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We consider the large-scale structure of freely decaying incompressible homogeneous anisotropic helical turbulence, whose energy spectrum E(k) is given by E(k)=Ck^2+o(k^2) at k→0. Here k=|k|,k is the wave vector, and C is a dynamical invariant. The helicity spectrum H(k) is given by H(k)=Chk^3+o(k^3) at k→0, where Ch is in general nonzero in helical turbulence. By generalizing Saffman's argument for nonhelical turbulence [Saffman, J. Fluid Mech. 27, 581 (1967)] to helical turbulence, it is shown that Ch is another dynamical invariant. We present a theoretical analysis based on the time independence of the O(k^0) term of the velocity correlation spectral tensor at k→0 and a self-similarity assumption of flow evolution at large scales including the energy containing range scales. The analysis suggests that if the O(k^0) term is reflection asymmetric at an initial instant, the turbulence does not relax to any reflection symmetric state at the large scales. A simple dimensional analysis yields the decay rates of the helicity and kinetic energy in the fully developed turbulence state. The theoretical results agree with results obtained by direct numerical simulation of incompressible helical turbulence in a periodic box.

Nonlinear interactions between waves and turbulence cause systematic frequency shifts in Rossby waves. The frequency shifts in the inertial subranges of statistically steady β-plane turbulence were examined theoretically and numerically. The theoretical analysis is based on the Lagrangian closure called the Lagrangian renormalized approximation and predicts that when the β effect is small, the frequency shifts of Rossby waves are proportional to k_x /k^{4/3} in the inverse energy transfer range, while they are proportional to k_x with or without a log-correction term in the enstrophy transfer range, depending on the flow conditions, where k_x is the wave number in the eastward direction. Numerical simulations using 1024^2 grid points of forced β-plane turbulence that exhibit the inertial subranges, show fairly good agreement with theoretical predictions.

Numerical simulations of two-dimensional forced turbulence suggest that the enstrophy transfer range energy spectrum E(k) a little steeper than k^-3 is robust in the sense that it may be realized in a wave number range under different run conditions. It is shown that such energy spectra fit well E(k)=C_Kη^{2/3} k^-3[ln(k/k_1)]^{-1/3}, where C_K is a dimensionless constant, η the enstrophy transfer rate per unit mass and k_1 a wave number at the bottom of the range. The simulations give C_K ≈1.9 in fairly good agreement with the existing theoretical estimates.

A spectral approximation for diffusion of passive scalar in stably and strongly stratified turbulence is presented. The approximation is based on a linearized approximation for the Eulerian two-time correlation and Corrsin's conjecture for the Lagrangian two-time correlation. For strongly stratified turbulence, the vertical component of the turbulent velocity field is well approximated by a collection of Fourier modes (waves) each of which oscillates with a frequency depending on the direction of the wavevector. The proposed approximation suggests that the phase mixing among the Fourier modes having different frequencies causes the decay of the Lagrangian two-time vertical velocity autocorrelation, and the highly oscillatory nature of these modes results in the suppression of single-particle dispersion in the vertical direction. The approximation is free from any ad hoc adjusting parameter and shows that the suppression depends on the spectra of the velocity and fluctuating density fields. It is in good agreement with direct numerical simulations for strongly stratified turbulence.

A class of exact solutions of the Navier–Stokes equations is derived. Each of them represents the velocity field v=U+u of a thin vortical layer (a planar jet) under a uniform strain velocity field U in three-dimensional infinite space, and provides a simple flow model in which nonlinear coupling between small eddies plays a key role in small-scale vortex dynamics. The small-scale structure of the velocity field is studied by numerically analysing the Fourier spectrum of u. It is shown that the Fourier spectrum of u falls off exponentially with wavenumber k for large k. The Taylor expansion in powers of the coordinate (say y) in the direction perpendicular to the vortical layer suggests that the solution may be well approximated by a function with certain poles in the complex y-plane. The Fourier spectrum based on the singularities is in good agreement with that obtained numerically, where the exponential decay rate is given by the distance of the poles from the real axis of y.

A study is made about the energy spectrum E(k) of turbulence on the basis of high-resolution direct numerical simulations (DNSs) of forced incompressible turbulence in a periodic box using a Fourier spectral method with the number of grid points and the Taylor scale Reynolds number Rλ up to 122883 and approximately 2300, respectively. The DNS data show that there is a wave-number range (approximately 5×10−3

International audience; Coherent vorticity simulation (CVS) is a multiscale method to compute incompressible turbulent flows based on the wavelet filtered Navier-Stokes equations. At each time step the vorticity field is decomposed into two orthogonal components using an orthogonal wavelet basis: the coherent vorticity, corresponding to the coefficients whose modulus is larger than a threshold, and the remaining incoherent vorticity. The threshold value only depends on the total enstrophy, which evolves in time, and on the maximal resolution, which remains constant. The induced coherent velocity is computed from the coherent vorticity using the Biot-Savart kernel. To compute the flow evolution one only retains the coherent wavelet coefficients and some of their neighbors in space, scale, and direction, which define the safety zone. Two different strategies are studied to minimize at each time step the number of degrees of freedom to be computed, either by increasing the threshold value or by reducing the width of the safety zone. Their efficiency is compared for a three-dimensional forced homogeneous isotropic turbulent flow at initial Taylor microscale Reynolds number Rλ=153. The quality of the results is assessed in comparison to a direct numerical simulation of the same flow. It is found that, as long as a safety zone is present, CVS well preserves the statistical predictability of the turbulent flow (even the vorticity and velocity probability distribution functions) with a reduced number of degrees of freedom.

A numerical study is made of the Lagrangian and Eulerian velocity autocorrelations RL and RE, and the eddy viscosity νeddy in two‐dimensional anisotropic turbulence with or without mean flow of simple shearing motion. A short‐time analysis is made of RL and RE in anisotropic turbulence, and also of νeddy in initially isotropic turbulence. The Lagrangian renormalized approximation is numerically integrated to compute RL and νeddy, and the results are compared with direct numerical simulations.

The question whether the vortex dynamics and structure at small scales have significant influence on the statistics at large scales is addressed on the basis of quantitative comparison of two turbulent fields. One is a reference field generated by direct numerical simulation of turbulence of an incompressible fluid obeying the Navier-Stokes (NS) equation in a periodic box. The other is an artificial field in which the coherent vortical structures at small scales (∼η) that could be formed by the NS dynamics in the reference field are destroyed by an artificial computational operation, where η is the Kolmogorov micro-length scale. The comparison of the two fields suggests that the statistics at larger scale (≫η) are not sensitive to the exact vortex dynamics and structure, at least in the case studied here.

The drag on a sphere in a random array of fixed spheres of volume concentraiton c (≪1)at Reynolds number R (≪1)is discussed. In the case when c and R^2 are of the same order of magnitude, the drag is determined up to terms of O (R).

The method of classical statistical mechanics is applied to an inviscid truncated model system of two‐dimensional turbulent shear flow. The idea of canonical equilibrium distribution is extended to treat a time‐dependent Liouville equation governing the evolution of probability distribution function in the phase space. Results of numerical simulations supporting the theoretical conjectures based on the canonical distribution are shown.