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Early Work in Numerical Hydrodynamics

by Francis H. Harlow

I

met Stan Ulam shortly after coming to Los Alamos in 1953. As an eager youngster chasing new dreams, I was inspired and encouraged (and sometimes properly chastised) by the older resident scientists and Laboratory consultants. Several stand out especially for their powerful encouragement; one of these is Stan.

Some of my associates, especially during the first six years, didn't like many of my wild ideas about fluid dynamics and the techniques for solving such problems by high-speed computers. Stan continually took the time to see what was going on and had the faith (not always justified) to tell others positive things about my explorations. I shall always be grateful for Stan's, as well as Conrad Longmire's, crucial influence in establishing our fluid dynamics group in the Theoretical Division.

Stan and I had many talks, especially on the stochastic behavior of complex systems. He seemed to feel how these systems worked: their collective properties were very real to him. He was intrigued by the almost-cyclic properties they sometimes could exhibit and participated in pioneering numerical experiments on fluid-like, many-particle dynamics.

His early work with John Pasta* created the grandaddy of the free-Lagrangian method of modeling turbulence and, in the sixties, led ultimately to the Particleand-Force technique for the calculation of shock formation and interaction problems. Although couched in terms of hydrodynamics, the pioneering work has had significant impact on many branches of numerical analysis, especially in terms of the interpretation and meaning of results. The main thrust of their thinking is

captured in the following excerpts.

"Our approach to the problem of dynamics of continua can be called perhaps "kinetic"-the continuum is treated, in an approximation, as a collection of a finite number of elements of "points;" these "points" can represent actual points of the fluid, or centers of mass of zones, i.e., globules of the fluid, or, more abstractly, coefficients of functions, representing the fluid, developed into series."

One of the motivations behind the freeLagrangian approach was the computational difficulties for fluid flow with large internal shears in which elements that were initially close later found themselves widely separated.

"It was found impractical to use a "classical" method of calculation for this hydrodynamical problem, involving two independent spatial variables in an essential way This "classical" procedure, correct for infinitesimal steps in time and space, breaks down for any reasonable (i.e., practical) finite length of step in time. The reason is, of course, that the computation... assumes that "neighboring" points, determining a "small" area-stay as neighbors for a considerable number of cycles. It is clear that in problems which involve mixing specifically this is not true... the classical way of computing by referring to initial (at time t = 0) ordering of points becomes meaningless."

The next point is one that Stan emphasized repeatedly, illustrating what he felt to be a potential power of their approach.

"The meaningful results of the calculations are not so much the precise positions of our elements themselves as the behavior, in time, of a few functionals of the motion of the continuum.

"Thus in the problem relating to the mixing of two fluids, it is not the exact position of each globule that is of interest but quantities such as the degree of mixing (suitably defined); in problems of turbulence, not the shapes of each portion of the fluid, but the overall rate at which energy goes from simple modes of motion to higher frequencies."

As it turned out, the behavior, in time, of the functionals of the motion that they calculated was very smooth despite the complicated, turbulent nature of the fluid's motion. Thus, an important perspective on the modeling of complex phenomenon had been established. Indeed, turbulence transport theory, the subject of the following article, depends upon the strong tendencies in nature towards universal behavior that are the basis for the observed smoothness in their functionals. This theory is an excellent example of Stan's idea that wonderful numerical results can emerge from averaging discrete-representations over a set of possible scale sizes. But the theory goes further in providing an analytic formulation of turbulence transport.

Stan lived to see the realization of some of his ideas others are still being investigated but I always had confidence that if Stan had a feeling for something, it was sure to be signifcant. He was a friend I shall long remember. ■

*John Pasta and Stan Ulam. 1953. Heuristic studies in problems of mathematical physics on high speed computing machines. Los Alamos report LA-1557.

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Instabilities & Turbulence

W

-hen the interface between two materials experiences strong accelerative or shearing forces, the inevitable results are instability, turbulence, and the mixing of materials, momentum, and energy. One of the most important and exciting breakthroughs in our understanding of these disruptive processes has been the recent discovery that the features of the processes often are independent of the initial interface perturbations. This discovery is so important that scientists at Los Alamos National Laboratory, the California Institute of Technology, the Atomic Weapons Research Establishment in Great Britain, Lawrence Livermore National Laboratory, as well as scientists in France, and no doubt in the Soviet Union, are working hard to confirm and extend this new understanding experimentally.

by Didier Besnard,

Francis H. Harlow, Norman L. Johnson,

Rick Rauenzahn, and Jonathan Wolfe

Theoretical analyses are likewise showing a firm basis for this astonishing discovery. Two types of theory are being employed, gradually combined, and even proved essentially equivalent. These are the multifield-interpenetration approach and the single-field turbulence approach. Even brute-force hydrodynamics calculations are demonstrating this same property of independence from initial perturbation.

