Purely irrotational theories of the effect of the viscosity on the decay of free gravity waves

Abstract

A purely irrotational theory of the effect of viscosity on the decay of free gravity waves is derived and shown to be in excellent agreement with Lamb’s (1932) exact solution. The agreement is achieved for all waves numbers k excluding a small interval around a critical k=kc where progressive waves change to monotonic decay. Very detailed comparisons are made between the purely irrotational and exact theory. 1. Introduction Lamb (1932, §348, §349) performed an analysis of the effect of viscosity on free gravity waves. He computed the decay rate by a dissipation method using the irrotational flow only. He also constructed an exact solution for this problem, which satisfies both the normal and shear stress conditions at the interface. Joseph and Wang (2004) studied Lamb’s problem using the theory of viscous potential flow (VPF) and obtained a dispersion relation which gives rise to both the decay rate and wave-velocity. They also computed a viscous correction for the irrotational pressure and used this pressure correction in the normal stress balance to obtain another dispersion relation. This method is called a viscous correction of the viscous potential flow (VCVPF). Since VCVPF is an irrotational theory the shear stress cannot be made to vanish. However, the corrected pressure eliminates this uncompensated shear stress from the power of traction integral arising in an energy analysis of the irrotational flow. Here we find that the viscous pressure correction of the irrotational motion gives rise to a higher order irrotational correction to the irrotational velocity which is proportional to the viscosity and does not have a boundary layer structure. The corrected velocity depends strongly on viscosity and is not related to vorticity; the whole package is purely irrotational. The corrected irrotational flow gives rise to a dispersion relation which is in splendid agreement with Lamb’s exact solution, which has no explicit viscous pressure. The agreement with the exact solution holds for fluids even 10 4 times more viscous than water and for small and large wave numbers where the cutoff wave number kc marks the place where progressive waves give rise to monotonic decay. We find that VCVPF gives rise to the same decay rate as in Lamb’s exact solution and in his dissipation calculation when k kc. The effects of vorticity are sensible only in a small interval centered on the cutoff wave number. We will do a comprehensive comparison for the decay rate and wave-velocity given by Lamb’s exact solution and Joseph and Wang’s VPF and VCVPF theories. 2. Irrotational viscous corrections for the potential flow solution The gravity wave problem is governed by the continuity equation 0 u ,