But is Prandtl’s Boundary Layer Theory a 20th century paradox? I argue that the answer is yes, since for quantitative agreement with experiment BLT will be outgunned by computational ﬂuid dynamics in the 21st century.(S. Cowley in Laminar boundary layer theory: A 20th century paradox )
Ludwig Prandtl’s work and achievements in fluid dynamics resulted in equations that were easier to understand than others…It is for this reason that he is referred to as the father of modern aerodynamics. (US Centennial Flight Commission )
Laminar and Turbulent Boundary Layers
The start of modern fluid mechanics is commonly connected to the “discovery” of the boundary layer by the German physicist
Ludwig Prandtl in 1904, as a thin region connecting the flow of a fluid to a solid boundary. A boundary layer can be
laminar or
turbulent as depicted by NASA:
Turbulent boundary layer.
If the fluid is viscous, like syrup, then the boundary layer is laminar. If the fluid is slightly viscous like air or water, then the boundary layer is usually turbulent. Prandtl focusses on laminar boundary layers understanding very well that this does not cover the case of a turbulent boundary layer with the following motivation
[1]:
- It is nevertheless useful to consider laminar flow because it is much more amenable to mathematical treatment.
However, turbulent and laminar flow have different properties,and drawing conclusions about turbulent flow from studies of laminar flow can be grossly misleading.
Paradoxes from Misunderstanding Mathematics
Since mathematics is such a difficult language for most people, there is a high risk that mathematical expressions and statements are misúnderstood and misinterpreted, in particular by scientists and mathematicians making the interpretations. Such misunderstandings result in mathematical paradoxes including:
- Twin paradox of special relativity
- Ladder paradox of special relativity
- Ehrenfest’s paradox of special relativity
- d’Alembert’s paradox of fluid mechanics
- Gibbs paradox of statistical mechanics
- Sommerfeld’s paradox of fluid mechanics
- Loschmidt’s paradox of gas theory
- Schrödinger’s cat paradox in quantum mechanics
- Blackbody radiation paradox of wave mechanics.
A paradox is lethal to a scientific theory and must be resolved, in one way or the other. If the paradox is mathematical in nature, like all the above, then the resolution must be mathematical. It cannot be solved simply by “scientific consensus” or “from a practical point of view”, as suggested in the
presentation of d’Alembert’s paradox on Wikipedia.
Prandtl’s Boundary Layer
d’Alembert’s paradox formulated by the mathematician d’Alembert in 1752 compares observation of substantial drag (resistance to motion) in nearly incompressible and inviscid (small viscosity) fluids such as water and air at subsonic speeds, with the theoretical prediction of zero drag of inviscid potential flow. Mathematics seems to say that you should be able to move through water or air without resistance, but all experience shows that this is not so, at all.
Evidently something must be fundamentally wrong with the potential solution, but nobody could figure out what. With the new era of aerodynamics of manned powered flight starting in the beginning of the 20th century, the pressure to come up with a resolution grew stronger, which encouraged the young engineer Prandtl to suggest that the trouble with the potential solution possibly could be that it does not satisfy a no-slip boundary condition, only a slip condition allowing fluid particles to slide along the boundary. That opened a way of discriminating the potential solution, and thereby resolving the paradox, but to start with nobody payed any attention, because the resolution lacked credibility since fluid particles really seemed to slide along a boundary with small friction in slightly viscous flow. No-slip seemed to require coupling of atomistic models to the continuum models of fluid mechanics, a daunting task.
But with the help of his two forceful students
Schlichting in Germany and
von Karman in the US, Prandtl was in the 1920s crowned as the “
father of modern fluid mechanics ” based on his idea of the boundary layer as a thin region connecting a no-slip zero-velocity boundary condition to a free-stream non-zero velocity, an idea which has come to dominate modern fluid dynamics. It is commonly believed that dominating contributions of drag and lift emanate from thin boundary layers.
Slip or No-Slip?
But how can you tell which boundary condition is correct, slip or no-slip, and more precisely which boundary condition is correct in what sense in what context? Prandtl said no-slip for a viscous laminar boundary layer, but said nothing about the turbulent boundary layers dominating in applications as shown in the
Flow Separation and Divorce Cost.
