idea is to associate the heat flux ratio Q with the pressure ratio P in the interaction area. However, some recent studies have shown that the canonical QP85 theory has some deviations in some cases (Priebe & Martín Reference Priebe and Martín2021; Helm & Martín Reference Helm and Martín2022; Tong et al. Reference Tong, Yuan, Lai, Duan, Sun and Dong2022b). To evaluate the QP85 theory more comprehensively, the data of Q and P have been collected from as much public literature as the authors are aware of, which supplements the highly cited but old datasets of Holden (Reference Holden1972). As shown in figure 16, the open symbols represent data from the collection of Holden (Reference Holden1972) that originally came from Levin & Fabish (Reference Levin and Fabish1962), Sayano, Bausch & Donnelly (Reference Sayano, Bausch and Donnelly1962), Magnan & Spurlin (Reference Magnan and Spurlin1966), while the solid symbols represent data from recent studies (Coleman & Stollery Reference Coleman and Stollery1972; Holden Reference Holden1972, Reference Holden1977; Hayashi, Sakurai & Aso Reference Hayashi, Sakurai and Aso1986; Delery & Coet Reference Delery and Coet1991; Schülein Reference Schülein2006; Bernardini et al. Reference Bernardini, Asproulias, Larsson, Pirozzoli and Grasso2016; Volpiani, Bernardini & Larsson Reference Volpiani, Bernardini and Larsson2018; Volpiani et al. Reference Volpiani, Bernardini and Larsson2020; Priebe & Martín Reference Priebe and Martín2021; Helm & Martín Reference Helm and Martín2022; Tong et al. Reference Tong, Yuan, Lai, Duan, Sun and Dong2022b). The symbol colour reflects the Mach number (except for the red one that represents the present DNS result). It is easy to find that the classical QP85 prediction is obviously higher than the heat flux ratio in recent studies, especially for hypersonic flows. Quantitatively, the maximum relative error exceeds 70 %. According to recent studies, the deviation can be attributed to the different mainstream Mach number and the wall temperature. Therefore, in this work we have conducted a theoretical analysis based on the boundary layer thickness change to develop a more elaborate prediction method of the wall heat flux ratio in STBLIs.

Figure 16. The relationship between peak heat flux ratio and peak pressure ratio in turbulent STBLI for a wide range of $Ma$.

Assuming that the heat transfer theory in compressible turbulent boundary layers still remains valid after boundary layers pass through shock waves, then the Stanton number at the reference temperature (Eckert Reference Eckert1960) should be proportional to the dimensionless boundary thickness, which leads to

(3.1)\begin{equation} \frac{St_d^*}{St_u^*}=\frac{\delta_d/x_d}{\delta_u/x_u}, \end{equation}

where $St \equiv q_w / [\rho _e U_e (H_r-H_w)]$ , $\delta$ is the boundary layer thickness and $x$ is the effective developing length of the boundary layers. The subscript $u$ represents the parameters in the undisturbed boundary layer before shock waves while the subscript $d$ represents the parameters in the disturbed boundary layer after shock waves, as also illustrated in figure 17. The subscript $e$ represents the parameters at the edge of boundary layers while the subscript $w$ represents the parameters on the wall. The superscript $*$ represents the parameters at the reference temperature that can be calculated as (Eckert Reference Eckert1960)

(3.2)\begin{equation} \frac{T^*}{T_e}=\frac{1}{2} \left(\frac{T_w}{T_0}+1\right) + \frac{\gamma-1}{4}Ma_e^2 \left(0.44r+\frac{T_w}{T_0}\right), \end{equation}

where $r$ is the recovery coefficient and $r=Pr^{1/3}$, $T_0$ is the total temperature, $\gamma$ is the specific heat ratio and set to 1.4. The left-hand side of (3.1) includes the heat flux ratio that is the target quantity and the right-hand side needs to be further handled.

Figure 17. Definition of parameters in the STBLI for the theoretical analysis.

