4.1.1. Turbulent structures

Figure 4 depicts, for the three controlled cases, the instantaneous turbulent structures identified by means of isosurfaces of the imaginary part of the velocity gradient tensor's complex eigenvalues $\lambda _{ci}h/U_\infty$ (swirling strength criterion (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999)), coloured by the streamwise velocity component (a supporting video about the fluid field generated by a supersonic turbulent boundary layer impinging on a microramp is available in high resolution at https://youtu.be/o8olmjiWSl8 (Salvadore et al. Reference Salvadore, Memmolo, Modesti, Della Posta and Bernardini2022)). First of all, it is possible to observe the fine turbulent structures upstream of the microramp, which become smaller by increasing the Reynolds number. At high Reynolds number, the large-scale streamwise streaks are evident, corresponding to regions of alternating high- and low-speed flow. When the flow encounters the ramp, the slope of the top surface induces a first shock that generates a small region with reduced velocity at the foot of the ramp. On the top ramp surface, the flow accelerates as streamlines are forced to converge, then the flow deflects laterally and diverges towards the sides of the ramp, up to its edges. From here, the accelerated flow leaves the top surface and naturally generates the primary vortices on the sides of the ramp, together with smaller secondary vortices close to the top and bottom edges. The primary vortices increase in strength and radius proceeding downstream until they converge on the trailing edge of the ramp where they deflect, becoming nearly parallel and aligned with the streamwise direction. The convergence of the primary vortex pair and the accelerated flow from the top edge of the ramp initiate vortex roll-up in the arc region above the primary vortices. Soon after the edge of the ramp, arc-shaped vortical structures are distinguishable in all the three cases, especially for higher Reynolds numbers. In accordance with the model proposed by Sun et al. (Reference Sun, Schrijer, Scarano and van Oudheusden2014b), the lift-up of the wake and the entrainment of lateral flow in the near-wall region tend to close the arc-shaped structures into almost-toroidal vortex rings that enclose the decaying primary vortex pair. Vortex rings are then convected downstream and, together with the primary vortex pair inside, induce velocity fluctuations that are responsible for the meandering and, eventually, the breakdown of the wake.

Figure 4. Isosurfaces of the swirling strength coloured by the streamwise velocity component: (a) $Re_\tau = 500$ ($\lambda _{ci}h/U_\infty =3.2$); (b) $Re_\tau = 1000$ ($\lambda _{ci} h/U_\infty =5.1$); (c) $Re_\tau = 2000$ ($\lambda _{ci}h/U_\infty =7.1$).

4.1.2. Shock system

Besides the highlighted vortical structures, a series of complex, 3-D shocks is also generated in the field as a consequence of the numerous changes in direction of the supersonic flow around the ramp. To better understand the complex structure of the shocks, we report in figure 5 an isosurface of the magnitude of the mean density gradient, coloured by the streamwise velocity for the intermediate Reynolds number. Slices of the isosurface at constant $x$ help us to understand its 3-D curvature and the interior structure of the field. Figure 6, instead, shows the gradient of the mean density in logarithmic scale on several planes at constant $x$ from two different 3-D views and on a wall-parallel plane at $y/h=5.7$. For symmetry reasons, only half of the domain is shown. Time averaging is conducted over a period of approximately $340h/U_\infty.$

Figure 5. Isosurface of the magnitude of the mean density gradient ($\|\boldsymbol {\nabla } \rho \delta _0/\rho _\infty \| = 0.25$), coloured by the streamwise velocity component. Vertical $yz$ slices of the surface, spaced by $h$, highlight the 3-D shape of the shocks, whereas the isosurface at the trailing edge region is hidden to show the internal convolution of the primary vortices.

Figure 6. Contours of the non-dimensional mean density gradient in logarithmic scale on several $yz$ transparent planes at different streamwise locations spaced by $2h$ (a,b) and on a $xz$ plane at $y/h=5.7$ (c). Only half of the domain is shown for symmetry.

We note that the density gradient is able to highlight better the shocks and the dominant vortical structures delimiting the wake. As mentioned above, the slope of the MVG induces a first shock from the foot of the ramp, which we will refer to as foot shock. Since the ramp has a finite extension, the originally planar foot shock must curve in the spanwise direction while evolving in the longitudinal direction, as can be seen from its curvature on the top of the domain.

After the foot shock, in the final region of the developing primary vortices at the sides of the ramp, we observe the formation of an oblique shock wave inclined at approximately $45^\circ$ right on top of the vortices. These shocks, which we will refer to as vortex shocks, are stronger for flow cases at higher Reynolds numbers. Although in part related to the reduced viscous dissipation, we will discuss in § 4.2.1 that if the ramp geometry is fixed with respect to the height of the boundary layer, increasing Reynolds number leads to an increase in the circulation of the vortex pair and in the velocity magnitude of the flow captured in the side vortices in general, which increases the strength of the compression in the end.

