Multiple scales analysis of the turbulent undular hydraulic jump

Abstract

The undular hydraulic jump in turbulent open channel flow is considered in the double limit of very large Reynolds numbers, and Froude numbers approaching the critical value, i.e. Fr = 1 + 3/2 ε with ε → 0+. The undular jump is associated with a distinguished limit, which is characterized by the similarity parameters A and a. The square root of the first parameter √A is essentially the ratio of the dimensionless friction velocity and the difference of the Froude number to its critical value. The second parameter a is a scaled measure of the difference of the incident turbulent flow to the fully developed turbulent flow. 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 ε, ε3/2, ε2, and ε5/2 have to be considered in order to obtain a complete first-order solution. In case of fully developed incident flow analytical results for the amplitude and wave length of the first wave are obtained. They are compared with measured data and reasonable agreement is observed.