2.1. Numerical method

DNS achieved by a finite-difference method of high-order accuracy in space and time and with shock capturing ability is applied to study the hypersonic compression-ramp flow problem. The 3-D Navier–Stokes equations for unsteady, compressible flow are employed in a conservative form

(2.1)\begin{equation} \frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} + \frac{\partial G}{\partial y} + \frac{\partial H}{\partial z} = \frac{\partial F_\nu}{\partial x} + \frac{\partial G_\nu}{\partial y} + \frac{\partial H_\nu}{\partial z}, \end{equation}

where F, G and H are the inviscid fluxes, and $F_{\nu}$, $G_{\nu}$ and $H_{\nu}$ denote the viscous fluxes. $U = (\rho, \rho u, \rho v, \rho w, \rho e)^{\rm T}$ is the vector of conservative variables, $\rho$ is the density, $u$, $v$ and $w$ are the flow velocities, $e$ is the total energy per unit mass and T denotes the transpose of the matrix. The equation system is closed by the perfect gas law relating pressure, density and temperature, as well as Sutherland's law for calculating the viscosity.

In terms of the numerical methods, time integration is performed by an explicit third-order total variation diminishing Runge–Kutta scheme. A weighted essentially non-oscillatory finite-difference scheme of fifth-order is applied for the discretisation of the inviscid fluxes, based on the work of Jiang & Shu (Reference Jiang and Shu1996). A sixth-order central-difference scheme is used to approximate the viscous fluxes. Details about the numerical schemes may be found in Hermes, Klioutchnikov & Olivier (Reference Hermes, Klioutchnikov and Olivier2012), Gageik, Klioutchnikov & Olivier (Reference Gageik, Klioutchnikov and Olivier2015) and Cao (Reference Cao2021). The DNS solver has been validated and successfully applied to study hypersonic compression-ramp flows (Cao, Klioutchnikov & Olivier Reference Cao, Klioutchnikov and Olivier2019; Cao Reference Cao2021; Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier, Heufer and Wen2021a,Reference Cao, Hao, Klioutchnikov, Olivier and Wenb).

2.2. Two-dimensional base flow

The numerically considered flow conditions and compression-ramp geometry are based on those used in the experimental campaign conducted in the shock tunnel TH2 at the Shock Wave Laboratory of RWTH Aachen University. Details about the experimental facility and set-up can be found in Roghelia et al. (Reference Roghelia, Olivier, Egorov and Chuvakhov2017). The compression ramp comprises a flat plate with a sharp leading edge and a ramp with a deflection angle of $15^\circ$. The length of flat plate is $L$ = 100 mm. To examine the complete transition process, the length of ramp used in the present numerical simulation is 220 mm. Table 1 lists the flow conditions at TH2 (A. Roghelia, private communication 2017). The free-stream Mach number ($M_\infty$) and Reynolds number ($Re_{\infty,L}$) are 7.7 and $8.6\times 10^5$, respectively. The total enthalpy $h_0$ is relatively low allowing the use of the calorically perfect gas assumption. Owing to the short running time of the shock tunnel, an isothermal wall condition is applied, and the wall temperature ($T_w$) is given by 293 K, which corresponds to a wall-to-total temperature ratio of 0.18.

Table 1. Flow conditions at the shock tunnel TH2 (Roghelia et al. (Reference Roghelia, Olivier, Egorov and Chuvakhov2017) and A. Roghelia, private communication 2017).

