IJPAM: Volume 36, No. 4 (2007)

THE TEST FOR ADEQUACY OF SHELL MODELS
FOR A TURBULENCE PROBLEM

Igor M. Gaissinski$^1$, Vladimir Y. Rovenski$^2$
$^1$ Aerospace Engineering Department
Technion - Israel Institute of Technology
Haifa, 32000, ISRAEL
e-mail: ricka1993@012.net.il
$^2$Department of Mathematics
University of Haifa
Haifa, 31905, ISRAEL
e-mail: rovenski@math.haifa.ac.il


Abstract.The nonlinear interactions between the base functions used in a shell model are described by the tensor $T_{NML}$, obtained using Galerkin projection of Navier-Stokes equations on the wavelets base. The coefficients $T_{NML}$ must satisfy the conservation laws for the energy and the enstrophy or the helicity (depending on the dimension 2 or 3). The paper deduces these relations and presents their solution as the product of the partial solution and $G(\eta_{_{LNM}},\eta_{_{NML}},\eta_{_{MLN}})$, where $G$ is any odd function invariant under transpositions of arguments, and a tensor $\eta$ has zero diagonal terms and satisfies the conditions $\eta_{_{NML}}=-\eta_{_{LMN}},\ \eta_{_{N+k,M+k,L+k}}=\eta_{_{NML}}$. One may use these relations and their solution to verify the adequacy of wavelet functions base to the turbulence problem.

Received: January 29, 2007

AMS Subject Classification: 76E20, 76J20, 76N99

Key Words and Phrases: Navier-Stokes equations, enstrophy, helicity, shell model, turbulence

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 36
Issue: 4