At the trailing edge of the ramp, the flow on the top surface undergoes an initial vertical deflection in the central plane induced by the converging primary vortices, giving rise to a shock from the ramp main edge. This flow deflection is significant only in the trailing edge region close to the symmetry plane, whereas it decreases laterally, thus resulting in a conical shock. Slightly downstream of the ramp trailing edge, the increased lift-up generated by the mutual interaction of the primary vortex pair develops fully and pushes the wake further up, inducing a second shock. We will refer to these shocks as the trailing edge shocks. Given the topology of the foot shock and the spanwise variation of the flow deviation, the trailing edge shocks also have a conical shape, as can be seen from the slice in the $xz$ plane at the top of the computational domain. After less than one ramp length, the two trailing edge shocks coalesce. Below the conical compression, it is possible to appreciate the development of the primary vortex pair, with two distinct convoluted vortices just after the ramp trailing edge. Proceeding downstream, the diffusion and the entrainment of flow from outside of the wake enlarges the cores of the vortices and ‘unroll’ the inner convolution: after slightly more than one ramp length from the main edge, the low-momentum region has an almost circular shape. Finally, we also observe the shocks coming from the side ramps outside of the computational domain, as a result of the periodic boundary conditions.

In order to obtain a clearer picture of the shock system, we also report three contours of the instantaneous density in the $xy$ planes at $z/h = -0.5 L_z/h$, $-0.5 b/h$ and $0$ for the case at high Reynolds number, where $b$ is the spanwise half-width of the ramp (figure 7). At the side surfaces of the domain, it is possible to see only the foot shock and the trailing edge shock. From the intermediate spanwise position, we can observe the significant compression induced by the foot shock. The core of the primary vortex is then highlighted by the region with low density close to the edge of the ramp. Slightly after, on top of the trace of the primary vortex pair, the trailing edge shock occurs, followed by a milder compression due to the shock coming from the side ramps. Farther downstream, low-density regions suggest the presence of the vortex rings observed in figure 4. Finally, the slice at the symmetry plane of the ramp shows all the main shocks of the flow field clearly. The foot shock and the trailing edge shocks have here their maximum intensities, and behind the trailing edge shocks, the shock stemming from the side ramp is also visible. The instantaneous density on this plane also makes evident how, after fewer than two ramp heights, KH instabilities generate vortical structures over the primary vortices, with cores highlighted by spots of reduced density.

Figure 7. Contours of the instantaneous density on vertical slices at different spanwise sections: (a) lateral wall ($z/h = - 0.5 L_z/h$); (b) intermediate spanwise location ($z/h = -0.5 b/h$); (c) centre of the ramp ($z/h = 0$). Low Reynolds number case.

Concerning the effect of the Reynolds number on the shock structure, a comparison between the cases did not reveal relevant qualitative differences with respect to the described picture.

4.2.1. Wall-normal profiles of velocity on the symmetry plane

Figure 8 shows the velocity profiles on the symmetry plane at $x/h = 10.7$ for the three different Reynolds numbers, together with the corresponding uncontrolled cases and the experimental results of Tambe et al. (Reference Tambe, Schrijer and van Oudheusden2021). From the streamwise velocity component $\tilde {u}/U_\infty$ in figure 8(a), it is evident that for increasing ${\textit{Re}} _\tau$, the microramp is more efficient, and is thus able to produce fuller streamwise velocity profiles in the near-wall region (see also § 4.4).

Figure 8. Velocity profiles at $x/h=10.7$: (a) streamwise and (b) wall-normal components. Dashed lines denote the velocity profiles in undisturbed conditions. Symbols denote experiments by Tambe et al. (Reference Tambe, Schrijer and van Oudheusden2021) (circles, $M_\infty = 2.0$, $Re_\theta = 2.4 \times 10^4$): (red lines) $Re_\tau = 500$; (blue lines) $Re_\tau = 1000$; (black lines) $Re_\tau = 2000$.

Moreover, the region in the wake characterised by low momentum with respect to the undisturbed profiles rises to higher relative wall-normal locations. An explanation of this phenomenon is given by the profiles of the wall-normal velocity reported in figure 8(b), where peak values of $\tilde {v}/U_\infty$ increase markedly with the Reynolds number. The numerical results of both the velocity components tend to the experimental results of Tambe et al. (Reference Tambe, Schrijer and van Oudheusden2021), which have been performed at ${\textit{Re}} _\tau \approx 5000$.

Observing the longitudinal and vertical components of the velocity at four sections on the symmetry plane in figure 9, we appreciate the lifting up effect on the wake due to the longitudinal primary vortex pair. The trace of the shock wave generated by the microramp trailing edge is also visible in the profiles at section $x/h = 5$. Farther downstream from the microramp, the profiles of both velocity components tend to return to the undisturbed conditions due to the turbulent mixing and flow entrainment from outside the wake.

Figure 9. Streamwise evolution of the velocity profile at four different stations $x/h=0, 5, 10, 15$: (a) streamwise component and (b) wall-normal component. (Red lines) $Re_\tau = 500$, (blue lines) $Re_\tau = 1000$; (black lines) $Re_\tau = 2000$.

