Luca Sciacovelli

Xavier Gloerfelt

Donatella Passiatore
email: donatella.passiatore@ensam.eu

Paola Cinnella
email: paola.cinnella@ensam.eu

Francesco Grasso
email: francesco.grasso@lecnam.net

Numerical Investigation of High-Speed Turbulent Boundary Layers of Dense Gases

Keywords: High-speed flow, Turbulent boundary layer, Dense gas Affiliations

High-speed turbulent boundary layers of a dense gas (PP11) and a perfect gas (air) over flat plates are investigated by means of direct numerical simulations and large eddy simulations. The thermodynamic conditions of the incoming flow are chosen to highlight dense gas effects, and laminar-to-turbulent transition is triggered by suction and blowing. In the paper, the behavior of the fully developed turbulent flow region is investigated. Due to the low characteristic Eckert number of dense gas flows ( Ec = U 2 ∞ ∕c p,∞ T ∞ ), the mean velocity profiles are largely insensitive to the Mach number and very close to the incompressible case even at high speeds. Second-order velocity statistics are also weakly affected by the flow Mach number and the velocity spectra are characterized by a secondary peak in the outer region of the boundary layer because of the higher local friction Reynolds number. Despite the incompressible-like velocity and Reynolds-stress profiles, the strongly nonideal thermodynamic and transport-property behavior of the dense gas results in unconventional distributions of the fluctuating thermo-physical quantities. Specifically, density and viscosity fluctuations reach a peak close to the wall, instead of vanishing as in perfect gas flows. Additionally, dense gas boundary layers exhibit higher values of the fluctuating Mach number and velocity divergence and a larger dilatational-to-solenoidal dissipation ratio in the near-wall region, which represents a major deviation from high-Mach-number perfect gas boundary layers. Other significant deviations are represented by the more symmetric probability distributions of fluctuating quantities such as the density and velocity divergence, due to the more balanced occurrence of strong expansion and compression events.

Introduction

High-speed flows of gases with complex thermodynamic behavior (often referred to as "real gases") are being paid growing interest from the scientific community due to the manifold applications in aerospace and power generation systems. In this work we focus more particularly on so-called dense gas flows found in several engineering applications, ranging from energy production to high-Reynolds wind tunnels. A typical application is represented by energy conversion cycles like Organic Rankine Cycles (ORC) [see [START_REF] Colonna | Organic Rankine cycle power systems: from the concept to current technology, applications, and an outlook to the future[END_REF] for a review] and heat pumps [START_REF] Zamfirescu | Performance investigation of high-temperature heat pumps with various BZT working fluids[END_REF], but dense gases are also of interest for other applications, e.g., high-Reynolds wind tunnels [START_REF] Bodenschatz | Variable density turbulence tunnel facility[END_REF][START_REF] Berry | Nasa langley experimental aerothermodynamic contributions to slender and winged hypersonic vehicles[END_REF][START_REF] Corliss | Heavy gas conversion of the nasa langley transonic dynamics tunnel[END_REF]. Dense gases are single-phase fluids with complex molecules, operating at pressure and temperature conditions of the same order of their thermodynamic critical point. Examples of fluids with dense gas behavior are heavy fluorocarbons and hydrocarbons, refrigerants and siloxanes [START_REF] Cramer | Negative nonlinearity in selected fluorocarbons[END_REF][START_REF] Colonna | Dense gas thermodynamic properties of single and multicomponent fluids for fluid dynamics simulations[END_REF]. The dynamic behavior of such fluids is characterized by means of the fundamental derivative of gas dynamics [START_REF] Thompson | A fundamental derivative in gasdynamics[END_REF]

, = 1 + c c | | |s
(where c is the speed of sound, the density and s the specific entropy), measuring variations of the sound speed under reversible density perturbations. In gases with sufficiently high molecular complexity (defined as c v,∞ (T cr )∕R , with R the gas constant and c v,∞ (T cr ) the isochoric heat capacity in the dilute gas limit, measured at the fluid critical temperature T cr ), is less than one over an extended range of thermodynamic conditions in the vapor phase. This leads to a reversed variation of the speed of sound in isentropic transformation with respect to perfect gases: c decreases when the gas is compressed and increases when the gas is expanded. Even more dramatic deviations from the perfect gas behavior are expected for gases of the so-called Bethe-Zel'dovich-Thompson (BZT) type, for which a region of negative values of the fundamental derivative (named the "inversion zone") is predicted to appear in the vicinity of the liquid/vapor coexistence curve, on the vapor side. In such conditions, nonclassical waves like expansion shocks become physically admissible [see, e.g., [START_REF] Cinnella | Inviscid and viscous aerodynamics of dense gases[END_REF] and the references therein]. Although a number of numerical studies have investigated in the past dense gas dynamics, most of them are based on simplified models neglecting viscous effects (potential flow or Euler equations) or relying on the Reynolds-Averaged Navier-Stokes equations supplemented with a standard turbulence model (i.e., without any specific calibration or compressibility correction for dense gases). On the other hand, experimental investigations of dense gas flows are particularly difficult and costly, and only very recently some local experimental measurements of pressure and temperature at selected stations in a dense-gas nozzle have been made available [START_REF] Spinelli | Experimental evidence of non-ideal compressible effects in expanding flow of a high molecular complexity vapor[END_REF], as well as Schlieren photographs [START_REF] Zocca | Experimental observation of oblique shock waves in steady non-ideal flows[END_REF]. Unfortunately, to the authors' best knowledge, no velocity or Reynolds-stress measurements of turbulent dense gas flows have been obtained yet, and the fine details of transitional and turbulent dense gas flows remain largely unknown. In previous research, the present authors have investigated the influence of dense-gas effects in compressible decaying isotropic turbulence [START_REF] Sciacovelli | Dense gas effects in inviscid homogeneous isotropic turbulence[END_REF](Sciacovelli et al. , 2017) ) and channel flow (Sciacovelli et al. 2017) configurations by using Direct Numerical Simulations (DNS). Previous studies [START_REF] Kluwick | Interacting laminar boundary layers of dense gases[END_REF][START_REF] Cinnella | Inviscid and viscous aerodynamics of dense gases[END_REF] show that, due to their high specific heat capacities, dense gases are characterized by a strong decoupling of dynamic and thermal fields for any flow configuration. In the turbulent channel flow, this results in temperature fluctuations sensibly smaller than in air, and to a strong correlation between density and pressure fluctuations, in contrast with perfect gases. The complex thermodynamic behavior of dense gases at thermodynamic conditions corresponding to ≈0 also modifies compressibility effects and flow topology: for instance, in freely decaying dense gas turbulence at sufficiently high turbulent Mach numbers, strong compressions are weakened and strong expansions enhanced. The latter may correspond to sheet-like (rather than tube-like) structures, suggesting the formation of expansion eddy shocklets. These phenomena are not observed in confined wall-bounded configurations like turbulent channel flows, because of the smaller turbulent Mach numbers. For dense-gas channel flows, the most significant differences are related to the peculiar evolution of the temperature-and density-dependent transport properties: the temperature variations being small, viscosity and thermal conductivity essentially follow the density profiles, resulting in a liquid-like behavior. At such conditions, variable property scalings like the van Driest or the Trettel and Larsson scaling [START_REF] Trettel | Mean velocity scaling for compressible wall turbulence with heat transfer[END_REF] were found to collapse adequately the wall-normal profiles even at large bulk Mach numbers.

In this work we report the first-ever DNS/LES (Large-Eddy Simulation) results for dense-gas (DG) turbulent boundary layers at supersonic Mach numbers. The objective is to understand the role of dense-gas effects for wall-bounded, non-confined and spatiallyevolving configurations. For this purpose, a quasi-DNS (a fine wall-resolved implicit LES) is carried out at M = 6 . This value is much higher than those that are typically encoun- tered in practical dense-gas applications [a remarkable exception being the high-Mach, high-Reynolds NASA Langley CF4 wind tunnel, dismantled in 2017, which used a light perfluorocarbon, CF4, as the working fluid up to M = 6 [START_REF] Berry | Nasa langley experimental aerothermodynamic contributions to slender and winged hypersonic vehicles[END_REF]]; never- theless, such severe operating conditions are of theoretical interest for highlighting dense gas effects. We also report wall-resolved LES results for a lower Mach number ( M = 2.25 ), more representative of values encountered in ORC applications, to investigate Mach number effects. In both cases, dense-gas results are compared with perfect-gas (PG) simulations at the same Mach number with the objective to show how the intricate DG behavior affects the turbulent flow statistics and flow topology, and to check the validity of compressibility scalings.

The paper is organized as follows. Details of the governing equations, numerical methods, and computational setup are provided in Sect. 2. The results are discussed in Sect. 3, including the description of first-and second-order statistics, the validity of classical correlations and scalings, and the study of compressibility effects. Lastly, conclusions are drawn in Sect. 4, along with some perspectives for future research.