The consequences for developments in such main-line Laboratory projects as inertial-confinement fusion are profound. Our entire view of material mixing, turbulence shear impedance, and energy transport has undergone a revolutionary shift to qualitatively different directions.

What is the physical essence of this new way of thinking? No matter how

TURBULENCE EFFECTS

Fig. 1. The effects of turbulence include increased mixing of initially separated materials, an increase in shear impedance of fluid near rough boundaries due to the turbulent viscosity, and increased transport of heat into surrounding cooler regions.

Increased Mixing of Materials

Increased Shear Impedance

Increased Heat Diffusion

carefully we attempt to achieve smoothness and homogeneity, any sufficiently strong destabilizing influence at a material discontinuity will inevitably be disruptive. Indeed, the disruptive effects will be manifested in essentially the same manner as if there were a considerable roughness or inhomogeneity at or near the interface. Add to this the effects of any long-wavelength asymmetries, and we have an immutable inevitability for major instabilities in virtually every experimental circumstance of accelerative or shearing dynamics of interest to the Laboratory. Reliable predictability of new weapons designs in a comprehensive test ban, the design of any locally intense energy source, the development of workable concepts in Strategic Defense, the achievement of successful inertial-confinement fusion devices, and the success of many other Laboratory programs will depend crucially on our ability to model these instability and turbulence effects realistically.

What Is Turbulence?

To describe the techniques we are using to model these effects, we must first consider in more detail the properties of turbulence itself. Turbulence is the random fluctuation in fluid motion that often is superimposed on the average course of the flow. The effects of turbulence can be highly significant (Fig. 1), increasing the fluid's effective viscosity and enhancing the mixing of initially separated materials, such as the mixing of dust into air or bubbles into a liquid. Turbulence is a significant factor in the wind resistance of a vehicle, in the dispersal of fuel droplets in an internal combustion engine, in mixing and transporting materials in chemical plants, indeed in virtually every circumstance of high-speed fluid flow.

It is easy to be deceived into thinking that turbulence is rare, because it often is not directly visible to the casual observer. Although water flowing rapidly through a transparent pipe may look completely smooth, touching the pipe can reveal large vibrations and the injection of dye through a tiny hole in the wall can demonstrate rapid downstream mixing. Both effects are a direct result of intense turbulent fluctuations.

Turbulence in air can be demonstrated-even in a relatively calm room-by holding one end of a long thread and watching its fluctuating response to air currents. Sunshine streaming over the top of a hot radiator creates shadow patterns on a nearby wall that dance restlessly in the never-ending turbulence that accompanies the upward flow of air.

Why is nature discontent with the smooth and peaceful flow of liquids and gases, especially at high flow speeds? What are the processes that feed energy into turbulent fluctuations? The answers lie in the behavior of energy. In contrast to momentum, energy has the peculiar ability to assume numerous and varied configurations. Momentum constraints, while restrictive, are helpless to prevent seemingly capricious energy rearrangements. In any real fluid flow, these rearrangements are triggered by inevitable perturbations that can be fed from the reservoir of mean-flow energy.

It is helpful at this point to compare turbulence with the random motion of simple gas molecules in a box because the approaches to both of these problems include much that is similar. However, the analogy becomes seriously misleading if pushed too far.

Molecular Systems. In a box of molecules the dynamics of each individual can

be described quite accurately by Newton's laws. Yet we seldom try to analyze the complex interactions of all the trajectories, which are seemingly capable of very chaotic behavior. Instead, we appeal to the remarkably organized mean properties of the motion, identifying such useful variables as density, pressure, temperature, and fluid velocity.

We cannot ignore the departure of the individual from the behavior of the mean; indeed, some of the most interesting properties of the gas are directly associated with these departures. Diffusion of heat energy, for example, represents transport of kinetic energy by fluctuations; pressure in a "stationary" gas is the result of continual bombardment of molecules against objects immersed in the gas (Fig. 2(a)); viscous drag between two opposing streams of gas (Fig. 2(b)) arises because of fluctuations from the mean-flow velocity that cause molecules to migrate from one stream to the other.

Turbulent Eddies and Mean Flow. Turbulent eddies in a fluid superficially resemble individual molecules in a gas. They likewise bounce around in random fashion, carrying kinetic energy in their fluctuational velocities. (Such turbulence kinetic energy is typically as much as 10 per cent of the mean-flow kinetic energy, or even more in regions where the mean flow stagnates at a solid surface.) Eddies also diffuse momentum (plus heat and any imbedded materials), exerting pressure through momentum transport and bombardment against walls.