In the 2-D simulation for the base flow, the number of grid points in the streamwise ($x$) and vertical ($y$) directions is 4080 and 320, respectively. Preliminary simulations showed a converged 2-D base flow for this mesh resolution. For the subsequent 3-D simulation, 480 grid points are equi-spaced in the spanwise ($z$) direction over a width of 30 mm, yielding a total number of grid points of $4080\times 320\times 480 \approx 627$ million. In the $x$ direction, the mesh is clustered near the leading edge and on the ramp. In the $y$ direction, the mesh is clustered near the wall, and the minimum mesh spacing is $\Delta y_w = 3.8\times 10^{-6}$ m. It is noted that the mesh over the ramp is perpendicular to the wall. The set-up described above yields the following non-dimensional wall units at a position near the outflow boundary ($x/L = 3.0$): $x^+ = 13.1$, $y_w^+ = 0.9$, $z^+ = 14.0$. It should be mentioned that a turbulent boundary layer is established prior to the outflow region (see below), and the number of grid points inside the boundary layer (in the $y$-direction) is approximately 160 at $x/L = 3.0$.

In terms of boundary conditions, free-stream parameters are prescribed at the inflow and upper boundaries. A zero-gradient extrapolation condition is used for the outflow boundary. For the no-slip wall, isothermal conditions are specified with the wall temperature being 293 K. Periodic boundary conditions are applied in the spanwise direction for the 3-D simulation.

The 2-D base flow is visualised in figure 1 using the Mach number contour and numerical schlieren. Owing to the pressure rise induced by the ramp shock, a separation bubble forms around the corner with the separation and reattachment points located at $x/L = 0.49$ and $x/L = 1.26$, respectively. The separation shock generated at the separation position interacts with the reattachment shock at the triple point ($T$), resulting in a slip line and an expansion fan that impinges on the wall. Note that the leading-edge shock resulting from the viscous interaction (Anderson Reference Anderson2006) is relatively weak compared with the separation, reattachment and ramp shocks, as seen in the Mach number contour. In the following, 3-D simulation is performed on the basis of the 2-D base flow.

Figure 1. Base-flow visualisation: (a) Mach number contour; (b) numerical schlieren. Here, $T$ denotes the triple point, where the separation shock interacts with the reattachment shock.

2.3. Establishment of 3-D flow

The initial 3-D flow field is generated by extending the above 2-D solution in the spanwise direction. The initial spanwise velocity is set to zero. It has been demonstrated in our previous work (Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier, Heufer and Wen2021a) that the growth of instability waves can arise from the extremely low-level perturbations provided by numerical round-off error. No external disturbances are introduced at the inflow or at the wall, which enables the examination of intrinsic instabilities in the fluid-dynamic system.

To ensure the stability of numerical simulation, the time step is set as $4.6 \times 10^{-9}$ s. Starting from $tU_\infty /L = 0$, the 3-D simulation is run up to $tU_\infty /L = 41$. The sampling time interval for collecting the data is $\Delta tU_\infty /L = 0.006$, corresponding to a sampling frequency of $f_s = 2.88$ MHz ($f_s L/U_\infty = 167$).

To capture the growth of three-dimensionality, we consider the temporal evolution of the quadratic mean of spanwise velocity at a streamwise position, which is defined as

(2.2)\begin{equation} A_w = \sqrt{\frac{1}{N_y N_z} \sum_{j=1}^{N_y} \sum_{k=1}^{N_z} (w/U_\infty)_{j,k}^2}, \end{equation}

where $N_y$ and $N_z$ denote the number of grid points in the $y$ and $z$ directions, respectively. Figure 2(a) shows the temporal history of $A_w$ at $x/L = 1.05$. Similar to Cao et al. (Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021b), this streamwise position was chosen because $A_w$ is largest there compared with all other streamwise positions. It should be mentioned that this position is located in the separated region as the instability core lies inside the separation bubble (Sidharth et al. Reference Sidharth, Dwivedi, Candler and Nichols2018; Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021b). The slope of the dotted line shown in figure 2(a) corresponds to the growth rate of the most unstable mode identified by GSA, which is shown later. After an initial transient period, $A_w$ starts to grow exponentially at $tU_\infty /L \approx 3.5$. The exponential growth ends at $tU_\infty /L \approx 6.5$, from which on $A_w$ reaches its asymptotic level. This indicates that a saturated flow is achieved in the separation bubble.

