Dressler's theory for curved topography flows: iterative derivation, transcritical flow solutions and higher-order wave-type equations

The Dressler equations are a system of two non-linear partial differential equations for shallow fluid flows over curved topography. The theory originated from an asymptotic stretching method formulating the equations of motion in terrain-fitted curvilinear coordinates. Apparently, these equations failed to produce a transcritical flow profile changing from sub- to supercritical flow conditions. Further, wave-like motions over a flat bottom are excluded because the bed-normal velocity component is not accounted for. However, the theory was found relevant for several environmental flow problems including density currents over mountains and valleys, seepage flow in hillslope hydrology, the development of antidunes, the formation of geological deposits from hyper-concentrated flows, and shallow-water flow modeling in hydraulics. In this work, Dressler's theory is developed in an alternative way by a systematic iteration of the stream and potential functions in terrain-fitted coordinates. The first iteration was found to be the former Dressler's theory, whereas a second iteration of the governing equations results in velocity components generalizing Dressler's theory to wave-like motion. Dressler's first-order theory produces a transcritical flow solution over topography only if the total head is fixed by a minimum value of the specific energy at the transition point. However, the theory deviates from measurements under subcritical flow conditions, given that the bed-normal velocity component is significant. A second iteration to the velocity field was used to produce a second-order differential equation that resembles the cnoidal-wave theory. It accurately produces flow over an obstacle including the critical point and the minimum specific energy as part of the numerical solution. The new cnoidal-wave model compares well with the theory of a Cosserat surface for directed fluid sheets, whereas the Saint-Venant theory appears to be poor.