But the concept of a turbulent eddy is nebulous at best. Gas molecules have an easily identifiable shape, size, mean separation, and mean free path between collisions. Turbulent eddies, in contrast, have a spectrum of sizes; they overlap each other; the constraint on their motion through the fluid by the immediate presence of neighboring fluid precludes the simple concept of a mean free path.

Moreover, identification of what part of the dynamics is turbulence and what part is mean flow is arbitrary. For molecules the distinction is essentially unique; in most circumstances, individual molecular fluctuations take place on a scale that is orders of magnitude smaller than the scale of collective, fluid-like motion. For turbulent eddies the fluctuational scale may be an appreciable fraction of the mean-flow scale. More to the point, the observer's experimental configuration itself establishes the distinction between turbulence and mean flow.

To put the matter succinctly, mean flow is that part of the dynamics directly associated with the macroscopic conditions established or measured by the observer, whereas turbulence is the more capricious part of the flow associated with finer-scaled perturbations not controlled by the observer but inevitably present in any real flow.

As an example, consider air flow around a parked automobile on a gusty day. With suitable instruments an observer can record variations in the approaching wind velocity. These measurements describe the source of the mean flow, and the macroscopic features of the car constitute the boundary conditions. Mean-flow patterns in the wake on the downwind side of the vehicle can be observed either with ribbon that stretches out with the average air velocity at each place it is held or with an upstream smoke generator emitting a thin filament of smoke that can be photographed as it passes over the car.

Both the ribbon and the filament have an average direction to their motion that varies on the same time scale as that of the monitored gusts of wind; the relationship between these two features is the correlation that our investigator is seeking. In addition,

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TANGENTIAL DRAG

Fig. 3. Fluid moving in a circular trough loses mean-flow kinetic energy because of tangential drag on the walls. Although this entire loss in energy will eventually appear as heat, a significant fraction may first appear as the kinetic energy of turbulence.

however, the ribbon flutters rapidly about that average (at the rate of many fluctuations per second), and the smoke filament diffuses in contorted kinks into the surrounding air. This capricious variation around the time-varying average is what our observer calls turbulence.

A second observer standing nearby, but paying no attention to the detailed observations of the first, feels buffeted by the gusts and, likewise, would agree that there is much turbulence. However, this observer can legitimately disagree as to which part of the air flow is mean flow and which part is turbulence, seeing an average southwesterly wind with turbulent variations that last several seconds. Meanwhile, an earth-orbiting satellite reveals that the southwesterly wind is simply a momentary fluctuation (of a half hour or so) from the general westerlies crossing the continent that day.

This example has three different fluctuational scales, all properly identified as turbulence on the basis of the observer's chosen viewpoint. The difference, however, is not merely one of semantics, and we discuss below the consequences of this multiple viewpoint to mathematical modeling of the flow processes. Important guidance is furnished by a careful consideration of interactions among the various dynamical scales.

There is thus a seemingly random nature to both molecular dynamics and turbulence. The detailed flow field of a group of molecules or eddies can vary by large amounts as a result of minor initial perturbations on a microscopic scale. But the remarkable feature of these dynamical systems is that the overall stochastic behavior is essentially independent of the manner in which the fluctuations are introduced.

However, not every fluid flow is sensitive to minor perturbations. Viscous or slowly moving fluids travel in a purely laminar fashion, responding negligibly to finescale perturbations. Why does flow remain stable for some conditions and exhibit turbulence for others? The answer lies in the ways in which energy is drawn from the mean flow as the motion gradually decays to quiescence.

Turbulence Energy: Sources and Sinks

The statements of mass, momentum, and energy conservation lie at the foundations of fluid dynamics. In particular, fluid flow implies the presence of energy, which can exist in any of various forms: kinetic, heat, turbulence, potential, chemical. For the moment we are concerned only with the first three. By kinetic energy we mean the motion energy carried by the mean flow; heat energy refers to the kinetic energy of molecular fluctuations. Turbulence energy is at a scale between these first two: it is the kinetic energy of fluctuations that are large compared with the individual molecular scale but small compared with the mean-flow scale.

As we said earlier, in contrast to mass and momentum, which are highly constrained by their conservation laws, energy behaves very capriciously. Although total energy is rigorously conserved, transitions among the many manifestations of energy occur continuously. It is a remarkable fact of nature that, as a result of such transitions, any system devoid of remedial influences inevitably tends to move from order to disorder. An egg hitting the floor turns to a mess as ordered kinetic energy is converted into splat. Cars break down, rust, and eventually end up as nondescript piles of metallic and organic compounds blowing in the wind or leached by groundwater into a progressively wider and less ordered distribution. Fluids in a nicely ordered state of mean flow likewise

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