Douglas G. Dommermuth

A HIGH-ORDER SPECTRAL METHOD FOR THE STUDY OF NONLINEAR GRAVITY WAVES

We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number ( N = O (1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M , and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness ( ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

Numerical simulations of nonlinear axisymmetric flows with a free surface

A numerical method is developed for nonlinear three-dimensional but axisymmetric free-surface problems using a mixed Eulerian-Lagrangian scheme under the assumption of potential flow. Taking advantage of axisymmetry, Rankine ring sources are used in a Greens theorem boundary-integral formulation to solve the field equation; and the free surface is then updated in time following Lagrangian points. A special treatment of the free surface and body intersection points is generalized to this case which avoids the difficulties associated with the singularity there. To allow for long-time simulations, the nonlinear computational domain is matched to a transient linear wavefield outside. When the matching boundary is placed at a suitable distance (depending on wave amplitude), numerical simulations can, in principle, be continued indefinitely in time. Based on a simple stability argument, a regriding algorithm similar to that of Fink & Soh (1974) for vortex sheets is generalized to free-surface flows, which removes the instabilities experienced by earlier investigators and eliminates the need for artificial smoothing. The resulting scheme is very robust and stable. For illustration, three computational examples are presented: (i) the growth and collapse of a vapour cavity near the free surface; (ii) the heaving of a floating vertical cylinder starting from rest; and (iii) the heaving of an inverted vertical cone. For the cavity problem, there is excellent agreement with available experiments. For the wave-body interaction calculations, we are able to obtain and analyse steady-state (limit-cycle) results for the force and flow field in the vicinity of the body.

The initialization of nonlinear waves using an adjustment scheme

Abstract A procedure for initializing nonlinear free-surface simulations is developed and validated. Numerical simulations of nonlinear progressive waves are prone to developing spurious high-frequency standing waves unless the flow field is given sufficient time to adjust. An adjustment procedure is developed that allows nonlinear free-surface simulations to be initialized with linear solutions. The adjustment scheme allows the natural development of nonlinear self-wave (locked modes) and wave–wave (free modes) interactions. The implementation of the adjustment procedure is illustrated using a high-order spectral method. Comparisons are made to fully-nonlinear Stokes waves and Schrodinger theory.

Numerical simulation of the wake of a towed sphere in a weakly stratified fluid

We present some preliminary results from using large-eddy simulation to compute the late wake of a sphere towed at constant speed through a non-stratified and a uniformly stratified fluid. The wake is computed in each case for two values of the Reynolds number: Re = 10 4 , which is comparable to that used in laboratory experiments, and Re = 10 5 . An important aspect of the simulation is the use of an iterative procedure to relax the initial turbulence field so that the normal and shear turbulent stresses are properly correlated and the turbulent production and dissipation are in equilibrium. For the lower Reynolds number our results compare well with existing laboratory experimental results. For the higher Reynolds number we find that even though the turbulence is more developed and the wake contains finer structure, most of the similarity properties of the wake are unchanged compared with those observed at the lower Reynolds number

A HIGH-ORDER SPECTRAL METHOD FOR NONLINEAR WAVE-BODY INTERACTIONS

A high-order spectral method originally developed to simulate nonlinear gravity wave-wave interactions is here extended to study nonlinear interaction between surface waves and a body. The method accounts for the nonlinear interactions among NF wave modes on the free surface and NHB source modes on the body surface up to an arbitrary order M in wave steepness. By using fast-transform techniques, the operational count per time step is only linearly proportional to M and NF. Significantly, for a (closed) submerged body, the exponential convergence with respect to M, NF (for moderately steep waves), and NB is obtained. To illustrate the usefulness of this method, it is applied to study the diffraction of Stokes waves by a submerged circular cylinder. Computations up to M=4 are performed to obtain the nonlinear steady and harmonic forces on the cylinder and the transmission and reflection coefficients. Comparisons to available measurements as well as existing theoretical/computational predictions are in good agreement. The most important result is the quantification of the negative horizontal drift force on the cylinder which is fourth order in the incident wave steepness. It is found that the dominant contribution of this force is due to the quadratic interaction of the first- and third-order first-harmonic waves rather than the self-interaction of the second-order second-harmonic waves, which in fact reduces the negative drift force.

The laminar interactions of a pair of vortex tubes with a free surface

A fully nonlinear numerical method is developed to study the viscous interactions of a pair of vortex tubes rising toward a free surface. The numerical theory uses Helmholtzs decomposition to treat the irrotational and vortical components of the flow as separate nonlinearly coupled equations. The laminar interactions of a pair of vortex tubes with a clean free surface at intermediate Froude and Weber numbers and a low Reynolds number show two distinct phases. During the rise phase of the vortex pairs, instabilities lead to the formation of helical vorticity

On the breaking of standing waves by falling jets

By direct numerical time stepping, using two different methods of calculation, we follow the development of a standing gravity wave on the surface of deep water when it is given an initial energy exceeding that of the most energetic periodic wave of the same wavelength. Two aspects of the motion are studied in detail: the sequence of wave profiles close to the instant when the maximum surface slope angle is close to 45° and, second, the conditions under which a sharp cusp is formed at the free surface. When a cusp is formed, it can fall vertically into the wave trough, enclosing “bubbles of air.”

CONDITIONALLY AVERAGED DYNAMICS OF TURBULENCE

The conditionally averaged Navier-Stokes equations with fixed vorticity in a point are considered. It is found in particular that the conditionally averaged rates of vortex stretching and dissipation increase exponentially with the vorticity magnitude. The local imbalance of these effects leads to the formation and destruction of twisted vortex strings.

The formation of U‐shaped vortices on vortex tubes impinging on a wall with applications to free surfaces

Numerical simulations of a pair of vortex tubes impinging on a no‐slip wall show the formation of U‐shaped vortices that are wrapped around the primary vortex tubes. The essential stages of U‐vortex formation are as follows: generation of helical vortex sheets due to the onset of an instability on the primary vortex tubes, attachment of the helical vortex sheets to the separated wall boundary layer, wrapping of U‐shaped vortices around the primary vortex tubes, and feeding of boundary‐layer vorticity into the U vortices. The generation of U vortices helps to explain the striations and whirls that form on a clean or contaminated free surface above a pair of trailing‐tip vortices.

Simulation of steep breaking waves and spray sheets around a ship: the last frontier in computational ship hydrodynamics

Breaking ship waves are one of the most challenging problems in the field of free-surface hydrodynamics. We aim to produce and implement unique, scalable parallel-computing capabilities for simulating turbulent breaking waves and the resultant formation of spray and entrainment of air. SAIC has developed a Cartesian-grid method for simulating breaking waves around ships. Body-force and finite-volume formulations are used to model the hull and an interface capturing method is used to model the free surface. As a result, minimal user input is required to simulate breaking waves, which makes the tool ideal for ship design studies. At MIT, a suite of codes has been developed based on advanced large-eddy simulation of coupled air-water flows. This suite uses time accurate interface capturing and interface tracking methods to model turbulent free-surface flows. The results of the numerical simulations are used to guide the development of turbulence models for the SAIC code and other codes, such as Reynolds averaged Navier-Stokes (RANS), which are currently being used by the Navy. Through large-scale computations on the IBM SP3 and Cray T3E using both Cray SHMEM and MPI and hybrid techniques, the numerical results and their analyses provide us with the framework to develop models of wave breaking and spray formation and air entrainment. Numerical simulations of various ship-like geometries moving with forward speed and spilling breaking waves have been performed. With these promising results, which were achievable only through high-performance computations, the last frontier of computational ship hydrodynamics will be breached in the near future.