Estimates for the LANS-$\alpha$, Leray-$\alpha$ and Bardina models in terms of a Navier-Stokes Reynolds number J. GibbonD. Holm 76D03Navier-Stokesturbulence modelsregularization Estimates for the three $\alpha$-models known as the LANS-$\alpha$, Leray-$\alpha$ and Bardina models are found in terms a Reynolds number associated with a Navier-Stokes velocity field. They are tabulated for comparative purposes and show clearly that all estimates for the Leray-$\alpha$ model are smaller than those for the LANS-$\alpha$ and Bardina models. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3701 10.1512/iumj.2008.57.3701 en Indiana Univ. Math. J. 57 (2008) 2761 - 2774 state-of-the-art mathematics http://iumj.org/access/