tisdag 17 december 2019

The True Explanation of the Coanda Effect

A common description of the generation of lift of a wing is through the Coanda effect as the "tendency of a fluid jet to stay attached to a convex surface" typically demonstrated by a teaspoon in the water jet from a faucet subject to a lift force:

The same argument is then used to explain the generation of lift of a wing or airfoil:

But why does a fluid have a "tendency to stay attached to a convex surface"? The standard answer going back to Prandtl's boundary layer theory, is that it is an effect of viscosity which gives a fluid a "stickiness" which makes it "stick" to a surface with zero relative velocity as a no-slip boundary condition.

The new theory of flight exposed on The Secret of Flight gives a different explanation of the generation of lift as instead an effect a slip boundary condition allowing the fluid to slide along the surface with vanishingly small skin friction and then stay attached without separation because of the following property of 2d potential flow (stationary incompressible flow without rotational motion in two dimensions as in the airfoil picture):
  • Potential flow can only separate a stagnation where the flow velocity is zero.  
The reason is that the boundary of the 2d airfoil section with a slip boundary is a streamline followed by fluid particles sliding along the boundary. This means that a nearby streamline in the fluid will stay close to the boundary as long as the flow velocity is positive. The nature of flow separating at stagnation can be seen in the potential flow over the half-plane $y\ge 0$ in a $(x,y)$ 2d coordinate system with boundary $y=0$ with velocity $(v_x,v_y)$ given by
  • $v_x=-x$ and $v_y=y$ with stagnation at $(0,0)$.      
We see that the flow is incompressible and is directed towards $x=0$ with $y=0$ serving as a streamline and that the flow velocity $v_y=y$ away from the boundary is small for $y$ small thus preventing separation in finite time and allowing separation only after long time as the flow approaches $x=0$ with very small velocity $v_x=-x$. We see stagnation in the separated flow at stall  in the airfoil picture above.

Summary:  Incompressible flow can stay attached to a convex surface without separation and thus generate lift of wing,
  • because it satisfies a slip boundary condition,
  • not because it sticks to the boundary with a no-slip.      
The consequences are far-reaching as concerns both computational simulation and understanding of slightly viscous incompressible fluid flow including flight aerodynamics.

In fact, in laminar flow with no-slip the pressure gradient vanishes close to the boundary and so the flow cannot stay attached to a convex boundary. 

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