Inertial flows over the Koch leaf

Abstract

The main hypothesis of this paper is that inertial effects known from porous-media flow can be observed even in a simple system consisting of a single channel. The strength of inertia is controlled by the Reynolds number, $Re$. We investigated fluid flow through a channel with a cavity in the shape of a Koch leaf. The Koch leaf of level-6 was chosen because it combines regularity, repeatability, and representativeness with a complex multiscale structure and non-trivial hydrodynamic interactions. We analyzed nonlinear effects through the so-called friction factor and show that its deviation from Darcy's law is very subtle. We further examined the influence of inertia on the flow structure, including the participation number $π$ and the backward-flow fraction $ρ^{-}$. In contrast to disordered porous media, the participation number decreases with $Re$, indicating that inertia concentrates kinetic energy in the main through-flow, while recirculation (backward flow) increasingly occupies the Koch leaf cavity. The tortuosity varies non-monotonically with $Re$, the apparent permeability decreases, and in the inertial regime the two are related by a power law with an exponent comparable to that found for natural rock. At the lowest and highest Reynolds numbers, the local kinetic energy followed a power-law distribution, $P(\varepsilon)\propto\varepsilon^{-m}$, over several orders of magnitude with the same exponent, $m\approx0.83$; inertia modified only the tails of the distribution. Our research can be regarded as an investigation of transport in an extremely porous system with porosity $φ\approx 1$.