text: 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.
  @cite_26: 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.
  @cite_4: 
  @cite_22: Publisher Summary This chapter discusses that the optimum theory represents an approach toward the understanding of turbulent flows without the introduction of heuristic assumptions that are commonly used in other theories of turbulence. As there does not seem to be a generally accepted definition of turbulence, this chapter shall regard, for the present purpose, a fluid flow as turbulent if the details of the velocity field are too complex to be of interest and only the information about certain average properties is needed. The chapter highlights that the optimum theory is hardly appropriate for all turbulent systems of this general nature and a more suitable mathematical definition of turbulence will be required later. The main point is that turbulent fluid systems are characterized by an information gap that is neither possible nor desirable to fill. The optimum theory represents an alternative approach in which the lack of information about properties of the fluctuating velocity field is reflected by the theoretical results. Instead of a theoretical prediction of the physically realized average properties, bounds on those properties are obtained. The idea of the optimum theory is to consider not the manifold of solutions of the basic equations for a particular problem, but a larger manifold of vector fields, which includes the actual solutions. This manifold of fields shares with the solutions kinematic relationships such as boundary conditions and the energy balance.
  @cite_7: 
  @cite_28: 
  @cite_9: 
  @cite_29: 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 .
  @cite_17: 
  @cite_6: 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
  @cite_24: 
  @cite_19: A model for the Reynolds number dependence of the dimensionless dissipation rate Ce was derived from the dimensionless Karman-Howarth equation, resulting in Ce = Ce,∞ +C RL + O(1 RL ), where RL is the integral scale Reynolds number. The coefficients C and Ce,∞ arise from asymptotic expansions of the dimensionless second- and third-order structure functions. This theoretical work was supplemented by direct numerical simulations (DNS) of forced isotropic turbulence for integral scale Reynolds numbers up to RL = 5875 (Rλ = 435), which were used to establish that the decay of dimensionless dissipation with increasing Reynolds number took the form of a power law RLn with exponent value n = −1.000 ± 0.009, and that this decay of Ce was actually due to the increase in the Taylor surrogate U3 L. The model equation was fitted to data from the DNS which resulted in the value C = 18.9 ± 1.3 and in an asymptotic value for Ce in the infinite Reynolds number limit of Ce,∞ = 0.468 ± 0.006.
  @cite_23: 
  @cite_16: 
  @cite_12: The equations of hydrodynamics are modified by the inclusion of additional terms which greatly simplify the procedures needed for stepwise numerical solution of the equations in problems involving shocks. The quantitative influence of these terms can be made as small as one wishes by choice of a sufficiently fine mesh for the numerical integrations. A set of difference equations suitable for the numerical work is given, and the condition that must be satisfied to insure their stabilty is derived.
  @cite_11: 
summary: 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 .