On a 1/2-equation model of turbulence

TL;DR

This paper introduces a simplified 1/2-equation turbulence model where the mean of the turbulent kinetic energy satisfies an ODE, offering analytical advantages and comparable velocity statistics to traditional 1-equation models in 2D and 3D tests.

Contribution

The paper proposes a novel 1/2-equation turbulence model based on a simplified ODE for the mean kinetic energy, enhancing analytical tractability and maintaining accuracy.

Findings

The 1/2-equation model has attractive analytic properties.

Velocity statistics from the 1/2-equation model are comparable to full 1-equation models.

The model performs well in 2D and 3D turbulence simulations.

Abstract

In 1-equation URANS models of turbulence the eddy viscosity is given by $\nu_{T}=0.55l(x,t)\sqrt{k(x,t)}$ . The length scale $l$ must be pre-specified and $k(x,t)$ is determined by solving a nonlinear partial differential equation. We show that in interesting cases the spacial mean of $k(x,t)$ satisfies a simple ordinary differential equation. Using its solution in $\nu_{T}$ results in a 1/2-equation model. This model has attractive analytic properties. Further, in comparative tests in 2d and 3d the velocity statistics produced by the 1/2-equation model are comparable to those of the full 1-equation model.