# Direct numerical simulation of supersonic boundary layers over a microramp:
effect of the Reynolds number

G. Della Posta1<EMAIL_ADDRESS>M. Blandino1  D. Modesti2  F.
Salvadore3  M. Bernardini1  1Department of Mechanical and Aerospace
Engineering, Sapienza University of Rome, via Eudossiana 18, 00184, Rome,
Italy 2 Aerodynamics group, Faculty of Aerospace Engineering, Delft University
of Technology, Kluyverweg 2, 2629 HS Delft, The Netherlands 3 HPC Department,
CINECA, via dei Tizii 6/B, 00185, Rome, Italy

###### Abstract

Microvortex generators are passive control devices smaller than the boundary
layer thickness that energise the boundary layer to prevent flow separation
with limited induced drag. In this work, we use direct numerical simulations
(DNSs) to investigate the effect of the Reynolds number in a supersonic
turbulent boundary layer over a microramp vortex generator. Three friction
Reynolds numbers are considered, up to $Re_{\tau}=2000$, for fixed free-stream
Mach number $M_{\infty}=2$ and fixed relative height of the ramp with respect
to the boundary layer thickness. The high-fidelity data set sheds light on the
instantaneous and highly three-dimensional organisation of both the wake and
the shock waves induced by the microramp. The full access to the flow field
provided by DNS allows us to develop a qualitative model of the near wake,
explaining the internal convolution of the Kelvin–Helmoltz vortices around the
low-momentum region behind the ramp. The overall analysis shows that numerical
results agree excellently with recent experimental measurements in similar
operating conditions and confirms that microramps effectively induce a
significantly fuller boundary layer even far downstream of the ramp. Moreover,
results highlight significant Reynolds number effects, which in general do not
scale with the ramp height. Increasing Reynolds number leads to enhanced
coherence of the typical vortical structures in the field, faster and stronger
development of the momentum deficit region, increased upwash between the
primary vortices from the sides of the ramp — and thus increased lift-up of
the wake — and faster transfer of momentum towards the wall.

###### keywords:

Supersonic flow, turbulent boundary layers, turbulence simulation, flow
control, vortex interactions.

##  1 Introduction

The interaction between shock waves and boundary layers has been widely
studied during the last 70 years and is still the subject of extensive
research, because of its frequent occurrence in many aerospace systems
(Dolling, 2001; Gaitonde, 2015). Shock Wave/Boundary Layer Interactions
(SBLIs) can be found in a wide range of flows including transonic airfoils,
high-speed inlets, control surfaces of high-speed aircraft, space launchers
base flows, and overexpanded nozzles. In the interaction region, extreme,
unsteady, mechanical and thermal loads are typically observed (Clemens &
Narayanaswamy, 2014), and the adverse pressure gradient imposed by the shock
often causes the boundary layer to separate intermittently (Bernardini et al.,
2023a). This problem represents, for example, a critical issue for high-speed
intakes, since the low-momentum separated flow affects the mass flow rate
entering the engine, producing flow distortions, total pressure losses, and
potential inlet unstarts (Herrmann & Koschel, 2002).

Given its pivotal importance, it is evident that alleviating the detrimental
effects of SBLI by means of control devices would have a significant impact on
the performance and the reliability of many aerospace engineering systems
(Babinsky & Ogawa, 2008). The most widespread solution to delay shock-induced
separation in high-speed intakes, and to control SBLI in turn, is the so
called boundary layer bleed, which consists in diverting the separated flow
near to the wall out of the engine. Removing low-momentum fluid makes the
boundary layer more resistant to shock-induced separation and reduces the
distortion of the flow ahead of the engine too. However, since the bled air is
not re-injected elsewhere, the flow rate is reduced up to the 15-20% (Loth,
2000), requiring an increase in the intake frontal area and, in turn, an
increase in the aerodynamic drag of the vehicle. Other solutions have thus
been explored, like classical Vortex Generators (VGs) (Davis & Mitchell,
1968), especially with vane-type form. Classical VGs, whose heights are larger
than the boundary layer thickness, generate passively trailing vortices that
bring high-momentum flow from outside the boundary layer to the near-wall
region. However, despite their benefits, VGs generate considerable additional
drag.

To leverage the same mechanism of VGs and energise the near-wall region of the
boundary layer upstream of the interaction, Microvortex Generators (MVGs) have
been proposed (Wheeler, 1984; McCormick, 1993), with heights smaller than the
boundary layer thickness. Compared to their traditional counterpart, MVGs
offer the same control efficiency but with a decreased wave drag, given their
reduced height (Lin, 2002). Among the various configurations proposed,
although microvanes seem to be slightly more effective, microramps are
typically preferred because of their superior mechanical robustness. In fact,
MVGs are usually mounted upstream of the SBLI, and thus, in case of failure,
their ingestion would bring catastrophic consequences for the inlet.

