The undular hydraulic jump in turbulent open channel flow at large Reynolds numbers

Plane flow over a bottom of constant slope is considered in the double limit of very large Reynolds numbers, i.e., Reτ→∞, and Froude numbers approaching the critical value, i.e., Fr=1+32ε with ε→0. Fully developed turbulent flow far upstream is assumed. Owing to the large Reynolds number the inclination angle of the bottom, α, is small. The undular jump is analyzed for a parameter regime that is characterized by values of α that are of the same order of magnitude as ε2, i.e., α/ε2=O(1). The first-order perturbation equations contain unknown functions that are determined from a solvability condition of the second-order equations. Without making use of a turbulence model or empirical parameters, the following equation is obtained for the shape of the free surface: (d3H1/dX3)+(H1−1)(dH1/dX)−βH1=0, with H1→0 as X→−∞. H1(X) denotes the first-order perturbation of the surface elevation as a function of the nondimensional longitudinal coordinate X, and the parameter β=13αε−3/2 characterizes the slow changes of amplitudes and wavelengths, respectively. Numerical solutions of this ordinary differential equation are compared with experimental data.