Simulation Details

Governing Equations

The numerical simulations are based on the single-phase compressible Navier-Stokes equations, solved in a conservative form on Cartesian grids and supplemented by suitable dense-gas constitutive relations. The dense fluid considered in the following simulations is the perfluoro-perhydrophenanthrene, (chemical formula C 14 F 24 ), called hereafter with its commercial name PP11. Its main thermodynamic properties, extracted from [START_REF] Cramer | Negative nonlinearity in selected fluorocarbons[END_REF], are provided in Table 1. The model of [START_REF] Martin | Development of an equation of state for gases[END_REF] is used for the thermal equation of state, which is reasonably accurate for the fluid of interest and requires a minimum amount of experimental information for setting the gas-dependent coefficients. The equation of state for the pressure p involves five virial terms and writes:

with b = v cr [1 -(20.533 -31.883Z cr )∕15] and f i (T) = A i + B i T + C i exp (-kT∕T cr ) ,
where v = 1∕ denotes the specific volume, v cr , T cr and Z cr are the critical specific vol- ume, temperature and compressibility factor, and k = 5.475 . The coefficients A i , B i and C i depend on the critical temperature and pressure, the critical compressibility factor, the Boyle temperature and one point on the vapor pressure curve. Variations of the low-density specific heat with temperature are modeled by means of a power law:

where n is a material-dependent parameter.

In addition to the equations of state, suitable models for the dynamic viscosity and thermal conductivity of dense gases, which depend on both the gas temperature and pressure (or density), have to be specified. The present results are based on the model of [START_REF] Chung | Generalized multiparameter correlation for nonpolar and polar fluid transport properties[END_REF], which incorporates a high-pressure correction term to account for the modification of transport properties values at high pressure (or density) conditions. Such a model has been used in previous works on dense gases (e.g. [START_REF] Cramer | Transonic flows of Bethe-Zel'dovich-Thompson fluids[END_REF][START_REF] Cramer | On the suppression of shock-induced separation in Bethe-Zel'dovich-Thompson fluids[END_REF] and is considered to be a reasonably accurate semi-theoretical model for calculating the viscosity based on the knowledge of a few thermophysical input parameters [see [START_REF] Poling | The Properties of Gases and Liquids[END_REF] for more details], provided that the thermodynamic conditions are far enough from the transcritical region [START_REF] Kawai | A robust and accurate numerical method for transcritical turbulent flows at supercritical pressure with an arbitrary equation of state[END_REF], which is the case for the present study. A full description of the model equations can be found in "Appendix" of ref. Sciacovelli et al. (2017).

For the purpose of comparison, numerical results are also presented for air, modeled as a polytropic perfect gas:

where c v = R∕( -1) is the specific heat at constant volume and the specific heats ratio. At the lowest Mach number considered in the study (M = 2.25), Sutherland's law is used to model the viscosity : with ref = 1.716 × 10 -5 Pa s, T ref = 273.15 K and S 0 = 110.4 K. At the highest Mach number (M = 6), viscosity follows a power law, = ∞ T∕T ∞ 0.7 , in order to allow direct validation with the DNS results of [START_REF] Franko | Breakdown mechanisms and heat transfer overshoot in hypersonic zero pressure gradient boundary layers[END_REF]. In the present dimensional

(1) p = RT (v -b) + 5 ∑ i=2 f i (T) (v -b) i , (2) c v ∞ (T) = c v ∞ (T cr ) T T cr n (3) p = RT and e = c v T, ( 4
) ref = T T ref 3∕2 T ref + S 0 T + S 0

Numerical Method

The inviscid flux derivatives in the governing equations are approximated in space by means of centered tenth-order accurate differences. In order to introduce a minimal amount of numerical dissipation while ensuring computational robustness for compressible flow simulations, the centered scheme is supplemented by a high-order nonlinear artificial viscosity term, similar to the approach of [START_REF] Jameson | Numerical solution of the euler equations by finite volume methods using Runge-Kutta time stepping schemes[END_REF] and [START_REF] Kim | Adaptive nonlinear artificial dissipation model for computational aeroacoustics[END_REF]. In our case, we use a blend of second-and tenth-order derivatives approximated by standard central differences. For a 1D problem and a regular Cartesian grid with constant mesh spacing x (so that x j = j x ), the semi-discrete tenth-order scheme in space writes:

where w is the vector of conservative variables, is the classical difference operator over one cell, (•

) j ∶= (•) j+ 1 2 -(•) j-1 2
, and F j+ 1 2 is the numerical flux at cell interface j + 1 2 :

where f = f (w) is the physical flux, is the cell average operator, (•) j+ 1 2 ∶= 1 2 (•) j+1 + (•) j , and D is the numerical dissipation term: with (A) the spectral radius of the flux Jacobian matrix A = f ∕ w , and with k 2 and k 10 adjustable dissipation coefficients. The activation of the low-order term rests on Jameson's pressure-based shock sensor [START_REF] Jameson | Numerical solution of the euler equations by finite volume methods using Runge-Kutta time stepping schemes[END_REF]) combined with Ducros' sensor [START_REF] Ducros | Large-eddy simulation of the shock/turbulence interaction[END_REF]). The sensor is O(1) in high-divergence regions and tends to zero in vortex dominated regions, which allows the capture of flow discontinuities with minimal damping of the vortical structures inside the boundary layer. Note that, when k 2 = 0 and k 10 = 1 1260 the preceding method becomes equivalent to a ninth-order accurate upwind scheme. For all computations, we set k 2 = 2 and k 10 = 1 1260 , except for Mach 6 dense-gas case where the value of k 10 is multiplied by a factor of three (due to the near- wall high density fluctuations that can undermine the robustness of the simulation). Far from flow discontinuities, the spatial scheme has very low phase and dissipation errors, as discussed in [START_REF] Sciacovelli | Dense gas effects in inviscid homogeneous isotropic turbulence[END_REF]. An assessment of the accuracy of the numerical scheme for freely decaying compressible homogeneous isotropic turbulence can be found in the same reference. The tenth-order dissipation is selective in the wavenumber space, and it affects only wavelengths discretized with less than 6 mesh points. As a consequence, it constitutes a suitable implicit subgrid regularization term for LES simulations (Gloerfelt

(w t ) j + ( F) j x = 0 F j+ 1 2 = I - 1 6 2 + 1 30 4 - 1 140 6 + 1 630 8 f -D j+ 1 2 D j+ 1 2 = (A) j+ 1 2 2 w + 10 9 w j+ 1 2 2j+ 1 2 = k 2 max( j j , j+1 j+1 ), 10j+ 1 2 = max(0, k 10 -2j+ 1 2 ),
and Cinnella 2019), with the capability of seamlessly converging to DNS in smooth flow regions as the grid is refined.

The viscous flux derivatives are approximated by fourth-order accurate central formulae. Time advancement is carried out by means of an explicit four-stage low-storage Runge-Kutta scheme. The non-uniformity of the wall-normal mesh spacing is taken into account by a suitable 1-D coordinate transformation. Near the non-periodic boundaries, the finite-difference stencil for the convective terms is progressively reduced down to the second order, whereas the viscous stress terms are evaluated from the interior points by using fourth-order backward differences.

Computational Setup

The spatial development of zero-pressure-gradient turbulent boundary layers on a flat plate is simulated for both PP11 and air flows. Two configurations have been selected: a hypersonic TBL at M = 6 (HTBL) and a supersonic TBL at M = 2.25 (STBL). The higher value of the Mach number does not correspond to a configuration of practical use, but is of theoretical interest for highlighting dense-gas effects in wall turbulence. The lower Mach number is more representative of flow conditions achievable in densegas turbines. The free-stream conditions are chosen such that the fundamental derivative of gas dynamics is negative outside of the boundary layer and remains lower than unity close to the wall. This allows to investigate strongly non-ideal thermodynamic effects, although such operating conditions are difficult to reproduce in an experimental facility.

In order to avoid uncertainties associated with turbulent inflow conditions like synthetic turbulence [START_REF] Wu | Inflow turbulence generation methods[END_REF] or rescaling/recycling [START_REF] Lund | Generation of turbulent inflow data for spatially-developing boundary layer simulations[END_REF]), which have not been tuned for flows governed by dense gas equations of state, laminar inflow conditions are set at the entrance of the computational domain and transition to turbulence is triggered by means of suction/blowing at the wall.

The inlet of the domain corresponds to the leading edge of the flat plate for HTBL simulations, hence unperturbed free-stream conditions are imposed at the upstream boundary. On the contrary, STBL simulations start with a finite laminar boundary layer thickness to save computational time; in this case, the compressible laminar similarity solution is imposed at the inflow. The inlet Reynolds number is Re * in = 1700 for air and Re * in = 3000 for PP11, with * in the inlet displacement thickness. At the wall, no-slip and isothermal conditions are imposed. The wall temperature value is set equal to the laminar adiabatic wall temperature, calculated by means of a generalized compressible Blasius solver [START_REF] Gloerfelt | Dense-gas effects on compressible boundary-layer stability[END_REF]. This corresponds to a pseudo-adiabatic condition, a common practice in high-speed perfect-gas boundary layer calculations [see, e.g., Table 1 of Ref. [START_REF] Wenzel | DNS of compressible turbulent boundary layers and assessment of data scaling-law quality[END_REF]]. We verified that, for the dense gas cases, the wall heat flux (made non-dimensional with the outer density, velocity and enthalpy) is of the order of 10 -6 or less at the plate wall. As a consequence, the flow is much closer to adiabatic conditions than in perfect gas cases and wall cooling/heating effects are not expected to play any significant role. To achieve a turbulent state, the laminar-to-turbulent transition is triggered using a blowing and suction device acting on a small strip of the wall plate, which consists in a time-and-space-dependent wall-normal velocity disturbance of the form:

where A, and are the amplitude, the pulsation and the spanwise wavenumber of the fundamental mode. The coefficients a 2D , a 3D and a H are used to specify the amplitude of the 2D fundamental mode, 3D fundamental mode and higher-order modes, respectively. Following [START_REF] Franko | Breakdown mechanisms and heat transfer overshoot in hypersonic zero pressure gradient boundary layers[END_REF], the distributions of the forcing in the streamwise (f) and spanwise (g) directions are defined as:

The Gaussian strip is centered at x 0 with a half-width (such that 2 = 0.87

2 U ∞ * forc
, with * forc the displacement thickness at the forcing location x 0 ). A small dissymmetry is introduced in the spanwise direction, centered at z c with width z w (in all cases z w ∕ * forc = 6 ). The forc- ing frequency and spanwise wavenumber, as well as the forcing Reynolds number are determined based on a preparatory linear stability analysis [START_REF] Gloerfelt | Dense-gas effects on compressible boundary-layer stability[END_REF] and are different for each of the cases. The forcing parameters used in each simulation are reported in Table 2. A validation of the forcing procedure against numerical data from the literature for a turbulent boundary layer at M = 6 (Franko and Lele 2013) is reported in "Appendix".

