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Finite difference method of transinet nonlinear free surface wave problems
Abstract
A finite difference method is developed for computing the two-dimensional transient potential flow generated by an impulse on the free surface. Both the dynamic and kinematic free surface conditions are considered in nonlinear version. the primary features of the present paper include the use of special coordinates transformations so that the geometry of the flow field is transformed into a time-invariant region, presents an iteration process, by which the velocity potential is computed as the solution of a Poisson equation, the application of fast Fourier transform (FFT) technique results in a tri-diagonal system of equations which can be readily solved by the Thomas algorithm, the computing time is significantly reduced. Thus an efficient technique for handling the transient potential problems is well justified. The feasibility of the present method has been verified by two examples including different initial disturbances respectively.
The present paper describes a numerical model for predicting the non-linear wave forces exerted on large coastal or offshore structures. The interaction is calculated from a known initial condition corresponding to still water in the immediate vicinity of the structure. A prescribed incident wave of large amplitude approaches the structure. The boundary constraints on the free surface are considered in the fully non-linear versions. Not far away from the structure, an artificial open boundary is considered with a suitable description of the radiation boundary condition for an incident wave propagating inwards, and in addition the scattering wave being absorbed. The finite difference method and time stepping are adopted for numerical calculation in the present model. For illustration the wave forces on a surface-piercing vertical cylinder subjected to two different incident wave trains were evaluated. Reasonably good agreement could be obtained between the numerical results and the analytical solution given by Isaacson.1.
Two numerical methods for solving boundary-value problems related to potential flows with a free surface are introduced in this paper. The first method, termed finite-element variational method, applies the conventional variational form for this type of boundary value problems. A modified variational method utilizing eigenfunctions expansions proves to be much more efficient. The second method, termed fundamental-singularity distribution method, works with the fluid boundary only. Both methods place no restrictions on the geometry of the body nor the bottom topography and are valid for either two- or three-dimensional problems. Test results obtained by these methods for the case of a heaving circular cylinder in water of finite or infinite depth agree well with those obtained by others, except for the behavior of the added-mass coefficient at low frequency in water of finite depth. For three-dimensional problems, test computations for a sphere in heave and sway motion by both methods agree well with others. Results obtained by the second method for an ellipsoid also agree very well with existing results.
This chapter discusses some free surface transient flow problems of seepage and irrotational flow. It discusses the transient free surface condition, finite element discretization, and some applications to transient seepage problems. Taylor and Finn approached the free surface problem iteratively amending the boundary position for steady state situations. France showed that in the case of seepage problems, a very realistic solution may be achieved by considering a transient boundary condition attached to this surface and incorporated this in the solution procedure. This process can also be applied to cases where such a transient solution has no physical meaning. Conversely, in cases of seepage this transient is extremely important and its determination of great interest in the solution of real problems.
An iterative finite element scheme is developed to solve potential flow problems. In this scheme, the field of computation is divided into sectors, each of which has an arbitrary number of basic elements. Computations are performed by sweeping all the sectors. This scheme benefits in data handling of the finite element computer programs for larger problems as well as in implementing non-Dirichlet boundary conditions. To study the effect of the choice of mesh systems on the computational error, a parametric study is made by applying the finite element method to potential flow problems around an elliptic cylinder. Treatment of non-Dirichlet boundary conditions is considered concerning its effect on the computational error, and computation of spatial derivatives in the finite element mesh is discussed.
In this paper a finite element method for free-surface problems is described. the method is based on two different forms of Hamilton''''s principle. To test the present computational method two specific wave problems are investigated; the dispersion relations and the nonlinear effect for the well-known solitary waves are treated. The convergence test shows that the present scheme is more efficient than other existing methods, e.g. perturbation scheme.
The flow of a gas in a shock tube is treated in the context of kinetic theory as an initial value problem for the one‐dimensional Krook equation. The only simplifying assumption made is that the gas is one‐dimensional. This corresponds to a gas with an adiabatic constant γ = 3. A finite difference method is proposed, utilizing the fact that, for sufficiently small time intervals, the gas experiences essentially free molecular flow, so that the collision effects can be treated as a first‐order correction. The conservation laws are not used. The computed solution agrees excellently with the classical solution, but in addition, has shock structure, diffusion of the contact discontinuity, and dispersion of the expansion wave, all incorporated. The same procedure is used to calculate steady‐state shock structure, to which the shock developed in the shock tube is compared. It is seen that for the strength of the shock calculated (Pressure ratio of tube 10:1, shock Mach number 1.43, γ = 3), the shock is essentially ``fully developed'' after about 20 mean collision times of the low‐pressure gas. In conclusion, an exact reduction procedure is given by which problems for a monatomic three‐dimensional gas (γ = ) are reduced to the solution of two simultaneous Krook equations, manageable by the present procedure without vastly increasing the need for computing capacity.
The marker and cell method for high-speed computer was used to investigate the motion of a fluid whose free surface has been perturbed by an impulsive sinusoidal pressure. For gravity pointing out of the fluid, the resulting motion exhibits Rayleigh-Taylor instability, whose progress is followed from low amplitude to the asymptotic bubble and spike stage. The effects of small and large viscosity are contrasted. For gravity pointing into the fluid, the large-amplitude oscillatory motion exhibits a shift of resonance frequency and other effects of nonlinearity. Comparisons of results are presented wherever previous analytical or experimental work was available, and many new aspects of these types of flow are discussed. The calculations are based upon the full, time-dependent, nonlinear Navier-Stokes equations for an incompressible fluid.
We discuss the creation of a finite-element program suitable for solving incompressible, viscous free-surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. We conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related ‘stick-slip’ problem and the tube inlet problem, both of which also contain stress singularities, are also given.
A numerical technique has been developed for calculating the three-dimensional, transient dynamics of incompressible fluid having a free surface. The Navier-Stokes equations are solved by a solution algorithm based on the Marker-and-Cell method. The flow may be calculated around variously shaped and spaced obstacles that are fully submerged or penetrate the surface. To illustrate the capability of this technique a variety of examples are presented.
Complete free surface stress conditions have been incorporated into a numerical technique for computing transient, incompressible fluid flows. An easy to apply scheme, based on a new surface pressure interpolation, permits the normal stress to be applied at the correct free surface location. Tangential stresses are applied through the assignment of appropriate velocities near the surface. To illustrate the influence of the complete stress conditions, a variety of examples are presented, including some with highly contorting and colliding surfaces. Several of the examples are compared with experimental and analytical results. The influence of these boundary conditions on numerical stability is discussed from a simple qualitative point of view.
The nonlinear properties of finite-amplitude water waves are modelled by a numerical method based on the Marker and Cell technique. The paper introduces the Stanford University Modified MAC (SUMMAC) code which is shown to be a valid tool for analyzing incompressible flows with a free surface under transient conditions. The method is applied as an example to the run up of a solitary wave on a vertical wall. The results are compared critically with available experimental data.
A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free. The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement. The primary dependent variables are the pressure and the velocity components. Also used is a set of marker particles which move with the fluid. The technique is called the marker and cell method. Some examples of the application of this method are presented. All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired.
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