Bypass Transition from No-Slip Laminar Boundary Layer to Slip Boundary Condition

The New Theory of Flight is supported by Direct Finite Element Simulation DFS as best possible computational satisfaction of Euler's equations expressing first principle physics in the form (i) incompressibility, (ii) momentum balance and (iii) slip boundary condition on solid walls.

Observations and experiments (connecting to the so-called drag crisis) indicate that at a Reynolds number Re of about $10^6$ the boundary condition at a solid wall changes from no-slip at the wall accompanied with a thin laminar boundary layer, to effectively a slip condition as a thin film without layer.

Let us now see if we can understand this transition from no-slip with laminar layer to slip from some simple mathematical considerations. We thus consider flow over a flat plate as $y\ge 0$ in a $(x,y,z)$-coordinate system with main flow in the $x$-direction with speed 1. We consider stationary parallel flow with velocity $(u(y),0,0)$ only depending on $y$ and pressure $p(x)$ only on $x$ modeled by the following reduced form of the Euler equations:

• $\frac{\partial p}{\partial x}+\nu\frac{\partial^2 u}{\partial y^2}=0$ for $y\gt 0$,
• $u(0)=0,\quad u(\infty )=1$.

Normalising to $\frac{\partial p}{\partial x}=1$, the solution takes the form

• $u(y)=1-\exp(-\frac{y}{\sqrt{\nu}})$ for $y\gt 0$.

We see that velocity $u(y)$ has a boundary layer of width $\sqrt{\nu}$ connecting the free flow velocity $1$ to the no-slip velocity $0$.

We now replace the no-slip condition $u=0$ by a friction boundary condition of the form

• $\beta u=\nu\frac{\partial u}{\partial y}$ for $y=0$,

where $\beta \gt 0$ is a (skin) friction parameter. The solution is now (with $\frac{\partial p}{\partial x}=a$):

• $u(y)=1-a\exp(-\frac{y}{\sqrt{\nu}})$ for $y\gt 0$,
• $a = \frac{\beta}{\beta +\sqrt{\nu}}$.

For $\beta$ small (small friction), the solution is $u(y)=1$ with full slip, and for $\beta $ large it is

the no-slip solution. The transition is anchored at $\beta =\sqrt{\nu}$. 

We now return to the observation of transition for $\nu = 10^{-6}$ if we normalise $Re =\frac{UL}{\nu}$ with $U=1$ and $L=1$, which gives $\sqrt{\nu}=0.001$. 

Observation thus supports an idea that transition from no-slip to effectively slip can take place when

• skin friction coefficient is $\approx 0.001$,
• boundary layer thickness is $0.1\%$ of gross dimension,
• shear exceeds 1000.      

We thus observe the free flow to effectively act as having a slip/small friction boundary condition when the width of a laminar boundary layer is smaller than $0.1\%$ of the length scale $L$ in the specification of the Reynolds number. For an airplane wing of chord 1 m this means a boundary layer thickness of 1 mm, for a jumbojet 5 mm.

Note that slip occurring when shear is bigger than 1000 connects to both friction between solids where slip occurs when tangential force is big enough (scaling with normal force), and to plasticity in solids with slip surfaces occurring for large enough stresses, both as threshold phenomena. For a fluid the threshold thus may relate to shear and for a solid to shear stress.

We view such a transition form laminar no-slip to slip as "bypass" of transition into a no-slip turbulent boundary layer, which may take place for a smaller Reynolds number.  We see the difference in skin friction coefficient between that of thin film limit of a laminar boundary layer (red curve) and various turbulent boundary layers:

We also see support of the conjectured level of skin friction of 0.001 for transition to slip. 

We recall that the generation of lift of a wing critically depends on an effective slip condition, to secure that the flow does not separate on the crest of the suction side of the wing, which connects to observation of gliding flight only for $Re\gt 5\times 10^5$, allowing birds and airplanes to fly without the intense flapping required for little fruit flies with much smaller Re.

We recall that the flow around a wing, or more generally around a streamline body, can be more favourable as concerns bypass to slip because of the accelerating flow after attachment, which has a stabilising effect on streamwise velocity, followed by deceleration after the crest with stabilising effect on streamwise velocity.

We also recall that forced tripping of flow into transition to a turbulent boundary is typically used in flat plate experiments, which when translated to streamline bodies without artificial tripping incorrectly attributes most of drag to skin friction. See more posts on skin friction.