Assuming that the separation is not significant so that the flow structures can be described by figure 17, then the following geometric relation can be obtained:

(3.3)\begin{equation} \frac{\delta_d}{\delta_u}=\frac{\sin(\beta-\theta)}{\sin(\beta)}. \end{equation}

Here $\beta$ is the shock angle and $\theta$ is the turning angle. According to the oblique shock relation, the angles can be substituted by the pressure ratio $p_d/p_u$ and the incoming Mach number $Ma_u$:

(3.4)\begin{align} &\frac{\sin(\beta-\theta)}{\sin(\beta)}\nonumber\\ &\quad =\frac{Ma_u\left[(\gamma+1)-(\gamma-1)\dfrac{p_d}{p_u}\right]} {\left[\! Ma^2_u\left((\gamma+1)\dfrac{p_d}{p_u}+(\gamma-1)\!\right) - 2\left(\!\left(\!\dfrac{p_d}{p_u}\!\right)^2-1\!\right)\!\right]^{1/2} \left[\!(\gamma+1)\dfrac{p_d}{p_u}-(\gamma-1)\!\right]^{1/2}}. \end{align}

The effective developing length can be solved by the reference temperature method

(3.5)\begin{equation} \frac{\delta_d/x_d}{\delta_u/x_u} = \left(\frac{Re^*_{x_d}}{Re^*_{x_u}}\right)^{{-}0.2}, \end{equation}

from which then

(3.6)\begin{equation} \frac{x_d}{x_u} = \left(\frac{\delta_d}{\delta_u}\right)^{1.25} \left(\frac{\rho_{ed}U_{ed}}{\rho_{eu}U_{eu}}\right)^{0.25} \left(\frac{\mu_d}{\mu_u}\right)^{{-}0.25} \left(\frac{(T^*/T_e)_d}{(T^*/T_e)_u}\right)^{{-}0.25(1+\omega)}, \end{equation}

where $\omega$ is the viscosity–temperature exponent based on the assumption that viscosity satisfies the power relation with temperature and $\omega$ is set to 0.78 here. Finally, incorporating (3.3) and (3.6) into (3.1) and ignoring the change of the recovery temperatures and the mainstream velocities, then

(3.7)\begin{align} \frac{\dot{q}_d}{\dot{q}_u} &= \left(\!\frac{\sin(\beta-\theta)}{\sin(\beta)}\!\right)^{{-}0.25} \left(\! \frac{\left(\!1+\dfrac{T_w}{T_0}\!\right)+0.2Ma_d^2 \left(\! 0.44r+\dfrac{T_w}{T_0} \!\right)} {\left(\!1+\dfrac{T_w}{T_0}\!\right)+0.2Ma_u^2 \left(\!0.44r+\dfrac{T_w}{T_0}\!\right)}\,{\bullet}\, \frac{1+0.2Ma_d^2}{1+0.2Ma_u^2} \!\right)^{0.25\omega-0.75}\nonumber\\ &\quad \times \left(\!~\frac{p_d}{p_u}\right)^{0.75}. \end{align}

Furthermore, the Mach number after shock waves, $Ma_d$, in (3.7) can also be substituted by the pressure ratio $p_d/p_u$ and the incoming Mach number $Ma_u$ according to the oblique shock relation:

(3.8)\begin{equation} Ma_d^2 = \frac{Ma_u^2\left[(\gamma+1)\dfrac{p_d}{p_u} + (\gamma-1)\right] - 2\left[\left(\dfrac{p_d}{p_u}\right)^2-1\right]}{\dfrac{p_d}{p_u}\left[(\gamma+1)+(\gamma-1)\dfrac{p_d}{p_u}\right]}. \end{equation}

In summary, (3.7) combined with (3.4) and (3.8) relates the wall heat flux ratio ${\dot {q}_d}/{\dot {q}_u}$ to the pressure ratio $p_d/p_u$, incoming Mach number $Ma_u$ and the wall temperature $T_w/T_0$. According to this formula, the peak heat flux ratio can be predicted for different peak pressure ratios at different $Ma_u$ and $T_w/T_0$ as shown in figure 18. The results indicate that the heat flux ratio is strongly related to the pressure ratio, although the relation between them in (3.7) is not a simple power relation. The results show that the heat flux ratio is approximately 0.85 power of the pressure ratio for the hot wall conditions and 0.8 power for the cold wall conditions.

Figure 18. Present heat flux ratio prediction according to the present correlation, (3.7), for sample points at $Ma_u$ of 3–15 and $T_w/T_0$ of 0.2–0.8 (for cold wall in blue) or 1.2–1.8 (for hot wall in red).

