The flow around a circular cylinder with long spanwise length has been investigated in the critical regime using a compressible wall-resolved Large Eddy Simulation (LES) for the first time. The flow at such a critical Reynolds number combines complex features: large favorable and adverse pressure gradient, separation and turbulence transition and flow reattachment. The results of the present simulation agree well with previous experimental and incompressible LES data, for the distribution of the mean wall-pressure coefficient that dominates the drag coefficient, and the skin-friction coefficient that illustrates the flow separation and reattachment behaviors. A weak reattachment is observed from the quasi-zero skin-friction in the reattachment region. A detailed study of the boundary-layer and shear-layer development around the cylinder with profiles of mean velocity and turbulence intensities confirms the transition, separation and reattachment behaviors shown by the skin-friction coefficient. The maximum tangential velocity and its location above the wall have also proven to be adequate measures of the edge velocity and associated boundary layer thickness. The Kelvin-Helmholtz instabilities have been observed in the shear layer and the ratio of the frequency of these instabilities and the fundamental vortex shedding frequency matches well with the existing scaling based on experimental data. The far-field noise obtained by both direct computation and acoustic analogy shows a dominant vortex shedding tone, but with additional broadband sources in the cylinder wake. These sound sources are evaluated from maps of filtered pressure signals and cross-correlation analysis of the pressure fluctuations around the cylinder and in the far-field.

## I. INTRODUCTION

Turbulent flow around a circular cylinder is a classical problem of fluid dynamics
for bluff-bodies^{1,2} and has wide
engineering applications, for instance in transportation systems (train pantograph,
automotive axles, aircraft landing gears).^{3–5} The flow comprises complex physical features:
boundary layer successively subjected to favorable and adverse pressure gradients,
flow separation, transition of the shear layer and boundary layer, vortex shedding
and turbulent wake containing large structures. The shedding of vortices
characterized by a particular Strouhal number *St*_{VK} (shedding frequency of the
von Kàrmàn street) based on the cylinder diameter D and the freestream velocity *u*_{∞} occurs over a wide range of Reynolds numbers and
can lead to severe structural vibrations, enhanced mixing, significant increases in
the mean drag and high level acoustic noise.^{6}

According to Zdravkovich^{2} and
Achenbach,^{7,8} the turbulent
flow past a smooth circular cylinder can be divided into four regimes depending on
the Reynolds number *Re*_{D} based on *D* and *u*_{∞}:

Sub-critical regime (400 <

*Re*_{D}< 10^{5}): the boundary layer is laminar throughout the circumference. Laminar separation occurs at about*θ*_{s}= 70° − 80° from the forward stagnation point and transition to turbulence occurs in the separated shear layers^{9}(*θ*is defined in Fig. 1). The early separation results in a high drag coefficient ($CD=2FD/\rho \u221eu\u221e2A\u22481.2$,^{1,10}with*F*_{D}the drag force,*ρ*_{∞}the free stream density and*A*the reference area) that remains quasi-constant. The frequency of the vortex shedding is about*St*_{VK}≈ 0.2.^{11}Critical regime (10

^{5}<*Re*_{D}< 3 × 10^{5}): it is a range of transition between laminar and turbulent separation. The separation point shifts downstream to about*θ*_{s}= 90° and at a certain Reynolds number; a laminar separation bubble (LSB) occurs due to the positive pressure gradient; the transition occurs in the shear layer above the LSB and followed by a turbulent reattachment.^{8}The shear-layer is closer to the cylinder surface compared with the sub-critical regime. The final separation is located at about*θ*_{s}= 130° − 140°. The LSB and turbulent reattachment are responsible for the steep drop of drag.^{9,12}A sharp increase of the vortex shedding Strouhal number*St*_{VK}is also observed. The Reynolds number range of the critical regime is extremely sensitive to the inlet flow conditions.Super-critical regime (3 × 10

^{5}<*Re*_{D}< 2 × 10^{6}): for smooth surface cylinder, the flow is characterized by an immediate transition from a laminar to turbulent boundary layer downstream of*θ*= 90°. The separation angle reduces from*θ*_{s}= 140° to*θ*_{s}= 120°, leading to an increase in the drag coefficient. The vortex shedding Strouhal number is again constant*St*_{VK}≃ 0.45 − 0.5.^{13,14}Trans-critical regime or fully turbulent (

*Re*_{D}> 2 × 10^{6}): the transition from laminar to turbulent boundary layer occurs on the front part of the cylinder. The drag coefficient reaches a new plateau lower than in the sub-critical range. The vortex shedding Strouhal number significantly drops.