During the last twenty years, research focused particularly on the
characterisation of the wake downstream of microramps and on its influence on
SBLI separation (Lu et al., 2012; Panaras & Lu, 2015). As a matter of fact,
the control mechanism of microramps relies on how these change the flow
topology upstream of the interaction. Therefore, a deep understanding of the
complex structure of the wake produced by a microramp in a high-speed boundary
layer is precious and preliminary to the assessment of any effect MVGs may
have on SBLI. The first extensive description of the mean flow behind an MVG
is due to Pitt Ford & Babinsky (2007) and Babinsky et al. (2009). The authors
analysed through surface oil visualisations the different large scale
structures emerging from the interaction between a microramp and a supersonic
boundary layer. Firstly, the incoming flow is subjected to compression on the
microramp leading edge, which leads to a small pocket of separated flow and to
the development of a horseshoe vortex. Then, the fluid flows on top of the
ramp, turning laterally toward the ramp edges, where it separates from the
device and generates a pair of counter-rotating vortices on the two ramp
sides, the so-called primary vortex pair. The vortex pair brings high-momentum
flow from the outer region of the boundary layer down towards the wall,
inducing a fuller boundary layer which is less prone to separation. In
addition to these primary structures, a series of secondary vortices also
originates from the bottom and top edges of the device (Lu et al., 2010).
Downstream of the microramp trailing edge, the two primary vortices become
parallel and auto-induce a vertical force in the plane of symmetry that causes
the wake and the vortex pair to lift up. Therefore, moving downstream, the
wake lift-up, together with viscous and turbulent dissipation, progressively
weaken the energising mechanism of the wake, which eventually returns to the
undisturbed conditions. A qualitative representation of the mean vortex
structure behind the microramp is reported in figure 1(a).

Figure 1: Mean vortical structures in the model of Babinsky et al. (2009) (a)
and instantaneous flow organisation according to the model of Sun et al.
(2014b) (b).

Other studies have instead investigated the instantaneous organisation of the
flow field behind the microramp. Blinde et al. (2009) analysed the effect of
an array of microramps on a boundary layer at $M_{\infty}=1.84$ through
stereoscopic Particle Image Velocimetry (PIV). From the distribution of the
velocity on two planes parallel to the wall, the authors inferred the presence
of a train of hairpin vortices surrounding the primary streamwise vortices.
Implicit Large Eddy Simulations (ILESs) of a microramp in a turbulent boundary
layer at $M_{\infty}=2.5$ by Li & Liu (2011) revealed instead the existence of
a train of ring-shaped vortices surrounding the low-momentum region that
induces significant wake meandering. According to the authors, these large
scale structures are generated by a Kelvin–Helmholtz (K–H) instability
originating from the shear layer that envelops the momentum deficit in the
microramp wake. In their experimental analysis of a boundary layer at
$M_{\infty}=2.7$, Bo et al. (2012) observed that these K–H vortices are
characterised by a frequency between $50$ KHz and $60$ KHz, which is
transferred to the reflected shock donwstream, as the intermittent structures
pass through the interaction zone and deform the shock system. Vortex rings
from microramps pass across the SBLI without losing their coherence and
disrupt the separation bubble according to their shedding frequency, which
thus makes them a key component in the control mechanism of SBLI (Grébert et
al., 2018). To better understand the evolution of the wake behind microramps
and to solve the controversy in the different vortex models proposed by many
authors, an extensive investigation on the flow generated by a microramp in a
turbulent boundary layer at $M_{\infty}=2.0$ was carried out through
tomographic PIV and ILES in Sun et al. (2012, 2013, 2014b). Based on the
previous findings and on their results, the authors proposed a conceptual
model (see figure 1(b)) in which arc-shaped K–H vortices surrounding the
primary vortex pair develop at first, right behind the ramp. As the lift-up
induced by the primary vortex pair acts on the wake, the arc legs are
subjected to an inward movement that leads to the formation of closed vortex
rings. These periodic entities modulate then the primary vortex pair and
finally undergo turbulent distortion that eventually breaks them down.

Even if comprehensive experimental studies like the ones presented above are
paramount, their inherent restrictions in terms of spatial accuracy and data
accessibility limit their ability to fully understand how MVGs may control
SBLI. For instance, in many studies, flow features are observed only on a
limited number of planes, like the vertical symmetry plane. However, the field
is highly three-dimensional, and the interpretation of the measurements on
these surfaces may be misleading with respect to the global flow structure.
High-fidelity numerical simulations represent thus a valid complementary tool
that has proved effective to overcome such limitations. As a result, detailed
numerical data may help address the physical questions remained in some way
open, like the effect of the Mach and Reynolds numbers on the strength and
decay of the time-averaged and instantaneous wake features or the precise role
that K–H instability has in the control mechanism of SBLI.

