Euler CFD vs Navier's Friction Boundary Condition

Navier's friction boundary condition in the Navier-Stokes equations for fluid flow takes the form

•  $\nu\frac{\partial u}{\partial n}=\beta u$,      (1)

where $\nu > 0$ is a small viscosity, $u$ tangential velocity, $n$ unit normal into the fluid and $\beta \ge 0$ a skin friction coefficient.  For $\beta =0$ this is a zero-friction slip condition, while for $\beta$ large it is a no-slip condition $u=0$. 

For given $\nu >0$ there is a break even between balancing fluid and friction forces when $\beta\approx\sqrt{\nu}$ with $\beta >\sqrt{\nu}$ causing substantial reduction of $u$ creating a (laminar) boundary layer of width $\approx\sqrt{\nu}$ with $\frac{\partial u}{\partial n}\approx \frac{1}{\sqrt{\nu}}$, and $\beta <\sqrt{\nu}$ causing little reduction of $u$ approaching a slip condition without boundary layer. 

The drag crisis at $\nu \approx 10^{-6}$ with Reynolds number $Re\approx 10^6$ signifies a change from laminar no-slip boundary layer to an effective slip condition from a turbulent boundary layer as explained here.  This corresponds to a friction coefficient $\beta\approx 10^{-3}$ indicating a switch to slip in accordance with measured skin friction coefficients:

So, for $Re\approx 10^6$ an observed skin friction $\beta \approx 10^{-3}$ gives experimental support that slip can be used as an effective boundary condition, since $\beta \approx 10^{-3}$ causes little reduction of tangential velocity and has small contribution to total drag in bluff body flow.

A connection with potential flow with slip and formally $\nu =0$ can be made by observing that potential flow satisfies on the boundary

• $\frac{\partial u}{\partial n}=-u$,

and so for any $\nu >0$

$\nu\frac{\partial u}{\partial n}=-\nu u$,

thus with formallly a very small negative friction coefficient very close to 0 and slip in (1). Navier-Stokes with slip thus formally tends to potential flow as the viscosity tends to zero, while in reality interior turbulent flow develops because of instability, a turbulent flow which is computable by Euler CFD with millions of mesh points without need to resolve thin boundary layers because of effective slip. 

Euler CFD can be supplemented with positive skin friction and independence of e. g. drag for small skin friction can be observed.

Note that (1) with $\beta =0$ forms a weak boundary layer to satisfy $\frac{\partial u}{\partial n}=0$, while residual stabilisation in Euler CFD does not create any boundary layer.