Note that the flow may be significantly changed and transitions can occur at lower
values of *Re*_{D} if surface roughness,
vibrations or freestream turbulence are present.^{15,16} In the present work, we focus on the cylinder flow
in the critical regime. A few experimental studies have been performed at these
Reynolds numbers. To examine the boundary layer and the transition to turbulence,
Fage and Falkner^{10} systematically
measured the drag coefficient, the distribution of the mean wall-pressure and the
skin-friction coefficient on the cylinder surface. They observed an inflexion point
on the mean wall-pressure distribution as the flow enters the critical regime. The
transition from laminar to turbulence takes place in the boundary layer near this
inflexion point (see for instance Achenbach’s measurements^{8}). It is now well known that the LSB observed on the
cylinder surface is a characteristic of the critical regime and that its footprint
is a plateau in the pressure distribution which is located slightly after the
minimum pressure. Tani^{12} showed
that the LSB above the cylinder surface was similar to that observed on airfoils
with incidence. Another interesting feature of the critical regime is the existence
of asymmetric flow, where one-side flow separates as in the sub-critical regime and
the other side transition takes place above the separation bubble, as observed by
Bearman,^{9} Achenbach and
Heinecke^{13} and Schewe.^{14} Schewe^{14} reported that this asymmetric flow state is
stable only for a small range of Reynolds numbers. For sufficiently high Reynolds
numbers, the perturbations on both sides of the cylinder yield a nearly symmetric
flow behavior with the existence of the LSB on both sides of the cylinder.

Experiments have played a critical role in the understanding of the complex flow
around circular cylinders. However, most of them have focused on measuring the
global loads, the vortex shedding characteristics and point-wise time-averaged data
of the wall-pressure distribution. Besides, the experimental probes can introduce
perturbations which can eventually affect the flow regime and the generated sound.
Only recently, new experimental methods have been developed to detect sound sources,
such as the cross-correlation method^{17} that relates the velocity fluctuations measured by
Particle Image Velocimetry (PIV) in the wake and the sound pressure fluctuations in
the far-field measured with microphones. Oguma *et al.*^{18} reconstructed the near-field
pressure fluctuations using the velocities measured by PIV coupled with a Poisson
equation solver. They then carried out the correlation between this reconstructed
near-field and the measured far-field acoustic pressure. However, such a methodology
has only been applied to the sub-critical regime. Accurate numerical simulations are
then required as they can provide more flow details with constructive insights.
Direct numerical simulation (DNS) is the most accurate approach. However it is still
limited to low Reynolds number flows due to the large range of scales to be
resolved. Only few DNS of the flow around circular cylinders have been achieved up
to now. They are mostly two-dimensional (see for instance Inoue and Hatakeyama^{19}), and at a maximum Reynolds
number *Re*_{D} = 10000, which remains well
below the critical regime. Only recently a three-dimensional incompressible DNS was
achieved by Dong *et al.*^{20} at *Re*_{D} =
3900/4000 and 10000. Even though Large eddy simulations (LES) seem to be more
affordable to tackle the cylinder flow in the critical regime and beyond, most of
them remain in the sub-critical regime.^{21–26} Only the incompressible
LES by Lehmkuhl *et al.*^{27} and Cheng *et al.*^{28} cover the present flow range and characterize the
flow field in the critical states. Lehmkuhl *et al.*^{27} predicted the asymmetric flow
behavior for their low Reynolds numbers cases. To the authors’ knowledge, they were
the first ones to capture this flow feature numerically. Unfortunately, the
evolution of the boundary layer and the shear layer were not discussed. Cheng *et al.*^{28} focused on the results of the mean wall-pressure and the skin-friction. Only limited
discussion on the development of the boundary layer was provided by the evolution of
the boundary-layer thickness. Only the mean velocity profiles in the reattached
region have been presented for the case with *Re*_{D} = 3.5 × 10^{5}. In
both above incompressible LES simulations, the study of the sound radiation and its
sources was not possible. Moreover, the connection between the development of the
boundary layer and the separated shear layer and the noise generation is not yet
fully understood. Besides, the detailed noise sources of the cylinder flow in the
critical regime have not yet been studied. Compressible numerical simulation can
provide both time-resolved and mean observations of the whole flow field to
investigate the link between the boundary layer and the shear layer, and also the
analysis of the noise sources. These fundamental results will then be beneficial for
the understanding of both aerodynamics and aeroacoustics phenomena on more complex
configurations such as the flow around landing gears. As already pointed out by
Hutcheson and Brooks,^{5} the
sub-critical, critical and turbulent states of flow are also the most relevant for
landing gear flow applications because of the Reynolds number range involved on its
components (struts, cables, axles, and wheels). The selected flow configuration has
a Reynolds number, *Re*_{D} = 2.43 ×
10^{5} in the critical regime, as found on the main leg of the
simplified LAGOON landing gear.^{29–31} The chosen test condition can then be considered as
a key to real world industrial applications.^{3,32,33}

The complex flow features around a circular cylinder in the critical regime are thus investigated by a wall-resolved compressible Large Eddy Simulation (LES) for the first time. The numerical method and parameters of this simulation are described in the next section. The flow characteristics at the selected operating condition are investigated in the following section, and compared with previous experimental and numerical studies. The detailed evolution of the boundary layer and the shear layer are then discussed. Finally, the last section considers the analysis of the noise and its sources.

