Turbulence spectra

The scaling invariance of the Navier-Stokes equations in the limit of infinite Reynolds number is used to derive laws for the inertial range of the turbulence spectrum. Whether the flow is homogeneous or not, the spectrum is chosen to be that given by a well-chosen biorthogonal decomposition. If the flow is hoogeneous, this spectrum coincides with the classical Fourier (energy) spectrum which exhibits Kolmogorov's k^–5/3 power law if the scaling exponent is assumed to be 1/3. In the more general case where the homogeneity assumption is relaxed, the spectrum is discrete and decays exponentially fast under the assumption that the flow is invariant (in a deterministic or statistical sense) under only one subgroup of the scaling coefficient of one scaling group of the equations (corresponding to one value of the scaling exponent). If the flow is invariant under two subgroups of scaling coefficients and, the spectrum becomes maximal, equal toR ₊. Finally, when a full symmetry, namely an invariance under a whole group, is assumed and the spectrum becomes continuous, the decaying law for the spectral density is derived and found to be independent of the specific value ofh These ideas are then applied to locally self-similar flows with multiple dilation centers (localized in space and time) and multiple scaling exponents, extending the concept of multifractals to space and time.     

Turbulence noise

We show that the large-eddy motions in turbulent fluid flow obey a modified hydrodynamic equation with a stochastic turbulent stress whose distribution is a causal functional of the large-scale velocity field itself. We do so by means of an exact procedure of statistical filtering of the Navier-Stokes equations, which formally solves the closure problem, and we discuss the relation of our analysis with the decimation theory of Kraichnan. We show that the statistical filtering procedure can be formulated using field-theoretic path-integral methods within the Martin-Siggia-Rose (MSR) formalism for classical statistical dynamics. We also establish within the MSR formalism a least-effective-action principle for mean turbulent velocity profiles, which generalizes Onsager's principle of least dissipation. This minimum principle is a consequence of a simple realizability inequality and therefore holds also in any realizable closure. Symanzik's theorem in field theory—which characterizes the static effective action as the minimum expected value of the quantum Hamiltonian over all state vectors with prescribed expectations of fields—is extended to MSR theory with non-Hermitian Hamiltonian. This allows stationary mean velocity profiles and other turbulence statistics to be calculated variationally by a Rayleigh-Ritz procedure. Finally, we develop approximations of the exact Langevin equations for large eddies, e.g., a random-coupling DIA model, which yield new stochastic LES models. These are compared with stochastic subgrid modeling schemes proposed by Rose, Chasnov, Leith, and others, and various applications are discussed.     

Trapped Electron Turbulence

The effect of trapped electron turbulence on the dynamics of a typical unstable mode is investigated. We establish that its principal effects are turbulence-induced additional collision frequencies for the trapped electrons and transit ions both of which are stabilizing. The resulting nonlinearly saturated fluctuation level is found to be considerably smaller than other estimates. The corresponding estimates of anomalous particle and thermal diffusion are also smaller than previous results.     

Diffusion by Turbulence

The turbulent diffusivity K of the atmosphere has first been studied by Richardson 1926 who empirically found that K depends on the scale l, K = Al^α, with α = 4/3 and A = 0.6 cm^2/3 s^?1. This empirical scaling law is derived here from a unified theory (based on the Navier-Stokes equation) together with an explicit result for the prefactor, A = 2.4 ?^1/3. The mean atmospheric dissipation rate compatible with this is ? = 0.016 cm² s^?3. For windtunnels with typical dissipation rates ? ? 1 m² s^?3 the turbulent diffusion coefficient should be K/cm² s^?1 = 52 (l/cm)^4/3.     

Role of Pressure in Turbulence

There is very limited knowledge of the kinematical relations for the velocity structure functions higher than three. Instead, the dynamical equations for the structure functions of the velocity increment are obtained from the Navier–Stokes equation under the assumption of the local homogeneity and isotropy. These equations contain the correlation between the velocity and pressure gradient increments, which is very difficult to know theoretically and experimentally. We have examined these dynamical relations by using direct numerical simulation data at very high resolution at large Reynolds numbers, and found that the contribution of the pressure term is important to the dynamics of the longitudinal velocity with large amplitudes. The pressure term is examined from the view point of the conditional average and the role of the pressure term in the turbulence dynamics is discussed.     

旋转体系中湍流的LBM模拟

采用基于多松弛时间因子的格子Boltzmann方法对旋转体系中的湍流进行数值研究,考察Rossby数和Ekman数对湍流的影响,包括湍流能量及其耗散率、速度、涡结构及湍流的耗散尺度即Kolmorogov尺度和积分尺度等.研究表明,系统的旋转延缓了湍流能量的衰减速率,逐步破坏初始涡结构的均匀性,与旋转方向相反的涡逐步被抑制,并最终形成若干与旋转同向的涡柱.结果还表明,系统旋转越快,湍流的耗散尺度越小而积分尺度越大.     

