Analysis of the Taylor dissipation surrogate in forced isotropic turbulence

TL;DR

This paper derives a new model for the dissipation rate in forced isotropic turbulence, combining analytical and numerical methods, and compares it with previous models, revealing the effects of finite forcing and differences from decaying turbulence.

Contribution

The paper introduces a new scale-independent model for the nondimensional dissipation rate in forced isotropic turbulence, validated through analytical derivation and numerical data.

Findings

The model fits numerical data with $ extCinf=0.47$ and $C_L=18.5$.

$R_L^{-1}$ correctly describes the Reynolds number behavior.

Finite forcing effects are balanced by inertial and viscous terms.

Abstract

From the energy balance in wavenumber space expressed by the Lin equation, we derive a new form for the local Karman-Howarth equation for forced isotropic turbulence in real space. This equation is then cast into a dimensionless form, from which a combined analytical and numerical study leads us to deduce a new model for the scale-independent nondimensional dissipation rate $\Ceps$, which takes the form $\Ceps = \Cinf + C_L/R_L$, where the asymptotic value $\Cinf$ can be evaluated from the third-order structure function. This is found to fit the numerical data with $\Cinf = 0.47 \pm 0.01$ and $C_L= 18.5 \pm 1.3$. By considering $\Ceps - \Cinf$ on logarithmic scales, we show that $R_L^{-1}$ is indeed the correct Reynolds number behaviour. The model is compared to previous attempts in the literature, with encouraging agreement. The effects of the scale-dependence of the inertial and viscous terms due to finite forcing are then considered and shown to compensate one another, such that the model equation is applicable for systems subject to finite forcing. In addition, we also show that, contrary to the case of freely decaying turbulence, the characteristic decline in $\Ceps$ with increasing Reynolds number is due to the \emph{increase} in the surrogate expression $U^3/L$; the dissipation rate being maintained constant as a consequence of the fixed rate of forcing. A long-time non-turbulent stable state is found to exist for low Reynolds number numerical simulations which use negative damping as a means of energy injection.