Energy transfer in rotating turbulence

Abstract

The influence of rotation on the spectral energy transfer of homogeneous turbulence is investigated in this paper. Given the fact that linear dynamics, e.g. the inertial waves regime found in an RDT (rapid distortion theory) analysis, cannot affect a homogeneous isotropic turbulent flow, the study of nonlinear dynamics is of prime importance in the case of rotating flows. Previous theoretical (including both weakly nonlinear and EDQNM theories), experimental and DNS (direct numerical simulation) results are collected here and compared in order to give a self-consistent picture of the nonlinear effects of rotation on turbulence.The inhibition of the energy cascade, which is linked to a reduction of the dissipation rate, is shown to be related to a damping of the energy transfer due to rotation. A model for this effect is quantified by a model equation for the derivative-skewness factor, which only involves a micro-Rossby number Roω=ω′/(2Ω) – ratio of r.m.s. vorticity and background vorticity – as the relevant rotation parameter, in accordance with DNS and EDQNM results.In addition, anisotropy is shown also to develop through nonlinear interactions modified by rotation, in an intermediate range of Rossby numbers (RoL<1 and Roω>1), which is characterized by a macro-Rossby number RoL based on an integral lengthscale L and the micro-Rossby number previously defined. This anisotropy is mainly an angular drain of spectral energy which tends to concentrate energy in the wave-plane normal to the rotation axis, which is exactly both the slow and the two-dimensional manifold. In addition, a polarization of the energy distribution in this slow two-dimensional manifold enhances horizontal (normal to the rotation axis) velocity components, and underlies the anisotropic structure of the integral length-scales. Finally a generalized EDQNM (eddy damped quasi-normal Markovian) model is used to predict the underlying spectral transfer structure and all the subsequent developments of classic anisotropy indicators in physical space. The results from the model are compared to recent LES results and are shown to agree well. While the EDQNM2 model was developed to simulate ‘strong’ turbulence, it is shown that it has a strong formal analogy with recent weakly nonlinear approaches to wave turbulence.

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

“…The realizability condition, or condition for the Hermitian covariance matrixR(k, t) to be definite-positive, can be written as (Cambon, Mansour & Godeferd 1997)…”

“…Here, λ, λ and λ are angles that characterize the rotation of the plane of the triad around k, p and q respectively; x, y and z refer to the cosines of the internal angles of the triangle formed by the triad. The above expressions for the transfer terms T (E ) (k, t) and T (Z) (k, t) are also derived in Cambon et al (1997) in a different way. Similarly, one can deduce a closed-form expression of the transfer term T (RTI) (k, t) defined by (2.18), which generates the 'slow' component of the pressure-strain rate tensor:…”

Experiments on rapidly rotating turbulent flows

“…The period of the oscillations is thus proportional to the Rossby number Ro= ͗uЈ͘ TA / ͑2⍀L͒, with L = D m so these slow dynamics are essentially governed by resonant triads. 23 Now consider the right panels in Fig. 11.…”

“…Numerical simulations have shown that the faster growth of certain integral length scales is also related to the inhibited energy cascade. [21][22][23] Among the most recent experimental studies are those by Baroud et al, 13,14 Morize and co-workers, 15,24 and by Davidson an co-workers. 16, 17 Baroud and co-workers studied the flow in a rotating annulus with forcing created by pumping water into ͑out of͒ the annulus through an inner ͑outer͒ ring of holes on the bottom of the annulus.…”

Dimensional transition in rotating turbulence

“…Experiments of decaying turbulence in rotating tanks show the growth of length scales aligned with the rotation axis, giving evidence that turbulence developing in rotating systems is highly anisotropic. Both linear [14,15] and nonlinear [16,17] mechanisms have been advocated to explain the observed growth. Numerical simulations of rotating flows have also indicated a general trend towards two-dimensionalization [8,17] and some studies [9] have reported a split of the energy cascade into a downscale and an upscale process.…”