Figure 2. (a) Temporal evolution of spanwise velocity (2.2) at $x/L = 1.05$. The slope of the dotted line represents the growth rate of the most unstable mode predicted by GSA, which is shown later. (b) Temporal history of the wall Stanton number at $x/L = 1.45$ and $z/L = 0.15$, which is located at the centre line of wall.

Figure 2(b) plots the temporal history of the wall Stanton number at $x/L = 1.45$ and $z/L = 0.15$. The wall Stanton number is defined as

(2.3)\begin{equation} St = \frac{q_w}{\rho_\infty U_\infty c_p (T_{aw} - T_w)}. \end{equation}

Here, $q_w$ denotes the surface heat flux, $c_p$ is the specific heat capacity and $T_{aw}$ is the adiabatic wall temperature. It is noted that the chosen position is the peak-heating position in the vicinity of reattachment. As seen, the Stanton number remains nearly constant until $tU_\infty /L \approx 6.5$, which corresponds to the end of the exponential growth for $A_w$. This means that the change of the wall Stanton number downstream of reattachment is closely linked to the separation bubble flow, as demonstrated by Cao et al. (Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021b). Subsequently, the Stanton number exhibits an unsteady feature due to the intrinsic instability (see below).

Figure 3 presents the instantaneous wall Stanton number distribution at $tU_\infty /L = 6$, 16 and 41. Separation and reattachment positions are highlighted by iso-lines of zero skin-friction coefficient ($C_f$). Weak three-dimensionality can be observed in the vicinity of flow reattachment at $tU_\infty /L = 6$. At $tU_\infty /L = 16$, the wall Stanton number distribution deviates significantly from figure 3(a) and appears similar to that at $tU_\infty /L = 41$. Based on the previous discussions, it can be concluded that the 3-D flow is fully established prior to $tU_\infty /L = 16$. To avoid potential transient effects, we use the time period from $tU_\infty /L = 16$ to 41 to conduct the time-averaging process in the following.

Figure 3. Instantaneous wall Stanton number distribution at (a) $tU_\infty /L = 6$, (b) $tU_\infty /L = 16$ and (c) $tU_\infty /L = 41$. Black solid lines denote iso-lines of $C_f = 0$.

2.4. Validation of DNS results

The DNS results are validated below by comparing with experimental data and theoretical predictions. Figure 4(a) compares the surface pressure coefficient $C_p = (p_w - p_\infty )/0.5\rho _\infty U_\infty ^2$ with the experimental measurement (A. Roghelia, private communication 2017) and the inviscid solution based on the oblique shock theory. In general, the pressure plateau induced by boundary-layer separation and the pressure rise downstream of reattachment are in good agreement with the experimental data. The pressure decrease at $x/L \approx 2$ is caused by the impingement of the expansion fan emanating from the triple point (see figure 1b). As expected, the surface pressure at the rear part of ramp matches the inviscid solution because this position is located far away from the interaction zone.

Figure 4. (a) Streamwise distribution of the spanwise- and time-averaged surface pressure coefficient $C_p$, in comparison with experimental data and the inviscid solution based on the oblique shock theory. (b) Streamwise distribution of the spanwise- and time-averaged $St$, the spanwise-averaged $St$ at $t/U_\infty /L = 41$ as well as the $St$ for the 2-D base flow. The shaded grey region represents the envelope of the spanwise variation of time-averaged $St$. All experimental data are measured along the centre line of the model (A. Roghelia, private communication 2017).

