DEFINICION DE LA NASA SOBRE CAPA LÍMITE

The theory given on this page describes boundary layer effects and was first presented by Ludwig Prandtl in the early 1900's. The general fluids equations had been known for many years, but solutions to the equations did not properly describe observed flow effects (like wing stalls). Prandtl was the first to realize that the relative magnitude of the inertial and viscous forces changed from a layer very near the surface to a region far from the surface. He first proposed the interactively coupled, two layer solution which properly models many flow problems. The Wright brothers did not know about the existence of boundary layers or the role of boundary layers in causing wing stalls.

As an object moves through a fluid, or as a fluid moves past an object, the molecules of the fluid near the object are disturbed and move around the object. Aerodynamic forces are generated between the fluid

and the object. The magnitude of these forces depend on the shape of the object, the speed of the object, the mass of the fluid going by the object and on two other important properties of the fluid; the viscosity, or stickiness, and the compressibility, or springiness, of the fluid. To properly model these effects, aerodynamicists use similarity parameters which are ratios of these effects to other forces present in the problem. If two experiments have the same values for the similarity parameters, then the relative importance of the forces are being correctly modeled.

Aerodynamic forces depend in a complex way on the viscosity of the fluid. As the fluid moves past the object, the molecules right next to the surface stick to the surface. The molecules just above the surface are slowed down in their collisions with the molecules sticking to the surface. These molecules in turn slow down the flow just above them. The farther one moves away from the surface, the fewer the collisions affected by the object surface. This creates a thin layer of fluid near the surface in which the velocity changes from zero at the surface to the free stream value away from the surface. Engineers call this layer the boundary layer because it occurs on the boundary of the fluid.

The details of the flow within the boundary layer are very important for many problems in aerodynamics, including the development of a wing stall, and the skin friction drag of an object. Unfortunately, the physical and mathematical details of boundary layer theory are beyond the scope of this beginner's guide and are usually studied in late undergraduate or graduate school in college. We will only present some of the effects of the boundary layer.

On the slide we show the streamwise velocity variation from free stream to the surface. In reality, the effects are three dimensional. From the conservation of mass in three dimensions, a change in velocity in the streamwise direction causes a change in velocity in the other directions as well. There is a small component of velocity perpendicular to the surface which displaces or moves the flow above it. One can define the thickness of the boundary layer to be the amount of this displacement. The displacement thickness depends on the Reynolds number which is the ratio of inertial (resistant to change or motion) forces to viscous (heavy and gluey) forces and is given by the equation : Reynolds number (Re) equals velocity (V) times density (r) times a characteristic length (l) divided by the viscosity coefficient (mu).

Re = V * r * l / mu

Boundary layers may be either laminar (layered), or turbulent (disordered) depending on the value of the Reynolds number. For lower Reynolds numbers, the boundary layer is laminar and the streamwise

velocity changes uniformly as one moves away from the wall, as shown on the left side of the figure. For higher Reynolds numbers, the boundary layer is turbulent and the streamwise velocity is characterized by unsteady (changing with time) swirling flows inside the boundary layer. The external flow reacts to the edge of the boundary layer just as it would to the physical surface of an object. So the boundary layer gives any object an "effective" shape which is usually slightly different from the physical shape. To make things more confusing, the boundary layer may lift off or "separate" from the body and create an effective shape much different from the physical shape. This happens because the flow in the boundary has very low energy (relative to the free stream) and is more easily driven by changes in pressure. Flow separation is the reason for wing stall at high angle of attack. The effects of the boundary layer on lift are contained in the lift coefficient and the effects on drag are contained in the drag coefficient.

3.- ARTICULO RELACIONADO A LAS INVESTIGACIONES RECIENTES SOBRE LA TEORIA DE PRANDT PAR A LA CAPA LÍMITE LAMINAR.

The Spell of Prandtl's Laminar Boundary Layer

how generations of fluid dynamicists were misled

20th century fluid mechanics has been obsessed with Prandtl's theory of (separation in) viscous laminar boundary layers, despite the fact that the fundamentally different case of most importance concerns (separation in) slightly viscous turbulent boundary layers.

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 viscouslike air or water, then the boundary layer is usually turbulent. Prandtl focusses on laminar boundary layersunderstanding 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 paradoxis mathematical in nature, like all the above, then the resolution must be mathematical. It cannot be solvedsimply 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 allexperience 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 boundarycondition 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 knol   Flow Separation and Divorce Cost closely related to this knol.

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 performanceto 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 drag of a car obtained by solving the Navier-Stokes equations with slip shows good agreement with wind tunnel experiments using less than

a million mesh points affordable right now on a laptop:

              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) turbulent boundary layer in slightly viscous flow, see Turbulent Separation with Slip 

(B) laminar boundary layer in slightly viscous flow, see Laminar Separation with No-Slip

(C) laminar boundary layer in viscous flow, see Laminar Non-Separation in Viscous Flow. 

(A) is the most frequent case and Prandtl's case (B) typically leads back into (A) by reattachment afterlaminar separation into a turbulent boundary layer, see Flow Separation and Divorce Cost.

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). 

Referencias

1. L. Prandtl and O Tietjens, Applied Hydro- and Aeromechanics, 1934.