NASA's Contributions to Aeronautics, Volume 1 by National Aeronautics & Space Administration. - HTML preview

CASE

7

Case-7 Cover Image: Langley CFD Shuttle study, showing the hypersonic flow pattern as it executes atmospheric entry from orbit. NASA.

The visitor to the Smithsonian Institution’s National Air and Space Museum (NASM) in Washington, DC, who takes the east escalator to the second floor, turns left into the Beyond the Limits exhibit gallery, and then turns left again into the gallery’s main bay is suddenly confronted by three long equations with a bunch of squiggly symbols neatly painted on the wall. These are the Navier-Stokes equations, and the NASM (to this author’s knowledge) is the world’s only museum displaying them so prominently. These are not some introductory equations drawn for a first course in algebra, with simple symbols like a + b = c. Rather, these are “partial derivatives” strung together from the depths of university-level differential calculus. What are the Navier-Stokes equations, why are they in a gallery devoted to the history of the computer as applied to flight vehicles, and what do they have to do with the National Aeronautics and Space Administration (which, by the way, dominates the artifacts and technical content exhibited in this gallery)?

The answers to all these questions have to do with computational fluid dynamics (CFD) and the pivotal role played by the National Aeronautics and Space Administration (NASA) in the development of CFD over the past 50 years. The role played by CFD in the study and understanding of fluid dynamics in general and in aerospace engineering in particular has grown from a fledgling research activity in the 1960s to a powerful “third” dimension in the profession, an equal partner with pure experiment and pure theory. Today it is used to help design airplanes, study the aerodynamics of automobiles, enhance wind tunnel testing, develop global weather models, and predict the tracts of hurricanes, to name just a few. New jet engines are developed with an extensive use of CFD to model flows and combustion processes, and even the flow field in the reciprocating engine of the average family automobile is laid bare for engineers to examine and study using the techniques of CFD.

The history of the development of computational fluid dynamics is an exciting and provocative story. In the whole spectrum of the history of technology, CFD is still very young, but its importance today and in the future is of the first magnitude. This essay offers a capsule history of the development of theoretical fluid dynamics, tracing how the Navier-Stokes equations came about, discussing just what they are and what they mean, and examining their importance and what they have to do with the evolution of computational fluid dynamics. It then discusses what CFD means to NASA—and what NASA means to CFD. Of course, many other players have been active in CFD, in universities, other Government laboratories, and in industry, and some of their work will be noted here. But NASA has been the major engine that powered the rise of CFD for the solution of what were otherwise unsolvable problems in the fields of fluid dynamics and aerodynamics.

The Evolution of Fluid Dynamics from da Vinci to Navier-Stokes

Fluid flow has fascinated humans since antiquity. The Phoenicians and Greeks built ships that glided over the water, creating bow waves and leaving turbulent wakes behind. Leonardo da Vinci made detailed sketches of the complex flow fields over objects in a flowing stream, showing even the smallest vortexes created in the flow. He observed that the force exerted by the water flow over the bodies was proportional to the cross-sectional area of the bodies. But nobody at that time had a clue about the physical laws that governed such flows. This prompted some substantive experimental fluid dynamics in the 17th and 18th centuries. In the early 1600s, Galileo observed from the falling of bodies through the air that the resistance force (drag) on the body was proportional to the air density. In 1673, the French scientist Edme Mariotte published the first experiments that proved the important fact that the aerodynamic force on an object in a flow varied as the square of the flow velocity, not directly with the velocity itself as believed by da Vinci and Galileo before him.[1] Seventeen years later, Dutch scientist Christiaan Huygens published the same result from his experiments. Clearly, by this time, fluid dynamics was of intense interest, yet the only way to learn about it was by experiment, that is, empiricism.[2]

This situation began to change with the onset of the scientific revolution in the 17th century, spearheaded by the theoretical work of British polymath Isaac Newton. Newton was interested in the flow of fluids, devoting the whole Book II of his Principia to the subject of fluid dynamics. He conjured up a theoretical picture of fluid flow as a stream of particles in straight-line rectilinear motion that, upon impact with an object, instantly changed their motion to follow the surface of the object. This picture of fluid flow proved totally wrong, as Newton himself suspected, and it led to Newton’s “sine-squared law” for the force on a object immersed in a flow, which famously misled many early aeronautical pioneers. But if quantitatively incorrect, it was nevertheless the first to theoretically attempt an explanation of why the aerodynamic force varied directly with the square of the flow velocity.[3]