## II. NUMERICAL PARAMETERS

### A. Numerical methods

*ρ*,

*ρ*

**u**,

*ρE*), are written in the conservative form as:

**w**contains conservative variables (

*ρ*,

*ρ*

**u**,

*ρE*)

^{T}and the flux tensor

**F**can be divided into two parts:

**F**

^{c}is the convective flux depending on

**w**,

**F**

^{v}is the viscous flux depending on both

**w**and its gradient ∇

**w**. The contributions of the unresolved turbulent scales are included in the viscous flux through the addition of the so called turbulent viscosity or Sub-Grid Scale (SGS) viscosity

*ν*

_{t}.

^{34}The present work uses the Wall-Adapting Local-Eddy model

^{35}(WALE) to model the turbulent viscosity by

*S*

_{ij}denotes the tensor of the resolved strain rate,

*C*

_{w}= 0.5 the model constant, and Δ the characteristic filter length usually set to be the cube-root of the cell volume. $sijd$ reads

*g*

_{ij}denotes the resolved velocity gradient

The governing equations are solved by the unstructured compressible LES solver
AVBP,^{36} using the two-step
Taylor-Galerkin finite element scheme TTG4A, which is 4th order in time and 3rd
order in space and presents very low dispersion and dissipation.^{37–39} Such a numerical
methodology has been extensively and successfully used in several previous
aerodynamic and aeroacoustic applications.^{40–46}

### B. Simulation setup

The diameter of the cylinder is *D* = 0.05 m, with a uniform
inflow of *u*_{∞} = 72 m/s, which leads to a Reynolds
number *Re*_{D} = 2.43 × 10^{5} at *T* = 293 K. The reference pressure equals to *p*_{∞} = 98900 Pa. Freestream density and dynamic
viscosity equal to 1.171 kg/m^{3} and 1.8 × 10^{−5} kg/ms
respectively. The computational domain shown in Fig. 1 extends from − 18*D* to 26*D* in the streamwise direction and − 20*D* to 20*D* in the crosswise direction. The spanwise width is 3.5*D*, which
should be enough according to the experimental data of Schlinker *et
al.*^{47,48} A
hybrid prisms/tetrahedral unstructured grid is used here similarly as Lehmkuhl *et al.*^{27} The first prismatic layer of the wall resolved LES (WR-LES) is at least ten
times smaller in the wall normal direction compared to previous wall-modeled
simulations of the similar flow conditions,^{49} and 30 prismatic layers are generated at the wall.
In Fig. 2, the grids around the cylinder of
the refined mesh are displayed by a crinkle cut view in the mid-span plane. The
resulting wall resolution of the wall-resolved LES yields a dimensionless wall
distance *y*^{+} mostly near and below 1 with a maximum
at 2 as Lehmkuhl *et al.*^{27} Similar to the observations reported by Cheng *et al.*,^{28} the results of the wall-resolved simulation with similar Reynolds number without
additional perturbation are in the sub-critical state. Therefore, two prismatic
layers at *θ* = 72° and 288° (total height equals to
0.04%*D*) are removed from the mesh above the cylinder to
form two cuboid tripping geometries to introduce disturbances as in the
wall-resolved LES of Cheng *et al.*,^{28} as shown in Fig. 2. This tripping method stems from the recently proposed
methodology for realistic jet noise applications.^{46} The final wall-resolved mesh has approximatively
160 million cells, which corresponds to about 42 million nodes. For the present
wall resolved case, the streamwise and spanwise discretization yield
dimensionless grid spacings *x*^{+} and *z*^{+} of about 30, which enables a proper
turbulence development.^{50} No-slip adiabatic boundary conditions are applied on the cylinder surface, and
periodic boundary conditions are imposed in the spanwise direction. Uniform
velocity is imposed at the inlet, and Navier-Stokes characteristic
non-reflective boundary conditions^{51} are used in the far-field, which reduce acoustic
reflections.

The time step is fixed to Δ*t* = 0.2 × 10^{−7} s to ensure
an optimized Courant-Friedrichs-Lewy number equal to 0.7 for the TTG4A scheme.
The numerical simulation is initialized with homogeneous flow on the coarse
mesh. The converged flow results from the coarse mesh is interpolated on the
fine mesh as initial solution. After a transient time of 36 ms, a statistically
converged flow is achieved and the initial transient is completely washed out of
the simulation domain. Statistics are then collected for a physical time of 46
ms, which equals to 65 cylinder-diameter flow-through times or 22 shedding
cycles. This physical time also corresponds to the acquisition time for
wall-pressure fluctuations as well as for density, velocity, and pressure
signals extracted from numerical probes placed in the numerical domain. The
computational cost corresponding to the simulated physical time is 1.2 M CPU
hours on the Niagara Cluster of Compute Canada using Intel Skylake cores.

## III. RESULTS

The drag coefficient of the present simulation which results from the wall-pressure
and the skin-friction distributions is first compared with several measurements
(from Fage and Falkner,^{10} Spitzer,^{52} Achenbach and
Heinecke,^{13} Schewe,^{14} Bearman,^{9} Fujita *et al.*,^{15} Hutcheson and Brooks^{5} and Vaz *et
al.*^{53}) and LES
results from Lehmkuhl *et al.*^{27} and Cheng *et al.*^{28} in Fig. 3(a). Within the critical regime, the experimental data show an important
scatter due to the aforementioned sensitivity to the flow conditions. The drag
coefficient of the present tripped LES is located in the critical regime as
expected, within the experimental range. Similarly, the frequency of the vortex
shedding varies strongly within the critical regime and the present LES result
(*St*_{VK} = 0.33) is close to
Achenbach’s experimental data, and to Lehmkulh *et al.* LES result
but at a higher Reynolds number (*Re*_{D} =
3.8 × 10^{5}).