To the authors’ knowledge, a limited number of numerical studies has been
carried out to fully comprehend the flow alteration induced by a microramp,
and mainly using Reynolds-Averaged Navier–Stokes Simulations (RANSs) (Ghosh et
al., 2010) or ILESs (Lee & Loth, 2009; Li & Liu, 2010, 2011; Lee et al., 2011;
Sun et al., 2014a), with momentum thickness-based Reynolds numbers
$Re_{\theta}\in[1400,5800]$. Both numerical techniques rely on closure
assumptions, which can lead to model-dependent results, especially in the case
of RANS of separated flows. Despite the higher uncertainty of RANS, however,
this tool remains the only possibility to explore a large parameter space. For
example, Anderson et al. (2006) carried out a multi-objective optimisation of
the microramp geometry using RANS, with the aim of minimising the total
pressure loss across the SBLI and the boundary layer shape factor $H$. The
result has been deliberately taken as the baseline, optimal case by the
majority of the cited studies.

Although similar methodologies allowed researchers to better describe the
evolution of the main vortical structures and to refine models based on
experiments, they are not fully able to capture the unsteady and multiscale
nature of the flow under consideration. However, thanks to the technological
developments in High Performance Computing (HPC) and to the advent of the
Graphics Processing Unit (GPU) technology for scientific simulations, we are
today able to perform Direct Numerical Simulations (DNSs) with parameters that
are comparable with those of real experiments.

For this reason, we present here the results of a set of DNSs of a supersonic
turbulent boundary layer over a microramp. The geometry considered is the one
of the optimal microramp defined by Anderson et al. (2006), and free-stream
Mach number $M_{\infty}=U_{\infty}/c_{\infty}$ is equal to 2, where
$U_{\infty}$ and $c_{\infty}$ are the free-stream velocity and speed of sound,
respectively.

We especially focus our attention on the effects of the Reynolds number. Three
flow cases are examined at friction Reynolds numbers
$Re_{\tau}=\delta_{99}/\delta_{v}=500$, $1000$, and $2000$, based on the
properties of the undisturbed boundary layers at the ramp location, where
$\delta_{99}$ is the boundary layer thickness, $\delta_{v}=\nu_{w}/u_{\tau}$
is the viscous length scale, $u_{\tau}=\sqrt{\tau_{w}/\rho_{w}}$ is the
friction velocity, $\rho_{w}$ and $\nu_{w}$ are the density and kinematic
viscosity at the wall, respectively. The topic has been scarcely considered in
the literature. In particular, only Giepman et al. (2016) described the
streamwise evolution of the main features of the mean wake velocity for
Reynolds number based on the boundary layer thickness in the range $4.9\cdot
10^{5}\div 10.5\cdot 10^{5}$ but the results did not show any relevant
difference in the interval considered. However, despite the potential
importance, no study is available at present concerning the effects of
microramps at small or moderate Reynolds numbers.

The paper is organised as follows: Section 2 presents the methodology and
numerical setup of the simulations; Section 3 describes the database generated
and the validation carried out; Section 4 presents the results of the
analysis; finally, Section 5 reports some final comments.

##  2 Methodology and numerical setup

The DNS database considered in this work has been generated using STREAmS
2.0111https://github.com/STREAmS-CFD/STREAmS-2 (Bernardini et al., 2021,
2023b; Sathyanarayana et al., 2023). STREAmS is an open-source finite-
difference compressible flow solver developed by our group, designed to solve
the compressible Navier–Stokes equations for a perfect, heat conducting gas.
The solver targets canonical wall-bounded turbulent high-speed flows, and it
is oriented to modern HPC platforms with the capability to run on both NVIDIA
(Bernardini et al., 2021, 2023b) and AMD (Sathyanarayana et al., 2023) GPUs.

The discretisation of the convective terms adopts a hybrid energy-
conservative/shock-capturing scheme in locally conservative form (Pirozzoli,
2010). In smooth regions, a sixth-order, central, energy-preserving flux
formulation ensures stability without the addition of numerical diffusivity.
Close to the shock waves, a fifth-order, Weighted Essentially Non-Oscillatory
(WENO) reconstruction (Jiang & Shu, 1996) is used to compute numerical fluxes
at the cell faces, using a Lax–Friedrichs flux vector splitting. A shock
sensor evaluates the local smoothness of the solution and identifies the
regions where discontinuities occur, and where the switch between the central
and the WENO scheme takes place. Viscous terms are approximated by means of
sixth-order, central finite-difference approximations. Time advancement is
carried out by means of a third-order, low-storage, Runge-Kutta scheme
(Spalart et al., 1991).

Figure 2: Domain with sizes and orthogonal projections of the microramp.

