A study of the NS-\(\overline w\) model of turbulence
Received: 17 October 2012
Accepted: 26 November 2012
Mathematics Subject Classification (2010): 76B03; 76F05; 76D05; 35Q30.
Keywords: Turbulence models, Regularity, Navier-Stokes equations.
Abstract: In this paper, the NS-$\overline$ω model with periodic boundary conditions is studied. This model is derived from the rotational Navier-Stokes equations by regularizing with an explicit spatial filter of width α the second term of the nonlinearity. It is first shown that the regular solution for NS-$\overline$ω system verifies a sequence of energy inequalities called "ladder inequalities". These ladder inequalities give rise to series of time-averaged inverse square length-scales. These latter quantities are estimated in terms of the Reynolds number. Moreover, it is shown that the NS-$\overline$ω model follows the usual κ-5/3 Kolmogorov power law spectrum for wavenumbers smaller than 1/α in the inertial range. However, this model has a steeper power law spectrum for wavenumbers greater than 1/α. Finally, the relation between the NS-$\overline$ω model and the Navier-Stokes equations is discussed by proving a convergence theorem as the length scale α tends to zero.