An overall representation of the mean and fluctuating pressure field is then shown in Fig. 4 in dimensionless forms as pressure
coefficients defined as $Cp=(p\u2212p\u221e)/0.5\rho \u221eu\u221e2$ and $Cp,rms=prms/0.5\rho \u221eu\u221e2$ where $prms=(p\xaf2\u2212p2\xaf)0.5$ ($\u2022\xaf$ represents a time averaged quantity) respectively. An almost symmetrical mean static
pressure distribution is obtained with a stagnation point near *θ* =
0° (lift coefficient close to zero). The root-mean-square (rms) of the pressure
fluctuations are slightly higher on the upper half of the cylinder and two peaks can
be seen around the cylinder surface.

### A. Wall pressure and skin friction distribution

The above distribution of pressure around the bluff body (Fig. 4) is not only an important parameter to yield the
above drag and the lift coefficients (Fig. 3), but also the longitudinal pressure gradient is known to affect
the development of both laminar and turbulent boundary layers.^{55,56} The mean wall-pressure
coefficient from the present LES is shown in Fig. 5(a), and is compared with the measurements of Fage and Falkner^{10} (*Re*_{D} = 2.12 ×
10^{5}), Achenbach^{54} (*Re*_{D} =
2.6 × 10^{5}) and the LES of Cheng *et al.*^{28} (*Re*_{D} = 2.6 × 10^{5} and 3.5 × 10^{5}, referred as Cheng-1 and Cheng-2 respectively). The
mean wall pressure of present simulation shows a quasi-symmetric distribution on
the top and bottom sides of the cylinder. It agrees quite well with all the
other data and the plateau between 90° and 100° (also present in Fage’s data) is
due to a laminar separation bubble (LSB). This plateau caused by the LSB is also
observed by Lehmkuhl *et al.*^{27} within the critical region, on both sides for
higher Reynolds numbers (*Re*_{D} = 5.3
× 10^{5} and 6.5 × 10^{5}), while for *Re*_{D} = 2.5 × 10^{5},
only one side shows the LSB and on the other side, the flow remains in the
sub-critical regime. In this asymmetric case, the pressure minimum occurs near
289.5° on the sub-critical regime side, and on the other side of the cylinder
presents a deep depression which reaches its minimum at about $\theta Cp,min=82\u25cb$.
A similar position of the minimum pressure is found in the intermediate Reynolds
number computed by Lehmkuhl *et al.*^{27} at *Re*_{D} = 3.8 × 10^{5},
which has two slightly asymmetric LSB as observed in the present simulation. In
the results of Cheng *et al.*^{28} with *Re*_{D} = 3.5 × 10^{5}, the
minimum pressure occurs at $\theta Cp,min=88\u25cb$,
which is closer to the minimum of the potential flow pressure coefficient, *C*_{p} = 1 − 4 sin *θ*, at *θ* = 90°. Separation then occurs
further downstream in this higher Reynolds number case. The corresponding
wall-pressure coefficient for the root mean square (rms) of pressure shown in Fig. 4(b), defined as $Cp,rms=prms/(0.5\rho \u221eu\u221e2)$,
is depicted in Fig. 5(b). The distribution
is slightly asymmetric with different peak values on both sides of the cylinder.
The wall-pressure fluctuations which can be seen as the noise sources smoothly
increase around the cylinder from the stagnation point until 90° where the
laminar separation occurs, then slightly decrease in the separation region up to
100°, and again strongly increase close to the reattachment point of the
recirculation bubble at about 110° where the wall-pressure fluctuations show a
peak value. This sudden increase of rms pressure is caused by the transition to
turbulence of the detached shear layer of the recirculation bubble.

