Author(s): Ben R. Hodges; Jose G. Vasconcelos; Sazzad Sharior; Vitor G. Geller

Linked Author(s): Sazzad Sharior

Keywords: Stormwater; Numerical method; Free surface flow

Abstract: The transition from open-channel to surcharged flow is a problem for numerical modeling of stormwater systems. The problem roots are in the discrete shock forming when the hyperbolic Saint-Venant equations meet the elliptic incompressible flow equations at the surcharge transition. In the free-surface flow the pressure celerity is the gravity wave speed, whereas the surcharged flow has near-instantaneous pressure transmission at an acoustic pressure wave celerity. To make matters worse, when the incompressibility approximation is used with rigid pipe walls and the hydrostatic approximation, the modeled pressure celerity becomes infinite. When single-equation solution techniques (i.e., shock-capturing) with implicit solvers are applied, the free-surface/surcharge shock creates a stiff problem that converges slowly. When explicit solvers are applied, the shock results in unphysical oscillations at the pressurization interface that lead to numerical instability. Formally, the problems are reduced if the unsteady surcharged flow is modeled using the slight compressibility of water and the elasticity of the pipe -- i.e., introducing a hyperbolic component to the surcharged equations using an acoustic pressure celerity. Transient-resolving models (such as the method of characteristics) that are used for water hammer in distribution systems are arguably the most rigorous approach for such problems, but are computationally expensive due to their small time-step. However, there is a long tradition of approximate hyperbolic solvers for incompressible surcharged flow, including (1) Preissmann Slot, (2) Two-Component Pressure Approach, and (3) Artificial Compressibility. We will present and discuss the three approaches for modeling surcharged flow with hyperbolic equations. Each of the methods provides approximating terms that are controlled by model coefficients to alter the pressure wave celerity through the surcharged system. They share a common goal in slowing the pressure celerity below the true acoustic pressure wave speed, which allows the numerical solution to dissipate the transition shock between the free surface and surcharged flows without resorting to extraordinarily small time-steps. The different methods provide different capabilities and numerical implementations that affect their behavior and suitability for different problems.

DOI: https://doi.org/10.3850/IAHR-39WC2521711920221370

Year: 2022