Do irrotational water waves remain irrotational in the limit of a vanishing viscosity?

TL;DR

The paper addresses whether irrotational water-wave theory remains valid in the limit of vanishing viscosity under no-slip bottom conditions. It numerically solves the full free-surface Navier–Stokes equations in 2D with non-flat bathymetry, using an ALE framework and adaptive mesh refinement to resolve bottom boundary layers, and compares results to the irrotational Euler limit. The main finding is that boundary-layer vorticity is shed into the bulk as modons, producing finite perturbations to the free surface that do not vanish with increasing $Re$, making the vanishing-viscosity limit singular. This effect persists for both sharp and smooth bathymetries, with a curvature-threshold controlling separation, implying that irrotational models may be inadequate in nearshore contexts where bed-induced vorticity significantly alters wave shape and dynamics.

Abstract

Theoretical results on water waves almost always start by assuming irrotationality of the flow in order to simplify the formulation. In this work, we investigate the well-foundedness of this hypothesis via numerical simulations of the free-surface Navier-Stokes equations. We show that, in the presence of a non-flat bathymetry, either angular or smooth, a gravity wave of finite amplitude can shed vortex pairs from the bottom boundary layer into the bulk of the flow. As these eddies approach the free surface they modify the shape of the wave. It is found that this perturbation does not vanish as the Reynolds number is increased. The vanishing viscosity limit of water waves is therefore singular when no-slip boundary conditions are enforced on the bottom.

Figures (9)

• Figure 1: Geometry of the domain $\Omega(t)$ in dimensional units. The bathymetry can be chosen arbitrarily. It is pictured here as a rectangular step of height $h_b$ and length $\ell_b$ as it shall be used in sec. \ref{['sec:rect_step']}.
• Figure 2: Snapshots of the interfaces for different values of $Re$ and the free-surface irrotational Euler solution, at times $t = 10$ and $t = 15$. In all these simulations, the initial condition has been computed setting $L = 2\pi$, $A = 0.1$, $k=1$, $h_b = 0.2$ and $\ell_b = L/3$. A significant difference remains in the limit of small viscosity. The Euler simulation has been obtained using the code of Dormy and Lacave DormyLacave2024. The numerical parameters used to obtain this result are given in table \ref{['tab:numerical_params']}.
• Figure 3: On the left panel, time evolution of the $L^2$ distance $d_E$ between the free surface obtained from the Navier-Stokes solution and the one corresponding to the inviscid irrotational Euler solution for different values of $Re$. The distance $d_5\, ,$ with the simulation whose Reynolds number is $Re = 10^5 ,$ is presented on the right.
• Figure 4: Evolution of the vorticity at $Re = 10^5$ (movie available as supplementary material). The color code is centred on $[-5,5]$ to highlight eddies in the flow.
• Figure 5: Vorticity field at $t = 10$ and $t=11.5$ for Reynolds numbers $Re$ ranging from $Re = 10^{3.5}$ to $Re = 10^5$.