The accuracy of (3.7) is further evaluated by comparing with the data in previous literature, in which the data source is the same as in figure 16. The results show that the present heat flux ratio prediction reduces the relative error from 72 % to 35 %, compared with the canonical QP85 correlation as shown in figure 19. More detailed comparisons are provided in figure 20 where the wall heat flux distribution along the streamwise direction is predicted by the wall pressure distribution for the representative experiments of Delery & Coet (Reference Delery and Coet1991), Coleman & Stollery (Reference Coleman and Stollery1972) and Schülein (Reference Schülein2006), respectively. In figure 20(a) there is no distinct separation area in the STBLI, the increase and the peak of the wall heat flux are well predicted compared with experimental results. In figure 20(b) there is a small separation bubble and the wall heat flux is also well reproduced by the present correlation. In figure 20(c) the prediction of the STBLI induced by a $10^\circ$ shock generator is consistent with the experimental result while the prediction of the STBLI induced by a $14^\circ$ shock generator also gives a reasonable heat flux peak but shows some deviation in the separation area. In conclusion, the present correlation can predict the wall heat flux ratio and its distribution in the STBLI with relatively high accuracy.

Figure 19. Present heat flux ratio prediction accuracy of the previous studies compared with the canonical QP85 correlation. (The data are from the same source as figure 16.)

Figure 20. Present heat flux distribution prediction, (3.7), for three representative experiments (Coleman & Stollery Reference Coleman and Stollery1972; Delery & Coet Reference Delery and Coet1991; Schülein Reference Schülein2006).

Reviewing the physical meaning of (3.7) from the derivation process, the term in the first bracket on the right represents the heat transfer enhancement caused by boundary layer compression due to $Ma$ decrease. The term in the second bracket reflects the wall temperature effect on the STBLI by influencing the compressible boundary layer before and behind the shock. The complex coupling effect of the wall temperature on the interaction area is not considered in the present theoretical derivation, which makes the prediction not sensitive enough to the wall temperature for some cases and requires improvements in future research. The term in the third bracket represents the heat transfer enhancement caused by the enhancement of the fluid heat transport capacity due to the density increase. Therefore, from this perspective, the sharp increase of the heat flux in the STBLI is caused by the compression of boundary layers and the density increase in the mainstream.

3.2. Turbulent aerodynamic heat transport equation

The generation and transport process of aerodynamic heat are further investigated, which are controlled by the aerodynamic heat transport equation. The average heat transport governing equation is introduced in this subsection, while the corresponding DNS statistic results are given and discussed in the next section.

The derivation begins with the enthalpy transport equation that can be derived from the momentum conservation and energy conservation equations, which is

(3.9)\begin{equation} \frac{\partial \rho h}{\partial t} + \frac{\partial \rho u_j h}{\partial x_j} = \frac{\partial p}{\partial t} + u_j \frac{\partial p}{\partial x_j} + \tau_{ij}S_{ij} + \frac{\partial}{\partial x_j} \left(\lambda \frac{\partial T}{\partial x_j}\right), \end{equation}

where according to Newtonian fluid constitutive relation,

(3.10)\begin{equation} \tau_{ij}=2\mu S_{ij}^d, \end{equation}

where $S_{ij}^d$ is the deviatoric part of the shear rate tensor $S_{ij}$, that is, $S_{ij}^d \equiv S_{ij}-(S_{kk}/3)\delta _{ij}$.

Reynolds averaging is carried out for (3.9) and the Favre averaging, i.e. density-weighted averaging, is used on the left to avoid producing too many terms. Following the common custom, $\bar {q}$ represents the Reynolds average value of $q$ while $q'$ represents corresponding fluctuation; $\tilde {q}$ represents the Favre average value of $q$ while $q''$ represents corresponding fluctuation. Since this work is a posteriori analysis based on DNS statistics, the existence of both Reynolds and Favre averages does not cause any problems regarding reaching a closed-form solution. Assuming the flow field to be statistically steady and the correlation between fluctuations of physical properties and flow variables can be ignored, then

(3.11)\begin{equation} \frac{\partial \bar{\rho} \widetilde{u_j} \tilde{h} }{\partial x_j} + \frac{\partial \bar{\rho} \widetilde{u_j'' h''}}{\partial x_j} + \frac{\partial}{\partial x_j} \left(-\bar{\lambda} \frac{\partial \bar{T}}{\partial x_j}\right) = \overline{u_j} \frac{\overline{\partial p}}{\partial x_j} + \overline{u'_j \frac{\partial p'}{\partial x_j}} + 2\bar{\mu}\overline{S_{ij}^d}\overline{S_{ij}^d} + 2\bar{\mu}\overline{S_{ij}^{d'} S_{ij}^{d'}}. \end{equation}

