Comparison of shallow water models for rapid channel flows

Abstract

To model shallow free surface flows, the Saint-Venant Equations (SVE) are a convenient simplification of the incompressible Navier–Stokes Equations (NSE). In the present study, we compare the two models for one-dimensional channel flow over a hump (cf. Behr (XNS simulation program, 2016 [5]), Küsters (Compar-ison of a Navier–Stokes and a shallow water model using the example of flow over a semi-circular bump, 2013 [8]), Noelle et al. (J Comput Phys 226(1):29–58, 2007 [10]), Sikstel (Comparison of hydrostatic and non-hydrostatic shallow water models, 2016 [13])). Our numerical experiments show that the SVE fail for some rather standard transcritical flows, where the two models compute different water heights ahead of and different shock speeds behind the hump. Using numerical computations as well as a formal Cauchy–Kowalevski argument, we give a qualitative explanation of the shortcoming of the SVE. In addition, we examine a recently developed non-hydrostatic shallow water model Sainte-Marie et al. (Discrete and Cont Dyn Syst Ser B 20(4):361–388, 2014 [12]) which proposes to produce physically more realistic results.