At the upper boundary, one-dimensional nonreflecting characteristic boundary conditions [START_REF] Thompson | Time dependent boundary conditions for hyperbolic systems[END_REF] are applied, whereas a simple extrapolation of variables is used at the supersonic/hypersonic outlet. No sponge zones are introduced at the inlet, outlet, and upper boundaries. Lastly, periodic boundary conditions are set in the spanwise direction. Figure 1 illustrates the computational domain and the boundary conditions. For all cases, the Cartesian grid is evenly spaced in the streamwise (x) and spanwise (z) directions, whereas a geometric stretching is applied in the wall-normal direction (y) with a rate of 1.5%. A summary of the numerical parameters and of the reference conditions is provided in Table 3. The reported boundary layer properties are evaluated at the end of the computational domain and are used to get insight into the resolution of the various simulations. The values of grid sizes in wall units indicate that the perfect-gas simulation at M = 6 can be classified as DNS following the standard terminology (see, e.g., [START_REF] Poggie | Resolution effects in compressible, turbulent boundary layer simulations[END_REF][START_REF] Wenzel | DNS of compressible turbulent boundary layers and assessment of data scaling-law quality[END_REF][START_REF] Zhang | Turbulent/nonturbulent interfaces in high-resolution direct numerical simulation of temporally evolving compressible turbulent boundary layers[END_REF] with the caution, commonly accepted in compressible turbulence studies, that eddy shocklets are captured and not resolved. In the dense gas simulations, the characteristic Reynolds numbers are larger than in perfect gas, due to the high density (two orders of magnitude higher than air). As a consequence, strict DNS resolution requirements are more difficult to achieve. For the DG HTBL, the spatial resolution in wall units is somewhat coarser than in the PG case, and corresponds to a quasi-DNS or a (5) 4, confirm the previous statement: despite similar mesh topologies and sizes, resolutions achieved in the DG simulations differ significantly from the PG case, being more unfavorable at the wall and more favorable at the boundary layer edge. The present results may serve as guideline for mesh resolution requirements in future studies of dense gas boundary layers. Dense gas behavior and its influence on mesh resolution requirements are further discussed in the next Section. A recent study of resolution requirements in compressible DNS of spatially-evolving TBL by [START_REF] Poggie | Resolution effects in compressible, turbulent boundary layer simulations[END_REF] showed that wall-resolved ILES seamlessly converges to DNS as the grid is refined, which gives us confidence in the interpretation of the present results, at least in terms of first and second-order statistics. The PG and DG simulations at M = 2.25 , on the other hand, are coarser and correspond to the criteria of a wall-resolved ILES. Of note, the perfect gas simulations were found to be in good agreement with reference numerical results of Pirozzoli and Bernardini (2011) (see "Appendix").

v w = Af (x)g(z) a 2D cos( t) + a 3D cos( t ± z) + 5 ∑ m=1 a H cos( t ± 2m z) f (x) = e -x-x 0 2 2 and g(z) = 1 + 0.1 e - z-z w z w 2 -e - z+z w z w 2
To provide a glimpse of the computational cost, we mention that the M = 6 simulation for air (924 million points) was run on 7 392 cores and represents approximately 2 millions CPU hours on Intel Xeon 8168 processors. The corresponding simulation for PP11 (1.13 billion points) was split on 16 384 cores and corresponds to approximately 6 millions CPU hours on similar processors.

Numerical Results

In the following, numerical results for dense-gas boundary layers are compared with perfect-gas simulations at the same Mach and Reynolds numbers. In the discussion, the main focus is set on the highest Mach number configuration, which exhibits the largest deviations from the PG behavior and for which DNS resolution has been achieved. Wallresolved LES results at M = 2.25 are also presented with the aim of discussing Mach num- ber effects. x and a classical empirical turbulent correlation C f ,inc = 0.024 Re -1∕4

Global Flow Properties and First-Order Statistics

,inc [START_REF] Smits | Low-Reynolds-number turbulent boundary layers in zero and favorable pressure gradients[END_REF]) are also reported for reference. In all cases, the distribution of C f initially matches the laminar correlation; then it rapidly departs from it and a significant overshoot is visible for PG flow at M = 6 . The transition path and inten- sity of the overshoot obtained for the PG flow at M = 6 are in very good agreement with [START_REF] Franko | Breakdown mechanisms and heat transfer overshoot in hypersonic zero pressure gradient boundary layers[END_REF], as shown in "Appendix". At higher Re, the results are close to the empirical correlation. Similar considerations hold for PG at M = 2.25 . The DG skin fric- tion distribution follows reasonably well the trend of the turbulent correlation but values are lower, because the van Driest II compressibility correction for the skin friction, derived by integrating the boundary layer equations under PG assumptions, is not suitable for the DG. In practice, the van Driest II law has almost no effect on DG results, since T w ∕T ∞ remains close to the unity, as shown later, leading to F c ≈ 1 . An extension of the transformation to dense gases is not straightforward, due to the complex relations between flow properties in the dense gas. Interestingly, the skin friction distributions at M = 2.25 and 6 for the dense gas are very close to each other, with no overshoot at the laminar-to-turbulent transition and almost no influence of the Mach number.

In Fig. 2b, we also report the variation of the incompressible versus compressible momentum-thickness-based Reynolds numbers along the plate for DG and PG boundary (7) ,inc [START_REF] Smits | Low-Reynolds-number turbulent boundary layers in zero and favorable pressure gradients[END_REF]. In panel (c), diamond: correlation Re = 1.13Re 0.843 [START_REF] Schlatter | Assessment of direct numerical simulation data of turbulent boundary layers[END_REF] layers. While the range of Re is reasonably similar for DG and PG flows, its incompressible counterpart ( Re ,inc ) differs substantially. For DG at M = 2.25 the compressible and incompressible Reynolds numbers are very close to each other, due to the negligible friction heating and small viscosity variations in the DG. For the M = 6 case, Re ,inc becomes somewhat larger than Re , unlike PG, for which Re ,inc becomes smaller and smaller as M is increased. Due to the opposite behavior of Re ,inc in dense and perfect gas flows and to the fact that PG flows remain transitional over a large portion of the flat plate, the region of overlapping Re ,inc values in the fully turbulent region for DG and PG is almost void. A similar effect is observed for the friction Reynolds number Re , reported in Fig. 2c along with the incompressible correlation of [START_REF] Schlatter | Assessment of direct numerical simulation data of turbulent boundary layers[END_REF], Re = 1.13Re 0.843 ,inc . The latter matches very well the DG results, especially at the lowest Mach number, for which the boundary layer evolution is closer to the incompressible behavior. For PG cases, Re is much lower due to the significant friction heating, so that they do not share any common value with DG cases in the fully developed turbulent flow region. For this reason, in the following we choose to compare the wall normal profiles of first-and second-order statistics at iso-Re .

C f = w 1 2 ∞ U 2 ∞ , F c = T w ∕T ∞ -1 arcsin 2 , = T w ∕T ∞ -1 √ T w ∕T ∞ (T w ∕T ∞ -1)
Figure 3 shows selected profiles of first-order statistics in the turbulent region at Re = 4000 . We adopt the semi-locally-scaled wall normal coordinate y * = yu * ∕ , based on the semi-local friction velocity u * = √ w ∕ [START_REF] Huang | Compressible turbulent channel flows: DNS results and modelling[END_REF], and focus on thermodynamic and transport properties first. Both for PG and DG, the mean density (panel a) increases from the wall towards the boundary layer edge and, as expected, this effect is more pronounced at the highest Mach number. However, the variations are much smaller in DG, where the ratio of the external to the wall density is about 1.3 at M = 6 , against a factor of more than 6 for PG at the same conditions. We also notice that the DG density profile exhibits a trend similar to the velocity profile (shown in Fig. 5), with an approximately linear variation in the viscous sublayer and a nearly logarithmic variation in the outer region. This is consistent with previous observations (Sciacovelli et al. 2017) that density is strongly correlated with pressure and, consequently, with velocity in DG flows, whereas it is more correlated with temperature in PG ones. The temperature profiles for PG and DG ( panel b) highlight the strong friction heating of the near-wall region for PG flows, where the wall-to-external temperature ratios are approximately 1.8 and 6.5 for the 10 0 10 1 10 2 10 3 M = 2.25 and M = 6 cases, respectively. These values are in excellent agreement with the predictions of the Walz law [START_REF] Walz | Boundary Layers of Flow and Temperature[END_REF]) (reported in "Appendix"). For DG, the temperature also increases toward the wall, but its variations are one to two orders of magnitude smaller, and do not exceed 3% at the highest Mach number. The average viscosity profiles (panel c) follow the trend of the temperature profiles in PG, with the well-known frictionheating increase of viscosity in the wall region. For DG, the viscosity is essentially correlated with density and decreases toward the wall. This opposite behavior has been already observed for compressible turbulent channel flows (Sciacovelli et al. 2017) although, in that case, temperature and density increase from the wall toward the channel centerline. As a consequence, the average viscosities exhibit opposite variations with the wall distance with respect to the present boundary layer flows.

