Application of a Simple Space-Time Averaged Porous Media Model to Flow in Densely Vegetated Channels

Abstract

Traditional flow modeling in open channels uses time-averaged turbulence models. These models are valid in clear fluid, but not if dense obstructions are present in the flow field. In this article we show that newly developed flow models can describe open channel flow as flow in a porous medium. Clear fluid models do not take into account drag due to the presence of the obstacles. Flow in rivers, channels, estuaries, and irrigation networks is often obstructed by vegetation, and coarse bedrock. In computer modeling applications, appropriate turbulence resistance models are either absent or empirically based In this article we develop a space-time averaged form of the Navier-Stokes equations, in order to improve modeling of flow in densely obstructed channels. We use a combination of Reynolds averaging for the turbulent flow and volume averaging in order to take into account the dense obstructions. We show that the obstacle density can be modeled by a porosity term if structural parameters of the vegetation are taken into account. In order to take these into account we develop a representative unit cell (RUC) concept, borrowed from volume averaging in porous media. Inside the RUC, local flow solutions for the Navier-Stokes equations are developed and used as closure terms in the space-time-averaged form of the Navier-Stokes equations. Our expression depends on measurable quantities such as average porosity and average vegetation diameter. It can be used in computational models to include vegetation characteristics directly, instead of approximate resistance factors. As an application, we use our theoretically derived model to compute resistance factors for Manning's equation from the structural properties of the vegetation modeled as a porous medium.