Mathematical Fluid Mechanics
It is now mathematics comes in. Mathematically, fluid mechanics (of an incompressible fluid) is described by the Navier-Stokes equations expressing Newton´s law F = ma and incompressibility in terms of the fluid velocity and pressure, combined with certain boundary conditions. From a mathematical point of there are different ways of choosing the boundary conditions depending on what data you may have: You may choose between conditions on fluid velocity or on viscous forces. On a solid boundary you may thus choose between
- no-slip: requiring the tangential fluid velocity to be zero on the boundary
- slip: requiring the tangential (friction) force to be zero
both combined with vanishing normal velocity on a solid boundary. In general, if you know the tangential friction, then you are free to combine this force boundary condition with the Navier-Stokes equations, instead of imposing the tangential velocity.
For a slightly viscous flow experiments show that the tangential force or skin friction, is small. So even if you don’t know the skin friction precisely, you know that it is small, and replacing it by slip/zero skin friction could be OK, under an appropriate assumption of stability that small perturbations (of skin friction) have small effects (on certain aspects of the flow). And there is a lot of evidence that this assumption is satisfied for a turbulent boundary layer, if one considers mean-value aspects such as drag and lift, as evidenced in
Why It Is Possible to Fly and references therein, also showing that for a laminar boundary layer it is not OK. But for slightly viscous flow boundary layers are normally turbulent, and modeling them by slip/small is very useful as we will now see.
Computational Fluid Dynamics
Prandtl’s boundary layer theory has led computational fluid dynamics into a deadlock, because computational resolution of thin no-slip boundary layers requires today impossible quadrillions of mesh points in many applications, commonly estimated to take 50 years of continued improvement of compute performance to become possible.
If one accepts the need for resolution of turbulent boundary layers to compute aerodynamic forces, one is lead to pessimistic predictions claiming that it will take 50 years of continued computer development to compute the drag of a car by solving the Navier-Stokes equations!
But this is possible already today, if slip is used instead of no-slip. The lift and drag of a airplane obtained by solving the Navier-Stokes equations with slip shows good agreement with wind tunnel experiments using a couple of million mesh points affordable right now on a supercomputer:
Turbulent flow around a car computed from Navier-Stokes equations with slip boundary condition.
This is connected to the new resolution of d’Alembert’s paradox based on discriminating potential flow because it is unstable to small perturbations and thus has no physical significance, not because it does not satisfy no-slip. Potential flow thus represents a mathematical solution of the Navier-Stokes equations, which from physical point of view is fictitious, because it is mathematically unstable and inevitably turns into a turbulent solution with substantial drag under always present small perturbations.
This represents a mathematical resolution of a mathematical paradox, while Prandtl´s resolution is a formalistic non-mathematical resolution. The new resolution opens to combine Navier-Stokes with a slip/small friction boundary condition, which does not create any unresolvable boundary layer and thus opens new possibilities of computational fluid dynamics. Right now and not in 50 years!
Separation in Laminar and Turbulent Boundary Layers
Prandtl applied his laminar boundary layer theory to the basic phenomenon of flow separation from a solid boundary in order the explain drag as an effect of flow separation creating a low-pressure (turbulent) wake.
By leaving out certain viscous terms in the Navier-Stokes equations, which seemed to be small in the case of small viscosity, Prandtl arrived at his boundary layer equations stating in particular that the pressure gradient normal to the boundary should vanish. Prandtl then sought to connect separation to the pressure gradient in the flow direction parallel to the boundary. He identified a pressure increasing in the flow direction with an adverse pressure gradient as the cause of separation by retarding the flow to stagnation and recirculation.
Generations of fluid dynamicists have been seeking adverse pressure gradients as the cause of separation, however with little success for reasons we now explain. We consider the following basic cases of separation
(A) is the most frequent case and Prandtl’s case (B) typically leads back into (A) by reattachment after
In Prandtl’s case (B) the flow separates on the crest because the normal pressure gradient vanishes,
not because of any adverse pressure gradient.
Altogether, Prandtl’s boundary layer theory seems to have little of interest to say about real flows. It was invented primarily to resolve d’Alembert’s paradox, but it missed the point for slightly viscous flow, and contributed to an unfortunate separation of mathematical fluid mechanics from realities (without reattachment).