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 18 th 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.[758] 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.[759]

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.[760]

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 18 th 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.[761]

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 dynam – icists 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.[762] In short, the world of the­oretical fluid dynamics in the 18 th and 19 th 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 Ecole natio­nals des ponts et chaussees, 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.[763] 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.[764] 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 thermo­dynamics is found in the history of the development of thermo­dynamics 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.