Turbulence cascade in an inhomogeneous turbulent flow

Abstract

The inhomogeneous, anisotropic turbulence downstream of a square prism is investigated by means of direct numerical simulations (DNS) and two-point statistics. As noted by Moffatt (2002) “it now seems that the intense preoccupation [...] with the problem of homogeneous isotropic turbulence was perhaps misguided” acknowledging there is now a revived interest in studying inhomogeneous turbulence. The full description of the turbulence cascade requires a two-point analysis which re- volves around the recently derived Kármán-Howarth-Monin-Hill equation (KHMH). This equation is the inhomogeneous/anisotropic analogue to the so-called Kolmogorov equation (or Kármán-Howarth equation) used in Kolmogorov’s 1941 seminal papers (K41) which are the foundation to the most successful turbulence theory to date. Particular focus is placed on the near wake region where the turbulence is anticipated to be highly inhomogeneous and anisotropic. Because DNS gives direct access to all ve- locity components and their derivatives, all terms of the KHMH can be computed directly without resorting to any simplifications. Computation of the term associated with the non-linear inter-scale transfer of energy (Π) revealed that this rate is roughly constant over a range of scales which increases (within the bounds of our database) with distance to the wake generator, provided that the orientations of the pairs of points are averaged-out on the plane of the wake. This observation appears in tandem with a near −5/3 power law in the spectra of fluctuating velocities which deteriorates as the constancy of Π improves. The constant non-linear inter-scale transfer plays a major role in K41 and is required for deriving the 2/3-law (which is real space equivalent of the −5/3). We extend our analysis to a triple decomposition where the organised motion associ- ated with the vortex shedding is disentangled from the stochastic motions which do not display a distinct time signature. The imprint of the shedding-associated motion upon the stochastic component is observed to contribute to the small-scale anisotropy of the stochastic motion. Even though the dynamics of the shedding-associated motion differs drastically from that of the stochastic one, we find that both contributions are required in order to preserve the constant inter-scale transfer of energy. We further find that the inter- scale fluxes resulting from this decomposition display local (in scale-space) combinations of direct and inverse cascades. While the inter-scale fluxes associated with the coherent motion can be explained on the basis of simple geometrical arguments, the stochastic motion shows a persistent inverse cascade at orientations normal to the centreline despite its energy appearing to be roughly isotropically distributed.