We investigate experimentally the influence of a background rotation on the energy transfers in decaying grid turbulence. The anisotropic energy flux density, F(r) = δu (δu) 2 , where δu is the vector velocity increment over separation r, is determined for the first time using Particle Image Velocimetry. We show that rotation induces an anisotropy of the energy flux ∇ · F, which leads to an anisotropy growth of the energy distribution E(r) = (δu) 2 , in agreement with the Kármán-Howarth-Monin equation. Surprisingly, our results prove that this anisotropy growth is essentially driven by a nearly radial, but orientation-dependent, energy flux density F(r).The energy cascade from large to small scales, and the associated Kolmogorov 4/5th law, are recognized as the most fundamental results of homogeneous and isotropic turbulence [1,2]. In the presence of a background rotation, a situation which is relevant for most geophysical and astrophysical flows, the scale-to-scale energy transfers are modified by the Coriolis force, yielding a gradual columnar structuring of turbulence along the rotation axis [3][4][5][6][7]. The Taylor-Proudman theorem is often invoked, however improperly, to justify the resulting quasi-2D nature of turbulence under rotation. Indeed, this theorem is a purely linear result, which applies only in the limit of zero Rossby number (i.e. infinite rotation rate), and is therefore incompatible with turbulence; it cannot describe the anisotropic energy transfers responsible for the non-trivial organization of rotating turbulence which are a subtle non-linear effect taking place only at non-zero Rossby number. To date, no direct evidence for these anisotropic energy transfers towards the 2D state in the physical space has been obtained. In this Letter, we report for the first time direct measurements of the physical-space energy transfers in decaying rotating turbulence using Particle Image Velocimetry (PIV), and provide new insight into the anisotropy growth of turbulence at finite, and hence geophysically relevant, Rossby number.If homogeneity (but not necessarily isotropy) holds, the energy distribution and energy flux density in the space of separations r are described by the fields E(r, t) = (δu) 2 andwhere u(x, t) is the turbulent velocity, δu = u(x + r, t) − u(x, t) is the velocity vector increment over r (Fig. 1), and · denotes spatial and ensemble averages. These key quantities satisfy the Kármán-Howarth-Monin (KHM) equation [1,8], which describes the evolution of the energy distribution in the space of separations, 1 2where R(r, t) = u(x, t) · u(x + r, t) = u 2 − E(r, t)/2 is the two-point velocity correlation and ν the kinematic viscosity. Importantly, this equation is still valid for homogeneous anisotropic turbulence [9], and in particular for axisymmetric turbulence in a rotating frame (here axisymmetry is to be understood in the statistical sense, with respect to r). For stationary (forced) turbulence, this equation reduces to ∇ · F = −4 in the inertial range, where stands for the rates ...
We report an experimental study of the decay of grid-generated turbulence in a confined geometry submitted to a global rotation. Turbulence is generated by rapidly towing a grid in a parallelepipedic water tank. The velocity fields of a large number of independent decays are measured in a vertical plane parallel to the rotation axis using a corotating particle image velocimetry system. We first show that, when a "simple" grid is used, a significant amount of the kinetic energy (typically 50%) is stored in a reproducible flow composed of resonant inertial modes. The spatial structure of those inertial modes, extracted by band-pass filtering, is found compatible with the numerical results of L. R. M. Maas [Fluid Dyn. Res. 33, 373 (2003)]. The possible coupling between these modes and turbulence suggests that turbulence cannot be considered as freely decaying in this configuration. We demonstrate however that these inertial modes may be significantly reduced (down to 15% of the total energy) by adding a set of inner tanks attached to the grid. These results suggest that it is possible to produce an effectively freely decaying rotating turbulence in a confined geometry.
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