Non-dimensionalize the above equation with corresponding incoming values (i.e. $\rho _\infty$, $U_\infty$, $T_\infty$, $\rho _\infty U_\infty ^2$, $\mu _\infty$) and unit length $L_{ref}=1$ mm, remain the original symbols for simplicity, and the dimensionless form of (3.11) can yield

(3.12)\begin{align} &\overbrace{\vphantom{\frac{\widetilde{u_j'' T''}}{x_j}} \frac{\partial \bar{\rho} \widetilde{u_j} \tilde{T} }{\partial x_j}}^{\boldsymbol{\nabla} \,\boldsymbol{\cdot}\, \boldsymbol{q}_{\boldsymbol{c}}} + \overbrace{\vphantom{\frac{\widetilde{u_j'' T''}}{x_j}} \frac{\partial \bar{\rho} \widetilde{u_j'' T''}}{\partial x_j}}^{\boldsymbol{\nabla} \,\boldsymbol{\cdot}\, \boldsymbol{q}_{\boldsymbol{t}}} + \overbrace{\vphantom{\frac{\widetilde{u_j'' T''}}{x_j}} \frac{1}{PrRe}\frac{\partial}{\partial x_j} \left(- \bar{\mu} \frac{\partial \bar{T}}{\partial x_j}\right)}^{\boldsymbol{\nabla} \,\boldsymbol{\cdot}\, \boldsymbol{q}_{\boldsymbol{\nu}}} \nonumber\\ &\quad =\underbrace{(\gamma-1)Ma^2 \left(\overline{u_j} \frac{\overline{\partial p}}{\partial x_j} + \overline{u'_j \frac{\partial p'}{\partial x_j}} \right)}_{H_{p,m} + H_{p,t}} + \underbrace{\vphantom{\left(\overline{u'_j \frac{\partial p'}{\partial x_j}} \right)} \frac{(\gamma-1)Ma^2}{Re} ( 2\bar{\mu}\overline{S_{ij}^d}\overline{S_{ij}^d} + 2\bar{\mu}\overline{S_{ij}^{d'} S_{ij}^{d'}})}_{\varPhi_m + \varPhi_{t}}. \end{align}

Equation (3.12) has a clear physical interpretation: the right-hand side is the generation terms of aerodynamic heat while the left-hand side is the transport terms of aerodynamic heat. Here $H_{p,m}$ is the directional derivative of the average pressure along the average streamlines and represents the aerodynamic heat generated by the average compression; $H_{p,t}$ is aerodynamic heat generated by fluctuation compression. The generation of compression aerodynamic heat will be dominant near the stagnation region or where there are obstacles. Here $\varPhi _m$ is aerodynamic heat caused by mean shear dissipation while $\varPhi _t$ is aerodynamic heat caused by the turbulent shear dissipation. Even though the turbulent fluctuations are not significant in certain situations, the turbulent shear dissipation is still considerable due to the turbulence on a very small spatial scale. Additionally, although both types of aerodynamic heat are converted from kinetic energy, the generation of compression aerodynamic heat is reversible while the generation of dissipation aerodynamic heat is irreversible. For the left-hand side of the equation, the first is the mean convective transport, the second is the turbulent transport and the third is the molecular viscous transport. Here $\boldsymbol {q}_{\boldsymbol {c}}$, $\boldsymbol {q}_{\boldsymbol {t}}$ and $\boldsymbol {q}_{\boldsymbol {\nu }}$ are the convective heat flux, turbulent heat flux and viscous heat flux, respectively. On the wall, the average convective and turbulent heat transport are both zero and, hence, the total heat flux is only contributed by viscous transport. However, the viscous flux on the wall is relatively passive and the analysis of the distribution of each heat transport term in the flow field is helpful to more comprehensively understand the heat transfer process.

In conclusion, (3.12) shows that the flow generates aerodynamic heat through compression and dissipation, which is then transported through mean convection, turbulent fluctuation and viscous conduction in the fluid. Although the heat is seemingly transferred only through viscous conduction on the wall, all generation and transport processes influence the viscous conduction and, hence, the wall heat flux.

