Author(s): M. Y. Lam; M. S. Ghidaoui; A. A. Kolyshkin
Keywords: Shallow Flows; Linearized shallow water equation; Linearized Energy Equations
Abstract: Shallow flows are turbulent flows bounded by the water depth, with the along-stream and cross-stream dimensions far exceed the vertical dimensions. Long waves and large scale turbulent structures are two examples of shallow flows. Long waves arise in the study of study of Tsunamis, floods, and tidal waves, to name a few only. Large scale turbulent structures arise in the study of compound and composite channels, island wake flows, channel junctions etc (Van Prooijen et al, 2004, Sobey, 2002). Basic techniques applied for the study of shallow flows include field observations, experiments, numerical simulations and stability analyses. The current paper focuses on stability analyses, which provide information about the onset of coherent structures as well as the onset of wave front steepening. Conventional stability analyses are based on the linearized shallow water equations (Chu et al, 1991, Chen and Jirka, 1997 and Singh, 1996), with the rigid-lid assumption and the normal-mode assumption. This paper derives stability criteria for shallow flows on the basis of the linearized energy equation, without the application of the rigid-lid assumption and the normalmode assumption. The energy criteria are successfully validated against a nonlinear shallow water model for Fr 2, the flow is always unstable due to the roll wave instability (Singh, 1996). In such a case, a nonlinear simulation is also performed to visualize the instability mechanisms of the flow.