**Hani Ali**

*A study of the NS-\(\overline w\) model of turbulence*

**Pages:** 151-174

**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.

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