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Ludwig Prandtl's Boundary Layer Ludwig Prandtl's Boundary Layer tions of Daniel Bernoulli (1700-82), Jean le In 1904 a little-.known physicist revolutionized fluid Rond d'Alembert (1717-83), and Leonhard dynamics with his notion that the effects of friction are Euler (1707-83)-all well-known heavy experienced only very near an object moving through a hitters in classical physics. fluid. Of the three, Euler was the most instrumental in conceptualizing the mathematical description of a fluid flow. He described flow in terms of spatially varying three- John D. Anderson Jr. dimensional pressure and velocity fields and modeled the flow as a continuous collection of infinitesimally small fluid elements. By uring the week of 8 August 1904, a small group of applying the basic principIes of mass conservation and Newton's Dmathematicians and scientists gathered in picturesque second law, Euler obtained two coupled, nonlinear partial Heidelberg, Germany, known for its baroque architecture, differential equations involving the flow fields of pressure and cobblestone streets, and castle ruins that looked as if they were velocity. Although those Euler equations were an intellectual still protecting the old city. Home to Germany's oldest university, breakthrough in theoretical fluid dynamics, obtaining general which was founded in 1386, Heidelberg was a natural venue for solutions of them was quite another matter. Moreover, Euler did the Third International Mathematics Congress. not account for the effect of friction acting on the motion of the One of the presenters at the congress was Ludwig Prandtl, a fluid elements-that is, he ignored viscosity. 29-year-old professor at the Technische Hochschule (equivalent to It was another hundred years before the Euler equations were a US technical university) in Hanover. Prandtl's presentation was modified to account for the effect of internal friction within a flow only 10 minutes long, but that was all the time needed to describe field. The resulting equations, a system of even more elaborate a new concept that would revolutionize the understanding and nonlinear partial differential equations now called the Navier- analysis of fluid dynamics. His presentation, and the subsequent Stokes equations, were first derived by Claude-Louis Navier in paper that was published in the congress's proceedings one year 1822, and then independently derived by George Stokes in 1845. later, introduced the concept of the boundary layer in a fluid flow To this day, those equations are the gold standard in the mathemati- over a surface. In 2005, concurrent with the World Year of cal description of a fluid flow, and no one has yet obtained a Physics celebration of, among other things, Albert Einstein and his general analytical solution of them. famous papers of 1905, we should also celebrate the 100th The inability to solve the Navier-Stokes equations for most anniversary of Prandtl's seminal paper. The modern world of practical flow problems was particularly frustrating to those aerodynamics and fluid dynamics is still dominated by Prandtl's investigators interested in calculating the frictional shear force on a idea. By every right, his boundary-layer concept was worthy of the surface immersed in a flow. This difficulty became acute at the Nobel Prize. He never received it, however; some say the Nobel beginning of the 20th century, with the invention of the first Committee was reluctant to award the prize for accomplishments practical airplane by Orville and Wilbur Wright and with the in classical physics. subsequent need to calculate the lift and drag on airplanes. Consider the flow over the airfoil-shaped body sketched in figure 1. Before Prandtl The fluid exerts a net force-the net aerodynamic force-on the air- To set the stage, let us take a quick journey back over the early foil. The figure shows the two sources of that force: the fluid development of fluid dynamics. Archimedes (287-212 BC) pressure and the shear stress that results from friction between the introduced some basic ideas in fluid statics, and Leonardo da surface and the flow.1 The pressure and shear-stress distributions Vinci (1452-1519) observed and drew sketches of complex flows are the two hands of Nature by which she grabs hold of the airfoil over objects in streams. But a quantitative physical and and exerts a force on it. mathematical understanding of fluid flow began-haltingly-only To determine the force, aerodynamicists need to calculate when Isaac Newton (1642-1727) devoted Book 11 of his Principia both the pressure and shear-stress distributions and then integrate Mathematica (1687) exclusively to the examination of fluid them over the surface of the airfoil. At the beginning of the 20th dynamics and fluid statics. Efforts to obtain a mathematical formu- century, pressure distributions could be obtained with the help of lation of a fluid flow took