Inviscid Channel Flows

Abstract

The computation of non-hydrostatic inviscid flows is considered here. For steady irrotational flows, the energy head is a constant in all the points of the domain of the flow, making use of the energy principle simple. However, both the energy and momentum principles for non-hydrostatic flows are presented and discussed. Extended Boussinesq-type energy and momentum equations are derived using Picard iteration method (Matthew in Proceedings of the ICE 91(3):187–201, 1991), based on an iterative solution of the Cauchy–Riemann equations. The first and second iterative cycles are detailed, and the ensuing analytical predictors for the velocity components are carefully tested using a complete two-dimensional potential flow model based on the x-ψ method. Another set of Boussinesq-type equations is derived based on an approximate treatment of the flow net formulating the Euler’s equations in intrinsic coordinates (Hager and Hutter in Acta Mechanica 51(3–4):31–48, 1984a; 53(3–4), 183–200, 1984b). Picard iteration method is also presented in topography following curvilinear coordinates. The development is used to derive a generalization of Dressler’s (Journal of Hydraulic Research 16(3):205–222, 1978) theory, in which cnoidal and solitary wavelike solutions are embedded. Applications for steady potential flows are detailed, including critical flows over spillways, flows in free overfalls, transitions from mild to steep slopes, and flows in vertical sluice gates. The inclusion of vorticity in non-hydrostatic models is presented with a specific approximation for flows in free overfalls. An introduction to the mathematical theory of irrotational water waves is given based on the Serre–Green–Naghdi equations. The frequency dispersion of non-hydrostatic water waves is explained using a small-amplitude wave. The cnoidal wave theory for finite-amplitude long waves is also discussed. The solitary wave theory is used to investigate the effect of linear and parabolic approximations for the vertical pressure distribution. Non-hydrostatic dam break waves over a rigid and horizontal bed are used to introduce the need of numerical models; a simple finite-difference scheme is employed to illustrate the behavior of these waves.