The increased vertical velocity at the symmetry plane suggests that the circulation of the two parallel primary vortices increases with the Reynolds number. Following for example the approach used in Mole, Skillen & Revell (Reference Mole, Skillen and Revell2022), the velocity field in the $yz$ plane produced by the primary vortex pair from a microramp can be modelled schematically by means of a couple of counter-rotating Batchelor vortices (Batchelor Reference Batchelor1964), with two additional mirror vortices with opposite circulation below the wall, to impose wall impermeability, and a linear damping of the azimuthal velocity close to the wall, to impose a no-slip condition. The azimuthal velocity $V_\theta$ of an isolated Batchelor vortex is defined as

(4.1)\begin{equation} V_\theta(r) = \frac{qR}{r} \left[ 1 - \exp\left( - \frac{r^2}{R^2} \right) \right] \end{equation}

where $q$ is the vortex strength, $r$ is the distance from the centre of the vortex and $R$ is the radius of the vortex core. Figure 10(a) shows only the vertical motion induced by a toy model with two parallel vortices of constant circulation on a slice at constant $x$ for two cases where the vortex strength – and hence the circulation – is increased ($q_2/q_1 = 1.25$). Except for the confinement induced by the trailing edge shocks and for the overall downward motion induced by the 3-D flow organisation at the observed station (which makes the DNS results more ‘greenish’), the distribution is very similar to the one observed in the wake of the simulated cases reported in figure 10(b), confirming that the toy model is able to reasonably represent the real case. From the comparison, we note that the effect of increasing the Reynolds number in the numerical simulations is equivalent to increasing the circulation in the analytical model.

Figure 10. Analytical and numerical vertical velocity. (a) Field generated by two parallel Batchelor vortices on a $yz$ plane. Vortex strength $q_1$ at negative $z$, vortex strength $q_2 \approx 1.25q_1$, at positive $z$. (b) Favre-averaged vertical velocity at $x/h = 6$ for low (negative $z$) and high (positive $z$) Reynolds numbers simulations.

4.2.2. Boundary layer thickness and shape factor

In order to evaluate the global properties of the boundary layer before and after the microramp, at the symmetry plane, we use the incompressible displacement and momentum thicknesses,

(4.2a,b)\begin{equation} \delta_i^* = \int_0^\infty \left(1-\frac{\tilde{u}}{U_\infty}\right) \mathrm{d}y \quad \mathrm{and}\quad \theta_i = \int_0^\infty \frac{\tilde{u}}{U_\infty} \left(1-\frac{\tilde{u}}{U_\infty}\right) \mathrm{d}y \end{equation}

which are reported in figure 11(a). Both $\delta _i^*$ and $\theta _i$ grow remarkably after the microramp, but at different rates, and thus the incompressible shape factor $H_i = \delta _i^*/\theta _i$ becomes lower than the upstream condition and also than the uncontrolled cases (figure 11b), indicating fuller boundary layers close the wall, which would be less prone to separation in the presence of SBLIs.

Figure 11. Streamwise evolution of (a) boundary layer displacement thickness (dash-dotted lines), momentum thickness (solid lines) and (b) incompressible shape factor for the uncontrolled case (dashed lines) and with the microramp (solid lines): (red lines) $Re_\tau = 500$; (blue lines) $Re_\tau = 1000$; (black lines) $Re_\tau = 2000$.

4.2.3. Wake velocity and maximum upwash

Another key element in the control efficiency of microramps is the strength and decay of the low-momentum deficit in the wake. In fact, in addition to the vertical motion induced by the primary vortices, the continuous entrainment of fluid with higher momentum from outside the wake, the interaction between the toroidal vortical structures and the primary vortex pair, and the effect of viscous and turbulent dissipation, gradually enlarge and restore the region with low momentum, leading the wake to disappear after some distance from the ramp.

The wake development along the $x$ direction is typically analysed by considering the following quantities on the basis of the vertical velocity profiles at the symmetry plane (Giepman et al. Reference Giepman, Srivastava, Schrijer and van Oudheusden2016): the wake velocity $\tilde {u}_{wake}$, defined as the minimum streamwise velocity in the low momentum region, and its corresponding vertical position $y_{wake}$; the maximum upwash velocity $\tilde {v}_{max}$ and its vertical position $y_{\tilde {v}max}$ (see figure 8).

Babinsky et al. (Reference Babinsky, Li and Ford2009) reported the streamwise evolution of the momentum deficit on several cross-stream planes and, inspired by the work of Ashill, Fulker & Hackett (Reference Ashill, Fulker and Hackett2005) in the incompressible regime, hypothesised that the flow development varies with the height of the device (for fixed boundary layer thickness). To test this hypothesis for geometrically similar ramps with different heights, the authors reported the wake location as a function of the streamwise distance from the main edge, both normalised with the ramp height. The collapse between the curves of the different cases, later confirmed also by other studies (Giepman et al. Reference Giepman, Schrijer and van Oudheusden2014), indicated that, at least for similar inflow conditions, the ramp height is able to scale the wake features successfully. For this reason, in the following, we try also in our case to scale wake lengths by the ramp height $h$.

