Third-order statistics and the dynamics of strongly anisotropic turbulent flows

Abstract

Anisotropy is induced by body forces and/or mean large-scale gradients in turbulent flows. For flows without energy production, the dynamics of second-order velocity or second-order vorticity statistics are essentially governed by triple correlations, which are at the origin of the anisotropy that penetrates towards the inertial range, deeply altering the cascade and the eventual dissipation process, with a series of consequences on the evolution of homogeneous turbulence statistics: in the case of rotating turbulence, the anisotropic spectral transfer slaves the multiscale anisotropic energy distribution; nonlinear dynamics are responsible for the linear growth in terms of t of axial integral length-scales; third-order structure functions, derived from velocity triple correlations, exhibit a significant departure from the 4/5 Kolmogorov law. We describe all these implications in detail, starting from the dynamical equations of velocity statistics in Fourier space, which yield third-order correlations at three points (triads) and allow the explicit removal of pressure fluctuations. We first extend the formalism to anisotropic rotating turbulence with 'production', in the presence of mean velocity gradients in the rotating frame. Second, we compare the spectral approach at three points to the twopoint approach directly performed in physical space, in which we consider the transport of the scalar second-order structure function (δq) 2 . This calls into play componental third-order correlations (δq) 2 δu (r) in axisymmetric turbulence. This permits to discuss inhomogeneous anisotropic effects from spatial decay, shear, or production, as in the central region of a rotating round jet. We show that the above-mentioned important statistical quantities can be estimated from experimental planar particle image velocimetry, and that explicit passage relations systematically exist between one-and two-point statistics in physical and spectral space for second-order tensors, but also sometimes for third-order tensors that are involved in the dynamics.

Decay and growth laws in homogeneous shear turbulence

“…Eddy-damping terms are kept isotropic, as done in previous EDQNM studies, [36][37][38] which showed that this was a reasonable assumption in the absence of rotation, which is the case here. More details about this assumption are given in Appendix 1, and such a choice presents also the advantage of not including new free parameters in the model.…”

“…Consequently, there is no guarantee that this model would work for rotating mean flows for instance: indeed, such a configuration involves turbulent waves which alter the third-order correlations. [37] Therefore, further investigations are needed, especially to fully understand the impact of rotation on the eddy-damping terms and its consequences on the RTI process.…”

A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically averaged descriptors

“…The derivation of the above expressions may be found in Cambon et al (2013), but with an error of sign in front of the rotation terms in (2.13). The nonlinear transfer terms on the right-hand sides of (2.12) and (2.13) are obtained by applying the (E , Z) decomposition to the transfer term T ij (k, t) in (2.3): 16) where the tensor τ ij (k, t) is defined by equations (2.5) and (2.6).…”

“…A more general representation, say (E , Z, H )(k, t), first introduced by Cambon & Jacquin (1989), also includes a helicity contribution generated by a helicity spectrum H (k, t). The helicity spectrum, which is associated with the imaginary part ofR ij (k, t), is not considered here because it remains zero in homogeneous turbulence if it is not initialized or forced, usually in an unphysical way (see also the trivial helicity equation in Cambon & Jacquin (1989) and Cambon et al (2013)). In addition to their mathematical origin, the recourse to the two different terms E (k, t) and Z(k, t) has physical meaning.…”

Statistical analysis of the velocity and scalar fields in reacting turbulent wall-jets

“…the so-called structure functions [7]. Indeed, the behavior of the higher order moments of small-scale fluctuations in turbulent flows has been the subject of both numerical and experimental studies for decade [8][9][10][11][12][13][14][15]. It is crucial to realize that not all possible structure functions are sensitive to anisotropy in the same way.…”