The size of the computational domain adopted for the simulations is
$L_{x}/\delta_{0}\times L_{y}/\delta_{0}\times L_{z}/\delta_{0}=100\times
15\times 10$, the reference length $\delta_{0}$ being the thickness of the
boundary layer at the inflow. The microramp geometry for the three flow
conditions considered is based on the optimal shape defined by Anderson et al.
(2006) (see figure 2), with a constant ratio between the ramp height and the
boundary layer thickness of the uncontrolled case, $h/\delta_{99}=2/3$. This
configuration has been studied experimentally by Tambe et al. (2021), at the
same Mach number but at a higher Reynolds number ($\Rey_{\theta}=2.4\times
10^{4}$, $\Rey_{\tau}\approx 5000$). The microramp is centred in the cross-
stream plane, and the ramp trailing edge is $60\,\delta_{0}$ from the inlet
section. The trailing edge of the ramp is also the origin of our reference
system. The mesh is uniform in the wall-parallel directions, corresponding to
a viscous-scaled spacing of $\Delta x^{+}\approx 6.9$ and $\Delta z^{+}\approx
5.5$, in the streamwise and spanwise directions, respectively. The mesh is
stretched in the wall-normal direction, corresponding to a wall-spacing in the
range $\Delta y^{+}\approx 0.75$–$0.95$. The microramp is simulated by means
of a ghost-point-forcing Immersed Boundary Method (IBM) (Piquet et al., 2016)
already validated in previous works (Modesti et al., 2022).

The boundary conditions are specified as follows. At the outflow and the top
boundaries, non-reflecting conditions are imposed by performing a
characteristic decomposition in the direction normal to the boundaries
(Poinsot & Lele, 1992). A characteristic wave decomposition is also employed
at the bottom no-slip wall, where the wall temperature is set to the recovery
value of the incoming boundary layer. In order to prescribe suitable turbulent
fluctuations at the inflow, a recycling-rescaling procedure (Lund et al.,
1998) is used, with a recycling station placed at $50\,\delta_{0}$ from the
inflow to guarantee a sufficient decorrelation between the inlet and the
recycling station. Finally, the flow is assumed to be statistically
homogeneous in the spanwise direction, thus periodic boundary conditions are
applied on the sides. Hence, simulations are representative of an array of
microramps with lateral spacing $L_{z}$.

##  3 Numerical data set and validation

The main parameters of the numerical database analysed in this work are
reported in table 1. The data set includes baseline simulations with
undisturbed boundary layer, which are used as a reference to assess the
effects of the microramps. We consider three values of friction Reynolds
number $\Rey_{\tau}=500$, $1000$, $2000$, based on the properties of the
undisturbed boundary layers. The corresponding Reynolds number based on the
momentum thickness is reported in table 1 together with the number of mesh
points. The free-stream Mach number $M_{\infty}$ is equal to 2 for all the
flow cases. In the following, unless otherwise specified, we will indicate
with the colours red, blue, and black the cases at low, intermediate and high
Reynolds number respectively.

Case | $M_{\infty}$ | $Re_{\tau}$ | $Re_{\theta}$ | $N_{x}\times N_{y}\times N_{z}$ | $h/\delta_{0}$
---|---|---|---|---|---
UBL - L | 2.00 | 500 | 2330 |  a4096 $\times$ 288 $\times$ a512 | -
UBL - M | 2.00 | 1000 | 4870 |  a8192 $\times$ 512 $\times$ 1024 | -
UBL - H | 2.00 | 2000 | 9880 | 16384 $\times$ 896 $\times$ 2048 | -
MBL - L | 2.00 | 500 | 2330 |  a4096 $\times$ 288 $\times$ a512 | 1.26
MBL - M | 2.00 | 1000 | 4870 |  a8192 $\times$ 512 $\times$ 1024 | 1.16
MBL - H | 2.00 | 2000 | 9880 | 16384 $\times$ 896 $\times$ 2048 | 1.12
Table 1: Main parameters of the numerical database. Reynolds numbers are
evaluated at the trailing edge of the microramp.
UBL: uncontrolled boundary layer, MBL: microramp-controlled boundary layer, L:
low Reynolds number, M: intermediate Reynolds number, H: high Reynolds number.

In order to characterise the incoming turbulent boundary layer, figure 3 shows
a comparison of the van Driest transformed mean velocity profiles and of the
density-scaled Reynolds stress components with the incompressible data from
the DNS database of Sillero et al. (2013). For the case with the highest
Reynolds number, an excellent agreement can be observed for the mean velocity
from the viscous sublayer up to the outer region, where the wake component
departs from the transformed profile slightly. The Reynolds stress components,
represented in wall units, are reported instead in figure 3(b). Increasing the
friction Reynolds number, all the components converge towards the
incompressible results.

