In this paper we present analytical studies of three-dimensional viscous and inviscid simplified Bardina turbulence models with periodic boundary conditions. The global existence and uniqueness of weak solutions to the viscous model has already been established by Layton and Lewandowski. However, we prove here the global well-posedness of this model for weaker initial conditions. We also establish an upper bound to the dimension of its global attractor and identify this dimension with the number of degrees of freedom for this model. We show that the number of degrees of freedom of the long-time dynamics of the solution is of the order of (L/l d ) 12/5 , where L is the size of the periodic box and l d is the dissipation length scale-believed and defined to be the smallest length scale actively participating in the dynamics of the flow. This upper bound estimate is smaller than those established for Navier-Stokes-α, Clark-α and Modified-Leray-α turbulence models which are of the order (L/l d ) 3 . Finally, we establish the global existence and uniqueness of weak solutions to the inviscid model. This result has an important application in computational fluid dynamics when the inviscid simplified Bardina model is considered as a regularizing model of the three-dimensional Euler equations.MSC Classification: 35Q30, 37L30, 76BO3, 76D03, 76F20, 76F55, 76F65

Inspired by the remarkable performance of the Leray-α (and the Navier-Stokes alpha (NS-α), also known as the viscous Camassa-Holm) subgrid scale model of turbulence as a closure model to Reynolds averaged equations (RANS) for flows in turbulent channels and pipes, we introduce in this paper another subgrid scale model of turbulence, the modified Leray-α (ML-α) subgrid scale model of turbulence. The application of the ML-α to infinite channels and pipes gives, due to symmetry, similar reduced equations as Leray-α and NS-α. As a result the reduced ML-α model in infinite channels and pipes is equally impressive as a closure model to RANS equations as NS-α and all the other alpha subgrid scale models of turbulence (Leray-α and Clark-α). Motivated by this, we present an analytical study of the ML-α model in this paper. Specifically, we will show the global well-posedness of the ML-α equation and establish an upper bound for the dimension of its global attractor. Similarly to the analytical study of the NS-α and Leray-α subgrid scale models of turbulence we show that the ML-α model will follow the usual k −5/3 Kolmogorov power law for the energy spectrum for wavenumbers in the inertial range that are smaller than 1/α and then have a steeper power law for wavenumbers greater than 1/α (where α > 0 is the length scale associated with the width of the filter). This result essentially shows that there is some sort of parametrization of the large wavenumbers (larger than 1/α) in terms of the smaller wavenumbers. Therefore, the ML-α

A new instrument for dynamic helical squeeze flow which superposes oscillatory shear and oscillatory squeeze flow Rev. Sci. Instrum. 83, 085105 (2012) The effects of hydrodynamic interaction and inertia in determining the high-frequency dynamic modulus of a viscoelastic fluid with two-point passive microrheology Phys. Fluids 24, 073103 (2012) MHD free convection flow of a visco-elastic (Kuvshiniski type) dusty gas through a semi infinite plate moving with velocity decreasing exponentially with time and radiative heat transfer AIP Advances 1, 022132 (2011) Transitional flow of a non-Newtonian fluid in a pipe: Experimental evidence of weak turbulence induced by shearthinning behavior Phys. Fluids 22, 101701 (2010) Effects of viscoelasticity on the probability density functions in turbulent channel flowWe consider passive scalar mixing by a prescribed divergence-free velocity vector field in a periodic box and address the following question: Starting from a given initial inhomogeneous distribution of passive tracers, and given a certain energy budget, power budget, or finite palenstrophy budget, what incompressible flow field best mixes the scalar quantity? We focus on the optimal stirring strategy recently proposed by Lin et al. ["Optimal stirring strategies for passive scalar mixing," J. Fluid Mech. 675, 465 (2011)] that determines the flow field that instantaneously maximizes the depletion of the H − 1 mix-norm. In this work, we bridge some of the gap between the best available a priori analysis and simulation results. After recalling some previous analysis, we present an explicit example demonstrating finite-time perfect mixing with a finite energy constraint on the stirring flow. On the other hand, using a recent result by Wirosoetisno et al. ["Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations," SIAM J. Numer. Anal. 50(1), 126-150 (2012)] we establish that the H − 1 mix-norm decays at most exponentially in time if the two-dimensional incompressible flow is constrained to have constant palenstrophy. Finite-time perfect mixing is thus ruled out when too much cost is incurred by small scale structures in the stirring. Direct numerical simulations in two dimensions suggest the impossibility of finite-time perfect mixing for flows with fixed power constraint and we conjecture an exponential lower bound on the H − 1 mixnorm in this case. We also discuss some related problems from other areas of analysis that are similarly suggestive of an exponential lower bound for the H − 1 mix-norm. C 2012 American Institute of Physics. [http://dx.

Abstract. We establish global existence and uniqueness theorems for the twodimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by Danchin and Paicu; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary "stream-function" associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations.

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n ≥ 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-α model, the Leray-α model, the modified Leray-α model, the simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-α-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. This family of models has become particularly important in the development of mathematical and computational models of turbulence. We give a unified analysis of the entire three-parameter family of models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α → 0 limit in α models. Next, we show existence of a global attractor for the general model, and then give estimates for the dimension of the global attractor and the number of degrees of freedom in terms of a generalized Grashof number. We then establish some results on determining operators for the two distinct subfamilies of dissipative and non-dissipative models. We finish by deriving some new length-scale estimates in terms of the Reynolds number, which allows for recasting the Grashof number-based results into analogous statements involving the Reynolds number. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, and determining operators for the well-known specific members of this family of regularized Navier-Stokes and MHD models, the framework we develop also makes possible a number of new results for all models in the general family, including some new results for several of the well-studied models. Analyzing the more abstract generalized model allows for a simpler analysis that helps bring out the core common structure of the various regularized Navier-Stokes and magnetohydrodynamics models, and also helps clarify the common features of many of the existing and new results. To make the paper reasonably self-contained, we include supporting material on spaces involving time, Sobolev spaces, and Grönwall-type inequalities.