Prandtl and the Boundary Layer

If Prandtl had never turned his attention to wing theory he would still have occupied a significant position in the history of fluid dynamics. In 1904, at the International Congress of Mathematicians, held that year in Heidelberg, Prandtl had delivered a brief paper called “Uber Flussigkeitsbewegung bei sehr kleiner Reibung” (On fluid motion in fluids with very small friction).4 In this paper he introduced the now famous concept of the boundary layer. At the time, the full significance of the work escaped most of the audience, though not Felix Klein.5 Much later the Heidelberg paper came to be seen as one of the most important contributions to science that was made during the twentieth century.6 It has been likened in its impact to Einstein’s 1905 paper on the theory of relativity.7 The significance of Prandtl’s work was that it provided a bridge—a long-sought-for bridge—that connected the behav­ior of real, viscous fluids and the unreal, inviscid fluid of previous math­ematical theory. There had always been a gap between the Stokes equations, which appeared to be true but unsolvable, and the Euler equations, which were known to be solvable but untrue. This logical gap had profound meth­odological consequences. It attenuated the link between the mathematical hydrodynamics of the lecture theater and the engineering hydraulics of the workshop. It undermined hope in the unity of theory and practice. Prandtl’s boundary-layer theory restored that hope. Figure 7.1 shows Prandtl at work on his boundary-layer research.

The theory of the boundary layer can be broken down into four parts: (1) an underlying physical model, (2) an implied technology of control, (3) a mathematical formulation of the model and the technology, and (4) a heuris­tic resource. I briefly describe each of these dimensions of the theory.

The physical model expressed the idea that, in a fluid of small viscosity, the effects of viscosity arise in, and are often confined to, a thin layer that is in contact with a solid boundary. In the vicinity of the boundary, the fluid layer possesses a sharp velocity gradient. On the actual surface of the body along which the fluid is moving (for example, a wing or the walls of a chan­nel), the fluid is stationary. A short distance away it achieves the velocity of the free stream. The velocity gradient in the Ubergangsschicht, or transition layer as Prandtl called it, is shown diagrammatically in figure 7.2 (taken from the 1904 paper). As long as the fluid within the layer has the kinetic energy to overcome any adverse pressure gradient, then the boundary layer will con­form to the surface along which it is flowing. If it meets too great a pressure, then a backflow will set in and the flow will separate from the surface. This process is shown in Prandtl’s diagram. The intense vorticity of the fluid in

Prandtl and the Boundary Layer

figure 7.1. Ludwig Prandtl (1875-1953). Prandtl is shown ca. 1904 at the technische Hochschule in Hanover demonstrating his hand-driven water channel used to take flow pictures of boundary-layer phenomena.

Prandtl and the Boundary Layer

figure 7.2. Separation of boundary layer according to Prandtl. From Prandtl 1904, 487. (By permis­sion of Herr Helmut Vogel)

the boundary layer will then diffuse into the surrounding flow and alter its general character.

The boundary-layer theory thus encompassed the phenomenon of flow separation, which had intrigued Prandtl from his early days as an engineer in industry when he had worked on suction machinery.8 For Prandtl, as an engineer, the question was how to stop separation and improve the ef­ficiency of the suction effect. A significant part of the 1904 paper implicitly bore upon this engineering problem because it was devoted to the question of boundary-layer control. Prandtl reasoned that if the boundary layer could be removed, then it could not detach itself and modify the rest of the flow. He therefore constructed an apparatus to explore this effect. It consisted of
a hollow cylinder with a slit along one side. The cylinder was inserted in a flow of water and, by means of a suction pump, some of the fluid from the boundary layer was drawn through the slit. The result was that on the side of the cylinder with the slit, the remaining flow stayed close to the surface of the cylinder. As predicted, it did not detach itself and cause vorticity and turbulence in the surrounding fluid. Prandtl presented his Heidelberg audi­ence with photographs of this process to show them the difference made by the intervention.9