4.1. Wall heat transfer enhancement

Across the interaction, the wall heat flux is six times stronger than the incoming value and the increase process is greatly different from the skin friction as shown in figure 21(a). There is a local minimum before the heat flux jump, which basically coincides with the average separation point position. The heat flux across the interaction jumps sharply and a wide peak appears downstream of the reattachment point. In the convective heat transfer investigation, the Reynolds analogy is a widely used method, which suggests that $St$ shows similar behaviours as $C_f$ or in other words the Reynolds analogy factor $S\equiv C_f/2St$ is constant. However, the DNS results show that the wall heat transfer increases while the skin friction decreases around the corner. As can be seen more clearly from figure 21(b), the Reynolds analogy factor deviates from the value of 0.86 in the turbulent boundary layer (Qu et al. Reference Qu, Liu, Zeng and Liu2001) and changes drastically across the interaction, especially in the recirculation region. Interestingly, although the Reynolds analogy factor reaches constant downstream, it does not recover to the undisturbed value and stays at 0.5. This phenomenon is also reported by Helm & Martín (Reference Helm and Martín2022), which is not well understood in current research and is open to be studied. As for the failure of the Reynolds analogy, it is not surprising due to the strong adverse pressure gradient and, thus, highlights the need to develop other wall heat flux prediction methods in the STBLI.

Figure 21. Stanton number and failure of Reynolds analogy through interaction.

As mentioned in the previous section, the heat flux change is strongly related to the pressure change and, hence, the DNS results are investigated from the present correlation (i.e. (3.7)). The wall heat flux is non-dimensionalized by $q_{u}$, the wall heat flux at the reference position of the undisturbed boundary layer, and is then converted to a ratio as shown in figure 22. At the same time, (3.7) is used to predict the heat flux ratio for the same conditions as the DNS in this work. The pressure $p_d$ is replaced by the wall pressure $p_w$ and the pressure $p_u$ is replaced by the reference pressure at $x=-60$ mm. The prediction result of the present theory is provided in figure 22 with the theoretical prediction of Coleman & Stollery (Reference Coleman and Stollery1972). The results show that both theories are in good agreement with the DNS result before the reattachment point, but the prediction results are higher downstream. The relative deviation of the peak heat flux ratio between the present theory and the present DNS is 23 % while the prediction of Coleman & Stollery (Reference Coleman and Stollery1972) is 34 %. The relative deviation is acceptable according to the analysis in § 3 where the paper claims that the relative error of the present theory is under 35 %. The quantitative analysis suggests that the present theory is able to explain the heat flux jump in the STBLI in present DNS. Therefore, the theoretical and computational results show that the key reasons of wall heat flux jumps in the STBLI are the boundary layer compression and the mainstream density increase caused by shock waves.

Figure 22. Heat transfer on the wall through interaction compared with previous theoretical predictions and present theoretical predictions.

4.2. Aerodynamic heat generation and near-wall transport

To further reveal the mechanisms of the wall heat flux changes in the STBLI, all generation and transport terms in the turbulent aerodynamic heat transport equation (3.12) are calculated statistically during the time advance of DNS, and then the generation and near-wall transport process of aerodynamic heat are quantitatively analysed in different regions in detail.

