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

Zeng Liu, Martin Oberlack, Vladimir N. Grebenev, Shijun Liao


The homotopy analysis method (HAM) is applied to a nonlinear ordinary differential equation (ODE) emerging from a closure model of the von Kármán–Howarth equation which models the decay of isotropic turbulence. In the infinite Reynolds number limit, the von Kármán–Howarth equation admits a symmetry reduction leading to the aforementioned one-parameter ODE. Though the latter equation is not fully integrable, it can be integrated once for two particular parameter values and, for one of these values, the relevant boundary conditions can also be satisfied. The key result of this paper is that for the generic case, HAM is employed such that solutions for arbitrary parameter values are derived. We obtain explicit analytical solutions by recursive formulas with constant coefficients, using some transformations of variables in order to express the solutions in polynomial form. We also prove that the Loitsyansky invariant is a conservation law for the asymptotic form of the original equation.



homotopy analysis method; von Kármán–Howarth equation; solutions in closed form; conservation law

DOI: http://dx.doi.org/10.21914/anziamj.v52i0.3215

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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.