The comparison of the wall Stanton number is illustrated in figure 4(b). The black line represents the spanwise-averaged Stanton number at $tU_\infty /L = 41$, and the red line represents the spanwise- and time-averaged Stanton number. Excellent agreement can be found upstream and inside the separation bubble. The size of the separation bubble is also well captured by the DNS. Downstream of reattachment, a heating peak is generated at approximately $x/L = 1.45$ as a result of flow reattachment. An evident increase in the wall Stanton number starting at $x/L = 1.8 {\sim} 1.9$ can be found both in the experimental and numerical data. This is indicative of transition from a laminar state to a turbulent state. In contrast to the 3-D flow, the 2-D base flow exhibits a purely laminar state in the entire domain. Moreover, owing to the presence of 3-D phenomena, the separation bubble is slightly enlarged in the 3-D simulation. Therefore, disagreement (e.g. in surface heat transfer and separation bubble size) may arise when one uses a 2-D solution to predict the flow in a real situation.

The discrepancy of $St$ downstream of reattachment between the experimental and 3-D numerical results could be due to the following reasons. First, while the experimental data were measured along the centre line of the model, the DNS data shown in figure 4(b) are averaged in the spanwise direction. Figure 5 provides the spanwise distribution of $St$ at $x/L = 1.45$ for the time-averaged and instantaneous flows. It is apparent that a considerable variation of $St$ exists in the spanwise direction, especially for the instantaneous flow. The shaded grey region in figure 4(b) represents the envelope of the spanwise variation of time-averaged $St$ in the DNS data. Obviously, most experimental data fall in the grey region. Note also that the total uncertainty of the experimental Stanton number is approximately 10 % (Roghelia et al. Reference Roghelia, Olivier, Egorov and Chuvakhov2017). Second, the presence of external disturbances in the experiment may affect the formation of surface heat-flux streaks, which is not considered in the DNS.

Figure 5. Spanwise distribution of the time-averaged wall Stanton number (red dash dotted line) and the instantaneous wall Stanton number at $tU_\infty /L = 41$ (black solid line). The numerical results are taken at $x/L = 1.45$.

In addition to the measurement of surface pressure and wall Stanton number along the centre line of the model, infrared imaging was used to obtain a surface temperature map on the ramp. Figure 6 compares this experimental surface temperature map with the numerical Stanton number distribution. Both the experimental and DNS data are time averaged. Similar streak pattern can be observed between the experimental and numerical results. The peak-heating position also matches, as implied in figure 4(b). It should be noted that there is a slight discrepancy in the spanwise wavelength of the streaks. The reasons might be as follows. First, the window used for time averaging is different in the experiment (0.25 ms) and DNS (1.45 ms). Note that the flow is highly unsteady. In addition, the external disturbances existing in the experiments may affect the formation of heat-flux streaks, as mentioned above.

Figure 6. (a) Infrared image and (b) numerical Stanton number map showing the heat-flux streaks on the ramp surface. Both the experimental and numerical maps are time averaged. The peak heating position ($x/L = 1.45$) is highlighted by triangles.

To further validate the DNS results from a theoretical point of view, we compare both the laminar and turbulent boundary-layer profiles with theoretical predictions. As mentioned previously, the undisturbed boundary layer upstream of separation is laminar, and the boundary layer on the rear part of ramp (e.g. at $x/L = 3.0$) is turbulent. Firstly, the laminar boundary-layer profile is compared with the similarity solution of compressible boundary-layer equations (White Reference White2006) in figure 7(a). Both velocity and temperature profiles agree well with the similarity solution. Note that the slight discrepancy in the profiles results from the fact that the streamwise pressure gradient is assumed to be zero in the boundary-layer equations, while a favourable pressure gradient exists in the DNS due to the viscous interaction near the leading edge of the flat plate.

Figure 7. (a) Velocity and temperature profiles for the laminar boundary layer upstream of separation, in comparison with the similarity solution of the compressible boundary-layer equations. (b) The van-Driest-transformed mean velocity profile at $x/L = 0.4$ and 3.0. The dotted lines represent the linear-sublayer and log-law relations. (c) Mean temperature profile at $x/L = 3.0$, in comparison with the relations proposed by Walz (Reference Walz1969) and Duan & Martin (Reference Duan and Martin2011).