Figure 4 reports the profiles of the mean molecular Prandtl number and of the fundamental derivative of gas dynamics for DG boundary layers at Re = 4000 . The former varies across the boundary layers and decreases monotonically from values higher than 2 in the outer region to 1.95 and 1.53 at the wall for M = 2.25 and M = 6 , respectively. a consequence, the thermal boundary layer is somewhat thinner than the dynamic one in DG cases. The fundamental derivative also varies across the boundary layer and reaches positive values at the wall, mostly due to density variations. Nevertheless, a region of negative values of persists in the outer region of the boundary layer, characterized by supersonic flow conditions (vertical lines on the same picture indicate the location of the sonic points in the boundary layer profiles). Such a region is more extended for the lowest Mach number case, for which density variations (hence, variations), are smaller.

The velocity profiles for all cases are plotted in Fig. 5 using the classical incompressible and van Driest scalings:

For PG, the velocity profiles depend on the Mach number and deviate from the classical incompressible law of the wall (panel a). The discrepancy of u * VD with respect to the incompressible logarithmic law (panel b) comes from the use of the semi-local wall coordinate y * . A better behavior is recovered when plotting against y + (see "Appendix", Fig. 16c), but the results at different Mach numbers do not fully collapse anymore. For DG, the velocity (8) profiles at various Mach numbers are very close to each other using the incompressible scaling, and weakly sensitive to the van Driest scaling, due to the weak variations of across the boundary layer. As noticed in previous studies [START_REF] Cinnella | Inviscid and viscous aerodynamics of dense gases[END_REF]Sciacovelli et al. 2017), this effect results from the very low Eckert number of DG flows

u + = u u = u √ w ∕ w , u * VD = u VD u * = 1 √ w ∕ ∫
(Ec ∞ = U 2 ∞ ∕(c p,∞ T ∞ )
, see Table 3), leading to substantial decoupling of the dynamic and thermal boundary layers and to significantly smaller friction heating of the boundary layer compared to PG. The slope of the logarithmic region matches well the standard value of the von Kármán constant = 0.41 but the intercept constant is higher than the incompress- ible value ( C = 5.2 ). A possible explanation for this discrepancy may reside in the defini- tion of the scaling variables: for PG, turbulent fluctuations of fluid properties tend to vanish at the wall, so that averages of products/ratios of flow properties do not differ much from products/ratios of the average values. This is not true for DG flows, as discussed in the following.

Second-Order Statistics

The wall-normal profiles of the streamwise turbulent intensity u rms (i.e., the root mean square-rms-streamwise velocity fluctuations) are reported in Fig. 6 using both the incompressible and semi-local scaling. As expected, the latter succeeds in collapsing results at different Mach numbers for PG and has little effect DG profiles, which are very close to each other in both scalings and for all Mach numbers. The peak location corresponds to y * ≈11 ÷ 12 in all cases. However, DG results are characterized by a higher peak in the inner region and by a plateau in the outer one, and exhibit a general trend reminiscent of high Reynolds incompressible boundary layers. Similar considerations hold for the other Reynolds stresses (not shown).

Second-order statistics for the fluctuating thermodynamic properties are reported in Fig. 7. The temperature fluctuations (panel a) exhibit similar trends for PG and DG, with a peak located at y * ≈12 , i.e. close to the turbulent intensity peak. As expected, the values observed in DG are much lower (two orders of magnitude) than in PG, so that the flow can be considered as isothermal to a first approximation and the variations of all thermodynamic quantities can be explained in terms of density fluctuations only. The latter exhibit striking differences with respect to PG flows (panel while rms ∕ w reaches a maximum in the outer region (the peak value being close to 1 at M = 6 ), it decreases smoothly and almost monotonically when departing from the wall for DG. A similar effect has been also observed in compressible turbulent channel flows of dense gases and can be explained by the wall-normal variation of the mean density and by the slope of the constant-temperature and constant-entropy curves in a Clapeyron ( p-v ) diagram. For gases governed by real- gas equations of state like the Martin-Hou equation used in this study it can be shown that rms ∕ ≈ (p rms ∕ )∕( p∕ ) T [see Sciacovelli et al. (2017) for more details]. Due to the different trend of the mean density, however, the rms density profile is flatter in boundary layer flows, whereas it exhibits a well-pronounced peak at the wall in channel flows. Overall, despite the DG flow occurring at thermodynamic conditions of the same order of magnitude of the critical point, the density fluctuations remain small compared to the average density and Morkovin's condition 𝜌 rms ∕𝜌 ≪ 1 is well respected. Viscosity fluctuations (panel c) are clearly correlated with temperature fluctuations for the PG and with density fluctuations for the DG. It is worth noting that, at the wall, they are strictly zero for the PG (for which w = (T w ) = const. ), whereas they are maximum for the DG. The strong den- sity and viscosity fluctuations at the wall have an impact on the wall friction w and friction velocity u , leading to different possible definitions. The unconventional variation of the fluid properties has an impact on the characteristic flow scales and, consequently, on the spatial resolution obtained the simulations. The smallest motions scale with the Kolmogorov length scale , reported in Fig. 8 in both incompressible and compressible scalings. For both gases, + is close to the incompressible value 1.5 at the wall. However, + reaches a minimum close to the boundary layer edge for PG, due to the decreasing mean viscosity, whereas it monotonically increases in DG, due to the liquid-like viscosity profile. Application of the semi-local scaling allows to recover a trend similar to the incompressible limit for PG, while it does not alter much the DG results, as already observed for other flow properties. As expected [START_REF] Smits | Spatial resolution correction for wall-bounded turbulence measurements[END_REF]), + remains almost constant in the near-wall region (say, y * < 30 ), then increases with the wall distance becoming quite large in the non-turbulent region.

For a better understanding of the near-wall turbulent structures, we report in Fig. 9 the fluctuating streamwise velocity and density premultiplied spanwise spectra across the boundary layer for M = 6 as a function of the spanwise wavelength * z . Using the semi-local scaling, the small-scale peak corresponding to the characteristic streaks spacing is centred around * z ≈ 120 and y * ≈ 12 in the velocity spectra for both PG and DG. A larger-scale, outer peak is visible for the DG at * z ≈ 10 3 and y * ≈ 200 (corresponding to y∕ 99 ≈ 0.2 ), in accordance with reference data for incompressible boundary layers at similar Reynolds 2014) for Re = 4400 ]. On the other hand, a different spectral behavior is obtained for the density fluctuations. Specifically, the peak of k z E ′ ′ is located at the boundary layer edge for PG, whereas it is at the wall for DG; this is with the different distributions of the wall-normal rms profiles, as shown in Fig. 7b.

Compressibility Effects

The preceding discussion of first and second-order statistics shows that, in high-speed DG boundary layers, most flow quantities are largely insensitive to the freestream Mach number and exhibit a behavior similar to incompressible boundary layers. Nevertheless, the flow is characterized by strong fluctuations of the fluid properties in the near-wall region. In order to investigate the role and nature of compressibility effects in DG TBL, we report in Fig. 10 the wall-normal profiles of the turbulent and rms Mach number, defined as M t = u rms ∕c and M rms = (u∕c) rms . The turbulent Mach number (panel a) is considerably higher in DG flows with respect to PG at the same M ∞ , due to the thinner boundary layer. In particular, the values reached in the neighborhood of the turbulent intensity peak at M = 6 are sufficiently high ( M t,max > 0.5 ) to enable the formation of eddy shocklets. Even when M t is lower than this threshold, the existence of eddy shocklets is not excluded [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number[END_REF], since the instantaneous Mach number can be much higher. The profiles of the rms Mach number (panel b) are considerably different for PG and DG at M = 6 . In PG, the peak in the outer region is greater than the one located in the production region, due to significant fluctuations of the temperature and speed of sound (not shown). The opposite is observed in DG since fluctuations of the thermodynamic properties are globally smaller and M rms is mostly driven by u rms . The outer peak is strongly attenuated for both DG and PG at M = 2.25.

In the following we focus on the hypersonic cases. Figure 11 shows the probability density functions (pdf) of selected quantities at a wall-normal location corresponding approximately to the peak of u rms and another one in the outer logarithmic layer. More precisely, the pdf are extracted along spanwise meshlines and for several time instants at y * = 11 and y * = 138 for DG, and at y * = 10 , y * = 157 for air. Kernel density estimates of the pdf are obtained by using a total of about 4 million samples. The pdfs of the Mach number (panel a) are significantly skewed toward supersonic values, especially for DG, the right tail being very close to the free-stream Mach value. As a consequence, the probability of supersonic local events is rather high even in the inner region of the TBL. In panel b we report the pdf of the fluid density (normalized with the local average) at the same locations. Contrary to the Mach number, the pdf are much more symmetric for DG, whereas states characterized by density values above the average have a much higher probability in PG. Additionally, the width of the pdf is approximately constant for DG, whereas it tends to become wider when y * increases for PG. If a normalization with respect to the wall density is adopted instead of the local average, then a wider spread of density values observed for the location nearest to the wall in DG, whereas the opposite is true for PG. This is consistent with the trends observed for the wall-normal profiles of rms ∕ w (Fig. 7b). The pdf of the funda- mental derivative for PP11, displayed in Fig. 11c, shows that, even if becomes positive near the wall, instantaneous negative values are still present at y * = 10 , i.e. close to the region where M t peaks. More generally, most thermodynamic states in the flow correspond to ≈0 . In such conditions, expansion and compression events tend to become equally probable even in the absence of nonclassical BZT effets, as discussed in [START_REF] Sciacovelli | Dense gas effects in inviscid homogeneous isotropic turbulence[END_REF]Sciacovelli et al. ( , 2017)). In Fig. 11d, we report the pdf of the velocity divergence ( = ∇ ⋅ ), often used in the literature to discriminate the preferential occurrence of compression/expansion events. The values are normalized with the rms divergence and the local mean and rms Mach number to ease comparisons between different fluids and locations. Given the low M t reached in PG, the pdf is obviously symmetric at both the wall distances. For DG, the local Mach numbers are higher, and they would correspond to a pdf skewed toward negative values in PG homogeneous isotropic turbulence. In the DG, however, the pdf remains well symmetric, consistently with preceding observations of DG homogeneous isotropic turbulence at a similar M t (Sciacovelli et al. 2017).

