Disentangling Scaling Properties in Anisotropic and Inhomogeneous Turbulence

Abstract

We address scaling in inhomogeneous and anisotropic turbulent flows by decomposing structure functions into their irreducible representation of the SO(3) symmetry group which are designated by j, m indices. Employing simulations of channel flows with Re l ഠ 70 we demonstrate that different components characterized by different j display different scaling exponents, but for a given j these remain the same at different distances from the wall. The j 0 exponent agrees extremely well with high Re measurements of the scaling exponents, demonstrating the vitality of the SO (3) decomposition. [S0031-9007(99)09384-9] PACS numbers: 47.27.EqMost of the available data analysis and theoretical thinking about the universal statistics of the small scale structure of turbulence assume the existence of an idealized model of homogeneous and isotropic flows. In fact most realistic flows are neither homogeneous nor isotropic. Accordingly, one can analyze the data pertaining to such flows in two ways. The traditional one has been to disregard the inhomogeneity and anisotropy and proceed with the data analysis assuming that the results pertain to the homogeneous and isotropic flows. The second, which is advocated in this Letter, is to take the anisotropy explicitly into account, to carefully decompose the relevant statistical objects into their isotropic and anisotropic contributions, and to assess the degree of universality of each component separately. We analyze here direct numerical simulations (DNS) of a channel flow with Re l ഠ 70 [1-3]. The main conclusion of this Letter is that this procedure is unavoidable; in particular, it highlights the universality of the scaling exponents of the isotropic sector which are presumably those governing the universal small scale statistics at very high Reynolds numbers. In agreement with recent studies of this subject [4,5] we report that different irreducible representations of the symmetry group (characterized by indices j, m) exhibit scalar functions that scale with apparently universal exponents that differ for different j. The exponents found at low values of the Reynolds number for the j 0 (isotropic) sector are in excellent agreement with high Re results; these exponents are invariant to the position in the inhomogeneous flow, leading to reinterpretation of recent findings of position dependence as resulting from the intervention of the anisotropic sectors. The latter have nonuniversal weights that depend on the position in the flow.We consider here channel flow simulations on a grid of 256 points in the streamwise directionx, and ͑128 3 128͒ in the other two directions,ŷ,ẑ. We denote bŷ z the direction perpendicular to the walls and byŷ the spanwise direction in planes parallel to the walls. We employ periodic boundary conditions in the spanwise and streamwise directions and no-slip boundary conditions on the walls. The Reynolds number based on the Taylor scale is Re l ഠ 70 in the center of the channel ͑z 64͒. The simulation is fully symmetric with respect to the central plane. The f...

Recovering isotropic statistics in turbulence simulations: The Kolmogorov 4/5th law

“…5, the integration over Ω extracts the isotropic component of a generally anisotropic flow. This is fully consistent with recent experimental [13,14] and numerical [17,18] efforts to quantify anisotropic contributions by projecting the structure function onto a particular irreducible representation, labeled by j = 0, 1, . .…”

The reflection-antisymmetric counterpart of the Kármán–Howarth dynamical equation

“…The decomposition of the structure function into rotationally invariant, irreducible subgroups of the SO(3) symmetry group S j=0 αβ (r)+S j=1 αβ (r)+... allowed the separation of the isotropic (indexed by j = 0) from the anisotropic (indexed by j > 0) contributions to the structure function. This procedure has allowed better quantification of the rate of decay of anisotropy of the small scales in turbulence [6,7,8]. These analyses considered homogeneous, isotropic and reflection symmetric flows.…”

Spectrum of anisotropic exponents in hydrodynamic systems with pressure

“…Instead, components of the structure functions that belong to different irreducible representations (sectors) of the SO(3) group possess different scaling exponents. Each of these sectors is characterized by the angular momentum indices ℓ and m. By projecting the structure function onto the different sectors, we could measure [2][3][4] the universal scaling exponents in each sector separately.…”

“…Experiments and simulations on Navier-Stokes (NS) turbulence also indicate that anisotropic sectors possess larger scaling exponents than the isotropic sector [2][3][4]. However, to date, the exponents for ℓ > 2 were not determined with sufficient accuracy.…”