Auxiliary Variable-based Balancing (AVB) for source term treatment in open channel simulations

Abstract

Practical engineering applications of open channel flow modelling involve geometric terms arising from variations in channel shape, bottom slope and friction. This paper presents the family of schemes that satisfy the generalised C-property for which static equilibrium is a particular case, in the framework of one-dimensional open channel flows. This approach, named Auxiliary Variable-based Balancing, consists of using an auxiliary variable in place of the flow variables in the diffusive part of the flux estimate. The auxiliary variable is defined so as to achieve a zero gradient under steady-state conditions, whatever the geometry. Many approaches presented in the literature can be viewed as a particular AVB case. Three auxiliary variables are presented in this paper: water elevation, specific force and hydraulic head. The methodology is applied to three classical Riemann solvers: HLL, Roe and the Q-scheme. The results are compared on five test-cases: three steady-state configurations including friction, singular head losses and variations in bottom elevation, channel width and banks slope and two transient test-case (dam-break problems on rectangular and triangular channel). In each case, the auxiliary variable that best preserves the steady-state configuration is the hydraulic head. Besides, using the head as auxiliary variable allows head loss functions due to singularities to be incorporated directly in the governing equations, without the need for internal boundaries. However, it is generally less accurate when sharp transients are involved.