Exact second-order structure-function relationships

By R. Hill. Journal of Fluid Mechanics, 2002

164 citations recorded by the source corpus.

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. 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.

Subfields: Aerospace Engineering, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/20de073b606d315c80d2c4ae002796710ca7bac0

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Citations in the corpus

Dissipation in Turbulent Flows (2015, methodology): “From these equations written at both points ξ = ξ+ and ξ = ξ−, Hill (2002) and Marati et al. (2004), followed by Danaila et al. (2012) and P. Valente & J.C. Vassilicos (submitted manuscript), derived the generalized Kármán-Howarth equation, which is valid with no homogeneity and no isotropy…”
Energy cascade and spatial fluxes in wall turbulence (2004, methodology): “After lengthy calculations the final equation is (for a detailed derivation see Hill 2002) ∂〈δu2δuj 〉 ∂rj + ∂〈δu2δUj 〉 ∂rj + 2〈δuiδuj 〉 ∂δUi ∂rj + ∂〈u∗j δu2〉 ∂Xcj + ∂〈δu2U ∗j 〉 ∂Xcj + 2〈u∗j δui〉 ∂δUi ∂Xcj = −4〈 ∗〉 + 2ν ∂ 2〈δu2〉 ∂rj ∂rj − 2 ρ ∂〈δpδui〉 ∂Xci + ν 2 ∂2〈δu2〉 ∂Xcj 2 , (3.3) where an…”
Cascades and transitions in turbulent flows (2018, methodology): “Following [586, 588] one can derive the generalization of the von Kármán-Howarth-Monin relation (24) to this non-homogeneous and anisotropic case.”