Figure 12(a) shows that the streamwise evolution of $\tilde {u}_{wake}$, normalised by the undisturbed velocity $U_\infty$, collapses reasonably on a single curve for the three cases, following the experimental results of Tambe et al. (Reference Tambe, Schrijer and van Oudheusden2021) and Giepman et al. (Reference Giepman, Schrijer and van Oudheusden2014). The wake position is much more sensitive to the Reynolds number and numerical results converge towards the experimental ones only for the case at high Reynolds number, which suggests that after a certain threshold the flow becomes independent of the Reynolds number. A similar trend is also observed for the wall-normal upwash velocity $\tilde {v}_{max}$ and its location in figures 12(c) and 12(d).

Figure 12. Streamwise evolution of (a) wake velocity and (b) wake location, (c) maximum upwash velocity, (d) maximum upwash location for different $Re_\tau$, scaled by $h$. Symbols in (a,b) denote experiments by Tambe et al. (Reference Tambe, Schrijer and van Oudheusden2021) (circles, $M_\infty = 2.0$, $Re_\theta = 2.4 \times 10^4$) and by Giepman, Schrijer & van Oudheusden (Reference Giepman, Schrijer and van Oudheusden2014) (squares, $M_\infty = 2.0, Re_\theta = 2.1 \times 10^4$). Symbols in (c,d) denote experiments by Giepman et al. (Reference Giepman, Srivastava, Schrijer and van Oudheusden2016) (circles, $M_\infty = 2.0$, $Re_\theta = 5.0 \times 10^4$ and squares, $M_\infty = 2.0$, $Re_\theta = 9.9 \times 10^4$): (red lines) $Re_\tau = 500$; (blue lines) $Re_\tau = 1000$; (black lines) $Re_\tau = 2000$.

4.2.4. Self-similarity of velocity profiles

In this section, we also examine the self-similarity of the velocity profiles following the formulation proposed by Sun et al. (Reference Sun, Schrijer, Scarano and van Oudheusden2014b). For the profiles of the streamwise component, the velocity is normalised by the quantity $\Delta \tilde {u}_{deficit}$, defined as the maximum difference between the velocity profile of the controlled case and that of the undisturbed boundary layer $u_{BL}$, while the vertical coordinate is normalised by the wake location $y_{wake}$. Thus, we have that

(4.3a,b)\begin{equation} \hat{U}(\eta_u) = \frac{\tilde{u}(\eta_u) - u_{BL}(\eta_u)}{\Delta \tilde{u}_{deficit}},\quad \eta_u = \frac{y - y_{wake}}{y_{wake}} . \end{equation}

On the other hand, for the wall-normal component, the velocity is normalised by the maximum upwash velocity $\tilde {v}_{max}$, while the vertical coordinate is normalised by $y_{\tilde {v}max}$, so that

(4.4a,b)\begin{equation} \hat{V}(\eta_v) = \frac{\tilde{v}(\eta_v)}{\tilde{v}_{max}},\quad \eta_v = \frac{y - y_{\tilde{v}max}}{y_{\tilde{v}max}} . \end{equation}

As also demonstrated by Sun et al. (Reference Sun, Chen, Li, Liu, Guo and Yuan2020), velocity profiles exhibit self-similarity, especially in the central region of the wake, whereas discrepancies are expected in the near wake and in the near-wall region, where the bottom secondary vortices induce a region of low velocity, and flow reversal can also occur.

The normalised velocity profiles at several equispaced streamwise locations in the region $x/h = 3$ to $x/h = 30$ are shown in figure 13. A very good agreement is observed for the scaled profiles, which is quantified in figure 14 by reporting the streamwise average of the scaled profiles along with the corresponding standard deviation, for the three Reynolds numbers.

Figure 13. Velocity profiles at: (a,b) $Re_\tau =500$, (c,d) $Re_\tau =1000$, (e, f) $Re_\tau =2000$. Non-normalised (a,c,e) and normalised (b,d, f) streamwise and wall-normal velocity. Colour indicates the streamwise location according to the colourbar. Dashed lines indicate the uncontrolled boundary layer.

Figure 14. Normalised velocity profiles for different $Re_\tau$: (a) streamwise component and (b) wall-normal component. The normalised profiles are averaged in the streamwise direction. Banded regions indicate the corresponding standard deviation from the streamwise-averaged profile. (Red lines) $Re_\tau = 500$, (blue lines) $Re_\tau = 1000$, (black lines) $Re_\tau = 2000$.

Excluding the very near-wall region where viscous effects are dominant, the maximum standard deviation for the streamwise velocity component $\sigma _{\hat {U}}$ is 0.078, 0.082, 0.091 for the low, intermediate and high Reynolds numbers, respectively, which confirms that $\tilde {u}$ is self-similar. Differences among the three flow cases are also of the same order of the standard deviation, and therefore are considered satisfactory.