Newton, through his second law contributed indirectly to the breakthroughs in theoretical fluid dynamics that occurred in the 18th century. Newton’s second law states that the force exerted on a moving object is directly proportional to the time rate of change of momentum of that object. (It is more commonly known as “force equals mass time acceleration,” but this is not found in the Principia). Applying Newton’s second law to an infinitesimally small fluid element moving as part of a fluid flow that is actually a continuum material, Leonhard Euler constructed an equation for the motion of the fluid as dictated by Newton’s second law. Euler, arguably the greatest scientist and mathematician of the 18th century, modeled a fluid as a continuous collection of infinitesimally small fluid elements moving with the flow, where each fluid element can continually change its size and shape as it moves with the flow, but, at the same time, all the fluid elements taken as a whole constitute an overall picture of the flow as a continuum. That was somewhat in contrast to the individual and distinct particles in Newton’s impact theory model mentioned previously. To his infinitesimally small fluid element, Euler applied Newton’s second law in a form that used differential calculus, leading to a differential equation relating the variation of velocity and pressure throughout the flow. This equation, simply labeled the “momentum equation,” came to be known simply as Euler’s equation. In the 18th century, it constituted a bombshell in launching the field of theoretical fluid dynamics and was to become a pivotal equation in CFD in the 20th century, a testament to Euler’s insight and its application.

There is a second fundamental principle that underlies all of fluid dynamics, namely that mass is conserved. Euler applied this principle also to his model of an infinitesimally small moving fluid element, constructing another differential equation labeled the “continuity equation.” These two equations, the continuity equation and the momentum equation, were published in 1753, considered one of his finest works. Moreover, these two equations, 200 years later, were to become the physical foundations of the early work in computational fluid dynamics.[4]

After Euler’s publication, for the next century all serious efforts to theoretically calculate the details of a fluid flow centered on efforts to solve these Euler equations. There were two problems, however. The first was mathematical: Euler’s equations are nonlinear partial differential equations. In general, nonlinear partial differential equations are not easy to solve. (Indeed, to this day there exists no general analytical solution to the Euler equations.) When faced with the need to solve a practical problem, such as the airflow over an airplane wing, in most cases an exact solution of the Euler equations is unachievable. Only by simplifying the fluid dynamic problem and allowing certain terms in the equations to be either dropped or modified in such a fashion to make the equations linear rather than nonlinear can these equations be solved in a useful manner. But a penalty usually must be paid for this simplification because in the process at least some of the physical or geometrical accuracy of the flow is lost.

The second problem is physical: when applying Newton’s second law to his moving fluid element, Euler did not account for the effects of friction in the flow, that is, the force due to the frictional shear stresses rubbing on the surfaces of the fluid element as it moves in the flow. Some fluid dynamic problems are reasonably characterized by ignoring the effects of friction, but the 18th and 19th century theoretical fluid dynamicists were not sure, and they always worried about what role friction plays in a flow. However, a myriad of other problems are dominated by the effect of friction in the flow, and such problems could not even be addressed by applying the Euler equations. This physical problem was exacerbated by controversy as to what happens to the flow moving along a solid surface. We know today that the effect of friction between a fluid flow and a solid surface (such as the surface of an airplane wing) is to cause the flow velocity right at the surface to be zero (relative to the surface). This is called the no-slip condition in modern terminology, and in aerodynamic theory, it represents a “boundary condition” that must be accounted for in conjunction with the solution of the governing flow equations. The no-slip condition is fully understood in modern fluid dynamics, but it was by no means clear to 19th century scientists. The debate over whether there was a finite relative velocity between a solid surface and the flow immediately adjacent to the surface continued into the 2nd decade of the 20th century.[5] In short, the world of theoretical fluid dynamics in the 18th and 19th centuries was hopelessly cast adrift from many desired practical applications.

The second problem, that of properly accounting for the effects of friction in the flow, was dealt with by two mathematicians in the middle 19th century, France’s Claude-Louis-Marie-Henri Navier, and Britain’s Sir George Gabriel Stokes. Navier, an instructor at the famed École nationale des ponts et chaussées, changed the pedagogical style of teaching civil engineering from one based mainly on cut-and-try empiricism to a program emphasizing physics and mathematical analysis. In 1822, he gave a paper to the Academy of Sciences that contained the first accurate representation of the effects of friction in the general partial differential momentum equation for fluid flow.[6] Although Navier’s equations were in the correct form, his theoretical reasoning was greatly flawed, and it was almost a fluke that he arrived at the correct terms. Moreover, he did not fully understand the physical significance of what he had derived. Later, quite independently from Navier, Stokes, a professor at Cambridge who occupied the Lucasian Chair at Cambridge University (the same chair Newton had occupied a century and a half earlier) took up the derivation of the momentum equation including the effects of friction. He began with the concept of internal shear stress caused by friction in the fluid and derived the governing momentum equation much like it would be derived today in a fluid dynamics class, publishing it in 1845.[7] Working independently, then, Navier and Stokes derived the basic equations that describe fluid flows and contain terms to account for friction. They remain today the fundamental equations that fluid dynamicists employ for analyzing frictional flows.

