Gravity Waves on Turbulent Shear Flow: Reynolds Averaged Approach

Gravity waves propagating over free-surface flows with shallow depth are well-known phenomena. The small-amplitude sinusoidal wave and Korteweg-de Vries (KdV) equations are based on potential-flow theory being widely used to describe water-wave propagation. The KdV equation leads to two basic flow patterns, as follows: (1) cnoidal waves, and (2) solitary waves. However, in case of a real Newtonian fluid, the bed resistance and the rapid motion of fluid generate turbulence (eddies) in the medium. The effects of turbulence are taken into account in this paper by using the equations for the surface elevation $\eta$ and depth-averaged flow velocity $U$ developed previously. These equations are based on the Reynolds-averaged Navier-Stokes (RANS) equations for turbulent flow in open channels. The wave profile can be approximated by a form $a\cos k\hat{\xi }/(1-b\cos k\hat{\xi })$, where $a$ and $b$ are constant amplitudes; $k$ = wave number; $\hat{\xi }$ = dimensionless horizontal distance given by $(x-ct)/h$; $x$ = horizontal distance; $c$ = wave velocity; $t$ = time; and $h$ = undisturbed flow depth. Such a profile has the characteristic that the peaks are narrower but higher compared to wider but shallower troughs. The effects of streamflow on wave propagations are also considered. If the waves travel in the direction of the streamflow, there is a lengthening effect on the peaks and troughs, whereas if the waves travel against the direction of streamflow, they become shorter.