Turbulence structure resulting from multi-fluid or multi-species, variable-density isotropic turbulence interaction with a Mach 2 shock is studied using turbulence-resolving shock-capturing simulations and Eulerian (grid) and Lagrangian (particle) methods. The complex roles density play in the modification of turbulence by the shock wave are identified. Statistical analyses of the velocity gradient tensor (VGT) show that the density variations significantly change the turbulence structure and flow topology. Specifically, a stronger symmetrization of the joint probability density function (PDF) of second and third invariants of the anisotropic velocity gradient tensor, PDF$(Q^\ast, R^\ast)$, as well as the PDF of the vortex stretching contribution to the enstrophy equation, are observed in the multi-species case. Furthermore, subsequent to the interaction with the shock, turbulent statistics also acquire a differential distribution in regions having different densities. This results in a nearly symmetrical PDF$(Q^\ast, R^\ast)$ in heavy fluid regions, while the light fluid regions retain the characteristic tear-drop shape. To understand this behavior and the return to "standard" turbulence structure as the flow evolves away from the shock, Lagrangian dynamics of the VGT and its invariants are studied by considering particle residence times and conditional particle variables in different flow regions. The pressure Hessian contributions to the VGT invariants transport equations are shown to be not only affected by the shock wave, but also by the density in the multi-fluid case, making them critically important to the flow dynamics and turbulence structure.
Video highlighting multiple features of the interaction between isotropic turbulence and a nominally planar shock wave, normal to the mean flow direction. The visualizations showcased in this video were obtained from a Direct Numerical Simulation.
The Quasi-Spectral Viscosity (QSV) method is a novel closure for a high-order finite-difference discretization of the filtered compressible Navier-Stokes equations capable of unifying dynamic sub-filter scale (SFS) modeling and shock capturing under a single mathematical framework. Its innovation lies in the introduction of a physical-space implementation of a spectral-like SFS dissipation term by leveraging residuals of filter operations, achieving two goals: (1) estimating the energy of the resolved solution near the grid cutoff; (2) imposing a plateau-cusp shape to the spectral distribution…
An initially planar shock wave propagating into a medium of non-uniform density will be perturbed, leading to the generation of post-shock velocity perturbations. Using numerical simulations we study this phenomenon in the case of highly-non-uniform density (order-unity normalized variance, $σ_ρ/\overlineρ \sim 1$) and strong shocks (shock Mach numbers $\overline{M}_s \gtrsim 10$). This leads to a highly disrupted shock and a turbulent post-shock flow. We simulate this interaction for a range of shock drives and initial density configurations meant to mimic those which might be presently achie…
The dynamics of the multi-dimensional randomly forced Burgers equation is studied in the limit of vanishing viscosity. It is shown both theoretically and numerically that the shocks have a universal global structure which is determined by the topology of the configuration space. This structure is shown to be particularly rigid for the case of periodic boundary conditions.
The aim of the present paper is to provide a comparison between several finite-volume methods of different numerical accuracy: second-order Godunov method with PPM interpolation and high-order finite-volume WENO method. The results show that while on a smooth problem the high-order method perform better than the second-order one, when the solution contains a shock all the methods collapse to first-order accuracy. In the context of the decay of compressible homogeneous isotropic turbulence with shocklets, the actual overall order of accuracy of the methods reduces to second-order, despite the u…
Turbulence in the interstellar medium (ISM) is crucial in the process of star formation. Shocks produced by supernova explosions, jets, radiation from massive stars, or galactic spiral-arm dynamics are amongst the most common drivers of turbulence in the ISM. However, it is not fully understood how shocks drive turbulence, in particular whether shock driving is a more solenoidal(rotational, divergence-free) or a more compressive (potential, curl-free) mode of driving turbulence. The mode of turbulence driving has profound consequences for star formation, with compressive driving producing thre…
We report on the experimental observation of a transition from a dispersive wave turbulence regime to a nondispersive regime involving shock waves on the surface of a fluid. We use a magnetic fluid in a canal subjected to an external horizontal magnetic field to tune the dispersivity of the system. For a low magnetic field, gravity-capillary wave turbulence is observed, whereas for a high enough field, random steep coherent structures arise which are found to be shock waves. These shock waves create singularities in the second-order difference of the surface elevation, leading to an $ω^{-4}$ f…
1-D scalar conservation laws with convex flux and Markov initial data are now known to yield a completely integrable Hamiltonian system. In this article, we rederive the analogue of Loitsiansky's invariant in hydrodynamic turbulence from the perspective of integrable systems. Other relevant physical notions such as energy dissipation and spectrum are also discussed.
