Exact second-order structure-function relationships

Abstract

Equations that follow from the Navier-Stokes equation and incompressibility but with no other approximations are "exact.". Exact equations relating second-and third-order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal, and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment, and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: i) the trace of the structure functions, ii) DNS that has periodic boundary conditions, and iii) an average over a sphere in r-space. The last case (iii) introduces the average over orientations of r into the structure function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

Opportunities for use of exact statistical equations

“…5 by use of the exact structure function equation. For the case of spatially periodic direct numerical simulation, similar results were obtained from the exact structure function equations in [2].…”

“…The Navier-Stokes equation has been used to derive exact equations relating velocity structure functions of velocity increments and other statistics [2]. Such exact equations have been obtained for all orders of velocity structure functions in [14].…”

“…Frisch [25] gives a definition that is equivalent to that of Monin and Yaglom [24], except that the translations are restricted to a domain the size of the spatial scale characteristic of the production of turbulent energy (which he calls the "integral scale"). Two-point structure function equations of all orders contain a statistic that is the product of not only factors of the velocity difference but also one factor of the sum of the two velocities, i.e., u n + u ′ n [2,14]. Because the definitions of local homogeneity by Monin and Yaglom [24] and Frisch [25] involve only the joint probability distribution of two-point differences, but do not involve u n + u ′ n , it follows that those definitions are not sufficient to simplify structure function equations to the same level of simplification as does homogeneity.…”

Cascades and wall-normal fluxes in turbulent channel flows

“…To provide a more complete view, the four-fifths law in the form of a balance equation for second-order structure function, originally proposed by Hill (2002), was used by Marati, Casciola & Piva (2004) to address the energy transfer in both spatial and scale spaces for a turbulent channel flow. The multidimensional and directional description provided by this equation was exploited by Cimarelli, De Angelis & Casciola (2013) to understand the formation and sustainment of long and wide turbulent fluctuations.…”

“…Hereafter, as anticipated in the introduction, we will often refer to the second-order structure function as the scale energy. The balance equation of δu 2 in wall flows is the generalized Kolmogorov equation (Hill 2002) which for a turbulent channel flow with longitudinal mean velocity U(y) (Marati et al 2004…”