Figure 12 reports the wall-normal distributions of the normalized rms dilatation , as well as the dilatational-to-solenoidal turbulent dissipation ratio ∕ , where = 4 3 2 and = 2 2 (Huang et al. 1995)

( = |∇ × | ).
Of note, this ratio tends rapidly to infin- ity outside the boundary layer, the vorticity of the freestream being zero. In DG, near-wall values of the dilatation fluctuations are almost twice those of the PG, which is consistent with the high density fluctuations in the vicinity of the wall. Overall, the contribution of the dilatational dissipation is more significant in a DG boundary layer, and the differences are particularly marked the inner region and in the wake region. This seems consistent with the highest instantaneous values of the Mach number observed in DG, which increase the probability of eddy shocklets with respect to PG.

In order to provide a global view of the flow topology, we show instantaneous snapshots of the magnitude of the density gradients and of the velocity divergence in Figs. 13 and14, respectively. The density gradients are normalized by ∕ 99,end to obtain comparable ranges for the two flows. The numerical Schlieren for the PG flow exhibits elongated, sheet-like structures corresponding to high-density-gradients ejection events moving towards the boundary layer edge. On the contrary, the high-|∇ | spots for the DG flow are located near the wall (in a region where the rms density is also at its maximum), whereas sheet-like structures in the outer region (where 𝛤 < 0 ) are more sporadic and intermittent. Such spots do not seem to be preferentially associated to expansion or compression events, which are shown to be almost equally likely for both flows. With the aim of relating the occurrence of strong density gradients with expansion/compression events, joint pdfs of the velocity divergence and magnitude of the density gradients are reported in Fig. 15. A weak correlation is noted, as indicated by the low values of the Pearson coefficients r p (reported in the figure), which give a measure of the linear relationship between the two datasets. The symmetry of the joint pdf for the DG shows that density gradients are not preferentially correlated to compression nor expansion events, whereas a slight correlation with expansion events is noted for the PG, more marked near the wall. Therefore, we conclude that the Schlieren-like illustrations are not representative of the presence of shocklets; for the PG, they rather show the sharp edges of the bulge structures in the outer flow. This picture is strongly altered for the DG, where the density variations are more uniform across the layer and intense events are located close to the wall.

Conclusions

In this paper, we reported for the first time high-fidelity numerical simulations of supersonic and hypersonic boundary layers of dense gas (DG) at M = 2.25 and M = 6 . The working fluid selected for the study is the heavy fluorocarbon PP11, already considered in previous studies of decaying homogeneous isotropic turbulence and turbulent channel flows. The calculations were conducted by using a high-order numerical solver and advanced thermodynamic and transport-property models. The dense-gas results are compared with perfect gas (PG) results generated at the same Mach and Reynolds numbers.

For the dense gas, the complex thermodynamics and transport-property variations play a crucial role. Most of the observed effects are due to the much higher molecular complexity of the dense gas, leading to an increase of more than one order of magnitude of the heat capacity compared to light gases like air and, consequently, to much smaller Eckert numbers. In such conditions, temperature variations are nearly negligible throughout the flow field and friction heating is very small even at hypersonic conditions. Due to the substantial decoupling of the thermal and dynamic boundary layer, the thickening rate of the boundary layer in the dense gas is close to that of an incompressible flow and dynamic quantities are weakly sensitive to the Mach number. On the contrary, wall-normal profiles of mean thermodynamic and transport properties depend on the Mach number, although the amplitude of variation across the boundary layer is much smaller than in perfect gas boundary layers. Specifically, the wall temperature is less than 3% greater than the free-stream temperature for the DG boundary layer at M = 6 , against a factor close to 7 observed for air at similar conditions. Similarly, the wall density is reduced by less than 25% in the dense gas, against a factor of more than 6 in the PG, and variations of the fluid viscosity across the boundary layer are also greatly reduced and amount to less than 30% for the highest Mach number. Additionally, since the viscosity mostly depends on density in DG conditions (instead of temperature, like in PG), it exhibits an opposite trend across the boundary layer; namely, it decreases when approaching the solid wall. Both of the above-mentioned effects lead to higher local Reynolds numbers, as shown by the velocity spectra, where an outer peak is present for the DG. The velocity profiles and turbulent intensities are almost unaffected by the Mach number, although further work is probably required to adapt compressible wall scalings, originally derived for perfect gases, in order to improve the matching of dense gas results with incompressible references. Indeed, significant differences are observed for the density and viscosity fluctuations, which reach a maximum near the wall in DG instead of vanishing as in PG. The reduced thickening of the DG boundary layer also leads to a more extended supersonic region than perfect gases and to higher turbulent and fluctuating Mach numbers. Probability density functions of the Mach number clearly show that more frequent supersonic events occur within the DG boundary layers. The analysis of the dilatational events provides a first view of the peculiar behavior in the dense-gas regime. In particular, inspection of the instantaneous divergence fields and their pdf distributions shows that expansion and compression events are rather balanced, differently from the perfect gas. Schlieren-like representations, often used to exhibit the compressive character of the flow, are weakly correlated to compression and expansion events. They, however, depict very different compressibility effects on the boundary layer structure. Sharp bulges dominate the outer flow in PG, whereas density variations are severely attenuated in the outer region of the DG boundary layer, where values of the fundamental derivative of the gas dynamics are close to zero. On the contrary, a strong dilatational activity is located in the nearwall region. Further work is required to better understand how the peculiar compressibility effects may affect the turbulent structures or the Mach wave radiation. The definition of appropriate scalings to ease comparisons with perfect gases and with the incompressible limit may help in this endeavour. numerical setup is that the inlet of the domain corresponds to the leading edge of the flat plate, whereas [START_REF] Franko | Breakdown mechanisms and heat transfer overshoot in hypersonic zero pressure gradient boundary layers[END_REF] started with a finite laminar boundary layer thickness, such that the Reynolds number based on the inflow displacement thickness, * in , F&L , is 3000. For the comparisons in Fig. 17 Finally, in Fig. 18, we compare temperature profiles from the present PG simulations at M = 2.25 and M = 6 at Re = 4000 with the classical temperature law of Walz (1969). The numerical results are found to match very well the analytical model.

Fig. 1

 1 Fig. 1 View of the computational domain for the M = 6 air (top) and PP11 (bottom) cases. Instantaneous snapshot of the Q-criterion colored by the distance from the wall, and isocontours of log |∇ | on the background planes. The white symbol " ∼ " denotes the location of the suction and blowing forcing

Figure 2

 2 Figure 2 provides an overview of the boundary layer evolution for DG and PG cases. The figure reports the skin friction coefficient distributions along the plate, transformed to equivalent "incompressible" values by means of the Van Driest II transformation (Van Driest 1956):

Fig. 2

 2 Fig. 2 Evolution of selected properties across the turbulent boundary layers. red dashed line Air M = 2.25 , blue line Air M = 6 , orange dotted line PP11 M = 2.25 , green dash-dotted line PP11 M = 6 . In panel (a), square: Blasius laminar solution, triangle: turbulent correlation C f ,inc = 0.024Re -1∕4

Fig. 3

 3 Fig. 3 Wall-normal profiles of selected first-order statistics for Air and PP11, normalized with respect to wall values. a, average density; b, average temperature; c, average viscosity. Red dashed line Air M = 2.25 , blue line Air M = 6 , orange dotted line PP11 M = 2.25 , green dash-dotted line PP11 M = 6

Fig. 4

 4 Fig. 4 Wall-normal profiles of the average Prandtl number a and fundamental derivative of gas dynamics b for PP11. Vertical dashed lines in panel b denote the sonic lines for the two cases. Orange dotted line PP11 M = 2.25 , green dash-dotted line PP11 M = 6

Fig. 5 Fig. 6

 56 Fig. 5 Velocity profiles for Air and PP11: (a), wall scaling; (b), semi-local Van-Driest scaling. Black dashed lines denote the linear and logarithmic laws, with = 0.41 and C = 5.2 . Red dashed line Air M = 2.25 , blue line Air M = 6 , orange dotted line PP11 M = 2.25 , green dash-dotted line PP11 M = 6

Fig. 7

 7 Fig. 7 Wall-normal profiles of selected second-order statistics for Air and PP11, normalized with respect to wall values. a, rms temperature; b, rms density; c, rms viscosity. Red dashed line Air M = 2.25 , blue line Air M = 6 , orange dotted line PP11 M = 2.25 , green dash-dotted line PP11 M = 6

