History of KZP

The Father of Modern Fluid Mechanics inspecting the Ho III 1938 Rhön Contest Challenger.

The historical survey Fluid Mechanics in the First Half of This Century (20th), by S. Goldstein illuminates the history of the formation of the KZP theory. We quote:

Prandtl Boundary Layer Theory

  • In 1904 Prandtl read his paper “On Motion of Fluids Flow with Very Little Viscosity” to the Third International Congress of Mathematicians at Heidelberg.
  • This was a most extraordinary paper of less than eight pages. In 1928 I asked Prandtl why he had kept it so short, and he replied that he had been given ten minutes for his lecture at the Congress and that, being still quite young, he had thought he could publish only what he had had time to say. The paper will certainly prove to be one of the most extraordinary papers of this century, and probably of many centuries.
  • However, for some years after it was published Prandtl’s lecture was almost, if not completely, unnoticed. Perhaps this is not surprising. It was so very short, and it was published where no one who was likely to appreciate it might be expected to look for it.      
  • The influence of Prandtl’s boundary-layer theory has been enormous. It has been used to make clear physical phenomena that were, or would have been, otherwise baffling or at least murky.

Slip or No-Slip Boundary Condition

  • I suppose that it is still correct that for practical purposes in most situa­tions our quantitative knowledge of resistance is mainly empirical. However, much more is understood now of the underlying physical processes, and we may ask what was the main cause of the difficulties and confusion. Certainly it did not lie in any lack of intellectual ability of the very distinguished scientists who wrestled with the problems from Newton to Stokes and Rayleigh. The real trouble was doubt about the boundary conditions to be applied at the dividing surface between a solid and a liquid or gas. In the theory of the irrotational motion of an inviscid fluid, the relative normal velocity at the surface of an impermeable solid must be zero, and no other boundary condition is required or can be imposed. For the motion of a viscous fluid, on the other hand, according to the dynamical differential equations published during the period 1822 to 1845 (Navier, 1822; Poisson, 1829; Saint-Venant, 1843; Stokes, 1845), another boundary condition is re­quired for a solution. There was doubt and vacillation for a long time. On the whole, one thing seems to have been agreed: that there is no slip, i.e., no relative tangential velocity, at the surface of a solid body in the case of a very slow motion in a viscous fluid; but all else was in doubt. 
  • The discrepancies between the actual motions of a real fluid of small viscosity, when laminar, and the results calculated for the irrotational motion of an inviscid fluid arise mainly, in most cases, from the condition in a real fluid of no slip at a boundary. If a fluid could slip freely over the surface of a solid body it would be a very different world. Those, among them Lamb and Levi-Civita, who have asserted in the past that viscosity cannot be considered a predominant cause of direct resistance, were correct in this sense in most ordinary circumstances.
  • In his 1904 lecture to the International Congress of Mathematicians Prandtl stated briefly but definitely that by far the most important question in the problem (of the flow of a fluid of small viscosity past a solid body) is the behavior of the fluid at the walls of the solid body. He continued :
  • “The physical processes in the boundary layer (Grenzschicht) between fluid and solid body can be calculated in a sufficiently satisfactory way if it is assumed that the fluid adheres to the wans, so that the total velocity there is zero-or equal to the velocity of the body. If the viscosity is very small and the path of the fluid along the wall not too long, the velocity will have again its usual value very near to the wall. In the thin transition layer (Ubergangsschicht) the sharp changes of velocity, in spite of the small viscosity coefficient, produce noticeable effects. 

Kutta-Zhukowsky Circulation Theory

  • Zhukovskii’s famous lift theorem, connecting the lift force with the cir­culation quite generally for the two-dimensional flow of an inviscid fluid, was published in two notes in 1906, one in Russian and one in French. 
  • The circulation theory of lift was not widely accepted with any rapidity. In the early 1920’s at least one distinguished aeronautical engineer was still expressing scepticism, which produced the experiment of Bryant & Williams (1925) at the National Physical Laboratory. This not only verified the existence of a circulation but confirmed the Kutta-Zhukovskii formula for the lift, even for a real fluid with the presence of a wake, if the contour around which the circulation is taken does not approach the airfoil too closely and cuts the trailing wake at right angles to the direction of the un­ disturbed relative motion.  

 The Father is Dead and His Ideas are Fading

Goldstein points to the confusion before Prandtl about the boundary condition at a solid boundary, slip or no-slip, and how Prandtl settled the problem once an for all by declaring that only no-slip was thinkable. Prandtl thereby took on the role as the Father of Modern Fluid Dynamics, because with his vanishingly thin boundary layer Prandtl offered a solution to all problems of slightly viscous flow including d’Alembert’s paradox, everything contained in a sparsely typed 8 page manuscript with only elementary mathematics.

Before Prandtl there was a real scientific discussion on the proper choice of boundary condition with slip as the first choice and Prandtl’s dictate to use no-slip was ignored well into the 1920s, when it finally took over. But science is not geared by dictate but by rationality and and so the old idea of slip has in recent years resurfaced as a rational model of the small skin friction of slightly viscous flow.  The Father of Modern Fluid Mechanics is gone since long and it is now time to let also his dictates fade away.

Prandtl’s boundary layer theory has led to a bewildering taxonomy of different types of drag including:

when there in fact is only one form of drag in slightly viscous flow.

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