Prandtl and the Boundary Layer

Prandtl was able to express the ideas underlying this process in a math­ematical form. He gave the equations of motion for the fluid elements in the boundary layer. He did so by reflecting on the orders of magnitude of the forces and accelerations of the flow in the boundary layer as the viscosity approached zero.10 This line of thought told him which quantities could be ignored in the original Stokes equations governing viscous fluids. It led to a simplification of the equations that did not involve wholly ignoring either the viscous forces or the inertial forces. It proved possible to keep them both in play. Prandtl thus managed to simplify the Stokes equations without simpli­fying them too much. Consider the two-dimensional flow of an incompress­ible fluid in a boundary layer that flows horizontally, that is, along the x-axis. After his simplification Prandtl was left with two equations that described the flow of fluid in the boundary layer by specifying the respective velocity com­ponents, u and v, in the x and y directions. If p is the density, p the pressure, and p the viscosity, then Prandtl was able to write

Prandtl and the Boundary Layer

and

On the basis of these two equations Prandtl worked out an approximate, but reasonable, value for the drag on a horizontal plate acting as the solid bound­ary along which the fluid was flowing. He was also able to arrive at an expres­sion giving the thickness of the boundary layer and show that the thickness approached zero as the viscosity approached zero. In 1908, in a Ph. D. thesis supervised by Prandtl, Blasius fully solved the boundary-layer equations for the case of the flat plate and improved on the original estimate of the drag.11 Other Gottingen doctoral students—Boltze, Hiemenz, and Toepfer—refined Blasius’ procedure and extended the analysis to circular cylinders and bodies
of rotation.12 Although work on the boundary layer began slowly and, for a decade, was confined to Gottingen and the circle around Prandtl, the theory gradually became the focus of extensive empirical and theoretical research in Europe and America. The idea of the boundary layer eventually found appli­cation in every branch of technology where fluid dynamics plays a role.13

Given this idea’s wide applicability, it is worth noting some of the logical characteristics of Prandtl’s equations and reflecting on their methodological status. I have written the equations in a way that brings out their similarities and differences with the Euler equations and the Stokes equations. It is easy to see that the first equation is more complicated than the corresponding Eu­ler equation but simpler than the corresponding Stokes equation. But notice in particular the second, and shorter, of the above equations. It indicates that, given the approximations that are in play, there is a zero rate of change of pressure perpendicular to the plate. The pressure is constant along the y-axis as it cuts through the boundary layer. Clearly, Prandtl’s picture of the bound­ary layer involved some ruthless idealizations. This fact was emphasized by Hermann Schlichting, another of Prandtl’s pupils, who would later write an authoritative monograph on the boundary layer.14 Commenting explicitly on the second of the above equations, Schlichting said:

Die hieraus folgende Vernachlassigung der Bewegungsgleichung senkrecht zur Wand kann physikalisch auch so ausgesprochen werden, dafi ein Teilchen der Grenzschicht fur seine Bewegung in der Querrichtung weder mit Masse behaftet ist noch eine Verzogerung durch Reibung erfahrt. Es is klar, dafi man bei so tief greifenden Veranderungen der Bewegungsgleichungen erwarten mufi, dafi ihre Losungen einige mathematische Besonderheiten aufweisen, und dafi man auch nicht in allen Fallen Ubereinstimmung der beobachteten und berechneten Stromungsvorgange erwarten kann. (121)

The disregard of the equation of motion at right angles to the wall that results from this can be expressed in physical terms by saying that, in its transverse motion, a fluid particle in the boundary layer has no mass and experiences no frictional retardation. It is clear that with such far-reaching changes in the equations of motion one must expect that their solutions will show some mathematical peculiarities and that one cannot in all cases expect agreement between the observed and calculated flow processes.

The fluid particles in the boundary layer, as described by Prandtl’s equa­tions, have zero mass and zero friction in the direction transverse to the layer. Clearly no one believes that a real, physical object could satisfy these specifications, at least not given all the assumptions about the world taken for granted by physicists. Thus Prandtl portrayed the fluid in his boundary layer in terms that are reminiscent of the idealized fluid of classical hydro­dynamics. Euler’s equations of inviscid flow generated false empirical pre­dictions, and these errors were usually explained by noting that the equa­tions neglected friction, whether between the fluid elements themselves or between the fluid and solid boundaries. One might therefore expect that a determined effort would be made to remove all such idealizations and un­realities concerning friction in the course of producing the improved, more realistic, boundary-layer equations. This appears not to have been the case. As far as friction is concerned, the particles of fluid in the boundary layer are hardly less exotic than the particles of an ideal fluid. More will be said later about the way in which idealization is an enduring feature of scientific progress in fluid dynamics.

