by Francis H. Harlow and Jacob E. Fromm

Lab. y. Scheepsbouwkunde
Technische Hogeschool Delft
MAART 1965
REPRINT UIT: SCIENTIFIC AMERICAN

The fundamental behavior of fluids has traditionally been studied in tanks and wind tunnels. The capacities of the modern computer make it possible to do subtler experiments on the computer alone.

The natural philosophers of ancient Greece liked to do experiments in their heads. Centuries later Galileo developed the "thought" experiment into a fruitful method of inquiry and in our own time the method appealed strongly to such men as Albert Einstein and Enrico Fermi. Now the arrival of the modern electronic computer has made the method immensely more powerful and versatile. The computer makes it possible to simulate nature with numerical models and to investigate it in ways that have never been practicable before. Physical processes of enormous complexity are being examined minutely and with considerable realism. New hypotheses are being proved true or false. In physics, engineering, economics and even anthropology the computer has become a revolutionary tool.

One of the great attractions of experiment by computer is that it can avoid some of the uncertainties of measurement. Moreover, it provides a technique that can be classed as both theoretical and experimental. It is theoretical because it deals with abstract (that is, mathematical) statements of how things relate to one another. It is experimental because the computer is given only data specifying the initial state of a system and a set of rules for calculating its state at some time in the future. The computer worker has no more idea how this future state will unfold than has the traditional worker who conducts a comparable experiment in an actual laboratory.

To demonstrate the power of computer experiments we have chosen a single example involving the dynamic behavior of fluids. The particular experiment is a study of the flow of air past a rectangular rod.

At first thought the use of a computer for calculating this flow may seem to be a needlessly roundabout procedure. Would it not be simpler and more enlightening to put the rod in a wind tunnel and observe how air containing filaments of smoke flows around it? Actually it would not. For many of the questions to be investigated the physical experiment would be more complicated and costly, and it would not provide as much information as the experiment by computer.

For an example one can point to the problem of redesigning the Tacoma Narrows Bridge after it had been shaken to pieces by wind-induced vibrations soon after it was built. For the rebuilding of the bridge many elaborate models were made and tested again and again before a safe design was finally developed. Without doubt much of the cost and time spent on the problem could have been saved by computer calculations if the computers and appropriate numerical techniques had then been available. Experiments with numerical models can show the interaction of winds and a bridge in detail and produce answers in far less time than it takes to prepare a physical experiment.

The Soviet physicist A. A. Dorodnitsyn has remarked about such problems that computer calculation "can give a solution that is not only more rapid and cheaper but also more accurate" than the physical experiment itself.

Experimentation by computer also allows the investigation of many phenomena that are either inaccessible to direct study or involve factors that cannot be measured accurately. In the flow problem that we shall discuss, for example, it is difficult to measure directly in a wind tunnel the temperature distribution in the complicated downstream wake. Computer experiments, however, can yield a reliable description of the temperature distribution.

Another benefit of a computer experiment is that it usually affords far better control of the experimental conditions than is possible in a physical experiment. In wind tunnel studies, for instance, the experimenter must modify his interpretations to include the consideration of such effects as those due to the compressibility of the working fluid, variations in fluid viscosity and uncertainties in flow velocity. In a computer experiment such properties often can be excluded or included at will. Moreover, the computer program can isolate crucial features for examination, can eliminate irrelevant factors and can often assess the experimental uncertainties.

Finally, and most importantly, experiments by computer provide a test of the applicability of theory to the complicated phenomena under investigation. Do the equations of fluid dynamics really represent the correct theoretical description when applied to phenomena as complicated, say, as the oscillatory flow that develops in the wake of a retreating rectangular rod? For such problems the mathematician would like to obtain what he calls an analytical solution—the kind of exact solution that can be obtained by the processes of mathematical analysis. For problems in fluid dynamics, however, the necessary mathematical techniques for obtaining the complete solution have not yet been developed. The detailed results provided by a computer can actually help in the development of analytical solutions to the basic equations of fluid dynamics. Usually in the mathematical model of a complex problem some of the factors can only be approximated, and obtaining a realistic solution depends on finding out which features are crucial for a reasonable representation. With the help of computer experiments one tries to discover workable approximations that will simplify the mathematics needed to solve complicated problems—in this case a problem in oscillatory fluid flow.

