Sergio Pirozzoli

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 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.

On the spectral properties of shock-capturing schemes

In this short note we analyze the performance of nonlinear, shock-capturing schemes in wavenumber space. For this purpose we propose a new representation for the approximate dispersion relation which accounts to leading order for nonlinear effects. Several examples are presented, which confirm that the present theory yields an improved qualitative representation of the true solution behavior compared to conventional representations. The theory can provide useful guidance for the choice of the most cost-effective schemes for specific applications, and may constitute a basis for the development of optimized ones.

Generalized conservative approximations of split convective derivative operators

We propose a strategy to design locally conservative finite-difference approximations of convective derivatives for shock-free compressible flows with arbitrary order of accuracy, that generalizes the approach of Ducros et al. (2000) [1], and that can be applied as a building block of low-dissipative, hybrid shock-capturing methods. The approximations stem from application of standard central difference formulas to split forms of the convective terms in the compressible Euler equations, which guarantee strong numerical stability and (near) energy preservation in the inviscid limit. A convenient implementation of the high-order fluxes is suggested, which guarantees improved computational efficiency over existing methods. Numerical tests performed for isotropic turbulence at zero viscosity show stability of schemes with order of accuracy up to ten, and effectiveness of convective splitting of Kennedy and Gruber (2008) [2] in providing extra stability in the presence of strong density variations. Numerical simulations of compressible turbulent boundary layer flow indicate suitability of the method for non-uniform grids, and overall support superior computational efficiency of high-order schemes.

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.