The detailed distribution of aerodynamic heat generation around the corner are shown in figure 23 where the $y$ axis is stretched for better presentation of the near-wall characteristic. Dissipation aerodynamic heat is displayed in the first line while compression aerodynamic heat is displayed in the second line. For the dissipation aerodynamic heat as shown in figure 23(a), the generation rate increases and the generation area expands significantly, which leads to increased total heat generation. Interestingly, the peak location of the dissipation aerodynamic heat shows clear dependency on the detachment and reattachment of the shear layer. The dissipation aerodynamic heat is further decomposed by Reynolds averaging as shown in figure 23(b,c). It can be found that the dissipation heat in the undisturbed boundary layer is mainly generated from the strong mean shear near the wall, while the dissipation heat generation is dominated by turbulence in the interaction area. As the shear layer detaches and reattaches, two heating cores appear in the near-wall region, indicating that the turbulent dissipation enhancement in the recirculation region is related to the enhancement of the shear turbulence, which can further be attributed to the Kelvin–Helmholtz instability. For the compression aerodynamic heat, as shown in figure 23(d), the generation rate is extremely high in the interaction area due to the shock compression, which is the clear feature of the STBLI distinguished from the boundary layer. Note that the compression aerodynamic heat is relative to the local flow velocity, hence, the compression heat area is lifted up and away from the wall slightly due to the low-speed recirculation bubble in the corner. At downstream of the corner, strong compression heat is continuously generated near the wall because the flow is reattached and the hypersonic shock layer is close to the wall. Likewise, the compression aerodynamic heat is further decomposed by Reynolds averaging as shown in figure 23(e,f). It can be found that the compression heat is mainly generated from the deceleration and compression of the mean flow. The fluctuating compression heat between positive and negative values is generated around the theoretical shock, indicating that the shock wave is unstable, which also poses an interesting question but is outside the scope of this paper. In summary, in the STBLI area the generation of aerodynamic heat increases significantly, which is mainly due to the heat generated by the increased turbulent dissipation and the average flow deceleration and compression.

Figure 23. Aerodynamic heat with its Reynolds-average decomposition. Plots (a–c) are dissipation aerodynamic heat while plots (e–f) are compression aerodynamic heat. (White dash lines represent theoretical shock while the near-wall streamlines represent shear layer.)

The heat transport results in the near-wall region are presented by heat flux lines with intensity distribution as shown in figure 24 where the $y$-axis scale is not stretched and remains equal to the $x$ axis to ensure the orthogonality of the coordinate system. Note that the density of the heat flux lines in the figure cannot represent heat flux intensity because each heat flux is not conserved due to the generation of aerodynamic heat and the transformation of different transport modes. The hot area is the near-wall area and separation area as shown in the temperature field. In the figure the gradient field is also shown and a distinct feature of the distribution is that an adiabatic line divides the temperature field into two regions, which corresponds to the single-peak distribution feature of the near-wall temperature profile. The distribution of the conduction and turbulent heat flux line also has an adiabatic line and shows a similar feature. For the conduction transport, the heat is transported to the mainstream above the adiabatic line while the heat is transported to the wall under the line. The magnitude of the conduction heat flux has a peak band above and below the adiabatic line, showing a bimodal feature. For the turbulence transport, heat is transported upstream in most areas while heat is transported to the wall in the extreme near-wall region. The reason for this is that the turbulent heat flux lines are affected by both the mean flow field and the temperature gradient field, and the turbulence intensifies the heat exchange in the streamwise direction. The magnitude of the turbulent heat flux is strong in the compression area and the extreme near-wall area on the ramp. For the convection transport, the mean convection heat flux lines are close to the mean flow lines, which also indicates the Favre-averaged velocity field is similar to the Reynolds averaged velocity field in the present study. The aerodynamic heat is transported downstream with mean convection in the mainstream. In the recirculation near the wall, the aerodynamic heat is transported downstream at the outer edge and transported upstream at the bottom, which strengthens the mixing of the aerodynamic heat near the wall. In summary, focusing on the near-wall region within the adiabatic lines in the STBLI, the conduction dominates the heat transport to the wall in the extreme near-wall region, the turbulence in the near-wall region transfers heat to the extreme near-wall region and the mean convection strengthens the mixing of upstream and downstream aerodynamic heat.

Figure 24. Temperature distribution and three transport mechanisms presented by heat flux lines with their magnitude distribution through interaction in near-wall region.

To further investigate the aerodynamic heat generation and transport process, the heat budgets with the velocity and temperature profiles at five key positions are shown in figures 25 and 26 where the horizontal coordinate, $y$, is the normal distance away from the wall and dimensionless with the viscous scale $\delta _\nu$ at the reference position. Two key positions are selected in the turbulent boundary layer and the other three are in the STBLI area, which will be discussed respectively below. The black lines in the figure represent the sum of all the terms and the results indicate that the sum is around zero for every key position, which means the statistic results have reached balance.

Figure 25. Aerodynamic heat generation and transport budget in the turbulent boundary layer at (a) $x/\delta =-6$ and (b) $x/\delta =6$ with (c,d) the corresponding temperature and velocity profiles. Symbols in the legends of (a,b) are defined in (3.12).