(a)

(b)

Figure 3: Comparison of (a) van Driest–transformed mean velocity profile and
(b) density-scaled Reynolds stress components for the incoming boundary layer
with reference experimental data by Sillero et al. (2013) (circles,
$Re_{\tau}=1989$).

##  4 Results

###  4.1 Qualitative flow organisation

First, we describe the qualitative behaviour of the instantaneous flow in
terms of turbulent structures, and we outline the organisation of the shock
system taking place in the field.

####  4.1.1 Turbulent structures

Figure 4: Isosurfaces of the swirling strength coloured by the streamwise
velocity component. From top to bottom: low, intermediate, and high Reynolds
number cases.

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 (swirling strength criterion,
Zhou et al. (1999)), coloured by the streamwise velocity component222 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., 2022). . 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. (2014b), 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.

####  4.1.2 Shock system

Figure 5: Isosurface of the magnitude of the mean density gradient, coloured
by the streamwise velocity component. Only half of the domain is shown for
symmetry. Figure 6: Three-dimensional views of the contours of the non-
dimensional mean density gradient in logarithmic scale on several yz planes at
different streamwise locations. Intermediate Reynolds number case. Only half
of the domain is shown for symmetry.

Besides the highlighted vortical structures, a series of complex, 3D 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 at constant $x$ and transparency
help understand the 3D curvature and 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 3D views and on a wall-
parallel plane at the top of the computational domain. For symmetry reasons,
only half of the domain is shown.

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 of approximately 45 degrees 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 section 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 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, also the trailing edge
shocks 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 as a result of the periodic boundary conditions.

(a)

(a)

(b)

(c)

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

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 $z/h=-0.5L_{z}/h$,
$-0.5b/h$, and $0$ for the case at high Reynolds number, where $b$ is the
spanwise half-width of the ramp. 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. Further 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
less than two ramp heights, K–H instabilities generate vortical structures
over the primary vortices, with cores highlighted by spots of reduced density.

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

###  4.2 Streamwise evolution of the wake

In the following, we discuss various aspects concerning the streamwise
development of the wake, focusing in particular on the Favre-averaged
streamwise and vertical velocity components, for which also experimental data
are available. Favre-averaging of a generic variable $f$ is indicated with the
tilde and is defined as $\tilde{f}=\overline{\rho f}/\bar{\rho}$, where the
bar indicates Reynolds-averaging instead.

####  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. (2021). From
the streamwise velocity component $\tilde{u}/U_{\infty}$ in figure 8a, it is
evident that for increasing $\Rey_{\tau}$, the microramp is more efficient,
and is thus able to produce fuller streamwise velocity profiles in the near-
wall region (see also section 4.4). 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 to this phenomenon is
given by the profiles of the wall-normal velocity reported in figure 8b, 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. (2021), which have been performed at
$\Rey_{\tau}\approx 5000$.

(a)

(b)

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. (2021) (circles,
$M_{\infty}=2.0$, $Re_{\theta}=2.4\times 10^{4}$).

(a)

(b)

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.

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 donwstream 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.

(a)

(a)

(b)

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 $q_{2}\approx 1.25\,q_{1}$, at positive $z$. (b)
Favre-averaged vertical velocity at $x/h=6$ for low (negative $z$) and high
(positive $z$) Reynolds numbers simulations.

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 et al. (2022), 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, 1964), 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 no-slip condition. The azimuthal velocity $V_{\theta}$ of an
isolated Batchelor vortex is defined as

$V_{\theta}(r)=\frac{q\,R}{r}\left[1-\mathrm{exp}\left(-\frac{r^{2}}{R^{2}}\right)\right]$
(1)

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 10a 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 3D 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 10b,
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.

####  4.2.2 Boundary layer thickness and shape factor

(a)

(b)

Figure 11: Streamwise evolution of (a) boundary layer displacement thickness
(-.), momentum thickness (-) and (b) incompressible shape factor for the
uncontrolled case (- -) and with the microramp (-).

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,

$\delta_{i}^{*}=\int_{0}^{\infty}\left(1-\frac{u}{U_{\infty}}\right)\mathrm{d}y\quad\mathrm{and}\quad\theta_{i}=\int_{0}^{\infty}\frac{u}{U_{\infty}}\left(1-\frac{u}{U_{\infty}}\right)\mathrm{d}y$
(2)

which are reported in figure 11a. 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.

####  4.2.3 Wake velocity and maximum upwash

(a)

(b)

(c)

(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) and (b) denote experiments by Tambe et al. (2021) (circles,
$M_{\infty}=2.0$, $Re_{\theta}=2.4\times 10^{4}$) and by Giepman et al. (2014)
(squares, $M_{\infty}=2.0,Re_{\theta}=2.1\times 10^{4}$).
Symbols in (c) and (d) denote experiments by Giepman et al. (2016) (circles,
$M_{\infty}=2.0$, $Re_{\theta}=5.0\times 10^{4}$ and squares,
$M_{\infty}=2.0$, $Re_{\theta}=9.9\times 10^{4}$).

