The many disasters of recent years caused by floods, tidal waves, and the collapse of dams prove the need for a deeper understanding of the destructive power exerted by the most common and most useful substance on earth, water. In an age when man can unleash the fury of the atom and put himself into space, it is surprising how little is really known about such a seemingly simple process as the flow of water. However, if we look into a sparkling mountain stream and watch the swirling turbulent eddies, we realize that perhaps the processes involved are not as simple as one would first imagine.
The small store of knowledge we have concerning the details of fluid flow has come primarily from laboratory experiments. These experiments are limited in scope by size, cost, and lack of dependable measuring devices. It is absurd to even consider experiments to study the flow of water from an 80-foot dam or to study the effects of a 20-foot tidal wave. Even in an experiment of a reasonable size, the internal details of the flow often cannot be measured because the measuring device itself (if, indeed, an appropriate one exists) will change the flow pattern when it is inserted into the water.
The only alternative way to add to our knowledge of fluid flow is to develop dependable techniques for calculating mathematically the details of the processes involved. These processes are the same for all fluids, but water and air are the ones of interest to most people. The mathematical equations of motion which describe the flow of a fluid have been known for several centuries, but because of the complexity of these equations it was possible to solve them for only a limited number of simple cases where special assumptions could be made. Then, with the advent of the high speed electronic computer, a completely new and fascinating approach was taken toward the solution of these differential equations of motion. Scientists began to develop mathematical models that could be solved on a computer by numerical techniques.
The first big success came in the field of compressible, i.e., high velocity, fluid dynamics. In the last 15 years a large number of successful methods have been developed for finding the time-dependent solutions of a wide variety of interesting problems in this field. Accurate solutions have been obtained for the supersonic flow of air past a nose cone, the hypervelocity impact of a meteor with a space vehicle, the rise of a fireball through the atmosphere, and many other physical processes.
Techniques for solving incompressible, or low velocity, fluid flow problems were slower in coming. The first notable success was a method developed by Jacob Fromm [1] at the Los Alamos Scientific Laboratory. Fromm's method can be used to obtain the solution of problems involving a confined incompressible fluid. His success created a renewed interest in the field, and. several hydrodynamicists have adapted his technique to their own investigations.
Now a new technique has been developed by us at Los Alamos for the full time-dependent solution of the flow of an incompressible fluid with a free surface. For the first time, it is now possible to study by computer such violent processes as the destructive surge of water from a broken dam and the crashing of a wave against a breakwater (Figure 1). At the same time, the method has been applied to more gentle but equally interesting problems as, for example, the rise of an air bubble in water or the flow of water from a faucet into a sink. This technique is known as the Marker-And-Cell or MAC method. It is incorporated in a computer program which, for obvious reasons, we call SPLASH.
Using the SPLASH program, it is possible to set up computer experiments in much the same way that a scientist would set up a laboratory experiment. It is very easy, for example, to simulate a wind tunnel experiment for the study of low speed air flow past a rectangular obstruction. Figure 2 shows just such a computer experiment. The top and bottom boundaries represent the walls of the wind tunnel, with the base of the obstruction at the left boundary. Fluid is injected into the system from the left, and flows out of the system through the right boundary. The lines of particles shown in the plots are analogous to smoke lines which would be injected into an actual wind tunnel to better enable us to visualize the flow.
The wake behind the obstruction has always attracted the greatest interest because early experiments showed the creation of a regular sequence of eddies shed from the base. Not until the computer experiments, however, was it possible to see in detail exactly how the fluid was contorted into this strange and fascinating pattern.
Computer experiments serve several useful purposes. By means of a computer experiment it is possible to set up an idealized physical situation, calculate the changes which take place with time, and then compare the results with true physical phenomena.
In any laboratory experiment it is impossible to have complete control of the environment. Small variations in the density, temperature, or other physical properties of the fluid, or imperfections in the apparatus used for the experiment are always present. The computer eliminates all of these problems, and allows for a completely idealized system.
Another important use for computer experiments is to help in the design of useful laboratory experiments. By means of a few computer runs, the engineer can gain much useful information that will help him decide exactly what kind of laboratory experiment is needed. This eliminates considerable trial and error in setting up the experiment, and often results in a great saving of both time and money.