In figure 7(b), transformed streamwise velocity profiles at $x/L = 0.4$ and 3.0 are plotted in a rescaled wall-normal coordinate. Here, the streamwise velocity is normalised as

(2.4)\begin{equation} u^+= \frac{\bar{u}_s}{\bar{u}_\tau}, \end{equation}

with $\bar {u}_s$ being the velocity in the wall-parallel direction and $\bar {u}_\tau = \sqrt {\bar {\tau }_w/\bar {\rho }_w}$ the friction velocity. The overline ‘-’ denotes a spanwise- and time-averaged quantity. Then, the streamwise velocity is transformed using van Driest transformation as the flow is compressible

(2.5)\begin{equation} u^+_c = \int_0^{u^+} \sqrt{\frac{\bar{T}_w}{\bar{T}}}\mathrm{d}u^+. \end{equation}

The dotted lines represent the relations for the linear sublayer and the logarithmic overlap region

(2.6a,b)\begin{equation} u^+= y^+,\quad u^+= \frac{1}{\kappa} \mathrm{ln}(y^+) + C, \end{equation}

where $\kappa = 0.41$ and $C = 5.2$. At $x/L = 0.4$, the velocity profile is typical of a laminar profile. At $x/L = 3.0$, the velocity profile exhibits typical features for a turbulent boundary layer. Note that, in the viscous sublayer, $u^+ = y^+$ is only satisfied until $y^+ \approx 2$. This is because of the low wall-to-total temperature ratio and is consistent with the results of Duan, Beekman & Martin (Reference Duan, Beekman and Martin2010). They studied Mach 5 turbulent boundary layers at cold walls and found that the region of the viscous sublayer shrinks significantly with decreasing wall temperature. Nevertheless, the velocity profile at $x/L = 3.0$ overlays the laminar profile until $y^+ \approx 10$, indicating a well-resolved viscous sublayer. Furthermore, the velocity profile in the log-law region also matches the theoretical curve.

Figure 7(c) presents the spanwise- and time-averaged temperature profile at $x/L = 3.0$. One of the commonly used temperature–velocity relations is Walz's equation (Walz Reference Walz1969)

(2.7)\begin{equation} \frac{\bar{T}}{\bar{T}_e} = \frac{\bar{T}_w}{\bar{T}_e} + \frac{\bar{T}_{aw} - \bar{T}_w}{\bar{T}_e} \left( \frac{\bar{u}_s}{\bar{u}_e} \right) + \frac{\bar{T}_{e} - \bar{T}_{aw}}{\bar{T}_e} \left( \frac{\bar{u}_s}{\bar{u}_e} \right)^2. \end{equation}

As shown in figure 7(c), the DNS results do not match Walz's relation because this relation was originally built for the boundary layer over an adiabatic wall. By considering the effect of the wall temperature, Duan & Martin (Reference Duan and Martin2011) modified (2.7) using their DNS data

(2.8)\begin{equation} \frac{\bar{T}}{\bar{T}_e} = \frac{\bar{T}_w}{\bar{T}_e} + \frac{\bar{T}_{aw} - \bar{T}_w}{\bar{T}_e} f\left( \frac{\bar{u}_s}{\bar{u}_e} \right) + \frac{\bar{T}_{e} - \bar{T}_{aw}}{\bar{T}_e} \left( \frac{\bar{u}_s}{\bar{u}_e} \right)^2, \end{equation}

where

(2.9)\begin{equation} f\left( \frac{\bar{u}_s}{\bar{u}_e} \right) = 0.1741 \left( \frac{\bar{u}_s}{\bar{u}_e} \right)^2 + 0.8259 \left( \frac{\bar{u}_s}{\bar{u}_e} \right). \end{equation}

It is apparent that the present DNS data agree well with the modified relation of Duan & Martin (Reference Duan and Martin2011). The deviation is less than 2 %. To summarise, the laminar and turbulent boundary layers on the cold wall are well resolved by the present DNS. In the following, we start describing the transition process. As the transition to turbulence is triggered by the self-excited instability in the flow system, GSA is firstly employed to examine the intrinsic instability with respect to the 2-D base flow.

