Effect of vertically logarithmic steady currents on shallow surface waves

Abstract

The combined wave-current flow has been solved by researchers by assuming wave over either depthwise constant or linear current profile. Some complicated nonlinear current profiles have also been considered to simulate various shear currents. We consider a nonlinear current vertically logarithmic in nature and examine its interaction with a periodic surface wave. The Navier-Stokes equations for incompressible flow are solved for the current part and by using periodic boundary conditions. The effect of logarithmic current on wave components is assessed. The corresponding celerity and dispersion equation yields a close-form solution for the shallowwave approximation. Several comparative trends between wave-only, wave with log current, and wave with constant current for the wave following/opposing these currents have been discussed. The flow properties of the first order are presented which can be applicable to the real inland and coastal flows, where progressive waves are ubiquitous over a depthwise logarithmic current. The work is further extended to the second-order semiempirical wave component by using past experimental data on the wave spectrum of combined flow.

Frequency-wavenumber spectrum of the free surface of shallow turbulent flows over a rough boundary

“…16,17,19,39 The numerical solution proposed by Shrira 19 applies to an arbitrary shape of the velocity profile, but it has been determined for the infinite depth case only, while the solution derived by Patil and Singh 39 is valid for the logarithmic profile and for the long wavelength limit. The vertical velocity profile in a turbulent shallow flow with rough static bed can be approximated by a power law of the vertical coordinate (Equation (1)), such as the 1/7 power function considered by Fenton 17 and by Lighthill in the Appendix of the work of Burns.…”