Francesco Grasso

Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M = 2.25

A spatially developing supersonic adiabatic flat plate boundary layer flow (at M∞=2.25 and Reθ≈4000) is analyzed by means of direct numerical simulation. The numerical algorithm is based on a mixed weighted essentially nonoscillatory compact-difference method for the three-dimensional Navier–Stokes equations. The main objectives are to assess the validity of Morkovin’s hypothesis and Reynolds analogies, and to analyze the controlling mechanisms for turbulence production, dissipation, and transport. The results show that the essential dynamics of the investigated turbulent supersonic boundary layer flow closely resembles the incompressible pattern. The Van Driest transformed mean velocity obeys the incompressible law-of-the-wall, and the mean static temperature field exhibits a quadratic dependency upon the mean velocity, as predicted by the Crocco–Busemann relation. The total temperature has been found not to be precisely uniform, and total temperature fluctuations are found to be non-negligible. Consiste...

Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M=2.25

The interaction of a spatially developing adiabatic boundary layer flow at M∞=2.25 and Reθ=3725 with an impinging oblique shock wave (β=33.2°) is analyzed by means of direct numerical simulation of the compressible Navier-Stokes equations. Under the selected flow conditions the incoming boundary layer undergoes mild separation due to the adverse pressure gradient. Coherent structures are shed near the average separation point and the flow field exhibits large-scale low-frequency unsteadiness. The formation of the mixing layer is primarily responsible for the amplification of turbulence, which relaxes to an equilibrium state past the interaction. Complete equilibrium is attained in the inner part of the boundary layer, while in the outer region the relaxation process is incomplete. Far from the interaction zone, turbulence exhibits a universal behavior and it shows similarities with the incompressible case. The interaction of the coherent structures with the incident shock produces acoustic waves that prop...

Direct numerical simulations of isotropic compressible turbulence: Influence of compressibility on dynamics and structures

In the present paper the statistical properties of compressible isotropic turbulence are analyzed by means of direct numerical simulations. The scope of the work is to evaluate the influence of compressibility on the time evolution of mean turbulence properties and to quantify the statistical properties of turbulent structures, their dynamics and similarities with the incompressible case. Simulations have been carried out at various turbulent Mach numbers and compressibility ratios by using a conservative hybrid scheme that relies on an optimized weighted essentially nonoscillatory approach for the convective terms and compact differencing for the viscous contributions. In order to identify similarities with incompressible turbulence we have also carried out an analysis in the plane of the second (Q*) and third (R*) invariants of the anisotropic part of the deformation rate tensor. The simulations show that the joint probability density function (Q*,R*) has a universal structure, as found in incompressibl...

Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation

The interaction of a normal shock wave with a turbulent boundary layer developing over a flat plate at free-stream Mach number M ∞ = 1.3 and Reynolds number Re θ ≈ 1200 (based on the momentum thickness of the upstream boundary layer) is analysed by means of direct numerical simulation of the compressible Navier–Stokes equations. The computational methodology is based on a hybrid linear/weighted essentially non-oscillatory conservative finite-difference approach, whereby the switch is controlled by the local regularity of the solution, so as to minimize numerical dissipation. As found in experiments, the mean flow pattern consists of an upstream fan of compression waves associated with the thickening of the boundary layer, and the supersonic region is terminated by a nearly normal shock, with substantial bending of the interacting shock. At the selected conditions the flow does not exhibit separation in the mean. However, the interaction region is characterized by ‘intermittent transitory detachment’ with scattered spots of instantaneous flow reversal throughout the interaction zone, and by the formation of a turbulent mixing layer, with associated unsteady release of vortical structures. As found in supersonic impinging shock interactions, we observe a different amplification of the longitudinal Reynolds stress component with respect to the others. Indeed, the effect of the adverse pressure gradient is to reduce the mean shear, with subsequent suppression of the near-wall streaks, and isotropization of turbulence. The recovery of the boundary layer past the interaction zone follows a quasi-equilibrium process, characterized by a self-similar distribution of the mean flow properties.

Characterization of coherent vortical structures in a supersonic turbulent boundary layer

A spatially developing supersonic boundary layer at Mach 2 is analysed by means of direct numerical simulation of the compressible Navier–Stokes equations, with the objective of quantitatively characterizing the coherent vortical structures. The study shows structural similarities with the incompressible case. In particular, the inner layer is mainly populated by quasi-streamwise vortices, while in the outer layer we observe a large variety of structures, including hairpin vortices and hairpin packets. The characteristic properties of the educed structures are found to be nearly uniform throughout the outer layer, and to be weakly affected by the local vortex orientation. In the outer layer, typical core radii vary in the range of 5–6 dissipative length scales, and the associated circulation is approximately constant, and of the order of 180 wall units. The statistical properties of the vortical structures in the outer layer are similar to those of an ensemble of non-interacting closed-loop vortices with a nearly planar head inclined at an angle of approximately 20 ◦ with respect to the wall, and with an overall size of approximately 30 dissipative length scales.