Finally, in addition to the continuity and momentum equations, a third fundamental physical principle is required for any flow that involves high speeds and in which the density of the flow changes from one point to another. This is the principle of conservation of energy, which holds that energy cannot be created or destroyed; it can only change its form. The origin of this principle in the form of the first law of thermo-dynamics is found in the history of the development of thermo-dynamics in the late 19th century. When applied to a moving fluid element in Euler’s model, and including frictional dissipation and heat transfer by thermal conduction, this principle leads to the energy equation for fluid flow.

So there it is, the origin of the three Navier-Stokes equations exhibited so prominently at the National Air and Space Museum. They are horribly nonlinear partial differential equations. They are also fully coupled together because the variables of pressure, density, and velocity that appear in these equations are all dependent on each other. Obtaining a general analytical solution of the Navier-Stokes equations is much more daunting than the problem of obtaining a general analytical solution of the Euler equations, for they are far more complex. There is today no general analytical solution of the Navier-Stokes equations (as is likewise true in the case of the Euler equations). Yet almost all of modern computational fluid dynamics is based on the Navier-Stokes equations, and all of the modern solutions of the Navier-Stokes equations are based on computational fluid dynamics.

Computational Fluid Dynamics: What It Is, What It Does

What constitutes computational fluid dynamics? The basic equations of fluid dynamics, the Navier-Stokes equations, are expressions of three fundamental principles: (1) mass is conserved (the continuity equation), (2) Newton’s second law (the momentum equation), and (3) the energy equation (the first law of thermodynamics). Moreover, these equations in their most general form are either partial differential equations (as we have discussed) or integral equations (an alternate form we have not discussed involving integrals from calculus).

The partial differential equations are those exhibited at the NASM. Computational fluid dynamics is the art and science of replacing the partial derivatives (or integrals, as the case may be) in these equations with discrete algebraic forms, which in turn are solved to obtain numbers for the flow-field values (pressure, density, velocity, etc.) at discrete points in time and or space.[8] At these selected points in the flow, called grid points, each of the derivatives in each of the equations are simply replaced with numbers that are advanced in time or space to obtain a solution for the flow. In this fashion, the partial differential equations are replaced by a large number of algebraic equations, which can then be solved simultaneously for the flow variables at all the grid points.

The end product of the CFD process is thus a collection of numbers, in contrast to a closed-form analytical solution (equations). However, in the long run, the objective of most engineering analyses, closed-form or otherwise, is a quantitative description of the problem: that is, numbers. Along these lines, in 1856, the famous British scientist James Clerk Maxwell wrote: “All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers.”[9] Well over a century later, it is worth noting how well Maxwell captured the essence of CFD: operations with numbers.

Note that computational fluid dynamics results in solutions for the flow only at the distinct points in the flow called grid points, which were identified earlier. In a CFD solution, grid points are either initially distributed throughout the flow and/or generated during the course of the solution (called an “adaptive grid”). This is in theoretical contrast with a closed-form analytical solution for the flow, where the solution is in the form of equations that allow the calculation of the flow variables at any point of one’s choosing, that is, an analytical solution is like a continuous answer spread over the whole flow field. Closed-form analytical solutions may be likened to a traditionalist Dutch master’s painting consisting of continuous brush strokes, while a CFD solution is akin to a French pointillist consisting of multicolored dots made with a brush tip.

Generating a grid is an essential part of the art of CFD. The spacing between grid points and the geometric ways in which they are arrayed is critical to obtaining an accurate numerical CFD solution. Poor grids almost always ensure poor CFD solutions. Though good grids do not guarantee good CFD solutions, they are essential for useful solutions. Grid generation is a discipline all by itself, a subspecialty of CFD. And grid generation can become very labor-intensive—for some flows over complex three-dimensional configurations, it may take months to generate a proper grid.