When a planar shock wave interacts with a random pattern of pre-shock density non-uniformities, it generates an anisotropic turbulent velocity/vorticity field. This turbulence plays an important role at the early stages of the mixing process in the compressed fluid. This situation emerges naturally in shock interaction with weakly inhomogeneous deuterium-wicked foam targets in Inertial Confinement Fusion (ICF) and with density clumps/clouds in astrophysics. We present an exact small-amplitude linear theory describing such interaction. It is based on the exact theory of time and space evolution…
We consider two cases of interaction between a planar shock and a cylindrical density interface. In the first case (planar normal shock), the axis of the gas cylinder is parallel to the shock front, and baroclinic vorticity deposited by the shock is predominantly two-dimensional (directed along the axis of the cylinder). In the second case, the cylinder is tilted, resulting in an oblique shock interaction, and a fully three-dimensional shock-induced vorticity field. The statistical properties of the flow for both cases are analyzed based on images from two orthogonal visualization planes, usin…
In the Part 1 of the present paper the performance of several different low and high-order finite-volume methods were assessed by investigating how well they can capture the turbulent spectra of a compressible flow where small smooth turbulent structures interact with shocks and discontinuities. The comparisons showed that a second-order Godunov method with PPM interpolation provides results virtually the same as a fourth-order WENO scheme but at a significant lower cost. However, it is shown that the PPM method fails to provide an accurate representation in the high-frequency range of the spe…
In this paper, we introduce a relaxation filtering closure approach to account for subgrid scale effects in explicitly filtered large eddy simulations using the concept of anisotropic diffusion. We utilize the Perona-Malik diffusion model and demonstrate its shock capturing ability and spectral performance for solving the Burgers turbulence problem, which is a simplified prototype for more realistic turbulent flows showing the same quadratic nonlinearity. Our numerical assessments present the behavior of various diffusivity functions in conjunction with a detailed sensitivity analysis with res…
Large-scale turbulence in fluid layers and other quasi-two-dimensional compressible systems consists of planar vortices and waves. Separately, wave turbulence usually produces a direct energy cascade, while solenoidal planar turbulence transports energy to large scales by an inverse cascade. Here, we consider turbulence at finite Mach numbers when the interaction between acoustic waves and vortices is substantial. We employ solenoidal pumping at intermediate scales and show how both direct and inverse energy cascades are formed starting from the pumping scale. We show that there is an inverse …
We consider a three-dimensional acoustic field of an ideal gas in which all entropy production is confined to weak shocks and show that similar scaling relations hold for such a field as for forced Burgers turbulence, where the shock amplitude scales as $ (εd)^{1/3} $ and the $ p $:th order structure function scales as $ (εd)^{p/3} r/d$, $ ε$ being the mean energy dissipation per unit mass, $ d $ the mean distance between the shocks and $ r $ the separation distance. However, for the acoustic field $ ε$ should be replaced by $ ε+ χ$, where $ χ$ is associated with entropy production due to heat…
Small-scale fluctuating magnetic fields of order $n$G are observed in supernova shocks and galaxy clusters, where its amplification is likely caused by the Biermann battery mechanism. However, these fields cannot be amplified further without the turbulent dynamo, which generates magnetic energy through the stretch-twist-fold (STF) mechanism. Thus, we present here novel three-dimensional magnetohydrodynamic (MHD) simulations of a laser-driven shock propagating into a stratified, multiphase medium, to investigate the post-shock turbulent magnetic field amplification via the turbulent dynamo. The…
Turbulence models attempt to account for unresolved dynamics and diffusion in hydrodynamical simulations. We develop a common framework for two-equation Reynolds-Averaged Navier-Stokes (RANS) turbulence models, and we implement six models in the Athena code. We verify each implementation with the standard subsonic mixing layer, although the level of agreement depends on the definition of the mixing layer width. We then test the validity of each model into the supersonic regime, showing that compressibility corrections can improve agreement with experiment. For models with buoyancy effects, we …
The interaction between an incident shock wave and a Mach-6 undisturbed hypersonic laminar boundary layer over a cold wall is addressed using direct numerical simulations (DNS) and wall-modeled large-eddy simulations (WMLES) at different angles of incidence. At sufficiently high shock-incidence angles, the boundary layer transitions to turbulence via breakdown of near-wall streaks shortly downstream of the shock impingement, without the need of any inflow free-stream disturbances. The transition causes a localized significant increase in the Stanton number and skin-friction coefficient, with h…