Experimental and Numerical Study of the Laminar Separation in Hypersonic Flow

Abstract This article is devoted to an experimental and numerical study of shock wave/boundary layer interaction in hypersonic laminar flow (M = 10). The experimental was performed in the ONERA R5Ch wind tunnel on a hollow cylinder flare. The flow stream delivered by the R5Ch wind tunnel produces physical conditions which justify both the theoretical approach using classical Navier-Stokes equations and the approach by Direct Simulation Monte-Carlo. So the aim of this study is to improve the capacity of Navier-Stokes and DSMC codes to predict high Mach number interactions. Pressure and heat flux have been measured at the wall and compared with the results obtained with Navier-Stokes and DSMC codes. Two different meshes have been considered with the three Navier-Stokes codes used. The conclusion is that the codes can give a good evaluation of the physical quantities at the wall.

The wall pressure signature of transonic shock/boundary layer interaction

The structure of wall pressure fluctuations beneath a turbulent boundary layer interacting with a normal shock wave at Mach number M ∞ = 1.3 is studied exploiting a direct numerical simulation database. Upstream of the interaction, in the zero-pressure-gradient region, pressure statistics compare well with canonical low-speed boundary layers in terms of fluctuation intensities, space―time correlations, convection velocities and frequency spectra. Across the interaction zone, the root-mean-square wall pressure fluctuations attain very large values (in excess of 162 dB), with a maximum increase of about 7 dB from the upstream level. The two-point wall pressure correlations become more elongated in the spanwise direction, indicating an increase of the pressure-integral length scales, and the convection velocities (determined from space―time correlations) are reduced. The interaction qualitatively modifies the shape of the frequency spectra, causing enhancement of the low-frequency Fourier modes and inhibition of the higher ones. In the recovery region past the interaction, the pressure spectra collapse very accurately when scaled with either the free-stream dynamic pressure or the maximum Reynolds shear stress, and exhibit distinct power-law regions with exponent ―7/3 at intermediate frequencies and ―5 at high frequencies. An analysis of the pressure sources in the Lighthills equation for the instantaneous pressure has been performed to understand their contributions to the wall pressure signature. Upstream of the interaction the sources are mainly located in the proximity of the wall, whereas past the shock, important contributions to low-frequency pressure fluctuations are associated with long-lived eddies developing far from the wall.

Shock wave–thermal inhomogeneity interactions: Analysis and numerical simulations of sound generation

In the present paper the interaction of a moving planar shock wave with a cylindrical thermal inhomogeneity is analyzed. Due to the interaction, both vorticity and sound waves are generated. The main focus of the paper is on the generation and propagation of the acoustic waves and their dependence on shock strength and inhomogeneity intensity. An acoustic analogy of the problem has been formulated as a basis of comparison for the structure of the acoustic pressure field in the limit of either weak shocks or weak inhomogeneities. An extensive numerical study has been carried out in order to understand the influence of both shock strength and intensity of the inhomogeneity. The results show that the acoustic field obeys a two-stage evolution that occurs on a time scale that is independent of shock and inhomogeneity intensities. The comparison of the computed far-field pressure with the theoretical result shows that for weak inhomogeneities the sound is well predicted by the acoustic analogy regardless of th...

On the dynamical relevance of coherent vortical structures in turbulent boundary layers

The dynamical relevance of vortex tubes and vortex sheets in a wall-bounded supersonic turbulent flow at Mach number M = 2 and Reynolds number Re θ ≈ 1350 is quantitatively analysed. The flow in the viscous sublayer and in the buffer region is characterized by intense, elongated vorticity tongues forming a shallow angle with respect to the wall, whose characteristic length is O (200) wall units and whose size in the cross-stream direction is O (50) wall units. The formation of vortex tubes takes place starting from y + ≈ 10, and it is mainly associated with the roll-up and the interaction of vortex sheets. The analysis of the non-local dynamical effect of tubes and sheets suggests that the latter have a more important collective effect, being closely associated with low-speed streaks, and being responsible for a substantial contribution to the mean momentum balance and to the production of turbulence kinetic energy and enstrophy.

Interaction of a shock wave with two counter-rotating vortices: Shock dynamics and sound production

In the present paper the interaction of a planar shock wave with a pair of counter-rotating vortices is studied by means of extensive numerical simulations. The main objective of the study is to characterize the shock pattern and the nature of the acoustic field. For the purpose of shedding some light into the mechanism of sound generation, the acoustic analogy developed by the present authors for the interaction of a shock wave with an isolated vortex has been extended to the case of a vortex pair, so as to obtain a closed form solution for the acoustic far field pressure in the limit of weak vortices. The effect of various parameters, such as the shock and vortex strengths and the initial vortex separation distance has been investigated, and their influence on the dynamics has been assessed. In addition, we have also considered both the case of passing and colliding pairs. In the case of a colliding pair five types of interactions are found to occur, depending upon the value of the interaction parameter...