[Paper Review] A penalization method for calculating the flow beneath travelling water waves of large amplitude

Adrian Constantin, Konstantinos Kalimeris+1 more|arXiv (Cornell University)|Aug 8, 2014

Ocean Waves and Remote Sensing13 references3 citations

TL;DR

This paper presents a penalization method to numerically simulate large-amplitude traveling water waves with constant vorticity, reformulating the governing equations as a constrained optimization problem. The method accurately captures flow characteristics—velocity, pressure, and streamline patterns—beneath the wave surface, including in rotational flows with positive or negative vorticity and in irrotational cases.

ABSTRACT

A penalization method for a suitable reformulation of the governing equations as a constrained optimization problem provides accurate numerical simulations for large-amplitude travelling water waves in irrotational flows and in flows with constant vorticity.

Motivation & Objective

• To develop a robust numerical method for simulating large-amplitude traveling water waves in flows with constant vorticity, including both irrotational and rotational cases.
• To address the challenge of accurately computing the sub-surface flow dynamics beneath steep, nonlinear water waves where traditional methods may fail.
• To ensure the numerical solution satisfies the physical constraints of no-flux through the bed and free-surface kinematic conditions.
• To provide a stable, convergent iterative algorithm that selects genuine wave solutions from a broader family of potential solutions.

Proposed method

• Reformulate the steady water wave problem as a constrained optimization problem using a stream function formulation to reduce the number of unknowns.
• Introduce a penalization technique to enforce the governing partial differential equations and boundary conditions through a variational formulation.
• Use a finite-difference discretization of the stream function $ h(q,p) $ to represent the fluid domain and solve the system iteratively.
• Implement an iterative algorithm that alternates between updating the height function $ h^{(k)} $ and minimizing residuals of the PDE and boundary conditions.
• Apply a stopping criterion based on residual norms and proximity to stagnation points (i.e., $ 1/h_p o 0 $), with adaptive reinitialization when $ h_p^{(k+1)} o 0 $.
• Use an initial approximation derived from analytical formulas for non-laminar, large-amplitude waves with constant vorticity, with the irrotational case as a special case ($ u = 0 $).

Experimental results

Research questions

• RQ1 How can large-amplitude traveling water waves with constant vorticity be accurately simulated numerically despite the nonlinearity and lack of small-amplitude assumptions?
• RQ2 Can a penalization method effectively enforce the complex boundary conditions and PDE constraints in a rotational flow setting without stagnation points?
• RQ3 What is the behavior of velocity components and pressure distribution beneath large-amplitude waves in irrotational and rotational flows?
• RQ4 How does the presence of vorticity—positive or negative—affect the shape of the free surface and the streamline structure beneath the wave?

Key findings

• The penalization method successfully computes accurate numerical solutions for large-amplitude traveling water waves in both irrotational and rotational flows with constant vorticity.
• For the irrotational case ($ u = 0 $), the method reproduces known wave profiles and velocity patterns consistent with existing theoretical and numerical studies.
• In the case of positive vorticity ($ u = 2.95 $), the wave profile becomes asymmetric with a steeper front and broader crest, and the horizontal velocity $ c - u $ decreases near the bed.
• For negative vorticity ($ u = -1 $), the wave profile exhibits increased asymmetry with a steeper front, and the pressure distribution shows a more pronounced maximum beneath the wave crest.
• The vertical velocity $ v $ is strongest near the free surface and decreases toward the bed, with the largest values occurring near the wave crest in all cases.
• The pressure deviation from atmospheric pressure increases monotonically from the free surface to the bed, reaching a maximum just below the wave crest, consistent with hydrostatic expectations.