Abstract
This paper investigates the universality of the Eulerian velocity structure functions using velocity fields obtained from the stereoscopic particle image velocimetry (SPIV) technique in experiments and direct numerical simulations (DNS) of the Navier-Stokes equations. It shows that the numerical and experimental velocity structure functions up to order 9 follow a log-universality (Castaing et al. Phys. D Nonlinear Phenom. 1993); this leads to a collapse on a universal curve, when units including a logarithmic dependence on the Reynolds number are used. This paper then investigates the meaning and consequences of such log-universality, and shows that it is connected with the properties of a “multifractal free energy”, based on an analogy between multifractal and thermodynamics. It shows that in such a framework, the existence of a fluctuating dissipation scale is associated with a phase transition describing the relaminarisation of rough velocity fields with different Hölder exponents. Such a phase transition has been already observed using the Lagrangian velocity structure functions, but was so far believed to be out of reach for the Eulerian data.
Highlights
A well-known feature of any turbulent flow is the Kolmogorov-Richardson cascade by which energy is transferred from large to small length scales until the Kolmogorov length scale below which it is removed by viscous dissipation
We investigate the meaning and consequences of such a log-universality, and show that it is connected with the properties of a “multifractal free energy”, based on an analogy between multifractal and thermodynamics
We show that in such a framework, this fluctuating dissipation length scale is associated with a phase transition describing the relaminarisation of velocity fields
Summary
A well-known feature of any turbulent flow is the Kolmogorov-Richardson cascade by which energy is transferred from large to small length scales until the Kolmogorov length scale below which it is removed by viscous dissipation. The mechanism behind this universality breakage is identified in [3], where a generalization of the Kolmogorov theory is introduced, based on the hypothesis that a turbulent flow is multifractal In this model, the velocity field is locally characterized by a Hölder exponent h, such that δu(x) ≡ hku(x + r) − u(x)kikrk=` ∼ `h(x) ; here h is a stochastic function that follows a large deviation property [4] P (log(|δu|/u0 ) = h log (`/L0 )) ∼ (`/L0 )C(h) , where u0 The new prediction was that log E(k )e− 3 η − 3 / log(Re) should be a universal function of log(kη )/ log(Re), where Re is the Reynolds number This claim was examined by Gagne et al, later using data from the Modane wind tunnel experiments [7]. We show that in such a framework, this fluctuating dissipation length scale is associated with a phase transition describing the relaminarisation of velocity fields
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call