Non-Equilibrium Statistical Mechanics of Turbulence

The macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow for a suitable mechanical system (Ruelle, PNAS 109:20344–20346, 2012). Turbulent fluctuations (intermittency) then correspond to thermal fluctuations, and this allows to estimate the exponents \(\tau _p\) and \(\zeta _p\) associated with moments of dissipation fluctuations and velocity fluctuations. This approach, initiated in an earlier note (Ruelle, 2012), is pursued here more carefully. In particular we derive probability distributions at finite Reynolds number for the dissipation and velocity fluctuations, and the latter permit an interpretation of numerical experiments (Schumacher, Preprint, 2014). Specifically, if \(p(z)dz\) is the probability distribution of the radial velocity gradient we can explain why, when the Reynolds number \(\mathcal{R}\) increases, \(\ln p(z)\) passes from a concave to a linear then to a convex profile for large \(z\) as observed in (Schumacher, 2014). We show that the central limit theorem applies to the dissipation and velocity distribution functions, so that a logical relation with the lognormal theory of Kolmogorov (J. Fluid Mech. 13:82–85, 1962) and Obukhov is established. We find however that the lognormal behavior of the distribution functions fails at large value of the argument, so that a lognormal theory cannot correctly predict the exponents \(\tau _p\) and \(\zeta _p\) .     

Turbulence of a free surface

We study the free surface of a turbulent channel flow, in particular, the relation between the statistical properties of the wrinkled surface and those of the velocity field beneath it. For an irregular flow shed off a vertical cylinder, surface indentations are strongly correlated with vortices in the subsurface flow. For fully developed turbulence this correlation is dramatically reduced. This is because the large eddies excite random capillary-gravity waves that travel in all directions across the surface. Both their predominant wavelength and their anisotropy are determined by the subsurface turbulence.     

Gibbsian Hypothesis in Turbulence

We show that Kolmogorov multipliers in turbulence cannot be statistically independent of others at adjacent scales (or even a finite range apart) by numerical simulation of a shell model and by theory. As the simplest generalization of independent distributions, we suppose that the steady-state statistics of multipliers in the shell model are given by a translation-invariant Gibbs measure with a short-range potential, when expressed in terms of suitable spin variables: real-valued spins that are logarithms of multipliers and XY-spins defined by local dynamical phases. Numerical evidence is presented in favor of the hypothesis for the shell model, in particular novel scaling laws and derivative relations predicted by the existence of a thermodynamic limit. The Gibbs measure appears to be in a high-temperature, unique-phase regime with paramagnetic spin order.     

Turbulence without strange attractor

It is shown that pipe-flow turbulence consists of transients. The fractal dimensions of the dynamical process are thus all zero. Nevertheless, this is compatible with Grassberger-Procaccia analyses suggesting the existence of a high-dimensional strange attractor. The usefulness of the Grassberger-Procaccia method to detect the aging of transients is demonstrated.     

Turbulence Driven Tearing Modes

The effect of a turbulent background of density, electrostatic, and magnetic fluctuations on the growth of tearing modes is considered assuming three-wave interaction. It is found that, depending on the background conditions, the turbulence can cause anomalous electron resistivity or viscosity, leading to enhanced growth rate.     

Turbulence in nuclear reactions

An explanation of the emergence of turbulence in nuclear collisions is proposed on the basis of the assumption that, in an excited nucleus, there arise nonequilibrium steady-state distributions n(?) of occupation numbers.     

Path Lengths in Turbulence

By tracking tracer particles at high speeds and for long times, we study the geometric statistics of Lagrangian trajectories in an intensely turbulent laboratory flow. In particular, we consider the distinction between the displacement of particles from their initial positions and the total distance they travel. The difference of these two quantities shows power-law scaling in the inertial range. By comparing them with simulations of a chaotic but non-turbulent flow and a Lagrangian Stochastic model, we suggest that our results are a signature of turbulence.     

Taming Drift Wave Turbulence

Drift waves in magnetized plasmas often occur in a turbulent form and are often considered as being responsible for anomalous cross‐field particle transport. It is thus very appealing to achieve active control of the drift wave dynamics. A control scheme acting both in space and in time is developed to synchronize drift wave turbulence. It consists of an arrangement of eight electrodes (octupole exciter) in flush‐mounted geometry in the edge region of the magnetized plasma column. The electrodes of the octupole exciter are driven by sinusoidal signals. Between each two neighbouring electrodes, the phase shift of the exciter signal is kept fixed. It is demonstrated experimentally that the exciter signals have strong influence on the different drift wave states, i.e. the turbulent states can be synchronized to a single preselected drift wave mode. The efficiency of the spatiotemporal synchronization is sensitively dependent on both driver frequency and phase shift.