Fig. 8 Fig. 9

 89 Fig. 8 Kolmogorov length scale with incompressible (a) and semi-local (b) wall scaling across turbulent boundary layers at M = 6 : blue line Air, green dash-dotted line PP11

Fig. 10

 10 Fig. 10 Wall-normal evolutions of turbulent Mach number (a) and rms Mach number (b). Red dashed line Air M = 2.25 , blue line Air M = 6 , orange dotted line PP11 M = 2.25 , green dash-dotted line PP11 M = 6

Fig. 11

 11 Fig. 11 Probability density functions of the Mach number a, normalized density b, fundamental derivative of gas dynamics c and normalized dilatation d for air and PP11. Blue line Air M = 6 , green dash-dotted line PP11 M = 6 . Two vertical locations are considered: near the wall ( y * = 10 for air and y * = 11 for PP11), circle symbols; in the logarithmic region ( y * = 157 for air and y * = 138 for PP11), triangle symbols

Fig. 12 Fig. 13

 1213 Fig. 12 Wall-normal distributions of normalized dilatation (a) and ratio of dilatational to solenoidal turbulent dissipation for M = 6 cases. Blue line Air M = 6 , green dash-dotted line PP11 M = 6

Fig. 14

 14 Fig. 14 Instantaneous snapshots of normalized velocity divergence ∕ rms M∕M rms for Air (top) and PP11 (bottom) at M = 6

Fig. 15

 15 Fig. 15 Joint probability density functions for Air (a, c) and PP11 (b, d) at M = 6 near the wall (a, b; y * = 10 for Air and y * = 11 for PP11) and in the logarithmic region (c, d; y * = 157 for Air and y * = 138 for PP11)

  , we use this displacement thickness as reference ( * in , F&L = * ref ). Panel a shows that the distribution of the skin friction coefficient for the present and the reference calculation are in excellent agreement. The present simulation finally reaches a fully developed turbulent state where C f follows the trend of classical skin friction correlations (see Sect. 3). Selected velocity profiles at various stations in the laminar, transitional, and nearly turbulent flow regimes are reported on panel b of the same figure. Once again, the present results match remarkably well the reference data, thus confirming the quality of the present simulations.

Fig. 16 Fig. 17

 1617 Fig. 16 Wall-normal profiles of Reynolds stresses (a), normalized wall pressure (b) and Van Driest-scaled streamwise velocity profile (c), for current ILES of Air at M = 2.25 (lines) and Pirozzoli and Bernardini (2011) (symbols), extracted at Re = 580

Table 1

 1 Thermodynamic properties of PP11 ( C 14 F 24 ): molecular weight ( M ), critical temperature ( T cr ), critical density ( cr ), critical pressure ( p cr ), critical compressibility factor ( Z cr ), boiling temperature ( T b ), ratio of ideal-gas specific heat at constant volume over the gas constant ( c v (T cr )∕R ) at the critical point, and parameter for the low-density specific heat power law (n) We have noted (•) ∞ the incoming flow quantities. At both Mach numbers, a constant Prandtl number assumption is used to determine the thermal conductivity = c p ∕Pr (with Pr = 0.71).

	M	T cr	cr	p cr	Z cr	T b	c v (T cr )∕R	n
	g mol -1	K	kg m -3	MPa	-	K	-	-
	624.11	650.2	627.14	1.46	0.2688	488.15	97.3	0.5776

Table 2

 2

	Numerical parameters of the forcing function in	Case	A∕U ∞ a 2D a 3D a H	* forc	* forc ∕U ∞ Re * forc
	equation (5) for the different cases	Air M = 2.25 Air M = 6	0.01 0.002 0 0	1 1	0 0	0.2 0.3	0.12 0.15	2000 3000
		PP11 M = 2.25 0.035 0	1.05 0.05 0.3	0.6	3100
		PP11 M = 6	0.02	1	0.05 0.05 0.3	0.6	1200

fine wall-resolved ILES (implicit LES). The spatial resolution has been also evaluated with respect to the wall-normal distribution of the Kolmogorov length-scale = [ 3 ∕(

3

)]

Table 3

 3 Simulation parameters and boundary layer properties at the end of the domain

	Fluid	Air		PP11
	M	2.25	6	2.25	6
	U ∞ [m/s]	363.6	969.7	78.99	198.8
	∞ [kg/m 3 ]	0.13	0.13	348.4	348.4
	T ∞ [K]	65	65	650.8	646.83
	T w [K]	120.18	422.5	653.4	663.2
	Ec ∞	2.03	14.4	5.42 × 10 -3	3.31 × 10 -3
	N x	4000	7700	10000	14336
	N y	210	300	270	320
	N z	140	400	176	280
	Re * in	1700	0	3000	0
	Re * forc	2000	3000	3100	1200
	Re	4600	5720	9200	4402
	99 [mm]	5.43	5.98	0.132	0.032
	[mm]	0.46	0.23	0.015	0.0031
	H	3.66	13.81	1.51	2.12
	x +	11.57	3.76	19.57	8.12
	y + w	0.82	0.26	0.98	0.65
	y + e	10	5.1	24	18
	z +	5.63	2.09	12.07	9.77
	L x ∕ 99	62.5	75.2	75.1	90.1
	L y ∕ 99	1.75	2.13	1.48	1.75
	L z ∕ 99	1.06	2.17	0.85	1.56

Table 4

 4 Grid resolutions for M = 6 simulations, evaluated at the wall and at the boundary layer edge, normalized with respect to the local averaged Kolmogorov scale

	Case	( x) w	( y) w	( z) w	( x) e	( y) e	( z) e
	Air M = 6	2.7 w	0.2 w	1.5 w	4.3 e	3.5 e	2.4 e
	PP11 M = 6	6.3 w	0.5 w	7.5 w	1.6 e	2.8 e	2.1 e

w Re , being Re = ∞ U ∞ ∞ and the momentum thickness. The Blasius laminar solution C f ,inc = 0.664 Re -1∕2

Acknowledgements This work was granted access to the HPC resources of IDRIS and TGCC under the allocation 2018-7332 and 2018-1736 made by GENCI (Grand Equipement National de Calcul Intensif). We also acknowledge TGCC for awarding access to the Joliot-Curie supercomputer under the allocation "Grands Challenges" gch032.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

Appendix: Validations

In this appendix, we present validations of our numerical solver against well-established literature results for perfect-gas (air) high-speed boundary layers at M = 2.25 and 6. The simulation at M = 2.25 is compared to the DNS data of [START_REF] Pirozzoli | Turbulence in supersonic boundary layers at moderate Reynolds number[END_REF]. This study focused on the fully turbulent flow behavior and adopted a rescaling/recycling strategy to shorten the computational domain required to achieve fully-developed turbulence. Sutherland's law was used to model the viscosity, along with a constant Prandtl number hypothesis. Figure 16 shows wall-normal profiles of selected flow statistics for the present PG ILES and the DNS of [START_REF] Pirozzoli | Turbulence in supersonic boundary layers at moderate Reynolds number[END_REF]. An excellent agreement is observed.

The calculation at M = 6 has been performed in the same conditions and with the same thermodynamic and transport-property models as the DNS study of [START_REF] Franko | Breakdown mechanisms and heat transfer overshoot in hypersonic zero pressure gradient boundary layers[END_REF], except that our computational domain is much longer to achieve a fully turbulent state. These authors focused their analysis on the transition mechanisms, so that a comparison is possible only in the transitional regime. Another difference in our


Hatem A Rashwan

Sylvie Chambon

Pierre Gurdjos

Géraldine Morin

Vincent Charvillat

TOWARDS MULTI-SCALE FEATURE DETECTION REPEATABLE OVER INTENSITY AND DEPTH IMAGES

Keywords: Feature detectors, Curviness saliency, 2D-3D matching, Repeatability

Object recognition based on local features computed at multiple locations is robust to occlusions, strong viewpoint changes and object deformations. These features should be repeatable, precise and distinctive. We present an operator for repeatable feature detection on depth images (relative to 3D models) as well as 2D intensity images. The proposed detector is based on estimating the curviness saliency at multiple scales in each kind of image. We also propose quality measures that evaluate the repeatability of the features between depth and intensity images. The experiments show that the proposed detector outperforms both the most powerful, classical point detectors (e.g., SIFT) and edge detection techniques.

INTRODUCTION

A growing trend in recognition applications is to require 3D object models to overcome the limitations due to variations in viewpoint, texture or lighting that may modify the object appearance in the images. Since 3D capturing process is cheaper and faster, accurate 3D models become more available [START_REF] Aubry | Seeing 3D Chairs: Exemplar Part-Based 2D-3D Alignment Using A Large Dataset of CAD models[END_REF][START_REF] Choy | Enriching Object Detection with 2D-3D Registration and Continuous Viewpoint Estimation[END_REF][START_REF] Toshev | Shape-Based Object Recognition in Videos Using 3D Synthetic Object Models[END_REF][START_REF] Lee | Multi-Scale Feature Matching Between 2D Image and 3D Model[END_REF]. In addition, 3D models can be integrated through separate dense depth images captured by range scanners (e.g., laser and IR) [START_REF] Campbell | A Survey Of Free-Form Object Representation and Recognition Techniques[END_REF].

