Undular hydraulic jumps arising in non-developed turbulent flows

Abstract

Turbulent plane flow over a bottom of constant slope is considered for very large Reynolds numbers, very small slopes of the bottom, and Froude numbers close to the critical value 1. In contrast to a previous work (Grillhofer and Schneider, Phys Fluids 15:730–735, 2003), it is not assumed that the flow far upstream is fully developed. The first-order perturbation equations contain unknown functions that are determined from a solvability condition of the second-order equations. Without making use of turbulence modeling or empirical parameters, a third-order ordinary differential equation is obtained for the shape of the free surface. Slow changes of amplitudes and wave lengths, respectively, associated with a small damping parameter are described by a multiple-scales solution, which also reveals the source of peculiarities of numerical solutions. A universal diagram of solutions and a universal map of the initial conditions that lead to undular jumps are given. Both numerical and multiple-scales solutions are compared with experimental data.