Regarding the normalised vertical velocity profiles, minor deviations are visible in the lower region of the wake, and the standard deviation is generally very small. Vertical self-similarity is more troublesome to infer in the upper part of the wake and has a different behaviour especially for higher Reynolds number. In particular, the outer region is also influenced by the trailing edge shock, and thus perfect self-similarity is difficult to achieve. Additionally, the increased lift-up and the faster rise of the wake for the higher Reynolds number case is strong enough to induce a region of negative $\tilde {v}$ at the symmetry plane (see figure 10b), which further aggravates the differences with respect to the other two cases.

4.3.1. The near wake

We have seen in the previous sections that the captured flow from the incoming boundary layer generates two symmetric vortical structures at the sides of the ramp, which converge at the trailing edge and then proceed approximately in parallel. The behaviour and decay of the primary vortex pair far from the edge has been widely considered in several experimental studies (Giepman et al. Reference Giepman, Schrijer and van Oudheusden2014, Reference Giepman, Srivastava, Schrijer and van Oudheusden2016; Sun et al. Reference Sun, Schrijer, Scarano and van Oudheusden2014b; Tambe et al. Reference Tambe, Schrijer and van Oudheusden2021), and although some aspects still remain unclear, several studies provided valuable insights into the development of the far wake.

On the other hand, less is known about the near wake, as this region is difficult to study experimentally due to the presence of the solid walls. Despite its reduced extent, in this critical region important changes take place that influence the evolution of the entire wake downstream. In order to draw a qualitative picture of the flow in the near wake, we report in figure 15 the Favre-averaged streamwise velocity component and the three orthogonal components of the Favre-averaged vorticity $\boldsymbol {\zeta }$ on equispaced cross-stream planes in the region $x/h \in [-0.4, 2]$ for the flow case at intermediate Reynolds number. White isolines indicate a constant value of the density gradient, highlighting the edges of the main vortices and shocks, as reported in figure 6. Due to symmetry, only half of the domain is reported and discussed.

Figure 15. Vertical slices at $x/h \in [-0.4, 2]$ for the case at intermediate Reynolds number. Favre-averaged: (a) streamwise velocity; (b) streamwise; (c) wall-normal; (d) spanwise vorticity components. White lines indicate the points with $||\boldsymbol {\nabla } \rho \delta _0/\rho _\infty || = 0.25$, while yellow lines in (a) and its zoomed region indicate the points with $\tilde {u}/U_\infty = 0$.

The velocity field in figure 15(a) clearly highlights the core of the primary vortex, from which we can infer its internal organisation. Close to the wall and the symmetry plane, especially near the trailing edge of the ramp as shown in the zoomed region, we note an extended region of low-speed reversed flow, corresponding to a small separation and a near-wall secondary vortex. This region shrinks progressively downstream and flow reversal is almost absent at the end of the considered interval. The isolines of the density gradient on top of the main vortex show the oblique shock on the side vortex propagating downstream and disappearing approximately one $h$ from the ramp. The isoline of the density gradient also highlights the spreading and internal convolution of the main vortex, represented by the negative values of $\zeta _1$ in figure 15(b).

The distributions of the wall-normal and spanwise vorticity components in figure 15(c,d) complements the understanding of the above-described vortices. In particular, before the main edge, the two components help us to understand the orientation of the primary and secondary vortices and are able to highlight the internal convolution of the side vortices, superposed to the main streamwise helical motion. After the main edge, where secondary vortices become less important, the transversal components $\zeta _2$ and $\zeta _3$ are particularly useful to capture the trace of the vortical structures inside the wake.

On the basis of the behaviour of the observed quantities, in figure 16 we propose a qualitative description summarising the evolution of the vortical structures in the near wake ($x/h \in [0.4, 2]$). Before the trailing edge, the presence of the vertical lateral wall of the ramp and the inclination of the primary vortices imposes the formation of two asymmetric secondary vortices rotating in the opposite direction of the primary vortices. As soon as the fluid from the top of the ramp flows over the converging vortex pair, the strong shear induces the roll-up of KH instabilities. The KH vortices on top of the wake thus join laterally the counter-rotating helical motion of the two primary vortices, and the typical almost-toroidal vortices are generated as a result. Given its reduced intensity and the significant activity in the top part of the wake, the upper secondary vortex first moves upwards and laterally, following the rotation of the primary vortex, and then disappears soon after the ramp. The bottom secondary vortex instead persists for a longer distance. However, once the primary vortices align and the mutual interaction between them is fully established, the upward vertical motion in between the vortex pair reduces drastically the intensity of the bottom vortex.

Figure 16. Model sketch of the vortical structures in the near wake ($x/h \in [0.4, 2]$). Yellow arrows indicate the orientation of the helical flow in the primary and secondary vortices; blue arrows indicate the flow rotation in the $yz$ planes; red arrows indicate the orientation of the transversal rotation of the vortical structures superposed to the main helical motion; black lines indicate the approximate boundaries of the primary, secondary and inner vortices. Finally, the region with negative (positive) streamwise vorticity related to the primary vortex (secondary vortices) is indicated in orange (light blue).