Recent approaches are based on 3D model databases [START_REF] Choy | Enriching Object Detection with 2D-3D Registration and Continuous Viewpoint Estimation[END_REF]. These models can be represented by intermediate representations [START_REF] Irschara | From structure-from-motion point clouds to fast location recognition[END_REF], like artificial images generated by rotating around the object through varying yaw, pitch and roll angles and focal length [START_REF] Campbell | A Survey Of Free-Form Object Representation and Recognition Techniques[END_REF]. To cope with the shape variation independently of texture and lighting, an adapted representation is range (i.e., depth) images, which represent the object shape rather than its texture [START_REF] Choy | Enriching Object Detection with 2D-3D Registration and Continuous Viewpoint Estimation[END_REF][START_REF] Song | Sliding Shapes for 3D Object Detection in RGB-D Images[END_REF]. The key features in both depth and intensity images are then detected to be matched. A key requirement on these features, as in 2D-2D matching, is to be computed with a high degree of repeatability (i.e., the probability that key features in the intensity image are found close to those extracted in the depth image must be high). In this paper, we require such assessments of repeatability, as shown in figure 1. Our objective is to introduce a detector robust to color, texture and illumination changes. To detect repeatable features in 2D, edges [START_REF] Canny | A Computational Approach To Edge Detection[END_REF], corner detectors [START_REF] Harris | A Combined Corner and Edge detector[END_REF], eigenvalue analysis [START_REF] Shi | Good Features to Track[END_REF], multi-scale detectors (like SIFT [START_REF] Lowe | Distinctive Image Features from Scale-Invariant Keypoints[END_REF] and SURF [START_REF] Bay | Speeded-Up Robust Features (SURF)[END_REF]) and curvature detection [START_REF] Fischer | Image Descriptors Based on Curvature Histograms[END_REF] are the four most important and the most used techniques. Recently, [START_REF] Deng | Principal Curvature-Based Region Detector for Object Recognition[END_REF] presented curvature-based detector that use the structural cues to find the curvature in a multi-scale space. All these techniques are robust to lighting changes and translation; multi-scale approaches are also robust to scale and rotation. However, they depend on texture and/or color changes. Thus, the question is: what happens if we use these detectors for 2D-3D matching.

Recently, 2D-3D matching have been developed in the context of pose estimation. In [START_REF] Toshev | Shape-Based Object Recognition in Videos Using 3D Synthetic Object Models[END_REF], a sequence of silhouettes has been extracted from 3D models and input images. Then, shape similarity is measured between these silhouettes. In our paper, structural cues (e.g., curvilinear shapes) are extracted instead of only considering silhouettes since they are more robust to intensity, color, and pose variations. In fact, they not only represent outer contours (silhouette), but also inner (self-occluding) contours that also characterize the object. In addition, the histogram of gradients (HOG) detector [START_REF] Aubry | Seeing 3D Chairs: Exemplar Part-Based 2D-3D Alignment Using A Large Dataset of CAD models[END_REF][START_REF] Lim | Fpm: Fine Pose Parts-Based Model With 3D CAD Models[END_REF] or a fast version of HOG [START_REF] Choy | Enriching Object Detection with 2D-3D Registration and Continuous Viewpoint Estimation[END_REF] have been also used to extract the features from rendering views and real images. All of these approaches give interesting results, however, they do not evaluate the repeatability between the set of points detected in an intensity image and those detected in an image rendered from the 3D model. In fact, they all use a learning phase with manual matching but, in the context of our paper, we want to avoid a learning phase. Finally, in [START_REF] Lee | Multi-Scale Feature Matching Between 2D Image and 3D Model[END_REF], the authors match the image with the 3D models by using SIFT in 2D and surface variation in 3D, but they assume that the object in the input image has no or poor internal texture.

As illustrated in figure 1, this work addresses the problem of aligning two images generated differently: an intensity image and a depth image that we assume to be taken from the same viewpoint. Curvature features do highlight geometric characteristics of an object. We propose a new detector based on curviness saliency that is a function of the eigenvalues of the Hessian matrix, an estimation of curvature. Furthermore, as relevant details exist only over a restricted range of scale, we further consider these features in a multi-scale analysis. This detection yields more repeatable key points between intensity and depth images than the classical detectors. Finally, an intensive evaluation and comparison have been conducted to highlight the quality of the proposed interest points.

PROPOSED MULTI-SCALE CURVINESS SALIENCY DETECTOR

Given an intensity image I, we define the surface S such that S(x, y) = (x, y, I(x, y)) under the assumption that I is twice differentiable. We denote by ∇I the gradient vector of I and

H α H, where α = 1/ 1 + ∇I 2 . ( 1
)
We aim at detecting "curvilinear features" in a representation common to depth and intensity images. By curvilinear features, we refer to points lying on elongated structures at which one principal curvature strongly dominates the other one. The Laplacian-energy is often used for image representation. So why not to use it here? Usually, the discrete form

∇ 2 I(p) = I xx (p)+I yy (p) ≈ (1/h 2 ) h I(p + h)-I(p),
where p = (x, y) ⊤ and h varies in {±h} × {±h}, is applied to each image and the obtained values are then squared. The problem is that the Laplacian operator clearly behaves like a mean rate of local intensity change so important directional information is lost. Linking this to curvatures, using the rotation invariance of the Laplacian, it can be shown that

∇ 2 I = α trace H = α(κ 1 + κ 2 )
which means that ∇ 2 I computes twice an "intensity-weighted" mean curvature of S.

The image representation proposed in this work is the so-called curviness saliency representation which relies on a function computing the difference between principal curvatures. We will now justify such a choice. Remind that, on the tangent plane T S to S at point p, for all unit directions t in T S , (i) the normal curvatures κ t (p) at p associated with t are the curvatures of the curves obtained by slicing S with the planes containing p and parallel to t; (ii) ρ t (q) 1/|κ t (q)| are the radii of curvature of these curves. A nice geometry result [START_REF] Boehm | Geometric Concepts for Geometric Design[END_REF] is that all the points q = p + ρ t (q)t on T S , located at distance √ ρ t from p, lie on a conic known as the Dupin indicatrix at p. When p is the origin, the conic equation writes

(x, y) H(x, y) ⊤ = ±1. (2)
Let choose a sign for ± H such that its eigenvalues be λ 1 , λ 2 ensuring λ 1 ≥ λ 2 and λ 1 > 0 (which are in fact equal to the two principal curvatures up to a common sign). Semi majorand minor-axes are

r 2 = |λ 2 | -1/2 and r 1 = |λ 1 | -1/2 (since r 2 ≥ r 1 )
respectively. The conic specializes to an ellipse if

λ 1 λ 2 > 0, or an hyperbola if λ 1 λ 2 < 0 i.e., if λ 2 is negative.
The Dupin indicatrix yields a local information on the surface as the conic shape describes the 'distribution' of all normal curvatures at p (or more exactly of the squared roots of all radii of curvature). Various measures can describe this conic shape and we select the linear eccentricity E∓ r 2 2 ∓ r 2 1 , also called half-focal separation, which is the distance between the center and one focus, with '-' for ellipses and '+' for hyperpolas. Indeed, it can be easily shown that:

√ λ 1 -λ 2 = E∓ r1r2
, which provides a unified way of treating ellipses and hyperbolas (due to lack of space, the proof is omitted). The function:

CS λ 1 -λ 2 , (3)
is large when λ 1 ≫ λ 2 , which means distant foci and so a highly elongated ellipse or a "squashed" hyperbola. This occurs e.g., when the point is located on a ridge (either curved or straight). In turn, when λ 1 ≃ λ 2 , the conic approaches a circle and the distance between foci becomes very small. Therefore after computing (3) for every image pixel, any point with high value can be considered as a potential keypoint. Noting that the two eigenvalues λ 1 λ + and λ 2 λ -of the scaled Hessian matrix H can be directly computed as:

λ ± = α 2 I xx + I yy ± (I xx -I yy ) 2 + 4I 2 xy , (4)
the curviness saliency is then defined as

CS CS 2 = α 2 (I xx -I yy ) 2 + 4I 2 xy . (5)
In figure 2, we show the different detections obtained using the minimum or the maximum eigenvalue, as proposed by Deng et al. [START_REF] Deng | Principal Curvature-Based Region Detector for Object Recognition[END_REF]; the maximum eigenvalue provides a high response only for dark lines on a light background, while the minimum gives the high answer for the light lines on a dark background. Our proposition, the difference of the eigenvalues, improves robustness as it responds in both settings.

Computing the curviness saliency in a single-scale can only detect points that have high curvature in one scale and high curvature points in other scales are missed. In consequence, in this paper, we compute the curviness saliency images in a multi-scale space. To build the scale pyramid, an edge-preserving smoothing approach, anisotropic diffusion filter [START_REF] Paris | A Fast Approximation of the Bilateral Filter Using a Signal Processing Approach[END_REF] is used.

Contrary to depth images which represent textureless 3D shapes, intensity images are composed of shape and texture components. Consequently, the CS estimated from intensity images is affected by the textured regions. Our idea is to put forward the assumption that multi-scale analysis can discriminate between keypoints (those with high CS) due to shape and keypoints due to texture. As shown in figure 3.(b), at a coarse level, curves detected are reliable with poor localization and they miss small details. At a fine levels, details are preserved, but detection suffers greatly from clutters in textured regions. In addition, the CS values of small details and textures are high in the coarse level, whereas these values become lower in the finest levels. To combine the strengths of each scale, the CS value of each pixel over n scales is analyzed. If this value in all scales is higher than a threshold T , which is a function of the number of the smoothed images, m, (i.e., T = e -m ), the maximum curviness saliency (MCS) value of this pixel over all scales is then kept, see figure 3.(a). However, if the CS value is lower than T in one level, it is considered as texture (or small detail) point, thus it is removed from the final multiscale curviness saliency (MCS) image.