To summarize, the Navier-Stokes equations, the governing equations of fluid dynamics, have been in existence for more than 160 years, their creation a triumph of derivative insight. But few knew how to analytically solve them except for a few simple cases. Because of their complexity, they thus could not serve as a practical widely employed tool in the engineer’s arsenal. It took the invention of the computer to make that possible. And because it did so, it likewise permitted the advent of computational fluid dynamics. So how did the idea of numerical solutions to the Navier-Stokes equations evolve?

The Concept of Finite Differences Enters the Mathematical Scene

The earliest concrete idea of how to simulate a partial derivative with an algebraic difference quotient was the brainchild of L.F. Richardson in 1910.[10] He was the first to introduce the numerical solution of partial differential equations by replacing each derivative in the equations with an algebraic expression involving the values of the unknown dependent variables in the immediate neighborhood of a point and then solving simultaneously the resulting massive system of algebraic equations at all grid points. Richardson named this approach a “finite-difference solution,” a name that has come down without change since 1910. Richardson did not attempt to solve the Navier-Stokes equations, however. He chose a problem reasonably described by a simpler partial differential equation, Laplace’s equation, which in mathematical speak is a linear partial differential equation and which the mathematicians classify as an elliptic partial differential equation.[11] He set up a numerical approach that is still used today for the solution of elliptic partial differential equations called a relaxation method, wherein a sweep is taken throughout the whole grid and new values of the dependent variables are calculated from the old values at neighboring grid points, and then the sweep is repeated over and over until the new values at each grid point converges to the old value from the previous sweep, i.e., the numbers “relax” eventually to the correct solution.

In 1928, Richard Courant, K.O. Friedrichs, and Hans Lewy published “On the Partial Difference Equations of Mathematical Physics,” a paper many consider as marking the real beginning of modern finite difference solutions; “Problems involving the classical linear partial differential equations of mathematical physics can be reduced to algebraic ones of a very much simpler structure,” they wrote, “by replacing the differentials by difference quotients on some (say rectilinear) mesh.”[12] Courant, Friedrichs, and Lewy introduced the idea of “marching solutions,” whereby a spatial marching solution starts at one end of the flow and literally marches the finite-difference solution step by step from one end to the other end of the flow. A time marching solution starts with the all the flow variables at each grid point at some instant in time and marches the finite-difference solution at all the grid points in steps of time to some later value of time. These marching solutions can only be carried out for parabolic or hyperbolic partial differential equations, not for elliptic equations.

Courant, Friedrichs, and Lewy highlighted another important aspect of numerical solutions of partial differential equations. Anyone attempting numerical solutions of this nature quickly finds out that the numbers being calculated begin to look funny, make no sense, oscillate wildly, and finally result in some impossible operation such as dividing by zero or taking the square root of a negative number. When this happens, the solution has blown up, i.e., it becomes no solution at all. This is not a ramification of the physics, but rather, a peculiarity of the numerical processes. Courant, Friedrichs, and Lewy studied the stability aspects of numerical solutions and discovered some essential criteria to maintain stability in the numerical calculations. Today, this stability criterion is referred to as the “CFL criterion” in honor of the three who identified it. Without it, many attempted CFD solutions would end in frustration.

So by 1928, the academic foundations of finite difference solutions of partial differential equations were in place. The Navier-Stokes equations finally stood on the edge of being solved, albeit numerically. But who had the time to carry out the literally millions of calculations that are required to step through the solution? For all practical purposes, it was an impossible task, one beyond human endurance. Then came the electronic revolution and, with it, the digital computer.

The Critical Tool: Emergent High-Speed Electronic Digital Computing

During the Second World War, J. Presper Eckert and John Mauchly at the University of Pennsylvania’s Moore School of Electrical Engineering designed and built the ENIAC, an electronic calculator that inaugurated the era of digital computing in the United States. By 1951, they had turned this expensive and fragile instrument into a product that was manufactured and sold, a computer they called the UNIVAC, which stands for Universal Automatic Computer. The National Advisory Committee for Aeronautics (NACA) was quick to realize the potential of a high-speed computer for the calculation of fluid dynamic problems. After all, the NACA was in the business of aerodynamics and after 40 years of trying to solve the equations of motion by simplified analysis, it recognized the breakthrough supplied by the computer to solve these equations numerically on a potentially practical basis. In 1954, Remington Rand delivered an ERA 1103 digital computer intended for scientific and engineering calculations to the NACA Ames Aeronautical Laboratory at Sunnyvale, CA. This was a state-of-the-art computer that was the first to employ a magnetic core in place of vacuum tubes for memory. The ERA 1103 used binary arithmetic, a 36-bit word length, and operated on all the bits of a word at a time. One year later, Ames acquired its first stored-program electronic computer, an IBM 650. In 1958, the 650 was replaced by an IBM 704, which in turn was replaced with an IBM 7090 mainframe in 1961.[13]