4.1. Influence of low-frequency unsteadiness

The oscillatory unstable modes revealed by the GSA imply the presence of unsteady flow. According to the DNS results, the saturated flow exhibits a strong unsteadiness both inside and downstream of the separation bubble. For a similar compression-ramp flow with a lower Reynolds number, Cao et al. (Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021b) demonstrated that the flow unsteadiness can be attributed to the intrinsic instability of the flow system. With this view, the flow unsteadiness and its effects on the transition process are first addressed in this section.

To facilitate the following discussion, it is necessary to highlight some critical positions in the flow field. Figure 15 shows the time-averaged wall Stanton number on the ramp and marks several positions; C denotes the corner ($x/L = 1$); R corresponds to the spanwise-averaged reattachment position ($x/L = 1.27$); P represents the peak-heating position ($x/L = 1.45$) and T indicates the onset of transition ($x/L = 1.76$, estimated from figure 4b).

Figure 15. Contour of the time-averaged wall Stanton number. Here, C denotes the corner ($x/L = 1$), R corresponds to the spanwise-averaged reattachment position ($x/L = 1.27$), P represents the peak-heating position ($x/L = 1.45$) and T indicates the onset of transition.

Figure 16(a) presents the temporal history of the reattachment position ($x_R$) in the centre line of the wall, in comparison with the wall Stanton number signal shown in figure 2(b). In general, there exists an opposite trend for the two signals. When the reattachment position moves downstream, the Stanton number decreases, and vice versa. This is consistent with the results of Cao et al. (Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021b), who found that the pulsation of the reattachment position is accompanied by the variation of downstream surface heat flux. Figure 16(b) compares the temporal history of the wall Stanton number and the boundary-layer displacement thickness ($\delta _1$) at the same position ($x/L = 1.45, z/L = 0.15$). Similar to figure 16(a), a generally opposite trend can also be observed. In other words, a high heat flux on the wall corresponds to a thin boundary layer above this position, and vice versa. This is not surprising because the thinning of the boundary layer induces a high temperature gradient on the cold wall. Concerning the spatial distribution of the surface heat flux, its streak pattern indicates that there exist streamwise-elongated high- and low-momentum streaks for the boundary layer above the wall. Therefore, like the surface heat-flux streaks, the boundary-layer streaks also exhibit an unsteady feature. The complicated spatio-temporal distribution of the reattached boundary layer has a significant influence on the transition process, as is shown later.

Figure 16. (a) Temporal history of the reattachment position (solid line) in the centre line of the wall ($z/L = 0.15$), in comparison with the wall Stanton number signal (dash dotted line) at $x/L = 1.45$, $z/L = 0.15$ (extracted from figure 2b). (b) Temporal history of the boundary-layer displacement thickness (solid line) and wall Stanton number signal (dash dotted line) at $x/L = 1.45$ and $z/L = 0.15$.

To characterise the streamwise propagation of the unsteadiness, a two-point spatio-temporal correlation is employed to evaluate the wall Stanton number signal

(4.1)\begin{equation} R_{nm}(\tau)=\frac{\overline{St'_{n}(t)\cdot St'_{m}(t+\tau)}}{\sqrt{\overline{St'^{2}_{n}}} \cdot \sqrt{\overline{St'^{2}_{m}}}}, \end{equation}

where the subscript $n$ denotes the reference point: $x/L = 1.6$, $z/L = 0.15$; the subscript $m$ corresponds to the position varying in the streamwise or spanwise direction; $St' = St-\overline {St}$ is the fluctuation quantity ($\overline {St}$ is the time-averaged $St$ at local streamwise or spanwise position) and $\tau$ is the time delay. The resulting correlation maps are shown in figure 17. The vertical coordinates are given by $\Delta x = x/L - 1.6$ and $\Delta z = z/L - 0.15$. A strong correlation is found in the streamwise direction, whereas the correlation in the spanwise direction is quite weak. This means that the unsteadiness primarily travels in the streamwise direction. The slope of the black line shown in figure 17(a) indicates the travelling speed of the unsteadiness, which is evaluated approximately by $0.5U_\infty$.

