C.W Hirt

Volume of Fluid (VOF) MeThod for the Dynamics of Free Boundaries

Several methods have been previously used to approximate free boundaries in finite-difference numerical simulations. A simple, but powerful, method is described that is based on the concept of a fractional volume of fluid (VOF). This method is shown to be more flexible and efficient than other methods for treating complicated free boundary configurations. To illustrate the method, a description is given for an incompressible hydrodynamics code, SOLA-VOF, that uses the VOF technique to track free fluid surfaces.

Calculating Three-Dimensional Flows around Structures and over Rough Terrain

Abstract A computing technique for low-speed fluid dynamics has been developed for the calculation of three-dimensional flows in the vicinity of one or more block-type structures. The full time-dependent Navier-Stokes equations are solved with a finite-difference scheme based on the Marker-and-Cell method. Effects of thermal buoyancy are included in a Boussinesq approximation. Marker particles that convect with the flow can be used to generate streaklines for flow visualization, or they can diffuse while convecting to represent the dispersion by turbulence of particulate matter. The vast amount of data resulting from these calculations has been rendered more intelligible by perspective-view and stereo-view plots of selected velocity and marker-particle distributions.

Heuristic stability theory for finite-difference equations☆

Abstract A simple method is proposed for investigating the computational stability of finite-difference equations. The technique is especially powerful because of its applicability to nonlinear equations with variable coefficients. The method, which is based on an examination of certain kinds of truncation errors, is illustrated by applying it to a simple linear difference equation. Then it is used to explain the origin of instabilities observed in calculations of one- and two-dimensional fluid flows.

A lagrangian method for calculating the dynamics of an incompressible fluid with free surface

Abstract A new method is presented for the numerical solution of the transient flow of viscous incompressible fluids having free surfaces. The method is based on Lagrangian coordinates in contrast to other methods, which use the Eulerian representation. Lagrangian coordinates permit the accurate treatment of fluid interfaces and free surfaces, and make it a simple matter to include the effects of surface tension. Several examples illustrate the properties of this new technique.

Improved free surface boundary conditions for numerical incompressible-flow calculations

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.

Free-surface stress conditions for incompressible-flow calculations

Abstract The numerical study of transient incompressible fluid flows is greatly complicated by the presence of free surfaces. One method of treating such problems is the Marker-and-Cell technique, which in its original form, used simple approximations for the free-surface boundary conditions. These approximations are found to be inaccurate at low Reynolds numbers (R ≲ 10). With a simple modification it is possible to approximate the complete normal stress condition. This modification is shown to have a pronounced effect on some low-Reynolds-number flows.

Calculating three-dimensional free surface flows in the vicinity of submerged and exposed structures

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.

A simple scheme for generating general curvilinear grids

Abstract An intuitively simple approach is presented for the computer generation of two-dimensional curvilinear grids suitable for finite difference solutions of problems in the field of continuum dynamics. An iterative process is employed to transform uniform networks of rectangular zones into more complex configurations. Ease of use and optimal adjustment are stressed, and numerous examples are given.

A general corrective procedure for the numerical solution of initial-value problems☆

Abstract In many circumstances the finite difference equations used in the solution of an initial-value problem must be solved by an iteration process at each time step. This paper proposes a technique that permits crude iteration to be used without leading to a disastrous accumulation of error after many cycles of time advancement. The technique, which is quite general, is discussed for two examples. The second of these applications accounts for the success of the Marker-and-Cell computing method for the solution of incompressible fluid flow problems.

Adding limited compressibility to incompressible hydrocodes

Abstract A simple modification is described that may be used to add limited compressibility effects to incompressible hydrodynamics computer codes. Several sample calculations are discussed and used to compare the relative advantages of implicit and explicit time integration methods. It is also shown that the use of an artificially reduced speed of sound is not, in general, a good approximation for low speed fluid problems.