Figure 26. Aerodynamic heat generation and transport budget in the STBLI area at (a) separation point ($x/\delta =-0.82$), (b) recirculation centre ($x/\delta =-0.05$) and (c) reattachment point ($x/\delta =0.21$) with (d–f) the corresponding temperature and velocity profiles. Symbols in the legends of (a–c) are defined in (3.12).

In the turbulent boundary layer the aerodynamic heat is mainly generated by dissipation and transported by turbulence and viscous conduction as shown in figure 25. Figure 25(a,b) shows the aerodynamic heat budgets in the boundary layer upstream ($x/\delta =-6$) and downstream ($x/\delta =6$) of the shock wave. Note that the $y$-coordinate scale in figure 25(b) is one order of magnitude larger than figure 25(a). Due to the high shear in the hypersonic flow near the wall, the aerodynamic heat is mainly generated by mean dissipation in the viscous sublayer. Since there is no significant pressure change in the flat plate boundary layer, the aerodynamic compression heat is around zero. In the viscous sublayer, viscous conduction dominates the heat transport and the output energy from the viscous conduction is transferred to the wall. In the buffer layer the generated aerodynamic heat is transported mainly by turbulent fluctuation toward the wall and the main flow, which is because the buffer layer is a region of active turbulence and peak temperature as shown in figure 25(c). In the logarithmic layer the aerodynamic heat generation and transport are both less active. Before and behind the shock wave, the aerodynamic heat generation and transport in the boundary layer are qualitatively similar, but quantitatively quite different. Post reattachment, the heat generated by dissipation is closer to the wall and the intensity increases by nearly one order of magnitude while the turbulent transport and the viscous transport near the wall are also greatly strengthened. All of this can be attributed to the compression of boundary layers that make the shear increase dramatically after shock waves.

In contrast to the turbulent boundary layer, there exist significant compression aerodynamic heat generation and complex mean convective transport in the STBLI area that make the heat budget more complicated as shown in figure 26(a–c). The average compression aerodynamic heat at the three representative positions all show a peak value at the outer part of the boundary layer that indicates the position of the strong compression waves. The turbulent compression aerodynamic heat is not the dominant term near the wall and, hence, it will not be considered in this work. The clear differences in aerodynamic heat generation and transport between the separation region, recirculation region and reattachment region reveal the underlying different wall heat flux mechanisms in these three regions.

At the separation point $(x_S=-0.82\delta )$, the aerodynamic heat transport outward to the external flow is enhanced, which results in a local minimum of the wall heat flux. As shown in figure 26(a), compared with the undisturbed turbulent boundary layer, the average heat dissipation position at the separation point moves upward, which is associated with the upward shift of the shear layer. Meanwhile, the turbulent aerodynamic dissipation heat increases significantly with a peak value near $y^+=25$ and is larger than the average heat dissipation at all heights. The turbulent transport is active at all heights and becomes positive at the position of the compression wave, which means the heat is transported here by turbulence. Due to the shear layer uplift, more dissipation aerodynamic heat is transported to the outer boundary layer by the turbulent fluctuation and then balanced by the convective transport downstream together with the compression aerodynamic heat. Therefore, the aerodynamic heat transported to the wall decreases, which explains the local minimum of the heat flux at the separation point.

At the recirculation core $(x=-0.05\delta )$ near the corner vertex, the active turbulence transports aerodynamic heat to the wall, which is generated by compression and turbulent dissipation, leading to the sharp increase of the wall heat flux. As shown in figure 26(b), the dissipation aerodynamic heat near the wall is mainly contributed by turbulent dissipation, which is related to the recirculation instability of the separation bubble in the STBLI and the near-wall backflow can also be seen clearly in figure 26(e). At the same time as the turbulent dissipation enhancement, the turbulent transport near the wall is also significantly enhanced. The range of near-wall transport is expanded due to the separation effect, which can also be explained by the higher and wider temperature peak as shown in figure 26(e). In the extreme near-wall region, the heat input from turbulent transport with the turbulent dissipation heat is balanced with the output through viscous transport. The heat input from turbulent transport comes from the turbulent dissipation of the shedding shear layer and the compression aerodynamic heat of the outer layer. The above analysis suggests such a mechanism for the heat flux jump around the corner: to increase the pressure, the relatively low-speed near-wall shear layer lifts up so that the momentum exchange is enhanced between the near-wall fluid and the main flow, which also enhances the heat exchange leading to the high temperature recirculation bubble and the sharp rise of the wall heat flux. The heat is generated from the turbulent dissipation as well as the compression of the shear layer and the exchange way is mainly the turbulent transport.