Figure 17. Two-point temporal correlation map of the wall Stanton number in the (a) streamwise and (b) spanwise directions. The reference point is located at $x/L = 1.60$, $z/L = 0.15$.

Figure 18 presents the PSD of the wall Stanton number signal shown in figure 16 as well as the spanwise-averaged PSD at $x/L = 1.45$. Welch's method (Welch Reference Welch1967) is employed for the spectral estimation with three segments and 50 % overlap. A Hamming window is used for weighting the data on each segment prior to fast Fourier transform processing. The above setting yields the length of an individual segment as $12.5L/U_\infty$. It is apparent that the wall Stanton number signal has a broadband low-frequency feature. The dominant non-dimensional frequencies (Strouhal number) are of the order of 0.1. The spanwise-averaged PSD has a similar distribution to the PSD at $z/L = 0.15$, which indicates that the dominant frequencies at different spanwise positions are nearly the same.

Figure 18. PSD of the wall Stanton number signal shown in figure 16 (solid line) and the spanwise-averaged PSD (dash dotted line) at $x/L = 1.45$.

The influence of low-frequency unsteadiness on the transition process can be further illustrated by showing the wall pressure signal at different streamwise positions in figure 19. The chosen positions are $x/L = 1.45$, 1.80, 2.10 and 3.00 along the centre line of the wall. As seen, no high-frequency fluctuation exists at $x/L = 1.45$, indicating a laminar state for the boundary layer. At $x/L = 1.80$, high-frequency fluctuation packets intermittently occur in the pressure signal. By roughly counting the number of fluctuation packets, the frequency for the occurrence of a packet is approximately $fL/U_\infty = 7/25 = 0.28$, which is consistent with the typical frequency of the flow unsteadiness (see figure 18). At $x/L = 2.10$, the pressure signal exhibits a high-frequency feature at most time instants, although the intermittent appearance of fluctuation packets is still present. This means that the flow is highly transitional here. A turbulent boundary layer at $x/L = 3.00$ is indicated by the pressure signal shown in figure 19(d).

Figure 19. Temporal history of the wall pressure at different streamwise positions along the centre line of the wall. (a–d) Correspond to the signal at $x/L = 1.45$, 1.80, 2.10 and 3.00.

4.2. Breakdown of streamwise vortices

As mentioned previously, the unsteady heat-flux streaks on the wall are the footprint of unsteady boundary-layer streaks above the wall, and the unsteadiness can be traced back to the reattaching flow. The streak pattern for the reattached boundary layer is further visualised in the following. Figure 20 illustrates the streamwise velocity contour ($u/U_\infty$) in the wall-normal plane at $x/L = 1.27$, 1.35, 1.45, 1.50 and 1.60 for the time instant $tU_\infty /L = 41$ superimposed with the in-plane streamlines. Note that the contour outside the boundary layer is removed by cutting off the levels of $u/U_\infty > 0.92$. As $x/L = 1.27$ is the spanwise-averaged reattachment position, both reverse and reattached flows are present in figure 20(a). Downstream of reattachment (figure 20b–e), the distorted boundary layer exhibits spanwise corrugation, and the high- and low-momentum portions persist in the streamwise direction. This is consistent with previous GSA predictions (see figures 11 and 12). Therefore, it is concluded that the occurrence of streamwise-elongated boundary-layer streaks is triggered by the intrinsic instability in the flow system.

Figure 20. Instantaneous ($tU_\infty /L = 41$) streamwise velocity contour in the wall-normal plane at different streamwise positions superimposed with in-plane streamlines. (a–e) Correspond to $x/L = 1.27$, 1.35, 1.45, 1.50 and 1.60, respectively. The cutoff levels are $u/U_\infty < 0$ and $u/U_\infty > 0.92$.