While the wake expands, the lift-up in the central region also affects the internal convolution of the primary vortices. The combined distributions of $\zeta _2$ and $\zeta _3$ makes it possible to identify an internal ring-like structure with tangential vorticity oriented in the opposite direction compared with the external almost-toroidal vortices. This structure is superposed to the primary vortices and is progressively pushed upwards by the increased vertical motion induced by the primary vortex pair. In the space where the wake gradually starts to lift up, the intensity of these internal structures decreases and, as a result, there is no evident sign of them after less than $2h$ from the ramp trailing edge anymore. Close to the symmetry plane, on the other hand, convolution is still visible from $\zeta _2$ (around $z/h \approx -0.5$ in figure 15c), consistently with the observations of secondary vortices inside the wake in the PIV measurements of Sun et al. (Reference Sun, Schrijer, Scarano and van Oudheusden2012) and in the numerical results of Sun et al. (Reference Sun, Scarano, van Oudheusden, Schrijer, Yan and Liu2014a). Contrary to what is proposed by Sun et al. (Reference Sun, Scarano, van Oudheusden, Schrijer, Yan and Liu2014a), however, our observations suggest that the ‘curved legs’ of the KH vortices are already present from the start and are a consequence and evolution of the initial convolution of the primary vortices. Only far from the ramp (approximately $x/h > 15$), the tangential vorticity of the inner, vertical legs is almost completely dissipated and only a mild trace is visible inside the wake from $\zeta _2$ close to the symmetry plane. This suggests that in the first part of the wake, the KH vortices are not closed near the wall but are instead strictly tied to the helical motion of the primary vortices they surround. A stricter closure of the tangential vorticity takes place only far downstream, where instead we observed a significant decrease in coherence of the azimuthal structures (see figure 4), at least at the present Reynolds numbers.

4.3.2. The far wake

Once the primary vortices become parallel and the wake starts to lift up, the low-momentum region spreads and the intensity of the wake deficit progressively decays. In order to describe the evolution of the wake far from local effects due to the presence of the ramp, we report in figure 17 the behaviour of the Favre-averaged Mach number ($\tilde {M}=\sqrt {\tilde {u}^2 + \tilde {v}^2+ \tilde {w}^2}/\sqrt {\gamma R \tilde {T}}$), and of the Favre-averaged Reynolds stress components $\widetilde {\tau _{11}} = - \overline {\rho u'' u''}$, $\widetilde {\tau _{22}} = - \overline {\rho v'' v''}$, $\widetilde {\tau _{33}} = - \overline {\rho w'' w''}$, $\widetilde {\tau _{12}} = - \overline {\rho u'' v''}$, where the symbol $(\bullet )''$ indicates the fluctuations of a variable with respect to its Favre average. The figure shows the above-mentioned quantities on three $yz$ slices at $x/h = 4$, $8$ and $16$, for the intermediate Reynolds case only. Since we observed decay properties similar to the $\widetilde {\tau _{12}}$ component for the other two transversal components, $\widetilde {\tau _{13}}$ and $\widetilde {\tau _{23}}$, we do not report them in this figure and we report only their distributions at $x/h = 8$ (figure 18).

Figure 17. Contours for the intermediate Reynolds number case at $x/h = 4$ (a,d,g,j,m,p), $8$ (b,e,h,k,n,q) and $16$ (c, f,i,l,o,r) of Favre-averaged Mach number $\tilde {M}$ (a–c), Favre-averaged temperature $\tilde {T}/T_\infty$ (d–f) and Favre-averaged Reynolds stress components $\widetilde {\tau _{11}}/\rho _\infty U_\infty ^2$ (g–i), $\widetilde {\tau _{22}}/\rho _\infty U_\infty ^2$ (j–l), $\widetilde {\tau _{33}}/\rho _\infty U_\infty ^2$ (m–o), $\widetilde {\tau _{12}}/\rho _\infty U_\infty ^2$ (p–r). White isolines in the Mach and temperature contours indicate the points with $\tilde {M} = 1$.

Figure 18. Turbulent kinetic energy $\tilde {k}/\rho _\infty U_\infty ^2$ (a,e,i), $\widetilde {\tau _{12}}/\rho _\infty U_\infty ^2$ (b, f,j), $\widetilde {\tau _{13}}/\rho _\infty U_\infty ^2$ (c,g,k), $\widetilde {\tau _{23}}/\rho _\infty U_\infty ^2$ (d,h,l), at $x/h = 8$ for low (a,b,c,d), intermediate (e, f,g,h), high (i,j,k,l) Reynolds number.

