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

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.
In: ANZIAM Journal ISSN 1446-1811
[Article] , (2011)

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
Erschienen: 2011
Creators: Liu, Zeng and Oberlack, Martin and Grebenev, Vladimir N. and Liao, Shi-Jun
Title: Explicit series solution of a closure model for the Kármán-Howarth equation
Language: English
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.

Journal or Publication Title: ANZIAM Journal
Publisher: Australian Mathematical Society
Uncontrolled Keywords: Homotopy analysis method; von Kármán-Howarth equation; solutions in closed form; conservation law
Divisions: 16 Department of Mechanical Engineering > Fluid Dynamics (fdy)
16 Department of Mechanical Engineering
Date Deposited: 24 Aug 2011 18:17
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