Multiscale correlations and conditional averages in numerical turbulence

Siegfried Grossmann, Detlef Lohse, Achim Reeh
2000 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
The equations of motion for the nth order velocity differences raise the interest in correlation functions containing both large and small scales simultaneously. We consider the scaling of such objects and also their conditional average representation with emphasis on the question of whether they behave differently in the inertial or the viscous subranges. The turbulent flow data are obtained by Navier-Stokes solutions on a 60 3 grid with periodic boundary conditions and Re ϭ70. Our results
more » ... 70. Our results complement previous high Re data analysis based on measured data ͓A. L. Fairhall, V. S. L'vov, and I. Procaccia, Europhys. Lett 43, 277 ͑1998͔͒ whose preference were the larger scales, and the analysis of both experimental and synthetic turbulence data by ͓R. Benzi and co-workers, Phys. Rev. Lett. 80, 3244 ͑1998͒; Phys. Fluids 11, 2215 ͑1999͔͒. The inertial range fusion rule is confirmed and insight is obtained for the conditional averages ͑the local dissipation rate conditioned on the velocity fluctuations͒. PACS number͑s͒: 47.27.Ϫi To analyze the structure of turbulent flow fields Lagrangean longitudinal nth order structure functions S n (R)ϭ͗v n (R)͘ are of utmost importance. Here v(x,R;t) ϭ͓u(xϩR,t)Ϫu(x,t)͔•R/R denotes the longitudinal velocity difference on scale R. The equation of motion ͑cf. ͓1,2͔͒ in the case of statistically stationary turbulence introduces correlation functions of another type, containing local gradient and curvature in addition to the scale R. The (u•")u nonlinearity in the Navier-Stokes equation gives rise to D n (R,t) ͑which needs no further specification here because we do not consider it in what follows͒ and the viscous term ⌬u leads to J n ϭ͗⌬uv nϪ1 (R)͘. The Laplacean probes the local behavior, while R is in the inertial range. This two-scale character of J n becomes explicit if the local curvature ⌬u is approximated by finite differences, ⌬ r u(x) ϭ͓u(xϩr)Ϫ2u(x)ϩu(xϪr)͔/r 2 . The necessity for this discretization arises both in numerical turbulence and in the analysis of measured flow signals. It motivates to study the following more general objects: Here rӶR is assumed, and r ͑instead of r→0) is allowed to vary in the viscous as well as in the inertial subranges VSR and ISR. In particular r may be below or above n , where n denotes the transition scale between VSR and ISR in an nth order correlation function. It is these J n on which we concentrate in this note. The two-scale correlators J n are of interest to check the validity of the so-called fusion rules ͓3,4͔. These describe the proper factorization of the general nth order correlation functions
doi:10.1103/physreve.61.5195 pmid:11031565 fatcat:wu7dopejs5fkvh47mb324oyexu