# Explicit series solution of a closure model for the Kármán-Howarth equation

## Abstract

The Homotopy Analysis Method (HAM) is applied to the nonlinear problem with ξ=0 : f(ξ)=1 and ξ \to +∞ : f(ξ) \to 0. The problem is associated with a closure model for the von Kármán-Howarth equation, in terms of the normalized two-point double velocity correlation in the limit of infinite Reynolds number. Though the latter differential equation admits no Lie point symmetry groups it is still integrable once for the values σ = 0 and σ = 4 by means of integrating factors. The case σ = 4 is the only case that is again integrable for the given boundary conditions. The key result is that for the generic case HAM is employed such that solutions for arbitrary σ are derived. By choosing the correct parameters in the frame of HAM, we obtain the explicit analytic solutions by recursive formulae with constant coefficients using some transformations of variables in order to deal with a polynomial type of equation. In the appendix, we prove that the Loitsyansky invariant is the conservation law for the asymptotic form of the original equation.

Item Type: Article 2011 Liu, Zeng and Oberlack, Martin and Grebenev, Vladimir N. and Liao, Shi-Jun Explicit series solution of a closure model for the Kármán-Howarth equation English The Homotopy Analysis Method (HAM) is applied to the nonlinear problem with ξ=0 : f(ξ)=1 and ξ \to +∞ : f(ξ) \to 0. The problem is associated with a closure model for the von Kármán-Howarth equation, in terms of the normalized two-point double velocity correlation in the limit of infinite Reynolds number. Though the latter differential equation admits no Lie point symmetry groups it is still integrable once for the values σ = 0 and σ = 4 by means of integrating factors. The case σ = 4 is the only case that is again integrable for the given boundary conditions. The key result is that for the generic case HAM is employed such that solutions for arbitrary σ are derived. By choosing the correct parameters in the frame of HAM, we obtain the explicit analytic solutions by recursive formulae with constant coefficients using some transformations of variables in order to deal with a polynomial type of equation. In the appendix, we prove that the Loitsyansky invariant is the conservation law for the asymptotic form of the original equation. ANZIAM Journal Australian Mathematical Society Homotopy analysis method; von Kármán-Howarth equation; solutions in closed form; conservation law 16 Department of Mechanical Engineering > Fluid Dynamics (fdy)16 Department of Mechanical Engineering 24 Aug 2011 18:17 ASCII CitationDublin CoreT2T_XMLSimple MetadataMultiline CSVEP3 XMLBibTeXEndNoteAtomJSONRDF+XMLHTML CitationReference ManagerMODS TUfind oder in Google
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