The distributions of the temperature and of the Mach number highlight the wake region behind the microramp, which is initially characterised by subsonic conditions, given the lower velocity – and the higher temperature – experienced by the momentum deficit area. In fact, the fluid in the wake is the one captured directly (fluid from the top of the ramp captured by the side vortices) or indirectly (fluid entrained in the near-wall region from outside of the primary vortices) by the MVG flow structures. In both cases, fluid inside the wake was previously close to the wall. However, as a consequence of the imposed adiabatic wall conditions, the aerodynamic heating raises the temperature of this fluid, which thus results in a wake hotter than the corresponding undisturbed flow. The sonic line indicated in white also suggests the thinning of the boundary layer below the wake, due to the energising exchange of momentum promoted by the microramp. Proceeding downstream, the entrainment of colder and faster flow from outside induces an increase in the Mach number, which tends to the undisturbed conditions far from the ramp. In the wake region, even after $15h$ from the trailing edge, we can see that the subsonic region – and the wake in general – is not circular and homogeneous inside. As a matter of fact, the lift-up close to the symmetry plane and the external flow summoning due to the primary vortices makes the magnitude of the Mach number at the centre higher than elsewhere in the wake.

The principal Reynolds stress components highlight instead that the wake region, with its shear layers and vortical structures, is characterised as expected by an increased turbulent kinetic energy that decays as soon as the wake spreads. Given the larger exposure to the incoming flow, the magnitude of the streamwise velocity fluctuations is more significant in the top part of the wake. At the same time, vertical fluctuations fed by the lift-up of the primary vortex pair decay slowly and are more important at the symmetry plane, where the upwash is maximum. Other regions of intense turbulent activity are noticeable in the field, especially at $x/h = 4$, and represent a trace of the near wake features that rapidly disappears. In particular, initially $\widetilde {\tau _{11}}$ exhibits large values in magnitude also close to the symmetry plane, probably due to the observed volutes, with their secondary KH vortices inside the wake. Moreover, in the lower part at the spanwise centre, strong spanwise fluctuations are highlighted by negative peak values of $\widetilde {\tau _{33}}$, likely caused by the convergence of the primary vortex pair and by the last effects of the bottom secondary vortices.

The Reynolds shear stress $\widetilde {\tau _{12}}$ confirms that the top part of the annular shear layer is related to the most important turbulent mixing, where the maximum value is very close to the experimental measurements of Sun et al. (Reference Sun, Schrijer, Scarano and van Oudheusden2012) ($-\overline {u'v'}/U_\infty ^2 \approx 0.007$, where $(\bullet )'$ indicates fluctuations of a quantity with respect to its Reynolds average). Furthermore, the negative regions in the bottom part of the wake suggest that the bottom shear layer is weaker than the top one, in accordance with the previous observations in § 4.2.4 about the difference in the vertical gradients close and far from the wall.

Figure 18 shows instead a comparison of the Favre-averaged turbulent kinetic energy ($\tilde {k} = - ( \widetilde {\tau _{11}} + \widetilde {\tau _{22}} + \widetilde {\tau _{33}})$) and of the transversal shear stress components between the three cases at different Reynolds numbers on an intermediate $yz$ plane at $x/h = 8$. Increasing the Reynolds number does not affect qualitatively the distribution and has more of a quantitative effect on the evolution of the far wake. In particular, the magnitude of the turbulent kinetic energy and of the transversal turbulent stress components in the wake increases with the Reynolds number, indicating an increased mixing especially at the external and internal edges of the wake, where velocity gradients become stronger.

4.4.1. Added momentum

To evaluate the efficiency of the momentum transfer towards the near-wall region induced by the presence of the microramp, Giepman et al. (Reference Giepman, Schrijer and van Oudheusden2014, Reference Giepman, Srivastava, Schrijer and van Oudheusden2015, Reference Giepman, Srivastava, Schrijer and van Oudheusden2016) introduced the so-called added momentum. By including also the density in the definition, the (compressible) added momentum is defined as

(4.5)\begin{equation} E = \int_{0}^{0.43\delta_{99}} \frac{\overline{\rho u^2} - \overline{\rho u^2}_{BL}}{\rho_\infty U_\infty^2} \mathrm{d}y \end{equation}

where the subscript $BL$ indicates the momentum of the uncontrolled boundary layer. An upper limit of integration equal to $0.43\delta _{99}$ is used, since the separation bubble of the downstream SBLI in the experiments of the cited works was found to be mostly sensitive to the momentum of the lower 43 % boundary layer. Similarly to the other wake features, $E$ is typically scaled by the ramp height.

Contrary to the previous experimental works, which were limited to the symmetry plane only, our numerical database makes it possible to examine the distribution of the added momentum on the entire surface of the bottom wall (figure 19), allowing us to characterise also the spanwise behaviour of the momentum transfer towards the wall.

Figure 19. Normalised added momentum in the wall-parallel plane. White lines indicate the points for which $E/h = 0$. Low (a), intermediate (b) and high (c) Reynolds number.