The reader will find the "computer wind tunnel" experiment easier to follow if we consider briefly how a fluid behaves when it flows around a fixed object such as a rectangular rod.

At low speed the airflow is smooth and steady, a condition described as laminar flow. At a certain critical speed, which depends on the size of the rod, the laminar flow breaks down. For a rod one inch in height the critical speed in air is about one inch per second; the smaller the rod, the higher the speed at which turbulence begins. If the fluid is more viscous than air, laminar flow is more easily maintained and the critical speed for turbulence becomes higher.

Above the critical speed the airstream breaks up into vortices that are similar to the small whirlpools seen when a cup of coffee is stirred. These vortices are shed alternately from the top and bottom of the object placed in the airstream. This oscillating wake was first extensively studied by the aerodynamicist Theodor von Kármán and is known as a "von Kármán vortex street."

The oscillating wake sends out pulses that react back on the object itself. The vibration so produced is responsible for the sound made by a golf club swung rapidly through the air and for the whine of a ship's rigging in the wind. It was resonant vibration produced by the wind that caused the Tacoma Narrows Bridge to break and fall into the bay.

As the air speed increases, the vortices in the vortex street become more and more ragged and eventually break up into tiny eddies whose motion is almost entirely random. At this stage fully developed turbulence has been reached.

The known patterns of air motion past an object, then, give us certain definite phenomena to look for in the computer experiments. If the computer reproduces a vortex street and, at a later stage, turbulence, it will show that the theoretical understanding of fluid dynamics is accurate and therefore can be relied on to predict what will happen when a fluid flows past objects of various shapes and at various speeds.

To set up the calculational experiment we must first translate the physical situation into the language of numbers for the computer. For bookkeeping purposes the experimental area in the computer wind tunnel is divided into many square cells, which form the basic computing mesh. A typical mesh requires at least 49 cells in the direction of horizontal flow and 24 cells in the vertical dimension, for a total of 1,176 cells. Each cell must contain two numbers representing the components of average air velocity in two directions, together with other numbers representing such variable quantities as "vorticity," "stream function" and, if heat flow is desired, temperature as well. Finally, the computer must be supplied with a set of operating instructions, or "code," that spells out in detail exactly how the computer must manipulate every number in every cell in order to calculate how the flow configuration will change from instant to instant. It can require billions of mathematical operations and anywhere from a few minutes to a few hours of computing time to carry out the calculations needed to represent the flow of air for a time interval of several minutes. In our studies we have used either an IBM 704 or the somewhat faster machine, also built by the International Business Machines Corporation, known as Stretch.

The actual development of a successful code is a time-consuming process and is carried out in three steps. The first involves the testing of detailed numerical methods and is strewn with pitfalls. It is no trick, for example, to invent methods that develop numerical instability: the computer results rapidly run askew and lead to complete nonsense. Like physical experiments, computer experiments are also subject to interference by gremlins. Just as the vibration of a motor may produce extraneous turbulence in a wind tunnel, so the numerical approximations fed into a computer may lead to equally unwanted "truncation turbulence."

The second step is to prepare a full-scale code. For our problem in fluid dynamics this required many months, most of them consumed by "debugging," or ferreting out errors in the step-by-step instructions. Such a code is written with sufficient generality so that it can be used to solve a wide variety of roughly similar problems. Thus a good code can be used for years and will often be a source of inspiration for workers in other laboratories.

