A numerical, fluid-dynamics technique for high-speed computers is described and illustrated. It applies to the solution of problems dealing with incompressible viscous fluids and involing nonsteady motions in several dimensions in space. The ability to handle free-surface boundary conditions allows waves to be studied through all phases of breaking and splashing, as well as a number of related phenomena.
We have developed a new technique for numerically solving problems in fluid dynamics. It is particularly applicable to studies of waves and of other phenomena that are associated with the motion of an incompressible fluid with a free surface. Examples are the flow of water from a broken dam, the generation of water waves by an explosion, the formation of breakers on a beach, and the splash of a jet of liquid hitting a plate.
The application illustrated in Fig. 1 is to the surge of water under a sluice gate. The initial frame (top left) shows the water at rest immediately after the gate has quickly opened; the deep reservoir (left) is subjected to a surface pressure in addition to that produced by gravity and the shallow pond (right) is initially quiescent. Subsequent frames show the formation of a backward breaker, in which the flow is partly smooth, partly irregular.
The problem was scaled to give unit density to the water. The downward gravitational acceleration was also of unit magnitude, while the scale of distance is determined by the initial height of the reservoir behind the sluice gate, 2.9 distance units. In these dimensions, the applied surface pressure was 2.5, the coefficient of kinematic viscosity was 0.01, and the times of the six frames are t = 0, 1.0, 1.5, 2.0, 2.5, and 2.73.
The elements of fluid are represented in the calculations by marker particles. Determination of the trajectories of the particles is based on a finite-difference approximation to the full, nonlinear, Navier-Stokes equation for a viscous, incompressible fluid. The finite-difference equations are related to a Eulerian mesh of cells not shown in the figure. The cells cover the entire region of interest, amounting to 1500 in this case.
The computing method has been designed for use with a high-speed computer. The present program is run on the IBM 7030 (Stretch) Computer; a typical problem requires approximately 20 to 60 minutes. Available computer memories limit our studies to two space dimensions, but the method is equally applicable to three-dimensional problems. Configuration plots, such as those in Fig 1, are obtained directly from the computer through the Stromberg-Carlson microfilm recorder and are not retouched.
The results are generated by the computer through a succession of small time steps or cycles, resembling the frames of a motion picture. The frames shown in Fig 1 were selected from among hundreds that were obtained in the complete run. The sequence of processes necessary to accomplish the solution for each new frame is as follows:
The equations, boundary conditions, and techniques for solution [1] are related to those used by Fromm [2] [3] [4], but they have required modifications appropriate to a different set of primary dependent variables; Fromm employed the stream function and vorticity, variables that never enter directly into this pressure-velocity technique.
The results of such calculations have been compared with experiments, showing excellent agreement in every case. A study of the water motion from a broken dam, for example, proved that all aspects of the flow can be obtained at least as accurately as experiments could measure them [1] . An advantage of the calculations is that they give more detailed data than can be obtained from experiments. In some cases, costly or time-consuming experimental studies can even be avoided by the careful use of such computer studies. Even more important, computer studies often provide a valuable basis for analytical studies of physical processes, giving both new ideas for models and bases for comparing the results of analysis of the models.
[1] F H Harlow, J E Welch, Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface, Phys Fluids 8,2182(1965).
[2] J E Fromm, A Method for Computing Non-steady Incompressible, Viscous Fluid Flows, Los Alamos Report, LA-2910, May 1963.
[3] J E Fromm, F H Harlow, Numerical Solution of the Problem of Vortex Street Development, Phys Fluids 6,975 (1963)
[4] F H Harlow, J E Fromm, Dynamics and heat transfer in the von Karman wake of a rectangular cylinder Phys Fluids 7,1147 (1964)
[5] This work was sponsored by the Atomic Energy Commission (14 June 1965).