shape during the century following the various approximations. Pressure, however, is less problematic than publication of the Principia with the contribu- shear stress, because in calculating the pressure distribution, one can assume the flow is inviscid, or frictionless. Calculating the shear-stress distribution requires the inclusion of internal friction and the consideration of viscous flow. That is, one John D. Anderson Jr is the curator of aerodynamics at the Smithsonian Institution's National Air and Space Museum in Washington, DC, and professor emeritus of aerospace engineering at the University of Maryland in College Park. 42 December 2005 Physics Today @2005 American Institute 01 Physics, 8-0031-9228-0512-020- 1 needs to tackle the Navier-Stokes equations-but 100 years ago they could not be solved.2 The boundary-Iayer concept Against this backdrop, along came Prandtl and his seminal presentation at Heidelberg. The companion paper, entitled "Über Flüssigkeitsbewegung bei sehr kleiner Reibung" ("On the Motion of Fluids with Very Little Friction"), was only eight pages long, but it would prove to be one of the most important fluid-dynamics papers ever written.3 Much later, in 1928, when asked by the fluid dynamicist Sydney Goldstein why the paper was so short, Prandtl replied that he had been given only 10 minutes for his presentation, and he had been under the impression that his paper could contain only what he had time to say.4 Prandtl's paper gave the first description of the boundary-layer concept. He theorized that an effect of friction was to cause the fluid immediately adjacent to the surface to stick to the surface-in other words, he assumed the no-slip condition at the surface-and that frictional effects were experienced only in a boundary layer, a thin region near the surface. Outside the boundary layer, the flow was essentially the inviscid flow that had been studied for the previous two centuries. The concept of the boundary layer is sketched in figure 2. In the types of flows associated with a body in flight, the boundary fore playing there the same part as the Helmholtz surfaces layer is very thin compared to the size of the body-much thinner of discontinuity. than can be shown in a small sketch. With the figure in mind, consider Prandtl's description of the Prandtl was referring to the type of flow in which, as sketched in boundary layer:3 figure 3, the boundary layer separates from the surface and trails downstream. A separated flow region with some low energy flow A very satisfactory explanation of the physical forms in the wake behind the body, but essentially the region is process in the boundary layer [Grenzschicht] between dead air. a fluid and a solid body could be obtained by the The pressure distribution over the surface of the body is hypothesis of an adhesion of the fluid to the walls, radically changed once the flow separates. The altered distribution that is, by the hypothesis of a zero relative velocity creates a pressure drag due to flow separation, that is, a large between fluid and wall. If the viscosity was very small unbalanced force that acts in the direction of the free-stream flow- and the fluid path along the wall not too long, the the drag direction. When the flow separation is extensive-that is, fluid velocity ought to resume its normal value at a when the separated flow region is large-the pressure drag is very short distance from the wall. In the thin transition usually much larger than the skin-friction drag. layer [Übergangsschicht] however, the sharp changes The type of external inviscid flow that promotes boundary- of velocity, even with small coefficient of friction, layer separation is a flow that produces an adverse pressure produce marked results. gradient-in other words, an increasing pressure in the flow One of those marked results is illustrated in figure 2: The direction. Prandtl explained the effect as follows:3 velocity changes enormously over a very short distance normal to On an increase of pressure, while the free fluid the surface of a body immersed in a flow. In other words, the transforms part of its kinetic energy into potential boundary layer is a region of very large velocity gradients. energy, the transition layers instead, having lost a According to Newton's shear-stress law, which states that the shear part of their kinetic energy (due to friction), have no stress is proportional to the velocity gradient, the local shear stress longer a sufficient quantity to enable them to enter a can be very large within the boundary layer. As a result, the skin- field of higher pressure, and therefore turn aside friction drag force exerted on the body is not negligible, contrary from it. 10 what some earlier 19th-century investigators believed. Indeed, for slender aerodynamic shapes, most of the drag is due to skin The phenomenon described by Prandtl is illustrated in figure 3.
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