The IBM 7090 had enough storage and enough speed to allow the first generation of practical CFD solutions to be carried out. By 1963, four additional index registers were added to the 7090, making it the IBM 7094. This computer became the workhorse for the CFD of the 1960s and early 1970s, not just at Ames, but throughout the aero-dynamics community; the author cut his teeth solving dissertation on an IBM 7094 at the Ohio State University in 1966. The calculation speed of a digital computer is measured in its number of floating point operations per second (FLOPS). The IBM 7094 could do 100,000 FLOPS, making it about the fastest computer available in the 1960s. With this number of FLOPS, it was possible to carry out for the first time detailed flow-field calculations around a body moving at hypersonic speeds, one of the major activities within the newly formed NASA that drove both computer and algorithm development for CFD. The IBM 7094 was a “mainframe” computer, a large electronic machine that usually filled a room with equipment. The users would write their programs (usually in the FORTRAN language) as a series of logically constructed line statements that would be punched on cards, and the decks of punched cards (sometimes occupying many boxes for just one program) would be fed into a reader that would read the punches and tell the computer what calculations to make. The output from the calculations would be printed on large sheets and returned to the user. One program at a time was fed into the computer, the so-called “batch” operation. The user would submit his or her batch to the computer desk and then return hours or days later to pick up the printed output. As cumbersome as it may appear today, the batch operation worked. The field of CFD was launched with such batch operations on mainframe computers like the IBM 7094. And NASA Ames was a spearhead of such activities. Indeed, because of the synergism between CFD and the computers on which it worked, the demands on the central IBM installation at Ames grew at a compounded rate of over 100 percent per year in the 1960s.

With these computers, it became practical to set up CFD solutions of the Euler equations for two-dimensional flows. These solutions could be carried out with a relatively small number of grid points in the flow, typically 10,000 to 100,000 points, and still have computer run times on the order of hours. Users of CFD in the 1960s were happy to have this capability, and the three primary NASA Research Centers—Langley, Ames, and Lewis (now Glenn)—made major strides in the numerical analysis of many types of flows, especially in the transonic and hypersonic regimes. The practical calculation of inviscid (that is, frictionless), three-dimensional flows and especially any type of high Reynolds number flows was beyond the computer capabilities at that time.

This situation changed markedly when the supercomputer came on the scene in the 1970s. NASA Ames acquired the Illiac IV advanced parallel-processing machine. Designed at the University of Illinois, this was an early and controversial supercomputer, one bridging both older and newer computer architectures and processor approaches. Ames quickly followed with the installation of an IBM 360 time-sharing computer. These machines provided the capability to make CFD calculations with over 1 million grid points in the flow field with a computational speed of more than 106 FLOPS. NASA installed similar machines at the Langley and Lewis Research Centers. On these machines, NASA researchers made the first meaningful three-dimensional inviscid flow-field calculations and significant two-dimensional high Reynolds number calculations. Supercomputers became the engine that propelled CFD into the forefront of aerospace design as well as research. Bigger and better supercomputers, such as the pioneering Cray-1 and its successor, the Cray X-MP, allowed grids of tens of millions of grid points to be used in a flow-field calculation with speeds beginning to approach the hallowed goal of gigaflops (109 floating point operations per second). Such machines made it possible to carry out numerical solutions of the Navier-Stokes equations for three-dimensional fairly high Reynolds number viscous flows. The first three-dimensional Navier-Stokes solutions of the complete flow field around a complete airplane at angle of attack came on the scene in the 1980s, enabled by these supercomputers. Subsonic, transonic, supersonic, and hypersonic flow solutions covered the whole flight regime. Again, the major drivers for these solutions were the aerospace research and development problems tackled by NASA engineers and scientists. This headlong development of supercomputers has continued unabated. The holy grail of CFD researchers in the 1990s was the teraflop machine (1012 FLOPS); today, it is the petaflop (1015 FLOPS) machine. Indeed, recently the U.S. Energy Department has contracted with IBM to build a 20-petaflop machine in 2012 for calculations involving the safety and reliability of the Nation’s aging nuclear arsenal.[14] Such a machine will aid the CFD practitioner’s quest for the ultimate flow-field calculations—direct numerical simulation (DNS) of turbulent flows,