The skin-friction coefficient ($Cf=\tau w/0.5\rho \u221eu\u221e2$)
is another important parameter that indicates the state of the boundary layer.
Indeed, the skin-friction coefficient of a laminar region on a flat plate
differs from a turbulent one. In cylinder flows, this parameter has different
shapes for different regimes. For example, in the sub-critical regime at *Re*_{D} = 10^{5}, the
skin-friction has a maximum at about *θ* = 50°, and drops rapidly
downstream and vanishes at about *θ* = 80°, where the boundary
layer separates.^{54} In the
critical regime, the maximum value rather appears near *θ* = 60°
and separation occurs close to 90°. The drop of the skin-friction and the
laminar separation in the sub-critical and critical regimes are caused by the
adverse pressure gradient.^{8}^{,} Fig. 6 compares the skin-friction of the
present LES with that of Achenbach’s measurements at *Re*_{D} = 2.6 × 10^{5} and Cheng’s LES results. The same increasing skin-friction as in the measurement
is obtained from the stagnation point up to about *θ* = 60°, and
the experimental rapid drop-off downstream of 60° is nicely recovered. The
skin-friction of the present LES also closely follows Cheng’s LES results up to
70°. The disturbance between 70° and 80° is caused by the tripping line, which
should also be present in Cheng’s results. The skin-friction vanishes in the
present LES at 90° where the boundary layer separates in a laminar state, which
is again close to Achenbach’s measurement. In Cheng’s LES with *Re*_{D} = 2.6 × 10^{5} (Cheng-1), this position is slightly downstream around 100°. After the
separation point, from about 100° to 110°, a negative peak appears which
indicates the presence of a recirculation bubble, in the same angular range as
found by Lehmkuhl *et al.*^{27} Advancing downstream, the free shear layer becomes
turbulent and reattaches at about 114°. There is no positive friction peak
created by the weak reattachment, which differs from the higher Reynolds number
cases of the critical regime as in Cheng’s LES at *Re*_{D} = 3.5 × 10^{5} (Cheng-2), or in the LES from Lehmkuhl *et al.*^{27} (at *Re*_{D} = 5.3 × 10^{5} and 6.5 × 10^{5}) and the measurements from Achenbach^{8} at *Re*_{D} = 4 × 10^{5}.
Indeed, Achenbach’s measurements have shown that the skin friction peak due to
the turbulent reattachment of the boundary layer grows with increasing Reynolds
number within the critical flow range. Fig. 7 shows an instantaneous (top) and the mean (bottom) skin-friction
coefficient from 90° to 180° along the cylinder span, which confirms the
conclusions drawn from Fig. 6. Besides, the
laminar separation and transition of the boundary layer can be clearly located
in the instantaneous skin friction contours, which can be considered as the
footprint of the vortices in the boundary layer. More details about the
separation, recirculation and reattachment of the present LES are discussed in
the next section.

### B. Boundary layer evolution

Having examined the wall results, the boundary-layer evolution is presented in
this section. Boundary-layer behavior is the key factor that determines the flow
properties around a cylinder. In Figs. 8,
the time and spanwise averaged flow fields near the separation and the
recirculation zone of the boundary layer are illustrated by different variables,
together with the streamlines. The tangential velocity *u*_{θ} in Fig. 8(a) confirms that the boundary layer separates (zero
and reverse velocity) at about 90°, as also shown by the velocity profiles in Figs. 9. The recirculation bubble
mentioned in the previous section from the negative skin friction peak is
clearly seen. In the region between 90° and 100°, the recirculation is weak and
consequently the skin friction remains close to zero. The reattachment occurs at
about 114°. In the reattached region, the tangential velocity near the wall
stays close to zero which is confirmed by the tangential velocity profiles in Figs. 9. The velocity gradient in the
reattachment region (*θ* = 115° and 120°) is weak, which is
consistent with the above quasi-zero skin friction. The iso-contours of mean
static pressure in Fig. 8(b) confirms the
plateau on the *C*_{p} curve between 90°
and 105°. After 105°, the pressure increases rapidly until the final separation
at about *θ*_{s} = 130° as found by
Achenbach.^{54} The
transition to turbulence of the shear layer is illustrated by the maximum of
velocity and pressure fluctuations in Figs. 8 (c–f), immediately after 100°.

The cylinder potential effect prevents an outer uniform flow and the definition
of the boundary layer thicknesses requires special care. Therefore, the boundary
layer velocity profiles are first normalized by the free-stream velocity *u*_{∞} and wall normal distance is scaled by the
diameter of the cylinder, as plotted in Fig. 9. Before 80°, the pressure gradient is favorable, the boundary flow
accelerates and the maximum *u*_{θ} increases with *θ*. From 80° until the final separation, the
maximum tangential velocity decreases from about 1.8 *u*_{∞} to *u*_{∞}.

Secondly, the radial location of the first maximum value of the tangential
velocity *u*_{θ} is considered as the
outer edge of the boundary layer, as in Cheng *et al.*^{28} This maximum tangential
velocity is denoted as *u*_{e}, which is
plotted in Fig. 10(a) and compared with
the potential flow velocity and results of Cheng *et al.*^{28} at *Re*_{D} = 3.5 × 10^{5}.
The velocity *u*_{e} of Cheng *et
al.*^{28} follows the
potential flow result up to the separation bubble while in the present LES at
lower *Re*_{D}, *u*_{e} departs from the
potential flow gradually and shows a lower maximum value at *θ* =
80°. Cheng’s higher value is consistent with the lower pressure coefficient in Fig. 5(a). As expected, the present
levels are also larger than those found in the sub-critical regime, which
according to Cheng *et al.* all collapse on the curve *u*_{e} = 1.5
sin(1.25*θ*). Finally, similar plateau as in Cheng’s result
is observed in the separation bubble region. The velocity profiles normalized by
these scales are displayed in Fig. 11. The
streamwise pressure gradient becomes positive after 80°, which thickens the
boundary layer significantly, and lifts the mean velocity profiles off the wall
till the separation point at 90° (Fig. 10(b)). Polhausen’s profiles are also shown for both an attached and
a separated cases. They are shown to agree reasonably well with the present
results for both flow conditions. The shape factor, *H*, which is
the ratio between the displacement thickness and the momentum thickness is
plotted in Fig. 12(a). It shows a modest
increasing trend from the stagnation point around *θ* = 0° to the
minimum pressure position at $\theta Cp,min=82\u25cb$.
Beyond this position, a rapid increase of the shape factor is observed. The same
conclusion can be drawn from the results of Cheng *et al.*^{28} at *Re*_{D} = 3.5 × 10^{5},
where the minimum wall pressure is located close to *θ* = 90° and
the shape factor increases rapidly after this position, when the adverse
pressure gradient starts. In the reattached region (114° to 126°) of the present
LES, *H* shows a value close to 2, which is higher than the
typical values of fully developed turbulent boundary layer (TBL) between 1.4 and
1.8.^{56} This is again
caused by the weak reattachment. In Cheng *et al.*^{28}’s results, where a strong
reattachment occurs, *H* takes a value of about 1.44, which is
similar to the one of a flat plate TBL. In both simulations, a second increase
of *H* is observed after the reattachment where the pressure
gradient is adverse.

