tisdag 26 november 2019

Flying Impossible with Prandtl No-Slip Flow Separation

Ludwig Prandtl is named Father of Modern Fluid Mechanics because of his proposed resolution in 1904 of d'Alembert's paradox from 1755 based on the concept of no-slip boundray layer as a thin region connecting free flow velocity with zero relative velocity at a solid wall.

Prandtl thus proposed that the drag or resistance to motion of a more or less streamlined body like an airplane wing moving through air, is an effect of boundary layer separation causing a turbulent wake. Prandtl's scenario which has dominated 20th century fluid mechanics is illustrated in the above generic text book picture with the following elements:
  1. No-slip: the flow velocity is zero on the surface of the (still) wing.
  2. The boundary layer starts laminar at the leading edge stagnation point, grows in thickness with the flow and quickly after the crest of the wing turns turbulent and even thicker.
  3. The flow decelerates after the crest by increasing pressure in the flow direction (adverse pressure gradient), which ultimately leads to reverse flow followed by flow separation into a turbulent wake creating drag.         
But Prandtl's picture does not describe the actual flow dynamics around a wing, because this would not allow the wing to generate lift, which is the purpose of a wing. In short, this is because a flow with no-slip will separate already on the crest of the wing and little lift will be generated. The math is given below. You can see this effect in Prandtl's famous film of an airfoil dragged through a viscous fluid showing separation on the crest already at small angle of attack. Prandtl's wing would not fly.

Compare with DNS with heavily tripped turbulent boundary, which also shows separation quickly after the crest with loss of lift (real wings do not have such tripping devices).  

The New Theory of Flight shows that drag and lift do not originate from a thin no-slip Prandtl boundary layer, but instead from an effective slip boundary condition, which keeps the flow attached to the upper wing surface until the trailing edge (before stall) and thus creates lift by suction.

Prandtl has misled generations of fluid dynamicists to search for explanations in boundary layers so thin that they cannot be resolved computationally and thus cannot explain anything. 

The crucial difference between no-slip and slip is seen in mathematical terms as follows: Put a coordinate system with coordinates $x=(x_1,x_2,x_3)$ on top of the crest of the wing with the $x_1$-axis in the main flow direction, the $x_2$-axis perpendicular to the wing and the $x_3$-axis along the wing span. Consider momentum balance in the $x_2$ direction in velocity $u=(u_1,u_2,u_3)$ and pressure $p$ in the presence of vanishingly small viscosity, stationary state and no exterior forcing:
  • $u_1\frac{\partial u_2}{\partial x_1}+u_2\frac{\partial u_2}{\partial x_2}+\frac{\partial p}{\partial x_2}=0$ for $x_2\gt 0$,
with $u_2=0$ for $x_2=0$ for both no-slip and slip, and $u_1=0$ for $x_2=0$ in the case of no-slip, while $u_1$ is the free stream velocity with slip. The normal velocity $u_2$ is very mall close to the wall, and so the momentum balance can be reduced to
  • $\frac{\partial p}{\partial x_2}=-u_1\frac{\partial u_2}{\partial x_1}$ close to wall,     (1)
In order for the flow to not separate on the crest, the flow must be accelerated by a positive pressure gradient in the normal direction depending on the curvature of the crest, that is $\frac{\partial p}{\partial x_2}$ must be positive large enough. But with no-slip and $u_1=0$ on the surface, this is not compatible with (1) stating that
  • $\frac{\partial p}{\partial x_2}$ is vanishingly small close to wall.  
The effect is that flow with no-slip will separate on the crest and lift will be lost. Flying with no-slip is impossible.

Recall that Prandtl focussed on explaining drag, leaving lift to the (likewise unphysical) Kutta-Zhukovsky circulation theory, forgetting that it is incompatible with his boundary layer theory. Flying must have been a complete mystery to Prandtl.

On the other hand, flow with slip can separate only at stagnation, which cannot occur on the crest where the flow speed is maximal, and thus with $u_1 \gt 0$ the free flow velocity in the relation (1) (with a proper negative $\frac{\partial u_2}{\partial x_1}$) can be satisfied with required positive normal pressure gradient. Flying with slip is possible.

The New Theory of Flight thus is based a new theory for flow separation (see previous post) based on 3d rotational slip separation, which shows that the text book theory of Prandtl based on adverse pressure gradients does not correctly capture the true physics of flow separation. The consequences are far-reaching.

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