Grad-div stabilization, adding a term -gamma grad div u, has proven to be a useful tool in the simulation of incompressible flows. Such a term requires a choice of the coefficient gamma and studies have begun appearing with various suggestions for its value. We give an analysis herein that provides a restricted range of possible values for the coefficient in 3d turbulent flows away from walls.
The energy dissipation rate is a fundamental statistic in experimental and theoretical studies of turbulence, e.g., Sreenivasan @cite_24 , Frisch @cite_9 . In 1968, Saffman @cite_4 , addressing the estimate of energy dissipation rates, @math , wrote that In 1992 Constantin and Doering @cite_6 made a fundamental breakthrough, establishing a direct link between the phenomenology of energy dissipation and that predicted for general weak solutions of shear flows directly from the NSE. This work builds on Busse @cite_22 , Howard @cite_28 (and others) and has developed in many important directions. It has been extended to shear flows in Childress, Kerswell and Gilbert @cite_11 , Kerswell @cite_12 and Wang @cite_23 . For flows driven by body forces extensions include Doering and Foias @cite_19 , Cheskidov, Doering and Petrov @cite_17 (fractal body forces), and @cite_26 (helicity dissipation). Energy dissipation in models and regularizations studied in @cite_7 , @cite_26 , @cite_29 , @cite_16 .
@cite_26 @cite_4 @cite_22 @cite_7 @cite_28 @cite_9 @cite_29 @cite_17 @cite_6 @cite_24 @cite_19 @cite_23 @cite_16 @cite_12 @cite_11
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The Navier-Stokes equations are utilized to derive upper bounds on the turbulent energy dissipation rate for an incompressible Newtonian fluid confined between parallel comoving plates. These estimates provide a rigorous foundation for one of the basic scaling ideas of turbulence theory, namely, the independence of the dissipation rate and the viscosity at high Reynolds number. The bounds are compared to experiments on turbulence in the Couette-Taylor system This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. We consider a family of high-accuracy, approximate deconvolution models of turbulence. For body force driven turbulence, we prove directly from the model's equations of motion the following bounds on the model's time-averaged energy dissipation rate @math and helicity dissipation rate @math : where @math are global velocity and length scales, respectively, and @math is the large eddy simulation filter radius. We also give a partial result on the helicity dissipation rate of solutions of the Navier–Stokes equations. We consider the Navier-Stokes (NS)-alpha and the family of high-accuracy NS-alpha-deconvolution models of turbulence on Ω = [0, L Ω ] 3 subject to periodic boundary conditions. For body-force-driven turbulence, we prove directly from the model equations of motion the following bounds on the time-averaged modified energy dissipation rate, 〈e α,N (w α,N )〉, and unmodified helicity dissipation rate, 〈γ (w α,N )〉, for the Nth model (N = 0, 1, 2, ...): ≤ (N + 1)2 N+1 U 3 N L N and | | ≤ C 2 (N + 1)2 N+1 U 3 N L 2 N . Here, N is the degree of the approximate deconvolution operator, U N and L N are global velocity and length scales and C 1 and C 2 are constants that do not depend on U N .