Near the reattachment point $(x_R=0.21\delta )$, the enhancement of near-wall dissipation and turbulent transport as well as the compression aerodynamic heat caused by oblique impingement in the attachment process lead to the wall heat flux rise and the peak a little downstream of the attachment point. In the reattachment region there exists aerodynamic heat generated by the compression of the local oblique impingement flow together with the compression aerodynamic heat transported from upstream through the mean convection, which is further transported to the near wall through the active turbulent fluctuation. When transferred to the vicinity of the viscous sublayer, the input heat is balanced with the output by viscous conduction. At the downstream of the reattachment point, the average shear starts to increase and the average dissipation heat increases accordingly. Simultaneously, the near-wall fluid is compressed in a wide range resulting in continuous compression aerodynamic heat generation, due to the wall restriction. Subsequently, the wall heat flux reaches a relatively wide peak. Finally, with the thickening and relaxation of the boundary layer, the wall heat flux begins to decline slowly. The generation and transport processes of compression aerodynamic heat described here also reveal the underlying mechanism of the strong dependence of peak heat flux ratios to peak pressure ratios and these results provide theoretical support for the canonical Q-P correlation.

The above analysis indicates that the turbulence dominates the aerodynamic heating in the STBLI area and, hence, the turbulent vortices are further investigated, which provide a more fundamental perspective on the turbulent dissipation and turbulent transport. Turbulent vortices are extracted by the $Q$ criterion as shown in figure 27 where the instantaneous shock wave is also presented, which is represented by the isosurface of pressure. The instantaneous shock wave is not an ideal flat thin layer. Pits appears in the bottom of the shock wave when the near-wall streamwise vortices pass through and the whole structure appears as small spanwise ripples. The main shock surface has bulging structures on the boundary layer scale, which are caused by the motion of coherent turbulent structures in the outer boundary layer. As for the near-wall vortices, the small-scale streamwise vortex pairs dominate the near-wall turbulence in the upstream undisturbed boundary layer. When passing through the shock wave, the vortices are enhanced, which suggests that the turbulent intensity is significantly enhanced. The streamwise vortices lift up and there appear many spanwise vortices, which are the characteristics of free shear layers (Brown & Roshco Reference Brown and Roshco1974; Roshco Reference Roshco1976). The changes of the vortices shed light on the enhancement of turbulent dissipation and transport in the recirculation and reattachment regions.

Figure 27. (a) Near-wall turbulent vortices coloured by the vorticity in the $x$ direction and (b) instantaneous shock interacted with the vortices in the STBLI.

Finally, the mechanisms of aerodynamic heat generation and near-wall transport in different regions of the STBLI are summarized in figure 28. In the undisturbed boundary layer the aerodynamic heat is mainly generated by shear dissipation, which is transported to the wall by turbulent and viscous transport. However, the near-wall shear dissipation and turbulent transport are strengthened downstream of the shock wave, which causes the wall heat flux to reach a higher equilibrium level. Near the separation point, the enhanced heat transport to outer layers leads to the local minimum of the wall heat flux. In the recirculation region the high temperature separation bubble and the sharp rise of wall heat flux can be explained by the increasing aerodynamic heat generated by both dissipation and compression, and the enhancement of the heat exchange between the near-wall fluid and the outer fluid due to enhanced turbulence. The peak heat flux downstream of the reattachment point is due to the enhanced dissipation near the wall caused by shear layer reattachment, the compression heat brought from upstream and the local oblique impingement convection in the attachment process, and the near-wall turbulent transport enhancement as shown in the reattachment zone of the figure. The generation and transport of compression aerodynamic heat reveal the underlying mechanism of the strong dependence of peak heat flux ratios to peak pressure ratios. In addition, the enhancement of turbulent dissipation and transport in the recirculation and reattachment regions can be explained by the intensified near-wall turbulent vortices during the interaction.

Figure 28. Sketch of the aerodynamic heat generation and transport process in the hypersonic STBLI (Aerodynamic heat is generated from dissipation in red and compression in yellow; aerodynamic heat is transported from convection, turbulence and conduction as shown by the legend and the arrow thickness represents the transport intensity.)