Further evidence can be found by comparing the in-plane streamlines in figures 12 and 20. It is apparent that counter-rotating vortices are present downstream of reattachment. Between two adjacent vortex pairs, the boundary layer is compressed towards the wall, causing a large temperature gradient and thus a hot region on the wall. At the centre of a vortex pair, the upwash motion brings low-momentum flow away from the wall, producing a cold region on the wall. It should be mentioned that the change in the flow direction at the upper side of both figures 12 and 20 is induced by the flow reattachment and downstream growth of the boundary layer. The consistent vortical structure between the GSA and DNS results indicates that the global instability plays an important role in the formation of streamwise vortices and heat-flux streaks. It is further noted that there are three distinct vortex pairs in figure 20, whereas only one and a half vortex pairs can be observed in figure 12. In other words, the number of vortices or streaks in the saturated flow (DNS data) is twice of that in the flow governed by the linear instability (GSA data). This is due to the emergence of the second harmonic of the most unstable mode, as discussed previously.

To visualise the breakdown of the streamwise vortices, the instantaneous distribution of streamwise vorticity on the wall-parallel plane at $y_n/L = 0.005$ is shown in figure 21. The streamwise vorticity is defined as

(4.2)\begin{equation} \omega_s = \frac{\partial w}{\partial y_n} - \frac{\partial u_n}{\partial z}, \end{equation}

where $u_n$ denotes the wall-normal velocity component. Owing to the flow unsteadiness, the distributions of $\omega _s$ at $tU_\infty /L = 40$ and 41 differ from each other. The boundary layer tends to be homogeneous upstream of $x/L = 2.3$ at $tU_\infty /L = 41$, whereas it is still transitional up to $x/L = 2.5$ at $tU_\infty /L = 40$. In general, the typical breakdown position can be estimated at $x/L = 1.7 {\sim} 1.8$, which is consistent with the aforementioned position for the onset of transition (see figure 4). On the other hand, the streak patterns at the two time instants share some similarities. For example, the evolution of the streamwise streaks in both figures starts from the reattachment region (around $x/L = 1.27$). Moreover, each streak consists of a vorticity pair, which represents a pair of counter-rotating vortices.

Figure 21. Instantaneous distribution of the streamwise vorticity ($\omega _s$, non-dimensional) on the wall-parallel plane at $y_n/L = 0.005$ for the time instants (a) $tU_\infty /L = 40$ and (b) $tU_\infty /L = 41$.

Subsequently, the $Q$-criterion is employed to visualise the flow structures, Q is the second invariant of the velocity gradient tensor. As shown in figure 22, the vortical structure originates from the separation bubble flow and evolves in the streamwise direction. Affected by the flow unsteadiness, the onset of transition differs for each streamwise vortical streak. Interestingly, vortical structures with a ‘hairpin’ shape can be observed in the transitional region (see the enlarged views in figure 22b,d). This structure resembles the varicose mode of streak breakdown in a flat-plate boundary layer (Ovchinnikov, Choudhari & Piomelli Reference Ovchinnikov, Choudhari and Piomelli2008; Schlatter et al. Reference Schlatter, Brandt, De Lange and Henningson2008). In short, the above facts demonstrate that the transition process in the considered compression-ramp flow is initiated by its intrinsic instability and accomplished via the breakdown of streamwise vortices.

Figure 22. Instantaneous visualisation of the vortical structure on the basis of the $Q$-criterion. The iso-surface of $Q = 50$ coloured by the velocity magnitude $U = \sqrt {u^2 + v^2 + w^2}$ is shown in all panels. Panels (b,d) are enlarged views of panels (a,c), respectively. The time instant is $tU_\infty /L = 40$ for (a–b) and $tU_\infty /L = 41$ for (c–d). Numerical schlieren is added to highlight the separation bubble and shock structure.