On the determination of the grad-div criterion

Physica D: Nonlinear Phenomena

2020

The time relaxation model, which is family of high accuracy turbulence models, has proven to be effective in regularization of Navier-Stokes Equations. The model belongs to the class of Large Eddy Simulation models, and is derived by adding a linear time regularization term χu ⋆ to the Navier-Stokes Equations. The time relaxation operator truncates small solution scales by injecting an extra dissipation to a simulation, without altering appreciably the solution's large scales. Herein to evaluate the effect of the time regularization term on a simulation, the rate of energy dissipation of the model in body-forcedriven turbulence is studied. Our result, which agrees with Kolmogorov's conventional turbulence theory, is also consistent with the rate proven for the NSE. Moreover, employing the model requires a choice of the coefficient χ. It is known that the model's simulation is sensitive to the parameter. The analysis motivates a range of possible values for the coefficient χ in 3d turbulent flows away from walls.

Section: Introductionmentioning

confidence: 95%

On the long time behavior of time relaxation model of fluids

Physica D: Nonlinear Phenomena

2020

“…for the continuous model e.g. [9], [10], [24], [25], [29], [37], [38], [50] and [51]). The question explored herein is: What is the time-averaged energy dissipation rate ε(u h ) , when u h is an approximation of u on a given mesh h (fixed computational cost) for the Smagorinsky model?…”

Section: Introductionmentioning

confidence: 99%

Analysis of mesh effects on turbulent flow statistics

Journal of Mathematical Analysis and Applications

2019

Turbulence models, such as the Smagorinsky model herein, are used to represent the energy lost from resolved to under-resolved scales due to the energy cascade (i.e. non-linearity). Analytic estimates of the energy dissipation rates of a few turbulence models have recently appeared, but none (yet) study the energy dissipation restricted to resolved scales, i.e. after spatial discretization with h > micro scale. We do so herein for the Smagorinsky model. Upper bounds are derived on the computed time-averaged energy dissipation rate, ε(u h ) , for an under-resolved mesh h for turbulent shear flow. For coarse mesh size O(Re −1 ) < h < L, it is proven,

“…These works build on Busse [5], Howard [34], and others. Later, these approaches was substantially developed and extended by many authors (see, e.g., [3,25,22,42,43,47,48,49,53,54,55,58,63,64,65], etc.). For a good review on the recent results, including theoretical, computational and experimental, see the paper [60].…”

mentioning

confidence: 99%

On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations

Chow

2022

DCDS-B

<p style='text-indent:20px;'>We consider the three-dimensional stochastically forced Navier–Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper bounds of the mean value of the time-averaged energy dissipation rate are derived directly from the equations for weak (martingale) solutions. This estimate is consistent with the Kolmogorov dissipation law. Moreover, an additional hypothesis of energy balance implies the zeroth law of turbulence in the absence of a deterministic force.</p>