Why do moderate tapered wings stall first at mid-wing section and high tapered wings stall at tip first? The effective AoA should be always higher close to the root? (considering no geometric or aerodynamic washout)
I will try to make it simple without going into mathematical details.
Here, λ = ctip/croot
The important factors controlling the lift in the Tapered wing.
The ctip being too small affects the Reynolds Number (as the distance travelled is very small). Assuming the constant speed, density and viscosity, Reynolds number only varies with the distance travelled (x) directly. As the tip distance is too small, Reynolds number is not much increased for the boundary layer and therefore transition from laminar flow to turbulent can not happen. Also due to skin friction, the flow gets slow and becomes separated. The separation causes loss of Lift and thus Wing Tips stall first.
The sweepback effect makes the boundary layer tends to flow spanwise toward the tips and becomes separated near the leading edges of tip.
- Increasing Span Efficiency Factor,e as it is higher for tapered wings, it produces more Cl.
There may be other factors too, that I might have missed. But these are the most prominent. You can refer to below image. The Red Line shows seperated flows.
Yes, wingroot has a higher effective AoA. It is easier to understand this way. For a given AoA, the wingtips suffer from downwash and vortices which decreases AoAeff and increases Induced AoA (AoAi). As the wingroots do not suffer from the vortices, their AoAeff is higher.
Figure 2 & 3: FAA Pilot's Handbook of Aeronautical Knowledge
Figure 4: Introduction to Flight, Anderson
The main reason is the Reynolds number and the way it affects the airflow.
The stalling angle is more or less constant for all airfoils over a range of speeds. The shape and airspeed of the airfoil in themselves make very little difference to the stall angle. The stalling speed of a plane is dictated mainly by the angle of attack required to provide adequate lift at that speed.
Spanwise flow is also not a primary issue, as its direction varies markedly between types with a more sharply swept leading edge vs. types with a more forward-swept trailing edge. Yet both have the same problem with tip stall.
For a straight, constant-chord wing the lifting pressure causes an upflow in front of the leading edge near the root. This increases the effective AoA, so it reaches the critical angle and stalls first. The sideways spillage around the tip reduces the lift and hence also the AoA effect.
For a sharply tapered wing the Reynolds number becomes relevant. It is, for the present purpose, a function of the size of the airfoil and the speed and viscosity of the air. Since we are not discussing altitude we can take the viscosity as constant. Large size and high speed mean a large Reynolds number.
For a high Reynolds number, inertial effects of the air mass dominate and flow tends to be laminar for a long way back.
For a low Reynolds number, the viscosity of the air comes to dominate and it is this which tends to create turbulence as the pressure falls above the wing.
The angle of attack at which flow separation occurs thus depends critically on the Reynolds number.
A sharply tapered wing has a high Reynolds number at the root, so smooth flow is maintained to a relatively high AoA. But it has a low Reynolds number at the tip, so the air viscosity leads to flow separation and stall.
Spanwise flow seems to have a lot to do with it, and the wingtip vortices can affect both the local effective Angle of Attack and the airflow of the upper wing. (Credit to @Noorul Quamar)
Let's start with the rectangular wing. It stalls at the root first. Maximum pressure differential between upper and lower wing, and interference of the fuselage (and prop) with the airflow, cause flow separation here first as AOA increases. The wing tip vortices also help create a "weak spot" in the upper rear center of the wing by drawing airflow towards the wing tips.
Lower Reynolds number, from shorter chord, will also cause a wing section to stall at a lower AOA. With a moderately tapered wing, both effects tend to cancel each other out, and the onset of stall averages between the wing tip and fuselage.
With a highly tapered wing (without washout), the progressively shorter chord becomes the overriding factor. In spite of their excellent aerodynamic efficiency (as seen on sea gulls and DC 3s), these wings stall at the tips first, and are notoriously dangerous in "low and slow" turning flight.
Washout, and/or leading edge slats, makes these wings much safer. Bending a tapered tip up to form a "winglet" is also currently in vouge. Reverse sweep has also been studied.
Further information is available at airfoil tools