Matteo Bernardini

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.

Probing high-Reynolds-number effects in numerical boundary layers

We study the high-Reynolds-number behavior of a turbulent boundary layer in the low supersonic regime through very-large-scale direct numerical simulation (DNS). For the first time a Reynolds number is attained in DNS (Reτ=δ/δv≈4000, where δ is the boundary layer thickness and δv is the viscous length scale) at which theoretical predictions and experiments suggest the occurrence of phenomena pertaining to the asymptotic Reynolds number regime. From comparison with previous DNS data at lower Reynolds number we find evidence of a continuing trend toward a stronger imprint of the outer-layer structures onto the near-wall region. This effect is clearly manifested both in flow visualizations, and in energy spectra. More than a decade of nearly-logarithmic variation is observed in the mean velocity profiles, with log-law constants k ≈ 0.394, C ≈ 4.84, and a trend similar to experiments. We find some supporting evidence for the debated existence of a k−1 region in the power spectrum of streamwise velocity fluctu...

Inner/outer layer interactions in turbulent boundary layers: A refined measure for the large-scale amplitude modulation mechanism

The amplitude modulation (AM) imparted by the outer layer large-scale motions on the near-wall turbulence is studied through direct numerical simulation of compressible boundary layer flow at moderate Reynolds number. Mathis et al. [J. Fluid Mech. 628, 311 (2009)] introduced an amplitude modulation coefficient to quantify this effect, whereby carrier and modulated signals are decoupled through a procedure based on the Hilbert transform of the streamwise velocity signals. However, Schlatter and Orlu [Phys. Fluids 22, 051704 (2010)] have recently shown that a non-zero amplitude modulation coefficient is closely associated to a non-zero value of the velocity skewness, and therefore, it does not necessarily reflect genuine physics. In this paper, the analysis is extended through systematic use of the two-point amplitude modulation correlation, which is shown to be a refined measure of the top-down influence of large-scale outer events on the inner part of the boundary layer.

Wall pressure fluctuations beneath supersonic turbulent boundary layers

The structure of the wall pressure field beneath supersonic adiabatic turbulent boundary layers is analyzed by means of direct numerical simulations at free-stream Mach number M∞ = 2, 3, 4, spanning a relatively large range of Reynolds numbers. The data reported in the paper, which include wall pressure fluctuations intensities, frequency spectra, space-time correlations, and convection velocities, show that when pressure is scaled by the wall shear stress, most statistics well conform to low-speed findings, contradicting the conclusions of previous experimental studies. Genuine compressibility effects are found to provide a small contribution to the magnitude of the wall pressure fluctuations, their influence being limited to the high-frequency end of the spectra, where a systematic increase with the Mach number is observed.

A general strategy for the optimization of Runge-Kutta schemes for wave propagation phenomena

We analyze optimized explicit Runge-Kutta schemes (RK) for computational aeroacoustics, and wave propagation phenomena in general. Exploiting the analysis developed in [S. Pirozzoli, Performance analysis and optimization of finite-difference schemes for wave propagation problems, J. Comput. Phys. 222 (2007) 809-831], we rigorously evaluate the performance of several time integration schemes in terms of appropriate error and cost metrics, and provide a general strategy to design Runge-Kutta methods tailored for specific applications. We present families of optimized second- and third-order Runge-Kutta schemes with up to seven stages, and describe their implementation in the framework of Williamsons 2N-storage formulation [J.H. Williamson, Low-storage Runge-Kutta schemes, J. Comput. Phys. 35 (1980) 48-56]. Numerical simulations of the 1D linear advection equation and of the 2D linearized Euler equations are performed to demonstrate the validity of the theory and to quantify the improvement provided by optimized schemes.

GPU accelerated flow solver for direct numerical simulation of turbulent flows

Graphical processing units (GPUs), characterized by significant computing performance, are nowadays very appealing for the solution of computationally demanding tasks in a wide variety of scientific applications. However, to run on GPUs, existing codes need to be ported and optimized, a procedure which is not yet standardized and may require non trivial efforts, even to high-performance computing specialists. In the present paper we accurately describe the porting to CUDA (Compute Unified Device Architecture) of a finite-difference compressible Navier-Stokes solver, suitable for direct numerical simulation (DNS) of turbulent flows. Porting and validation processes are illustrated in detail, with emphasis on computational strategies and techniques that can be applied to overcome typical bottlenecks arising from the porting of common computational fluid dynamics solvers. We demonstrate that a careful optimization work is crucial to get the highest performance from GPU accelerators. The results show that the overall speedup of one NVIDIA Tesla S2070 GPU is approximately 22 compared with one AMD Opteron 2352 Barcelona chip and 11 compared with one Intel Xeon X5650 Westmere core. The potential of GPU devices in the simulation of unsteady three-dimensional turbulent flows is proved by performing a DNS of a spatially evolving compressible mixing layer.

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.

Parameterization of Boundary-Layer Transition Induced by Isolated Roughness Elements

The laminar-to-turbulent transition of boundary layers induced by isolated three-dimensional roughness elements is analyzed by mining a direct numerical simulation database, which covers the variation of many physical parameters, including Mach and Reynolds numbers, and obstacle shape and size. It is found that the transition process is approximately controlled by a Reynolds number based on the momentum deficit past the obstacle, which is proportional to the classical roughness Reynolds number and which approximately incorporates the effects of the roughness element shape. The analysis of the perturbation energy past the obstacle shows that the varicose mode of instability is always dominant in the close proximity of the obstacle, and it promotes transition in supercritical flow cases. On the other hand, the sinuous mode appears to dominate the evolution of marginally subcritical cases, which feature quasi-steady momentum streaks.

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.