EXPERIMENTS

To evaluate the repeatability of the detectors between intensity images and depth images, we apply a set of 9 detectors on a depth image, and its corresponding real images taken from the same point of view. The locations of features extracted on the depth images are considered as the ground-truth. Then, the extracted features on the real images are compared with the corresponding ground-truth. For all the tested 3D models, we generate depth images from approximately uniformly distributed viewing angles around a circle or a sphere 1 .

Datasets -First, we use 2 real objects (i.e., stairs), with respectively 4 and 2 different textures, with their 3D CAD models. A set of depth images from different view is rendered and for each viewpoint, real images with different texture are captured, see figure 4. Moreover, we use 3D textureless objects (available online2 ), and we collect a set of 15 real images of each object on the web by choosing views as close as possible to the views used for the generation of the depth images. Moreover, to highlight the robustness of the approach to different acquisition conditions, many real images of a similar model are taken. Furthermore, we used the PASCAL3D+ dataset [START_REF] Xiang | Beyond PASCAL: A Benchmark for 3D Object Detection in the Wild[END_REF] that contains real images corresponding to 12 rigid objects categories; we compute average results for all non occluded objects in each category i.e., around 1000 objects per category. The real images are acquired under different acquisition conditions (e.g., lighting, complex background, low contrast). We rendered the depth image of the corresponding 3D CAD model using the viewpoint information from the dataset. Comparison with existing detectors -The experiments include comparisons with these 9 following detectors3 : Edge-based detectors: Sobel, Laplacian of Gaussian (Log), Canny [START_REF] Canny | A Computational Approach To Edge Detection[END_REF] and Fuzzy logic technique [START_REF] Kiranpreet | Fuzzy Logic Based Image Edge Detection Algorithm in MATLAB[END_REF], corner detectors: Harris detector, Minimum Eigenvalues detector [START_REF] Shi | Good Features to Track[END_REF], multiscale detectors: SIFT, Scale Invariant Feature Transform [START_REF] Lowe | Distinctive Image Features from Scale-Invariant Keypoints[END_REF], SURF, Speeded Up Robust Features [START_REF] Bay | Speeded-Up Robust Features (SURF)[END_REF], a multi-scale Principal Curvature Image (PCI) detector [START_REF] Deng | Principal Curvature-Based Region Detector for Object Recognition[END_REF].

Setup -The images of the MCS results are calculated in scale space, similar to SIFT (see [START_REF] Paris | A Fast Approximation of the Bilateral Filter Using a Signal Processing Approach[END_REF] for details about the construction of the pyramid). What is important is that we set the number of smoothed images per octave to 5, and in consequence, we have only 1 image result per octave. Evaluation criteria -The behavior of the detectors is evaluated with 2 measures: Hausdorff distance, well known and used, and Intersection percentage, introduced in this paper and that evaluates the intersection between the set of points extracted from the 2 considered images. More precisely, for each point of the depth image, a corresponding point is sought at the same location in the real image. The intersection percentage (IP) counts the number of correspondences over the total number of points detected in the image.

Results -In figures 4 and 5, for the depth images, MCS gives points uniformly located on the silhouette of the objects and also some points inside the shape whereas the corner and the multi-scale detectors miss some specific parts of the shape, like the bottom of the car. Moreover, visually MCS performs best for detecting a set of points in the depth map close to those detected in the real images. All the detectors are affected by the background, i.e. they detect points that are not coherent with the depth map. For the edge operators, false contours are detected inside the car in the real images. MCS (column 1), PCI proposed in [START_REF] Deng | Principal Curvature-Based Region Detector for Object Recognition[END_REF] (2), SIFT (3), Harris (4) and Edge detector proposed in [START_REF] Kiranpreet | Fuzzy Logic Based Image Edge Detection Algorithm in MATLAB[END_REF] (5).

As shown in Table 1, MCS is able to find the highest number of features in the intersection, with real images captured under different textures and lighting conditions. For real images, as background is arbitrary, a perfect score (100%) can not be expected. For the edge-detectors, the intersection percentage is less than 30% for all the edge operators. However, MCS still outperforms the other edge operators. Moreover, PCI, the closest approach to our, yields to good repeatability results that are comparable to MCS results. However, MCS still provides the best results among the 9 tested detectors. In addition, MCS provides the lowest Hausdorff distance. We also have to notice that PCI and Edge detectors perform better than the corner and the multi-scale detectors.

CONCLUSION

We introduce a feature detector, MCS, based on a multi-scale curviness saliency estimation that can extract points more repeatable than classical detectors when it is used between an intensity image and a depth image. We also proposed a quality measure, the intersection percentage, to evaluate the repeatability of the extracted features. The experiments show that MCS yields the best repeatability score. Future work aim at introducing this detector in a robust 2D-3D matching for robust object recognition. The next step will be to introduce a descriptor based on shape and common to these different images to increase the quality of the recognition. In addition, we aim at using defocus maps [START_REF] Zhuo | Defocus Map Estimation from a Single Defocused Image[END_REF] to represent real images.

Fig. 1 :

 1 Fig. 1: The proposed detector based on curviness saliency to align intensity images to 3D models and the computation of the repeatability score between an intensity image and a corresponding depth image rendered from the same viewpoint.

  by H = I xx I xy I xy I yy the Hessian matrix of I, i.e., the order-2 matrix of second-order partial derivatives I xx , I xy and I yy . One key result is that the principal curvatures κ 1 (p), κ 2 (p) of S at point p are the eigenvalues of the matrix:

Fig. 2 :

 2 Fig. 2: Curviness saliency of two shapes (columns 1, 5) with minimum (2, 6), maximum (3, 7) and the difference between maximum and minimum eigenvalues (4, 8).

Fig. 3 :

 3 Fig. 3: (a) A pyramid for a curviness saliency image computation: green values are over a threshold, so, the maximum is kept, whereas red values are under and replaced by 0. (b) Curviness saliency at different scales, top to bottom: the original image, the coarsest and finest levels, the final result.

Fig. 4 :

 4 Fig. 4: Points detected with MCS on a depth image (column 1) and images with different textures rendered from the same viewpoint(column 2-5).

Fig. 5 :

 5 Fig. 5: Features extracted in the depth (row 1) and the real image (2) with:

Table 1 :

 1 Mean Intersection Percentage (IP) (higher is better) and Mean Hausdorff Distance (HD) (lower is better) of all depth image rendered from different viewpoints and all real images captured under different textures and lighting of, first, the two objects, second the 10 objects, and, third, the PASCAL3D+, with the proposed method (MCS) and 9 tested detectors. METime is the mean execution time in seconds of MATLAB codes of the MCS and 9 tested detectors executing on Intel Core(i7) 2.9 GHz.

	Methods	MCS		PCI	MinEig	Harris	SIFT	SURF	Sobel	Canny	LOG	Fuzzy
	Sequences	Ip	Hd	Ip	Hd	Ip	Hd	Ip	Hd	Ip	Hd	Ip	Hd	Ip	Hd	Ip	Hd	Ip	Hd	Ip	Hd
	Stairs1	63	29	43	37	39	48	35	41	32	52	42	39	25	67	30	59	31	61	21	75
	Stairs2	65	27	44	36	35	45	38	41	40	47	43	35	28	64	34	58	33	54	20	70
	Car	50	29	46	40	08	57	04	77	03	85	03	71	10	48	18	46	11	47	05	49
	Shoe	31	52	31	67	02	102	03	106	10	111	01	108	04	71	04	71	05	71	02	71
	Plane	55	23	38	19	06	37	04	43	10	46	03	47	18	26	21	26	21	24	14	24
	T-Rex	64	17	59	25	09	41	06	100	02	143	05	46	16	28	18	28	20	32	12	22
	Elephant	32	41	32	55	03	80	03	91	05	114	03	74	06	57	08	58	06	57	04	57
	Fhydrant	51	23	42	35	06	62	04	86	02	74	09	67	09	38	14	37	13	36	06	42
	Jeep	62	31	58	42	05	70	05	67	05	74	06	89	09	47	15	47	11	46	06	47
	Mug	54	56	50	65	02	129	03	133	04	134	03	145	08	72	12	76	07	75	08	75
	Teddy	39	24	32	31	04	72	05	69	09	77	04	101	07	47	14	44	08	47	07	47
	Pistol	67	16	61	26	09	34	09	96	09	44	04	73	13	30	23	65	14	29	07	26
	plane	50	48	37	59	15	61	09	63	08	68	13	73	10	68	13	65	11	69	10	71
	bicycle	61	75	57	79	25	90	08	101	16	93	24	100	13	83	15	84	18	82	14	87
	boat	36	68	28	75	09	79	10	77	06	87	10	76	09	75	14	71	11	78	09	76
	bus	24	110	17	117	05	128	06	123	02	131	04	127	04	121	06	118	04	122	04	123
	car	41	85	24	98	08	102	08	100	03	113	06	108	16	89	18	88	14	94	13	97
	chair	52	64	43	78	16	84	08	96	09	94	16	86	24	88	20	91	22	86	19	92
	table	38	85	19	96	06	117	05	118	04	118	08	111	11	117	12	114	11	116	07	120
	train	28	108	14	121	06	126	07	123	03	133	05	127	08	125	07	129	04	129	06	122
	METime		0.018	0.041	0.022	0.057	0.121	0.088	0.023	0.024	0.062	0.176

http://www.openu.ac.il/home/hassner/projects/poses/

http://tf3dm.com/

The proposed MCS detector is implemented in MATLAB. All tested point-like detectors and edge detection tech- niques are tested with the implementation given in MATLAB.