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Ludwig Prandtl's Boundary Layer

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