*θ*.

^{8}$Re\delta *$ is the Reynolds number based on the displacement thickness as:

The rms velocities and turbulent kinetic energy (TKE) between 80° and 130° are plotted in Figs. 13. At 80°, the rms velocities and the TKE are small. From the trip, the TKE grows to about 1.3% at the separation point at 90°, and 3% at 95°. A rapid increase can be observed from 95° to 105°, which indicates the transition to turbulence of the shear layer. The maximum TKE is found around 110°. After this angle, the peak value of TKE gradually decreases which is due to the adverse pressure gradient.

### C. Wall and shear-layer pressure spectrum

The PSD of the wall-pressure fluctuations at different angular positions are
plotted as a function of the Strouhal number *St* based on *D* and *u*_{∞} in Fig. 14. These spectra have been averaged in
the spanwise direction. All spectra show a quasi-tonal peak at *St*_{VK} = 0.33, which is the
fundamental oscillation frequency of the vortex shedding. From 60° to 105°, the
fluctuation levels in the middle and high frequency range increase. Above 110°,
these levels decrease.

In addition to the vortex shedding Strouhal number, the wall-pressure spectra can
be considered as the footprint of the instability of the shear layer, since the
transition occurs shortly after the separation where the shear layer remains
close to the cylinder surface. It is known that Kelvin-Helmholtz instabilities
in the shear-layer play a key role in the transition to turbulence. These
instabilities lead to the formation of small scale vortices which eventually
grow up and feed the large scale von Kàrmàn vortex street.^{27} The humps in the pressure PSD curves at 100°
and 105° at *St* between 20 and 30 (the peak frequency denoted as *St*_{KH}) are indeed caused by the
Kelvin-Helmholtz instabilities. At 110°, the shear layer becomes fully turbulent
and the KH instability hump merges with the turbulent background. These
instabilities have been studied back to Bloor,^{57} Wei and Smith^{58} and Kourta *et al.*^{59} The latter suggest a
dependency of the ratio *St*_{KH}/*St*_{VK} on the Reynolds number proportional to $ReD0.5\u2248493$,
whereas Prasad and Williamson^{60} suggest the following expression: $0.0235\xd7ReD0.67\u224895$.
The present simulation has *St*_{KH}/*St*_{VK} ≈ 84, which is close to these experimental based empirical values. On the curves
at 80°, 90° and 95°, some discrete peaks can be observed which are the initial
instabilities in the boundary layer and shear layer. Note that no direct
measurement of these instabilities has been reported so far for *Re*_{D} > 10^{5}.

Probes have been placed in the shear layer from 90° to 120°, as shown in Fig. 8(f). Generally, the corresponding
spectra of the crosswise velocity shown in Fig. 15 have similar forms as that of the wall pressure. However, the
peaks in the KH instability range from 100° to 107.5° are now comparable or
higher than the level near *St*_{VK}.
The first harmonic of the KH hump is also predicted for the positions from 95°
to 102.5°.

Figs. 16(a) and 16(b) show the flow field of the upper shear layer by an iso-surface of the Q-criterion, which is colored by the streamwise velocity. The transition of the separated boundary layer is characterized by the Kelvin-Helmholtz instabilities mainly in the form of rollers stretched in the spanwise direction. Below these structures, the presence of negative spanwise velocity confirms the existence of the LSB. These initial instabilities develop rapidly into small hairpins. The size of these hairpins increases quickly firstly due to the adverse pressure gradient shortly downstream and then caused by the vortex pairing downstream of the cylinder (after 0.5D).

