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EXPLICIT SERIES SOLUTION OF A CLOSURE MODEL FOR THE VON KÁRMÁN–HOWARTH EQUATION

Part of: Boundary value problems

Published online by Cambridge University Press:  05 September 2011

ZENG LIU ,
MARTIN OBERLACK ,
VLADIMIR N. GREBENEV  and
SHI-JUN LIAO
ZENG LIU
Affiliation:
State Key Lab of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, PR China (email: z_liu@sjtu.edu.cn, sjliao@sjtu.edu.cn)
MARTIN OBERLACK
Affiliation:
Chair of Fluid Dynamics, Department of Mechanical Engineering, Technische Universität Darmstadt, Petersenstr. 30, 64287 Darmstadt, Germany Center of Smart Interfaces, TU Darmstadt, Petersenstr. 32, 64287 Darmstadt, Germany GS Computational Engineering, TU Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany (email: oberlack@fdy.tu-darmstadt.de)
VLADIMIR N. GREBENEV
Affiliation:
Institute of Computational Technologies, Russian Academy of Science, Lavrentiev Ave. 6, Novosibirsk 630090, Russia (email: vngrebenev@gmail.com)
SHI-JUN LIAO*
Affiliation:
State Key Lab of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, PR China (email: z_liu@sjtu.edu.cn, sjliao@sjtu.edu.cn)
*
✉For correspondence; e-mail: sjliao@sjtu.edu.cn
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Abstract

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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.

Keywords

homotopy analysis methodvon Kármán–Howarth equationsolutions in closed formconservation law

MSC classification

Secondary: 34B15: Nonlinear boundary value problems
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 2 , October 2010 , pp. 179 - 202
Copyright
Copyright © Australian Mathematical Society 2011

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