Computers are also a useful aid in formulating analytical solutions. By carefully analyzing the data from a computer run, the theoretician can often see approximations which can be made, and can then develop dependable...
Missing pages. See also Welch's UAIDE 1965 paper for further details.
by each computer run. If the data were merely printed, there would be stacks of listings to analyze for each problem. Our solution to this dilemma has been to use the old adage, One picture is worth a thousand words. We, therefore, make extensive use of the Stromberg-Carlson SC-4020 microfilm recorder to display our results in a variety of ways, each of which illustrates particular aspects of the flow.
The easiest and perhaps the most useful display method is to make plots of particle configurations. Since each marker particle in the system has its coordinates stored in memory, it is a very simple matter to plot a point for each particle. This gives us a picture of the fluid very closely resembling a photograph taken of a laboratory experiment. Figure 5, for example, shows particle configuration plots taken at four different times for the flow of water from a faucet into a sink. The first frame shows the water just before it hits the quiescent shallow water in the sink and subsequent frames show the flow at later times.
Although particle plots have the advantage of giving a nice visual effect and of being relatively easy to produce, they do not convey any information concerning the internal properties of the flow. A single particle plot, for example, does not show the direction of the flow or any information about the pressures.
For showing the direction and speed of the fluid we make velocity vector plots. This is accomplished by plotting one velocity vector for each cell in the system. Each velocity vector is given a direction which is the local flow direction, and a length which is proportional to the velocity of the fluid in the cell. The first three frames in Figure 6 are velocity vector plots showing the flow of a liquid over an obstacle. The last frame is a particle configuration plot taken at the same time as the third velocity vector plot. Notice that the combination of the particle plot and the velocity plot give a picture of the flow that contains a considerable amount of useful information. The particle plot shows the shape and position of the fluid while the velocity plot shows the speed and direction of the flow. The only thing these plots fail to show is any information about the pressures.
The most useful techniques for visualizing the pressure field within the fluid is to plot lines of constant pressure. A given plot would contain lines for several different values of pressure, separated by a given pressure interval. This gives us a contour map very similar to geographical contour maps. As long as the pressure field is relatively smooth, the contour plots will provide us with a very useful and informative picture (see Figure 7). However, a knowledge of the problem under consideration or printed information is necessary in order to decide whether the lines show increasing or decreasing values of pressure.
A given computer run will require several hundred advancements through time. If we take a particle plot, velocity vector plot, or contour plot at the end of each of these time steps, we have a short movie which will last for several seconds [5]. Particle plot movies, in particular, give a visualization of the flow which is very close to a movie of the actual event. A short collection of these movies has been combined to form a silent film entitled Computer Studies of Fluid Dynamics (Y-154). This film is available on free short-term loan from the Report Library at the Los Alamos Scientific Laboratory.
The MAC method has a wide range of applicability, and has attracted world-wide attention in a variety of fields. The most obvious applications concern the study of waves on a beach, fluid flow through pipes, and other similar investigations. Enthusiastic responses have come from doctors interested in blood flow through the veins, engineers interested in using the method for ship design, and persons interested in solving problems involving liquid propellants in space craft.
Further advancement in computer technology will open fascinating new possibilities for this method in the fields of oceanography and weather prediction. The method as applied to problems in these fields is limited now by the memory size and speed of the present generation of computers. Such advanced problems would require 10-20 times the memory available on the Stretch computer, and would run from 10-100 times as long as the problems now being run. Use of the disc solves the problem of space, but the time requirement can only be solved by future generations of computers. It should also be pointed out that the SPLASH program is merely a starting point for the MAC method. The method is being refined and expanded to solve even more complex problems than those to which it is now applied. A super computer of the future, and a sophisticated version of the MAC techniques will probably some day be able to accurately predict weather for many days in advance, or compute the changing currents for an entire ocean. The only limit is man's imagination.
1. Fromm, J. E. A Method for Computing Nonsteady, Incompressible, Viscous Fluid Flows, Los Alamos Scientific Laboratory Report LA-2910 (1963).
5. J Eddie Welch, Moving Picture Computer Output, UAIDE October 1965.
