Kinematic Undertow Model with Logarithmic Boundary Layer

Abstract

A new kinematic undertow profile model is developed to relate the mean horizontal velocity, bottom shear stress, and boundary layer thickness in a simple but general manner. The model combines a logarithmic profile in the bottom boundary layer with a parabolic profile in the interior layer. Use of a logarithmic profile is justified using our laboratory measurements for regular waves spilling on a rough, impermeable slope. Two forms of the model are presented, each with one calibration coefficient associated with the mean bottom shear stress. By adjusting the calibration coefficient at each measuring line, the model is shown to be capable of predicting the measured undertow profiles both inside and outside the surf zone for our rough slope case and for smooth slope cases from the literature. The model does not predict the overshoot in the bottom boundary layer for the rough slope case outside the surf zone. The predicted velocity profile for the smooth slope case in the bottom boundary could not be verified due to a lack of data. The predicted boundary layer thickness agrees with the measurements for the rough slope case and appears to be reasonable for the smooth slope cases. The model predicts the shear velocity in the transition region and inner surf zone reasonably well for the rough slope case, and it underpredicts the shear velocity outside the surf zone. This model is shown to be simple and versatile, but it will need further validation using irregular wave data and varying bottom geometry before it can be used in practical applications.