Characteristic Sensitivity of Turbulent Flow within a Porous Medium under Initial Conditions

TL;DR

This paper investigates the sensitivity of turbulent flow within porous media using the $k-$ model, revealing a universal power-law behavior of eddy viscosity related to initial conditions, which differs from typical turbulent flows.

Contribution

It introduces a mathematical analysis of turbulence in porous media, highlighting a nullcline-based sensitivity and universal power-law behavior of eddy viscosity.

Findings

Nullcline is a line attractor for turbulence variables.

Eddy viscosity follows a power-law relation with initial conditions.

Sensitivity is unique to porous media turbulence, unlike standard flows.

Abstract

Flows within porous media play important roles in many scientific and industrial systems. However, the case wherein such flows become turbulent has not been completely understood, particularly, from a mathematical viewpoint. In this study, the $k-\varepsilon$ model (the variable $k$ denotes the turbulent kinetic energy per unit mass and $\varepsilon$ the dissipation rate for the turbulent kinetic energy), which has been widely used for the usual turbulent flow in a clear fluid, was applied to a turbulent flow through porous media. If a homogeneous, averaged flow and homogeneous isotropic turbulence are assumed, the governing equations for the variables $k$ and $\varepsilon$ describe a straight line as a nullcline common to both the variables. The nullcline was shown to be a line attractor. A temporal evolution of the eddy viscosity was also obtained, and the initial and final values of the eddy viscosity were observed to be related to a power law, indicating universal sensitivity. This sensitivity originates from the common nullcline and is not observed for the usual turbulent flow. Finally, nonlinear mathematical and seismological implications were provided using based on the results obtained.