Nonlinear Richtmyer--Meshkov instability and mixing transition induced by a Mach 1.45 shock and subsequent re-shock at an interface between two ideal gases (sulfur hexafluoride and air) with high Atwood number are studied with second-moment analysis using data from high-resolution compressible Navier--Stokes simulations. The analysis first addresses the importance of two second-order moments: turbulent mass flux and density-specific-volume covariance, together with their transport equations. These quantities play an essential role in the development of Favre-averaged Reynolds stress and turbul…
Effects of two types of shock topology, namely, small-scale shocklet and large-scale shock wave, on the statistics of temperature in compressible turbulence were investigated by simulations. The shocklet and shock wave are caused by the solenoidal and compressive modes of driven forces, respectively. Hereafter, the related two flows are called as SFT and CFT, respectively. It shows that in SFT the temperature spectrum follows the k^-5/3 power law, and the temperature field has "ramp-cliff" structures. By contrast, in CFT the temperature spectrum obeys the k^-2 power law, and the temperature fi…
We introduce corrections to the Navier-Stokes equation arising from the transitions between molecular states and the injection of external energy. In the simplest application of the proposed post Navier-Stokes equation, we find a multi-valued velocity field and the immediate possibility of velocity reversal, both features of turbulence.
The divergence theorem of Gauss plays a central role in the derivation of the governing differential equations in fluid dynamics, electrodynamics, gravitational fields, and optics. One is often interested in an evolution equation for the large scale quantities without resolving the details of the small scales. As a result, there has been a significant effort in developing time-averaged and spatially-filtered equations for large scale dynamics from the fully resolved governing differential equations. One should realize that by starting from these fully-resolved equations (e.g. the Euler or Navi…
Shocks form the basis of our understanding for the density and velocity statistics of supersonic turbulent flows, such as those found in the cool interstellar medium (ISM). The variance of the density field, $σ^2_{ρ/ρ_0}$, is of particular interest for molecular clouds (MCs), the birthplaces of stars in the Universe. The density variance may be used to infer underlying physical processes in an MC, and parameterises the star formation (SF) rate of a cloud. However, models for $σ^2_{ρ/ρ_0}$ all share a common feature -- the variance is assumed to be isotropic. This assumption does not hold when …
We propose a two-population lattice Boltzmann model on standard lattices for the simulation of compressible flows. The model is fully on-lattice and uses the single relaxation time Bhatnagar-Gross-Krook kinetic equations along with appropriate correction terms to recover the Navier-Stokes-Fourier equations. The accuracy and performance of the model are analyzed through simulations of compressible benchmark cases including Sod shock tube, sound generation in shock-vortex interaction and compressible decaying turbulence in a box with eddy shocklets. It is demonstrated that the present model prov…
Colliding hypersonic flows play a decisive role in many astrophysical objects. In this paper, we look at the idealized model of a 2D plane parallel isothermal slab (CDL) and at symmetric settings, where both flows have equal parameters. We performed a set of high-resolution simulations with upwind Mach numbers, 5 < M_u < 90. We find that the CDL is irregularly shaped and has a patchy and filamentary interior. The size of these structures increases with l_cdl, the extension of the CDL. On average, but not at each moment, the solution is about self-similar and depends only on M_u. We find the ro…
The behavior of the one-dimensional random-force-driven Burgers equation is investigated in the path integral formalism on a discrete space-time lattice. We show that by means of Monte Carlo methods one may evaluate observables, such as structure functions, as ensemble averages over different field realizations. The regularization of shock solutions to the zero-viscosity limit (Hopf-equation) eventually leads to constraints on lattice parameters required for the stability of the simulations. Insight into the formation of localized structures (shocks) and their dynamics is obtained.
The explosion of massive stars in core-collapse supernovae may be aided by the convective instabilities that develop in their innermost nuclear burning shells. The resulting fluctuations support the explosion by generating additional turbulence behind the supernova shock. It was suggested that the buoyant density perturbations arising from the interaction of the pre-collapse asymmetries with the shock may be the primary contributor to the enhancement of the neutrino-driven turbulent convection in the post-shock region. Employing three-dimensional numerical simulations of a toy model, we invest…