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., 2016): 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. (2009) reported the streamwise evolution of the momentum
deficit on several cross-stream planes and, inspired by the work of Ashill et
al. (2005) 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. (2014), 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 12a 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. (2021) and Giepman et al. (2014). 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 from the Reynolds
number. A similar trend is also observed for the wall-normal upwash velocity
$\tilde{v}_{max}$ and its location in figure 12c and 12d.

####  4.2.4 Self-similarity of velocity profiles

(a)

(a)

(b)

(c)

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

(a)

(b)

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.

In this section, we also examine the self-similarity of the velocity profiles
following the formulation proposed by Sun et al. (2014b). 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

$\hat{U}(\eta_{u})=\frac{\tilde{u}(\eta_{u})-u_{BL}(\eta_{u})}{\Delta\tilde{u}_{deficit}}\,,\qquad\eta_{u}=\frac{y-y_{wake}}{y_{wake}}$
(3)

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

$\hat{V}(\eta_{v})=\frac{\tilde{v}(\eta_{v})}{\tilde{v}_{max}}\,,\qquad\eta_{v}=\frac{y-y_{\tilde{v}max}}{y_{\tilde{v}max}}$
(4)

As also demonstrated by Sun et al. (2020), 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.

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 deviations 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 Mean wake organisation

In the following, we consider the evolution of the time-averaged flow
quantities in the very near ($x/h<2$) and far wake ($x/h>4$) behind the
microramp.

####  4.3.1 The near wake

(a)

(b)

(c)

(d)

Figure 15: Multiple vertical slices in the longitudinal range $x/h\in[-0.4,2]$
for the case at intermediate Reynolds number. Favre-averaged: (a) streamwise
velocity, (b) streamwise, (c) wall-normal, (d) spanwise vorticity components.

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 (Sun et al., 2014b;
Giepman et al., 2014, 2016; Tambe et al., 2021), 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.

The velocity field in figure 15(a) well highlights the core of the primary
vortex, from which we can infer its internal organization. Close to the walls,
especially right after the ramp, 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
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.

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).

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 K–H instabilities. The K–H 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.
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 to 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 $2\,h$ 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 15(c)), consistently with the observations
of secondary vortices inside the wake in the PIV measurements of Sun et al.
(2012) and in the numerical results of Sun et al. (2014a). Contrarily to what
proposed by Sun et al. (2014a), however, our observations suggest that the
“curved legs” of the K–H 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 K–H 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

(a)

(b)

(c)

(d)

(e)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 17: Contours of the Favre-averaged Mach number $\widetilde{M}$ (first
column), and of the Favre-averaged Reynolds stress components
$\widetilde{\tau_{11}}/\rho_{\infty}U_{\infty}^{2}$ (second column),
$\widetilde{\tau_{22}}/\rho_{\infty}U_{\infty}^{2}$ (third column),
$\widetilde{\tau_{33}}/\rho_{\infty}U_{\infty}^{2}$ (fourth column),
$\widetilde{\tau_{12}}/\rho_{\infty}U_{\infty}^{2}$ (fifth column) for the
intermediate Reynolds number at $x/h=4$ (first row), $8$ (second row), and
$16$ (third row). White isolines in the Mach contours indicate the points with
$\widetilde{M}=1$.

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
($\widetilde{M}=\sqrt{\tilde{u}^{2}+\tilde{v}^{2}+\tilde{w}^{2}}/\sqrt{\gamma
R\tilde{T}}\,$), and of the Favre-averaged Reynolds stress components
$\widetilde{\tau_{11}}=-\overline{\rho u^{\prime\prime}u^{\prime\prime}}$,
$\widetilde{\tau_{22}}=-\overline{\rho v^{\prime\prime}v^{\prime\prime}}$,
$\widetilde{\tau_{33}}=-\overline{\rho w^{\prime\prime}w^{\prime\prime}}$,
$\widetilde{\tau_{12}}=-\overline{\rho u^{\prime\prime}v^{\prime\prime}}$. 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).

The distribution of the Mach number highlights 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 $15\,h$ 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 K–H
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. (2012) ($-\overline{u^{\prime}v^{\prime}}/U_{\infty}^{2}\approx
0.007$). 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 section 4.2.4 about the difference in the
vertical gradients close and far from the wall.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(d)

(e)

(f)

(g)

(d)

(e)

(f)

(g)

Figure 18: Turbulent kinetic energy
$\widetilde{k}/\rho_{\infty}U_{\infty}^{2}$ (first column)
$\widetilde{\tau_{12}}/\rho_{\infty}U_{\infty}^{2}$ (second column),
$\widetilde{\tau_{13}}/\rho_{\infty}U_{\infty}^{2}$ (third column),
$\widetilde{\tau_{23}}/\rho_{\infty}U_{\infty}^{2}$ (fourth column), at
$x/h=8$ for low (first row), intermediate (second row), high (third row)
Reynolds number.

