A NOTE ON LOCAL ISOTROPY CRITERIA IN SHEAR FLOWS WITH COHERENT MOTION

Abstract

Whilst Local Isotropy (LI) is widely used, it is also necessary to test its validity, especially in shear flows, characterized by large-scale anisotropy. Important questions are whether the small scales are isotropic and how their properties depend on large-scale parameters (mean shear, the shear induced by a coherent motion, the Reynolds number etc.). We focus on two families of LI tests: i) classical, kinematic tests, in which time-averages are compared to their isotropic values. The large-scale parameters do not appear explicitly. We only use here one example of such tests. ii) Phenomenological tests, which explicitly account for the large-scale strain, as well as its associated dynamics. In flows populated by coherent motions in which phaseaverages are pertinent for describing the flow dynamics, we propose a Local Isotropy (LI) criterion based on the intensity of the turbulent strain rate at a given scale~r and a particular phase φ , sφ (~r,φ). The formulation is the following: ”If LI were to be valid at a vectorial scale~r and a phase φ , then the intensity of the turbulent strain rate sφ (~r,φ) should prevail over the combined effect of the mean shear S and of the shear S̃ associated with the coherent motion”. The mathematical expression of sφ (~r,φ) depends on the Laplacian of the total kinetic energy second-order structure function. Therefore, the proposed expression allows the eventual anisotropy to be taken into account. The new LI criterion is used together with data taken in the intermediate wake behind a circular cylinder. It is highlighted that (i) when S+ S̃ is important, LI only holds for scales smaller than the Taylor microscale (ii) when S+ S̃ is small, the domain in which LI is valid extends up to the largest scales. INTRODUCTION Local isotropy (LI) is seemingly one of the most important hypotheses on small-scale statistics. LI was first enunciated by Kolmogorov (1941), and further utilized and sometimes tested, in most of the laboratory flows. From the analytical viewpoint, LI leads to simplified expressions of e.g. the total kinetic energy, the dissipation rate of kinetic energy or scalar variance, structure functions at a given scale. Simple expressions of statistics are useful for the experimentalists, because of the limited possibilities to measure all the velocity components, as well as their spatial distribution. Although LI is extensively used, it is nonetheless necessary to test its validity, especially in shear flows, characterized by large-scale anisotropy. Important questions are whether the small scales are isotropic and if there is a clear dependence of their statistics on large-scale parameters (mean shear S, the shear induced by a coherent motion S̃, the Reynolds number etc.). Using a compilation of experimental and numerical data, Schumacher et al. (2003) showed that LI prevails for small values of the ratio S/Rλ (Rλ is the Taylor microscale Reynolds number). One should expect that the magnitude of the shear will play some role in determining how high an Rλ is required for LI to prevail. Whereas the conclusion of Schumacher et al. (2003) is optimistic quid the restoration of LI, the analytical study of Durbin & Speziale (1991) demonstrated that small scales cannot be isotropic in shear flows, independently of the values of Rλ and S. From a general viewpoint, the assessment of LI can only be done through specific criteria and a definitive conclusion about the validity of LI is unlikely to be realistic. The aim of this study is to understand how, in the context of shear flows, the anisotropy propagates across the scales from the largest to the smallest, how it evolves down the scales and finally, what the degree of anisotropy is at any given scale. To this end, we propose a phenomenological LI criterion based on the intensity of the turbulent strain rate at a given scale r. As a first step in answering the question of what the isotropy level is at any particular scale, we consider flows populated by a single-scale, persistent coherent motion (hereafter, CM). A good candidate is the cylinder wake flow, and this study focuses entirely on this flow. The other advantage of investigating the wake flow is that it allows to invoke phase averages. The latter operation results in a dependence of any statistical quantities on the phase φ characterizing the temporal dynamics of the CM. We focus on two families of LI tests: i) classical, kinematic tests, in which time-averages are compared to their isotropic values. The large-scale parameters (shear) does not appear explicitly. We only use here one example of such tests. ii) Phenomenological tests, which explicitly account for the