# Abstract

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

**laminar**or

**turbulent**as depicted by NASA:

If the fluid is viscous, like syrup, then the boundary layer is laminar. If the fluid is slightly viscous

**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

- 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.

## Prandtl’s Boundary Layer

**no-slip**boundary condition, only a

**slip**condition allowing fluid particles to slide along the boundary. That opened

### Slip or No-Slip?

### Mathematical Fluid Mechanics

**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

**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

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

**flow separation**from a solid boundary in order the explain drag as an effect of flow separation creating a

**low-pressure (turbulent) wake.**

**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.**

- (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.

**reattachm**

**ent**after