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R. Jurisits, W. Schneider:
"Undular hydraulic jumps arising in non-developed turbulent flows";
Acta Mechanica, 223 (2012), 8; 1723 - 1738.

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.

Undular Jump, Free Surface Flow, Turbulence, Asymptotic Theory, Multiple Scales Method

http://dx.doi.org/10.1007/s00707-012-0666-4