Not only did Prandtl’s boundary-layer equations involve physical unreali­ties, but the reasoning that generated them involved mathematical assump­tions for which no justifications were given. Certain mathematical questions had been passed over, for example, questions about the existence and unique­ness of solutions to the equations and the convergence of the approximation techniques that were employed. This left the precise relation between Prandtl’s equations and Stokes’ equations unclear. As one mathematician noted, even fifty years after the introduction of the boundary-layer equations, this de­ductive obscurity had still not been dispelled. But, he added, there has been a tendency to disregard it because of the great, practical success of Prandtl’s contribution.15

The boundary-layer equations, as such, played no explicit part in the mathematical apparatus employed in the early Gottingen aerodynamic work. The mathematics that Prandtl actually used for his theory of the finite wing was confined to the Euler equations of inviscid flow, but the idea of the boundary layer was always in the background and undoubtedly played a heuristic role.16 The interpretation of the theoretical results depended on qualitative reasoning that appealed to boundary-layer theory. For example, postulating the existence of the boundary layer effectively divided the fluid into two parts. One part demanded recognition of its viscosity, while the other could be treated as if it were an inviscid fluid. If the flow sticks closely to the surface of a solid body, and there is no separation, then the bulk of the flow can be treated as an exercise in ideal-fluid theory. This was the basis of Prandtl’s claims, discussed earlier, that for streamlined bodies the theory of perfect fluids had been dramatically confirmed. The viscosity assumed to be present in the boundary layer also provided a resource for explaining the ori­gin of the circulation around a wing. The viscous fluid in the boundary layer possesses vorticity, so that if fluid from the layer were to diffuse into the free stream, this occurrence might modify the overall structure of the flow and introduce a component of circulation, even if the circulating flow were then attributed to a perfect fluid.

The model of the boundary layer was itself subject to development both theoretically and experimentally. At first it had been assumed that the flow within the layer had a laminar character. Later, Prandtl relaxed this assump­tion and explored the idea of a turbulent boundary layer. Because turbu­lence implied an increased exchange of energy between the slower-moving boundary layer and the faster-moving free stream, a turbulent boundary layer would possess more energy than a laminar boundary layer because it would have absorbed energy from the free stream. The increased energy de­lays the separation that occurs when the boundary layer runs out of energy and brakes away from, say, the surface of the wing. The delay means the flow conforms more closely to the surface of the wing. This lowers the pressure drag and thus brings the behavior of the air closer to that of a perfect fluid. The idea of boundary-layer turbulence also explained some intriguing dis­parities between the wind-channel measurements of the resistance of spheres made in Gottingen and those from Eiffel’s laboratory in Paris. Strangely, in Paris resistance coefficients for spheres were about half the value of those in Gottingen: 0.088 compared with 0.22. In the course of a review of Eiffel’s wind-channel results, which were otherwise comparable with those in Got­tingen, Otto Foppl concluded that, in the case of the resistance of spheres, there was obviously some mistake in the French work: “Bei der Bestimmung des Widerstands einer Kugel ist offenbar ein Fehler unterlaufen.”17

Prandtl, however, was able to explain the result without attributing a mis­take to Eiffel. Rather than a trivial error, the anomaly indicated the presence of something deep. Prandtl argued that in Gottingen the flow in the wind channel was less turbulent than in Eiffel’s channel. He deliberately increased the turbulence in the Gottingen channel by means of a wire mesh and re­produced Eiffel’s results. What is more, Prandtl argued that the boundary layer itself may have been laminar in Gottingen, whereas in Paris it had been turbulent. This analysis was then subject to an ingenious experimental test in the Gottingen wind channel. Just as Prandtl had introduced the original idea of the boundary layer alongside a demonstration of how to remove the layer by suction, so he now showed how to manipulate the turbulence of the layer. He (counterintuitively) reduced the resistance of a sphere by wrapping a trip wire around it to render the boundary layer turbulent. Photographs taken by Wieselsberger provided further corroboration. Not only was the measured resistance reduced, but the introduction of smoke into the flow showed the separation points pushed toward the back of the sphere. The tur­bulent boundary layer must be clinging to the sphere longer than the laminar layer. In both of these cases, that of the laminar and the turbulent boundary layer, Prandtl’s engineering mind linked a novel theoretical idea to a novel technology of intervention.18