### D. Noise and sources

Flow results, especially the evolution of the boundary layer and shear layer,
have been intensively detailed so far, and compared with previous incompressible
simulations. We now focus on the analysis of the noise and its sources entailed
by the present compressible LES. Fig. 17 shows the iso-surface of Q-criterion colored by the streamwise velocity with the
dilatation in the background. The latter visualizes the sound field around the
cylinder. Large low-frequency lobes are first clearly visible, which corresponds
to the dominant dipolar vortex shedding tone. The dilation field then shows
high-frequency waves mainly centered near 100° where the boundary layer is
transitioning to turbulence. Some wave are noticeably radiated from the near
shear layer around 1.5D (shown by the white arrow). To investigate the noise and
its source, both direct noise computation and Ffowcs Williams and Hawking’s
(FWH) analogy have been considered for the prediction of the far-field
acoustics. The latter and its implementation in the in-house solver sherFWH, has
been detailed and validated in Fosso Pouangué *et al.*^{40} and Salas and Moreau^{61} for instance. Snapshots of
the unsteady simulation have been recorded in the mid-span plane of the
numerical domain, on the cylinder surface (termed as solid surface) and on a
surface surrounding the cylinder and its wake (termed as porous surface as shown
by the dashed-line in Fig. 1) every 1000
time-steps and 100 time-steps, resulting in sampling frequencies of *St* = 34.7 (50 000 Hz) and of *St* = 347 (500
000 Hz). Modes up to *St* = 17 and to *St* = 173
can therefore be obtained respectively. The physical time length of these
unsteady signals are given in Table I.

. | Sampling frequency . | |
---|---|---|

Signal length . | St =
34.7
. | St =
347
. |

Solid surface | 46 ms | 32 ms |

Porous surface | 46 ms | 32 ms |

Mid-span plane (z=0) | 32 ms | 16 ms |

. | Sampling frequency . | |
---|---|---|

Signal length . | St =
34.7
. | St =
347
. |

Solid surface | 46 ms | 32 ms |

Porous surface | 46 ms | 32 ms |

Mid-span plane (z=0) | 32 ms | 16 ms |

The PSD of the far-field acoustic pressure from both direct simulation and FWH
analogy are shown in Fig. 18. The direct
simulation results are extracted at 10D above the cylinder (*θ* =
90°) and have been scaled to 50D, which is the observer position used for the
solid and porous FWH analogies. The solid and porous calculations show similar
results around *St*_{VK}. The direct
computation shows slightly higher levels at *St*_{VK} and the broadband
level in the mid-frequency range agrees well with the porous surface results.
The drop at *St* = 4 in the direct simulation result is caused by
the grid cut-off. In the mid-frequency range, both the direct and porous sound
levels are higher than the solid ones because of additional noise sources in the
cylinder wake. At high frequencies, a broadband hump around *St* = 20 is shown in both spectra, which is caused by the KH instability and the
resulting vortex pairing. Beyond the frequencies of the KH instability, the
noise levels drop quickly. Even though these additional noise sources appear,
the vortex shedding yielding the aeolian tone at *St* = 0.33
remains the dominant noise source in this critical regime.

The snapshots of the flow field in the mid-span plane are considered to identify
the noise sources. Fourier transforms have been carried out on the time sequence
of pressure, crosswise velocity and density at each grid point. The frequency
signals are then filtered around *St* = 0.33
(*St*_{VK}), 0.66
(2*St*_{VK}), 3.0 (turbulent
broadband) and 26 (within the KH instabilities). An inverse Fourier transform is
performed on these filtered signals to reconstruct the temporal signals. Fig. 19 shows a snapshot of each of these
modes. Despite the high intensity of the pressure fluctuations in the shear
layer and in the wake, this method is able to clearly capture the propagative
sound pressure. In Fig. 19(a), the mode at *St* = *St*_{VK} behaves as a clear lift dipole centered on the final separation position, mostly
propagating in the crosswise direction (at 90° and 270°). As shown in the zoom
view in Fig. 19(a), the wave front is born
behind the cylinder (solid arrow), and then diffracted by the cylinder into a
wave with opposite phase and lobes slightly inclined upstream (dashed
arrow).

In Fig. 19(b), for the mode
2*St*_{VK}, the wavelength decreases
to about 7D, and most of the noise radiation occurs in the streamwise direction
(drag dipole). The origin of the propagating waves for this mode is slightly
shifted downstream in the shear layer at about 1.5D from the center of the
cylinder. This explains the higher sound level in the middle frequency range
using porous surface compared with the solid surface in Fig. 18. Fig. 19(c) shows the mode at *St* = 3 in the middle frequency range. The
waves for this mode have similar amplitude to the mode
2*St*_{VK} while their origin is
shifted further downstream at about 2D. Fig. 19(d) shows the mode at *St* = 26. At this frequency,
the wavelength equals roughly 0.15D. The waves are centered at the LSB region on
both sides of the cylinder.

