SMALL SCALE UNIVERSALITY AND SPECTRAL CHARACTERISTICS IN TURBULENT FLOWS

Abstract

A review is given on studies of statistics at small scales in turbulent flows from a view point of universality. It is assumed in the view that the statistics at sufficiently small scales in the absence of mean flow are at a certain kind of local equilibrium state, and the influence of the mean flow may be regarded as a perturbation added to the equilibrium state. This idea has been examined by comparison of spectral characteristics derived by the idea with those in turbulent boundary layers, mixing layers and direct numerical simulations (DNS) of homogeneous turbulent shear flow. The applicability of this idea to turbulent channel flows is discussed in the light of the data of the log-law region in DNS of turbulent channel flows with the friction Reynolds numberReτ up to 5120. INTRODUCTION Turbulence is a phenomenon involving a huge number of degrees of dynamical freedom. A paradigm of study dealing with systems consisting of such a huge number of degrees of freedom is the statistical mechanics of systems at or near thermal equilibrium state. In the statistical mechanics, it is known that although it is difficult to trace the trajectory of each of the molecules or atoms in the physical or phase space, there are certain kinds of simple relations between a few variables, the socalled macroscopic variables, such as the pressure, density and temperature characterizing the equilibrium state. The relations are universal in the sense that they are independent of the detail of the difference in the trajectories of the molecules or atoms. It is also known that there are another kind of universal relations characterizing the response of the thermal equilibrium system to the disturbance added to the system. It is attractive to assume that the similar idea is applicable to turbulence. In fact, underlying the celebrated Kolmogorov theory (Kolmogorov,1941), referred here as K41, is the idea of existence of universal local equilibriums state, the statistics of which can be characterized by a few variables. In this paper, a review is given on studies along this idea with an emphasis on the spectral characteristics. Discussions are also made on the applicability of this idea to turbulent channel flows in the light of the log-law region in recent DNS of turbulent channel flows with the friction Reynolds number Reτ up to 5120. UNIVERSALITY AT LOCAL EQUILIBRIUM STATE We consider here the motion of incompressible fluid obeying the Navier-Stokes (NS) equation. Although it has not been rigorously proved, nor neither is it trivial that there is universality in the statistics of small scales in high Reynolds number turbulence, evidences supporting the existence have been accumulated. Among them is the so-called 4/5 law. The NS equation is compatible with the statistical homogeneity and isotropy of turbulent flows. Under the assumption of the homogeneity and isotropy of the turbulence statistics, the NS equation with the incompressibility condition yields a rigorous relation called Ḱarmán-Howarth (KH) equation (Ḱ armán and Howarth, 1938). If (i) the external force is confined to only large scales∼ L f , (ii) the statistics is almost stationary at scales much smaller than the characteristic length scale LE of the energy containing eddies, and (iii) the viscosity works only at small scales ∼ η , then it is shown from the KH equation that B3(r) = −4/5⟨ε⟩ r, (1) for L f ,LE ≫ r ≫ η , whereB3(r) is the third order longitudinal velocity structure function, ⟨ε⟩ the average of the rate of energy dissipation ε per unit mass, andη the Kolmogorov micro length scale defined by η ≡ (ν3/⟨ε⟩)1/4 with ν being the kinematic viscosity. This 4/5 law has been confirmed by experiments and numerical simulations. Note that the law asserts that (1) holds irrespectively of the de-