Figure 18 shows instead a comparison of the Favre-averaged turbulent kinetic
energy
($\widetilde{k}=-(\widetilde{\tau_{11}}+\widetilde{\tau_{22}}+\widetilde{\tau_{33}})$)
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 Near-wall behaviour

As we demonstrated throughout the previous discussion, the wake behind a
microramp is highly three-dimensional and, as a result, streamwise and
spanwise modulation of the flow behaviour close to the wall is expected. In
order to discuss these modifications, we consider in the following the
distribution on the wall-parallel plane of the added momentum, introduced by
Giepman et al. (2014), and of the difference of the friction coefficient
between the controlled and uncontrolled cases.

####  4.4.1 Added momentum

(a)

(a)

(b)

(c)

Figure 19: Normalised added momentum in the wall-parallel plane. White lines
indicate the points for which $E/h=0$.

(a)

(b)

Figure 20:  Normalised added momentum distribution (a) along the spanwise
direction at $x/h=0$ (–), $10$ (-.-), $30$ ($\cdots$); (b) along the
streamwise direction at $z/h=-b/h$ (–), $-0.5b/h$ (-.-), $0$ ($\cdots$), with
$b$ being the half of the spanwise extent of the microramp. Circles denote
experiments by Giepman et al. (2016) at $M_{\infty}=2.0$ and
$Re_{\theta}=6.7\times 10^{4}$.

To evaluate the efficiency of the momentum transfer towards the near-wall
region induced by the presence of the microramp, Giepman et al. (2014, 2015,
2016) introduced the so-called _added momentum_. By including also the density
in the definition, the (compressible) added momentum is defined as:

$E=\int_{0}^{0.43\,\delta_{99}}\frac{\overline{\rho u^{2}}-\overline{\rho
u^{2}}_{BL}}{\rho_{\infty}U_{\infty}^{2}}\,\mathrm{d}y$ (5)

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,
allowing us to characterise also the spanwise behaviour of the momentum
transfer towards the wall.

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 among the three cases,
figure 20 shows the normalised added momentum for the three Reynolds number
along the spanwise direction at $x/h=0$, $10$, and $30$, and along the
streamwise direction at $z/h=-b/h$, $-0.5b/h$, and $0$. Although the results
agree well with the experimental results of Giepman et al. (2016) 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 among 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 Reynolds number is
increased. However, contrary to what was expected, the peak added momentum at
high Reynolds 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 side walls 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 this reasons, further analysis in
future work is needed to better understand the balance and scaling of the
different contrasting aspects.

####  4.4.2 Relative friction coefficient

(a)

(a)

(b)

(c)

Figure 21: Percentage difference of the skin friction coefficient between the
controlled and uncontrolled boundary layer. White lines indicate the points
for which $(\partial\tilde{u}/\partial y)_{w}=0$.

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 of a prolonged region of higher friction, which persists even
30$h$ downstream 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. (2009) showed that an important effect of microramp arrays upstream of the
interaction region in 2D SBLIs consists in the modulation of the separated
flow area, which is broken up in multiple individual pockets of three-
dimensional reverse flow, separated by regions of attached flow.

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.

##  5 Conclusions

Microramp MVGs are promising devices proposed to passively control flow
separation and mitigate SBLI adverse effects. This work presents a campaign of
DNSs which aims at examining the effects of a microramp immersed in a
supersonic turbulent boundary layer for different Reynolds numbers,
approaching the values achieved in experiments. Three conditions are
considered, with friction Reynolds number $Re_{\tau}$ equal to 500, 1000, and
2000, fixed free-stream Mach number $M_{\infty}$ equal to $2$, and fixed shape
and relative height of the ramp with respect to the uncontrolled boundary
layer thickness ($h/\delta_{99}=2/3$).

The high fidelity of the DNS database and the opportunity to access completely
the flow field make it possible to observe in detail the instantaneous
organisation of the turbulent structures, disclosing the formation of primary,
secondary, and almost-toroidal vortices in the wake, but also of the system of
shock waves interacting with the wake, hardly addressed in the literature. We
observe in particular two shocks nearly conical in shape, which originate at
the foot and trailing edge of the microramp and leave a visible signature on
the flow close to the wall, as revealed by the distributions of the skin
friction coefficient and of the added momentum.