Some correlations between the far-field acoustic pressure (at 10D above the
cylinder) and the near-field pressure fluctuations in the mid-span plane have
been performed, as was done experimentally by Oguma *et al.*^{18} in the sub-critical regime
(*Re*_{D} = 40000). They measured
the far-field acoustic with microphones and the near-field velocity field using
Particle Image Velocimetry (PIV). The near-field pressure was reconstructed
using the Poisson equation, which limits this method to incompressible flows.
They showed that the peaks of the correlation magnitude were near the separation
points (near 80°) and in the near wake. Fig. 20(a) shows the correlation of the far-field and near-field around
the cylinder from the present simulation. Highest peaks are found on the
cylinder surface around 107.5° and slightly lower peak levels are located in the
near-field shear layer from 1.0D to 1.8D. These features are very similar to the
contours of pressure fluctuations in Fig. 4(b). Compared with the sub-critical regime results of Oguma *et al.*,^{18} the peak spot near the separation of the critical regime is located closer to
the cylinder surface. The correlation between the far-field and the filtered
near-field pressure signal around *St*_{VK} is shown in Fig. 20(b). The patterns are similar to the
one using the complete signal however higher correlation coefficients are
observed using the filtered signal at *St*_{VK}. The distribution of
the correlation coefficients on the cylinder surface is plotted in Fig. 20(c). The correlation increases
downstream along the cylinder surface up to the separation point near 90° where
a drop occurs caused by the LSB. A clear maximum is observed at 107.5° where the
negative peak of the skin-friction coefficient is located and the transition to
turbulence of the shear layer occurs.

## IV. CONCLUSIONS

A compressible wall-resolved LES of the flow around a circular cylinder in the
critical regime (*Re*_{D} = 2.43 ×
10^{5}) has been achieved for the first time using the high-order
unstructured solver AVBP. As expected, the flow in this critical regime yields a
significant reduced drag compared to the sub-critical cases. The obtained vortex
shedding Strouhal number *St*_{VK} = 0.33 is
typical of the critical regime, and lies within the experimental and numerical data
range. The mean pressure coefficient and mean skin friction show good agreement with
both the previous measurements and incompressible LES references. For instance, the
boundary layer displacement thickness and shape factor compare fairly well with
Cheng *et al.*^{28}’s
LES up to the separation using the maximum tangential velocity as boundary layer
edge velocity. The latter choice overcomes the difficulty of determining *u*_{95} and *u*_{99}, and
recovers the classical boundary-layer mean velocity and TKE profiles. The boundary
layer profiles reveal more details about the flow: strong accelerated profiles are
observed up to *θ* = 80° where the static pressure near the wall is
the minimum. Above this position, the flow encounters an adverse pressure gradient
and some instabilities appear, which are shown by the wall-pressure spectra.
However, up to 90°, as shown by the profiles of the TKE for instance, the boundary
layer remains rather in a laminar state: the maximum turbulent intensity is less
than 0.4%. After 90°, the boundary layer separates on both sides of the cylinder. At
the beginning of the flow separation from 90° to 100°, the wall shear stress is
nearly zero. At 95° and 100°, the TKE reaches 1% and 2%, and at 105°, this flow
parameter reaches 5%, and typical TKE levels of fully turbulent boundary layer are
observed from 110° onward, which indicates that the transition to turbulence is
complete. The laminar separation and transition in the shear layer are typical
features of the critical regime, and the slight flow asymmetry with two LSB is also
consistent with the previous incompressible LES results by Lehmkuhl *et
al.*^{27} Yet, the
present laminar recirculation bubble with very weak reattachment (quasi-zero shear
stress in the reattachment zone), to the authors’ knowledge, is observed and
reported for the first time both experimentally and numerically.

The wall-pressure spectra confirm the different states of the boundary layer around
the cylinder, and additional details have been revealed as well. The footprint of
the transition to turbulence triggered by the shear-layer instability close to the
cylinder wall has been captured. The *Q*-criterion clearly shows that
Kelvin-Helmholtz instabilities (rollers) are the main cause of this transition. The
corresponding Strouhal number, *St*_{KH} ≈
27, is consistent with Prasad and Williamson^{60}’s curve fit of several experimental data at lower
Reynolds number *Re*_{D}. To our knowledge,
this is another original contribution as no direct measurement of these
instabilities have been reported so far for *Re*_{D} > 10^{5}, and
confirms the previous incompressible LES results of Lehmkuhl *et
al.*^{27}

Finally, this compressible LES provides the first insight into the noise sources and associated sound generated by a circular cylinder in the critical regime. The noise sources are qualitatively identified by the filtered time signal of the instantaneous pressure field at different frequencies. The major sources of the far-field sound are also identified by the cross-correlation between the near/far-field pressure fluctuations: the region above the cylinder close to the end of the transition of the shear layer and the shear layer in the near wake are significant additional broadband far-field noise sources in the critical regime. Yet, the dominant noise source remains the vortex shedding, which yields a tone broader and shallower than in previously reported sub-critical cases.

## ACKNOWLEDGMENTS

The authors acknowledge Compute Canada and Calcul Québec for providing technical support and necessary computational resources for this research. The first author’s PhD grant is funded by AIRBUS through the aeroacoustic industrial Chair at Université de Sherbrooke.

## REFERENCES

*Flow around circular cylinder: Volume 1: Fundamentals*

^{3}to 5 × 10

^{6}

*Re*

_{D}= 3900

*Re*

_{D}from 3.9 × 10

^{5}to 8.5 × 10

^{5}: A skin-friction perspective

*Large Eddy Simulation for incompressible flows: An introduction*

*Finite element methods for flow problems*

*Re*= 5 × 10

^{6}

*Boundary-layer theory*