The third step is to formulate the code in terms of a specific problem. In our oscillating-wake study an important part of the formulation was to determine the "initial" and "boundary" conditions. The initial condition describes the state of the air as turbulence progressively increases (four steps from left to right). In this experiment computational particles are introduced through a single cell (horizontal streaks), as though squirting a jet of colored water into a clear tank. The jet of air is unstable and soon breaks into expanding, irregular vortices like those exhibited by a plume of smoke. Similar but far more complex experiments can be used to test theories about aircraft jet engine noise suppression.

We could have assumed, for example, that the air was at rest, corresponding to the condition in a real wind tunnel before the fan is turned on. We found that it was simpler, however, to start with the fluid behaving as if it were flowing past the rod in a simple laminar manner without viscosity.

The boundary conditions refer to what is happening at the edges of the computational mesh. Our decision was to have the top and bottom edges represent the walls of the wind tunnel and to have the left edge represent an air input of uniform flow. The right edge gave us more trouble, but we finally arranged for the fluid to flow out and back into the computing region in a way that created a minimum of mathematical disturbance.

The computing process itself can be compared to the making of a motion picture. Starting with the initial conditions prescribed for each of the 1,176 cells in "frame" No. 1, the computer follows the coded instructions to determine the conditions in each cell a brief instant of time later, thereby producing frame No. 2 of the film. Each successive frame is similarly generated on the basis of numerical data computed for the preceding frame. The fastest computer available to us, Stretch, can generate about 10 frames a minute. When the calculation has proceeded far enough, the results are gathered up for study.

The computer's results can be presented in any of several different forms. One form of print-out consists of all the numbers describing the flow in each frame. Usually this form of print-out is restricted to samplings taken at selected intervals, because the complete data for every one of the hundreds or thousands of cycles in an experiment would be far too much for an analyst to digest, to say nothing of storing the reams of paper. Sometimes the computer is programmed to print certain calculations that supply particular points of information, such as the amount of air drag caused by the obstacle at specific wind speeds. The most useful and popular type of print-out, however, is the actual plotting of the flow in pictorial form.

The computer itself can generate plots of the flow configurations and put them on film by means of a microfilm recorder. Several selected frames from such recordings, exactly as they came from the computer, are among the illustrations on this page and preceding pages of this article. The sequence of all the frames of an experiment, combined in a film strip and run through a motion picture projector, gives a very vivid picture of the development of vortices and other features as a fluid flows around an obstacle.

From the numbers describing the flow in each cell of the computing mesh, the computer generates streamlines that show both the direction and the speed of flow throughout the space. The speed is indicated by the spacing between the streamlines: where the lines are close together the flow is fast; where they are farther apart the flow is slower. The computer can show the streamline patterns in either of two ways: as if a camera were photographing a stream of air flowing past it or as if the camera were moving along with the stream. The latter view shows the pattern of vortices in clear detail.

The computer can even simulate the motion of markers often used to make flow visible, such as filaments of smoke in air or of dye in water. In the computer the markers consist of "computational particles." At certain desired points in the computation these particles are thrown in (the magic of the computer allows their creation anywhere at will) and thereafter they are carried along wherever the flow of air goes. Their paths of motion produce lines called streak lines. The streak lines generated by the computer give a remarkably faithful impression of the behavior of smoke or dye filaments. Perhaps the most striking of these computer constructions is the configuration of streak lines emerging from a jet: it looks like a filament of cigarette smoke.

Usually the computer is programmed to furnish several different configuration plots, showing features of the flow from various points of view. These are by no means merely an interesting album of pictures. They show the qualitative features of the development of the flow and provide precise quantitative information about the flow at every point. In many cases the computer reveals important details that would be extremely difficult to obtain from physical experiments.

The example we have described of a computer technique for investigating fluid flow is only one of many successful efforts that have been made to carry out complex experiments by computer. Other workers have used computers to tell in detail what is going on inside a nuclear reactor and to assess in an instant the progress of a rocket soaring into space. Tomorrow the computer may give accurate forecasts of the weather, of the future of the economy and of the state of man's health.

n5321 | 2025年8月9日 21:01