We report a detailed study of Eulerian and Lagrangian statistics from high resolution Direct Numerical Simulations of isotropic weakly compressible turbulence. Reynolds number at the Taylor microscale is estimated to be around 600. Eulerian and Lagrangian statistics is evaluated over a huge data set, made by $1856^3$ spatial collocation points and by 16 million particles, followed for about one large-scale eddy turn over time. We present data for Eulerian and Lagrangian Structure functions up to ten order. We analyse the local scaling properties in the inertial range and in the viscous range. Eulerian results show a good superposition with previous data. Lagrangian statistics is different from existing experimental and numerical results, for moments of sixth order and higher. We interpret this in terms of a possible contamination from viscous scale affecting the estimate of the scaling properties in previous studies. We show that a simple bridge relation based on Multifractal theory is able to connect scaling properties of both Eulerian and Lagrangian observables, provided that the small differences between intermittency of transverse and longitudinal Eulerian structure functions are properly considered.
An explicit analitical description of the compressible turbulence, based on the exact solution of the one-dimensional Euler equations in the unbounded case is obtained. The Onsager dissipative anomaly is resolved. The exact universal -8/3 energy spectrum corresponding to the finite time collapse of Euler's solution is stated. That spectrum is relevant to strong acoustic turbulence and observation in the space turbulence.
We derive from first principles analytic relations for the second and third order moments of the velocity gradient mij = dui/dxj in compressible turbulence, which generalize known relations in incompressible flows. These relations, although derived for homogeneous flows, hold approximately for a mixing layer. We also discuss how to apply these relations to determine all the second and third moments of the velocity gradient experimentally.
Liouville Spectral Gap and Bifurcation Driven Lagrangian Eulerian Decoupling with Nondiffusive Turbulence Closures
"La Sapienza" University, Dipartimento di Ingegneria Meccanica e Aerospaziale, Via Eudossiana, 18, 00184 Rome, Italy, phone: +39--0644585268, \ \ fax: +39--0644585750, e-mail: , \ \
In our recent works we proposed a theory of turbulence in inertial gas flow via the mean field effect of an intermolecular potential. We found that, in inertial flow, turbulence indeed spontaneously develops from a laminar initial condition, just as observed in nature and experiments. However, we also found that density and temperature in our inertial flow model behave unrealistically. The goal of the current work is to demonstrate technical possibility of modeling compressible, turbulent flow at low Mach number where both density and temperature behave in a more realistic fashion. Here we foc…
We use particle tracking velocimetry to study Eulerian and Lagrangian second-order statistics of superfluid $^4$He grid turbulence. The Eulerian energy spectra at scales larger than the mean distance between quantum vortex lines behave classically with close to Kolmogorov-1941 scaling and are almost isotropic. The Lagrangian second-order structure functions and frequency power spectra, measured at scales comparable with the intervortex distance, demonstrate a sharp transition from nearly-classical behavior to a regime dominated by the motion of quantum vortex lines. Employing the homogeneity o…
The accelerations of tracer and light particles in compressible homogeneous isotropic turbulence (CHIT) is investigated by using data from direct numerical simulations (DNS) up to turbulent Mach number $M_t =1$. For tracer particles, the flatness factor of acceleration components, $F_a$, increases gradually for $M_t \in [0.3, 1]$. On the contrary, $F_a$ for light particles develops a maximum around $M_t \sim 0.6$. The PDF of longitudinal acceleration of tracers is increasingly skewed towards the negative value as $M_t$ increases. By contrast, for light particles, the skewness factor of longitu…
A positivity-preserving fractional algorithm is presented for solving the four-equation homogeneous relaxation model (HRM) with an arbitrary number of ideal gases and a liquid governed by the stiffened gas equation of state. The fractional algorithm consists of a time step of the hyperbolic five-equation model by Allaire et al. and an algebraic numerical thermal relaxation step at an infinite relaxation rate. Interpolation and flux limiters are proposed for the use of high-order Cartesian finite difference or finite volume schemes in a general form such that the positivity of the partial densi…
We study the properties of various Eulerian contributions to fluid particle acceleration by using well-resolved direct numerical simulations of isotropic turbulence, with the grid resolution as high as $12288^3$ and the Taylor-scale Reynolds number $R_λ$ in the range between 140 and 1300. The variance of convective acceleration, when normalized by Kolmogorov scales, increases linearly with $R_λ$, consistent with simple theoretical arguments, but very strongly differing from phenomenological predictions of Kolmogorov's hypothesis as well as Eulerian multifractal models. The scaling of the local…
An analytic model for steady state turbulence is employed to obtain the inertial range power spectrum of compressible turbulence. We assume that for homogeneous turbulence, the timescales controlling the energy injected at a given wavenumber from all smaller wave-numbers, are equal for each spatial component. However, the longitudinal component energy is diverted into compression, so the rate controlling the energy that is transferred to all larger wave-numbers by the turbulent viscosity is reduced. The resulting inertial range is a power law with index -2. Indeed such power spectra were obser…
For the first time, the non-equilibrium time-relaxation kinetic model (NTRKM) is proposed for compressible turbulence modeling on unresolved grids. Within the non-equilibrium time-relaxation framework, NTRKM is extended in the form of modified Bhatnagar-Gross-Krook model. Based on the first-order Chapman-Enskog expansion, NTRKM connects with the six-variable macroscopic governing equations. The first five governing equations correspond to the conservative laws in mass, momentum and total energy, while the sixth equation governs the evolution of unresolved turbulence kinetic energy Kutke. The u…
We study conformal invariance of vorticity clusters in weakly compressible two-dimensional turbulence at low Mach numbers. On the basis of very high resolution direct numerical simulation we demonstrate the scaling invariance of the inverse cascade with scaling close to Kolmogorov prediction. In this range of scales, the statistics of zero-vorticity isolines are found to be compatible with those of critical percolation, thus generalizing the results obtained in incompressible Navier-Stokes turbulence.
The scaling of acceleration statistics in turbulence is examined by combining data from the literature with new data from well-resolved direct numerical simulations of isotropic turbulence, significantly extending the Reynolds number range. The acceleration variance at higher Reynolds numbers departs from previous predictions based on multifractal models, which characterize Lagrangian intermittency as a naive extension of Eulerian intermittency. The disagreement is even more prominent for higher-order moments of the acceleration. Instead, starting from a known exact relation, we relate the sca…
We study the correlation function and mean linear response function of the velocity Fourier mode of statistically steady-state, homogeneous and isotropic turbulence in the Eulerian and Lagrangian coordinates through direct numerical simulation (DNS). As the Lagrangian velocity, we here adopt Kraichnan's Lagrangian history framework where Lagrangian particles are labelled with current positions and their velocity are measured at some time before. This Lagrangian velocity is numerically calculated with a method known as passive vector method. Our first goal is to study relation between the corre…
