The damping of viscous gravity waves

Using a solution of the linearized Navier–Stokes equations, an approximate formula has been derived for the damping rate of gravity waves in viscous fluids. The proposed solution extends the results found by Lamb (1932) [5] for waves propagating in deep-water conditions for large Reynolds numbers and those derived by Biesel (1949) [14] under more general hypotheses. Specifically, comparisons with the Lamb solution highlight large differences in intermediate and shallow depths and/or for moderate Reynolds numbers while significant discrepancies are observed with the Biesel solution in deep-water conditions. For these reasons, the proposed solution is of great importance for the estimation of the viscous dissipations during the wave motion and represents a useful benchmark for the validation of numerical solvers. With respect to this, the theoretical findings have been compared with numerical simulations obtained by means of a well-known Smoothed Particle Hydrodynamics solver.

► The linearized Navier–Stokes equations are solved to study viscous gravity waves.
► An approximate formula for the damping rate of viscous gravity waves is derived.
► The proposed solution reduces to Lamb’s one for high Reynolds deep-water waves.
► The estimation of dissipations may be useful for validating numerical solvers.