At the sides of the ramp and close to the symmetry plane, the upward motion induced by the primary and secondary vortices removes momentum from the region close to the wall, thus generating areas with negative $E$. However, even at the symmetry plane, the added momentum ends up being larger than zero not far from the trailing edge of the ramp (only marginally for the case at low Reynolds). On the other hand, approximately below the primary vortices, a noticeable region of positive $E$ indicates that the helical motion related to the primary vortices brings fresh fluid with high momentum from the sides of the wake towards the plane of symmetry, from higher wall-normal locations to the bottom portion of the domain.

In order to better capture the quantitative differences between the three cases, figure 20 shows the normalised added momentum for the three Reynolds numbers along the spanwise direction at $x/h = 0$, $10$ and $30$, and along the streamwise direction at $z/h = -b/h$, $-0.5 b/h$ and $0$. Although the results agree well with the experimental results of Giepman et al. (Reference Giepman, Srivastava, Schrijer and van Oudheusden2016) at the symmetry plane, confirming the trend for increasing Reynolds number, the plots highlight again that the only added momentum at the spanwise centre is insufficient to portray appropriately the overall transfer to the wall induced by the microramp wake. In fact, we can see that in the far wake, $E/h$ is limited at the symmetry plane but, at the same time, is high below the cores of the primary vortex pair, where the most important addition of momentum takes place. Concerning the differences between the cases, a clear trend for increasing Reynolds number is not easily recognisable. We showed in the previous sections that increasing the Reynolds number means dealing with stronger vortical structures, which lift up the wake faster and move – reasonably – more fluid. As a consequence, the region with negative added momentum at the spanwise centre shrinks progressively as the Reynolds number is increased. However, contrary to what was expected, the peak added momentum at high Reynolds number is smaller than the other cases. This result could be related to the fact that if the low-momentum region rises faster because of the increased upwash, the azimuthal motion of the primary vortices affects only fluid that is farther from the wall, and so, despite the circulation being larger, the resulting added momentum is smaller. Moreover, since for different Reynolds numbers, the primary vortices have different trajectories (vortices close to the sidewalls of the ramp and more abrupt alignment to the streamwise direction for larger Reynolds numbers) and the wakes have different streamwise evolutions, comparing the added momentum for the three conditions at fixed $x/h$ and $z/h$ can be misleading. For these reasons, further analysis in future work is needed to better understand the balance and scaling of the different contrasting aspects.

Figure 20. Normalised added momentum distribution (a) along the spanwise direction at $x/h = 0$ (solid lines), $10$ (dash-dotted lines), $30$ (dotted lines); (b) along the streamwise direction at $z/h = -b/h$ (solid lines), $-0.5 b/h$ (dash-dotted lines), $0$ (dotted lines), with $b$ being the half of the spanwise extent of the microramp. Circles denote experiments by Giepman et al. (Reference Giepman, Srivastava, Schrijer and van Oudheusden2016) at $M_\infty = 2.0$ and $Re_\theta = 6.7 \times 10^4$: (red lines) $Re_\tau = 500$; (blue lines) $Re_\tau = 1000$; (black lines) $Re_\tau = 2000$.

4.4.2. Relative friction coefficient

Figure 21 reports the percentage relative difference of the Favre-averaged skin friction coefficient at the wall of the flat plate ($C_f = 2 \tilde {\tau }_w/\rho _\infty U_\infty ^2$) between the controlled and the uncontrolled cases ($\Delta C_f = C_f - C_{f,BL}$). In particular, since the spanwise friction component $\tilde {\tau }_z = (\partial \tilde {w}/\partial y)_w$ was far smaller in magnitude than the streamwise component $\tilde {\tau }_x = (\partial \tilde {u}/\partial y)_w$, we consider in the following the approximation for the controlled case that $\tilde {\tau }_w \approx \tilde {\tau }_x$. In the $C_f$ contours, we find many flow features also observed in the wake analysis, confirming the insights from the added momentum. The primary and secondary vortices at the sides of the ramp are highlighted by regions of enhanced and reduced friction, whereas the trailing edge shock is visible in the transversal low-friction region right after the trailing edge of the ramp. After the near wake, the entrainment of high-momentum fluid induced by the primary vortices is responsible for a prolonged region of higher friction, which persists even 30$h$ downstream of the microramp. The contours point out the significant streamwise and spanwise variation of the boundary layer properties, which plays a major role in the SBLI control mechanism. In this regard, Babinsky et al. (Reference Babinsky, Li and Ford2009) showed that an important effect of microramp arrays upstream of the interaction region in two-dimensional SBLIs consists in the modulation of the separated flow area, which is broken up in multiple individual pockets of 3-D reverse flow, separated by regions of attached flow.

Figure 21. Percentage difference of the skin friction coefficient between the controlled and uncontrolled boundary layer. White lines indicate the points for which $(\partial \tilde {u}/\partial y)_w = 0$. Low (a), intermediate (b) and high (c) Reynolds number.

Finally, since the wake rises faster at higher Reynolds number, we note that increasing $Re_\tau$ the relative skin friction increase becomes weaker in the far wake.