Investigation of shallow mixing layers by BGK finite volume model

Abstract

Turbulent shallow mixing layers and their associated vortical structures are ubiquitous in rivers, estuaries and coasts. Examples of these flows can be found in compound/composite channels, at the confluence of two rivers, at harbour entrances and at groyne fields. A finite volume 2D model, based on the averaging of the 3D shallow water equations with respect to depth and in which the numerical fluxes are obtained from the Bhatnagar–Gross–Krook (BGK) Boltzmann equation, is applied to shallow mixing layers for which experimental results are available. This model is hereafter referred to as the BGK model or BGK scheme. The BGK scheme is explicit, second order in time and space and conserves both mass and momentum. The BGK relaxation time is locally evaluated from the classical turbulence model of Smagorinsky. The BGK model accurately represents the mean flow field such as mean velocity profile, mean spread of the mixing layer, mean position of the mixing layer centreline and mean surface water profile. In addition, the Kelvin–Helmholtz (KH) instability including inception, vortex roll up, vortex growth by pairing and the eventual decay of the vortices by bed shear is well represented by the model. On the other hand, the magnitude of the turbulence intensity is over-predicted by the shallow water model. This discrepancy is partly due to the fact that the turbulence forcing assumed may not represent the actual random perturbations that may exist in the laboratory experiments and partly due to the inability of the depth-averaged shallow water equations to allow for the redistribution of turbulent energy along the vertical direction, since these governing equations do not model the 3D turbulence. Thus, the depth-averaged shallow water equations are well suited for investigating the KH stability and for predicting mean flow field including velocity profiles and transversal mixing of mass momentum in shallow environments. Accurate prediction of turbulence statistics would require resolving the small 3D scales with respect to water depth.

Keywords:

• shallow mixing layers
• stability of shallow shear flows
• BGK modelling of shallow flows
• coherent structures
• quasi-2D turbulence

Acknowledgements

The authors wish to thank the Research Grants Council of Hong Kong for financial support under Project HKUST6227/04E and Project 613306.