A multiple scales analysis of the undular hydraulic jump in turbulent open channel flow

The undular hydraulic jump in turbulent open channel flow is considered in the double limit of very large Reynolds numbers, i.e. Re-tau --> infinity, and Froude numbers approaching the critical value, i.e. Fr = 1 + 3/2 epsilon with epsilon --> 0. Fully developed turbulent flow far upstream is assumed. Owing to the large Reynolds number the constant inclination angle of the bottom, a, is small. The undular jump is associated with a distinguished limit, which is characterized by the similarity parameter A = alpha/epsilon(2) = O( 1). Since a wavy solution with a slowly varying amplitude is expected, a multiple scales expansion is performed. A new independent variable is introduced such that the wave length becomes constant and normalized to one. The perturbation equations of the orders epsilon, epsilon(3/2), epsilon(2), and epsilon(5/2) are considered in order to obtain a complete first-order solution. Analytical results for the amplitude of the first wave and the first wave length are obtained. They are compared with measured data and reasonable agreement is observed. The analysis also shows that a previously given ordinary differential equation describing the shape of the free surface is uniformly valid. (C) 2003 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.