A High-Order Hybrid-Spectral Incompressible Navier-Stokes Model For Nonlinear Water Waves

TL;DR

The paper develops a high-order nodal hybrid-spectral solver for the incompressible Navier–Stokes equations with a free surface to simulate nonlinear, dispersive water waves. It combines a σ-coordinate transformation, a low-storage explicit Runge–Kutta time integrator, and a geometric $p$-multigrid–accelerated GMRES solver to efficiently enforce mass conservation via a Poisson pressure problem. The spatial discretization uses a Fourier basis horizontally and Chebyshev basis vertically, with aliasing control and a cranked pressure-Poisson solve at each RK stage. Results demonstrate spectral convergence across depths, accurate linear dispersion, boundary-layer resolution with limited grid points, and good agreement with experiments for waves over non-flat bottoms, highlighting both computational efficiency and physical fidelity.

Abstract

We present a new high-order accurate computational fluid dynamics model based on the incompressible Navier-Stokes equations with a free surface for the accurate simulation of nonlinear and dispersive water waves in the time domain. The spatial discretization is based on Chebyshev polynomials in the vertical direction and a Fourier basis in the horizontal direction, allowing for the use of the fast Chebyshev and Fourier transforms for the efficient computation of spatial derivatives. The temporal discretization is done through a generalized low-storage explicit 4th order Runge-Kutta, and for the scheme to conserve mass and achieve high-order accuracy, a velocity-pressure coupling needs to be satisfied at all Runge-Kutta stages. This result in the emergence of a Poisson pressure problem that constitute a geometric conservation law for mass conservation. The occurring Poisson problem is proposed to be solved efficiently via an accelerated iterative solver based on a geometric $p$-multigrid scheme, which takes advantage of the high-order polynomial basis in the spatial discretization and hence distinguishes itself from conventional low-order numerical schemes. We present numerical experiments for validation of the scheme in the context of numerical wave tanks demonstrating that the $p$-multigrid accelerated numerical scheme can effectively solve the Poisson problem that constitute the computational bottleneck, that the model can achieve the desired spectral convergence, and is capable of simulating wave-propagation over non-flat bottoms with excellent agreement in comparison to experimental results.

Figures (8)

• Figure 1: Illustration of a two-dimensional fluid domain, $\Omega$, and associated boundaries, $\Gamma_w$ and moving boundary $\Gamma_{FS}$. The free surface is denoted by $\eta(x, t)$ and the water depth by $h(x)$.
• Figure 2: Example of a Geometric multigrid V-cycle where the grid hierarchy is traversed through utilization of transfer operators.
• Figure 3: Convergence plots for $u$ for varying nonlinearity ($H/L$) across rows in the columns and varying dispersion ($kh$) across columns in each row.
• Figure 4: Linear dispersion error for $N_x = 2$.
• Figure 5: Wave boundary velocity profile