# On the long time behavior of time relaxation model of fluids

## TL;DR

This paper analyzes the long-term behavior of the time relaxation model for fluids, showing its effectiveness in turbulence simulation and its consistency with Kolmogorov's theory, while exploring optimal parameter choices.

## Contribution

It provides a theoretical analysis of the energy dissipation rate in the time relaxation model, validating its accuracy and guiding parameter selection for turbulent flows.

## Key findings

- Model agrees with Kolmogorov's turbulence theory
- Energy dissipation rate matches that of Navier-Stokes equations
- Parameter sensitivity analysis suggests optimal coefficient ranges

## Abstract

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 $\chi u^{\star}$ 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 force driven 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 $\chi$. It is known that the model's simulation is sensitive to the parameter. The analysis motivates a range of possible values for the coefficient $\chi$ in $3d$ turbulent flows away from walls.