In order to describe the evolution of the boundary layer and that of the wake,
and to assess the control efficiency of the device, we take into consideration
the streamwise development of the streamwise and wall-normal velocity
components at the symmetry plane. We find important Reynolds number effects in
the range considered, although the highest Reynolds number case shows good
agreement with recent experiments at $\Rey_{\tau}\approx 5000$, confirming the
relevance of our high-Reynolds number data set.

To complete the picture of the wake development, we then assess the self-
similarity of the velocity profiles in the far wake, following the work of Sun
et al. (2020). The results show that self-similarity is not strictly recovered
close to the wall, where the bottom secondary vortices have a prolonged
influence on the streamwise velocity. Similarly, self-similarity breaks down
in the outer part of the wake, where shock waves leave their imprint.

The three-dimensional distribution of the Favre-averaged flow statistics
highlights then the presence of a complex organisation of the vortical
structures inside the low-momentum region, especially in the near field. On
the basis of the mean vorticity components, we propose a model of the internal
structure of the primary vortices close to the trailing edge, which is
fundamental to understand the following development in the far field.
According to the model, primary vortices are initially convoluted at the sides
of the ramp and joins naturally the K–H vortices that take place on top of the
wake. The model is also able to explain the experimental observations of
internal K–H vortices inside the wake in wall-parallel planes in Sun et al.
(2012), and adds further details to the understanding of the almost-toroidal
vortices around the low-momentum area.

We also analyse the Reynolds stress components, which indicate regions of
intense turbulent activity and highlight the annular shear layer around the
wake. The distribution of the Reynolds stresses also confirms that the legs of
the external K–H vortices do not close at the bottom, but rather join the
motion induced by the primary vortices inside the wake.

Finally, we focus on the changes in the boundary layer close to the wall,
which allows us to assess the spanwise flow modulation induced by the
microramp. The added momentum, introduced by Giepman et al. (2014), and the
relative difference in skin friction coefficient between the controlled and
uncontrolled cases, confirm the control mechanism of the microramp, which
induces two counter-rotating vortices bringing high-momentum fluid closer to
the wall thus increasing the local skin friction coefficient.

Moreover, results allow us to advance a prediction about the control
effectiveness of microramps. Even though vortical structures are those
steering momentum transfer, shape factors show no sign of recovery in the mean
boundary layer properties even after $x/h=30$, where the intensity of the
vortices is low. This observation suggests that the effectiveness in
controlling SBLI could be rather insensitive to the position of the microramp,
for large relative distances. Hence, we expect good control performance in a
fairly wide range of shock positions. Simulations in the presence of SBLI are
currently ongoing to verify this conjecture.

As anticipated above, almost all the quantities considered in this study
reveal a non-negligible dependence on the Reynolds number. To sum up,
according to the results obtained, the main effects due to low Reynolds number
observed are:

  1. 1.

turbulent structures have a less coherent roll-up around the wake deficit,
leading to weaker almost-toroidal vortices.

  2. 2.

Wake features do not scale with the geometric size of the ramp and thus depend
also on the incoming flow properties. If one uses the height $h$ to scale
typical wake features, results indicate a slower and weaker development of the
wake for lower Reynolds numbers.

  3. 3.

The self-similarity of the wake, especially in its top portion, is less
strict.

  4. 4.

Smaller momentum captured by the microramp at lower Reynolds number means
smaller circulation of the side vortices, which leads to a reduction of the
induced upwash and thus a decrease in the lift-up of the wake.

  5. 5.

Side vortices become parallel more gradually. At higher Reynolds number, the
primary vortices remain close to the side walls of the ramp longer, the
separation near the trailing edge of the ramp starts more downstream and is
slightly more prolonged in the streamwise direction.

  6. 6.

Weaker vortices at low Reynolds number, with a slower wake development, lead
also to a slower transfer of momentum towards the wall, which is visible in
the smaller added momentum especially at the symmetry plane. At the same time,
weaker vortices means slower lift-up, which in the end prolongs the region for
which the wake is sufficiently close to the wall to influence the increase in
momentum transfer and in skin friction behind the ramp.

[Acknowledgements]We acknowledge CINECA Casalecchio di Reno (Italy) for
providing us with the computational resources required by this work.

[Funding]This research received no specific grant from any funding agency,
commercial or not-for-profit sectors.

[Declaration of interests]The authors report no conflict of interest.

[Data availability statement]The full data set of the DNS simulations is on
the order of several thousands of gigabytes. By contacting the authors, a
smaller subset can be made available.

[Author ORCID] G. Della Posta, https://orcid.org/0000-0001-5516-9338; M.
Blandino, https://orcid.org/0009-0008-3478-1446; D. Modesti,
https://orcid.org/0000-0003-2214-5799; F. Salvadore,
https://orcid.org/0000-0002-1829-3388; M. Bernardini,
https://orcid.org/0000-0001-5975-3734

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