Stochastic characteristics of orbital velocities of random water waves

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Abstract
This paper presents the stochastic properties of orbital velocities of random water waves in intermediate water depth. Both the emergence effect and weak nonlinear effects are studied; the theoretical predictions are compared with measured kinematics and the deviations from linear theory are quantified.This study includes new ideas in fluid dynamics. An analytic formula for probability distribution for velocities modified by the emergence effect as well as by nonlinearities of the wave motion in intermediate water depth is developed. This probability function gives us the first statistical moment, the second statistical moment for modified velocities in an analytical form, and by numerical integration the third statistical moment for modified velocities.The theoretical formulae for the statistical moments for surface elevation and for velocities up to third order, with nonlinearities of the motion taken into account, for the case when the emergence effect can be neglected, i.e. below the surface layer, have been developed. This includes a generalized formula for free-surface elevation setdown and calculation of the asymmetry of the horizontal velocity, which is found to be negative in agreement with measurements of Anastasiou et al. (1982b).From the first statistical moment of the modified horizontal velocity, the mean flux between any two levels in the wave flume may be calculated. When the integration is carried out from the bottom up to + ∞, it leads in approximation to the formula for total mean flux found by Phillips (1960). This agreement with Phillips’ formula encourages one to interpret the positive mean value of horizontal velocities as a ‘real current’. This interpretation also provides a new understanding of the fluid dynamic implications of results presented by Tung (1975).Theoretical prediction of the measured kinematics has allowed a better estimation of the return flow in the wave flume, and in the vicinity of the mean water level currents in two different directions are noted. Firstly, the emergence effect gives rise to a current at the mean water level in the direction of the wave advance. Secondly, a flow in the opposite direction, interpreted as a return current in the wave flume, is noticed just below that level.

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