Wings of finite aperture(lifting line approach)

stefano.gaggero@unige.it

2D models are not adequate to predict/calculate these phenomena

If we explain the wing lift as the results of the pressure difference between face (overpressure) and back (depressure), at the wing tips, where we lot the physical separation between back and face, pressure must be equal:

A recirculation (vortex) take places at the tips, with the flow moving from the overpressure regions towards the depressure regions, with a non zero component of the velocity alonf the span direction

Realistic distribution of load

Constant distribution of load

Load lost

Load (i.e. the distributed force perpendicular to the fplow direction, produiced by the difference in pressure) cannot be constant along the wing span: at the tips pressure jump is zero, so load must be zero!

Way to reduce this phenomenon

=

In percentual the influence of the extremities of the wing is stronger if the wing has a span comparable with its chord, i.e. the aspect ratio is low.

In percentual the influence of the extremities of the wing is stronger if the wing has a span comparable with its chord, i.e. the aspect ratio is low.

At tips

At Midspan

The slope of the lift coefficient with respect to the angle of attack is close to 2PI only for wings with very high AR

Da prove al tunnel su geometrie semplici

Skew Angle

vi vi

Seen from Back:

Vortex t the wing tips induce a downstream velocity that locally alters the angle of attack of each equivalent 2D sections

Seen from Back:

Vortex t the wing tips induce a downstream velocity that locally alters the angle of attack of each equivalent 2D sections

Kutta-Joukowsky theorema still holds but the LIFT is perpendicular to the real EFFECTIVE incoming flow, thus to the flow as a superposition of the inflow velocity plus the induced downward velocity. Lift, as a consequence, has a component along the direction of the undisturbed velocity. This is an INDUCED DRAG, only due to the 3D of the wing.

We need:

Helmholtzs theorems: Each vortex has constant intensity Each vortex can exist only as a closed (ring) line (infinite).

Il pedice indica laDirezione del vettore

We need:

The Biot-Savart Law: (as for magnetic/electric fields):

We need:

The Biot-Savart Law for vortexes placed on a plane, with induction points (pointswhere the induced velocity are calculated) placed on the same plane:

Induced velocity is perpendicular to the plane:

By integration:

By integration, with alfa = 90:

Semi-Infinite Vortex (that cannot exist!!!)

We need:

The Biot-Savart Law for vortexes placed on a plane, with induction points (pointswhere the induced velocity are calculated) placed on the same plane:

Induced velocity is perpendicular to the plane:

Free vortex at the tips

Starting vortex (very, veryfar, in accordance withHelmholtz and Kelvin)

Bound vortex on the lifting line (needed to have circulation and lift, according to Kutta-Joukowsky)

Each wing can be substituted by an horseshoe vortex satisfying the Helmholtz theorem. But load cannot be constant along the span (at the tips) so a single horseshoe is not a proper model.

This Load/circulation disytribution has to satisfy the Zero value at the tips!

Induced velocity (on a generic point y on the bound vortex) by a single horseshoevortex

Prandtl soolution: the load of the wing can be substituted by a superposition of horseshoe vortexes, each with a constant intensity and a different bound vortex length, in order to realize each kind of load distribution able to satisfy the zero load condition at the tips using (Helmholtz) vortexes of constant strenght.

From discrete to continua:

Taking into account the continuous distribution of vortexes on the wake, on a genericy0 point on the bound vortex the induced vertical (normal to the plane) velocity is:

Two possible approaches:

PRANDTL Lifting Line approach (based on angles identities)

FULLY NUMERICAL APPROACH (Based on Neumann BoundaryConditions)

For the Prandtl Lifting Line approach we have to recall the 2D THIN PROFILE THEORY:

= 2 + 0 + 1

From the Analitical Solution of the problem usingFourier Series

0 + 1 0

Lifting Coefficient depends only by angle of attack and Profile curvature!

0

As usual the solution can be defined as an opportune distribution of vortexes that satisfies the zero normal velocity condition on the wing.

It can be demonstrated that the induction point should be placed not on the bound vortex itself but with an appropriate inset (3/4 of chord from the leading edge).

However, if the AR of the wing is sufficiently high, the exact induction point can be simply placed on the bound vortex

Induced angle:

The unknown is represented by the load/circulation distribution that, of course, will depends on the wing geometry and on the aerodynamic characteristics of each sections. Once the circulation is known, all the wing characteristics (Lift, Induced Drag) can be determined.

Hypothesis: the Thin profile theory is applicable for each section, i.e. the lift coefficient linearly depends from the angle of attack, with a slope of 2PI

0 =0

But:

By the application of the Kutta-Joukowsky theorem at each section (section y0, with chord c(y0) )

=(0)

2+ =0(0)

The geometrical angle of attack depends on the effective angle of attack and on the induced angle of attack. Once the distribution of chord, of the angle of zero lift and of the gemetrical angle of attack are known along the wing span it is possible to determine the circulation distribution from which:

Viscositycoefficient

ChordGeometrical angle of attack minusangle of zero lift

ELLIPTIC DISTRIBUTION OF CIRCULATION

AR

Greater AR = less induced drag

WING SHAPE?

Tende al 2D per AR infinti

If the angle of attack is constant along the span (the induced angle of attack is constant if the wing is elliptic) and the circulation has an elliptic distribution, then the chord distribution is elliptic

GENERALIZED DISTRIBUTION OF CIRCULATION

GENERALIZED DISTRIBUTION OF CIRCULATION

GENERALIZED DISTRIBUTION OF CIRCULATION

Skewed Wings

Also the bound vortexesinduce on the control points

Changing the loaddistribution producesstronger vortexes at the tip, with higher inducedvelocities

Skewed Wings

If the AR is not sufficiently high, it is not more possible to assume that the exact induction point (3/4 of chord) coincides with the middle point of the bound vortex

Inverse Problem design of a wing for a given load

Assigned distribution of circulation along the span

The solution is not unique: infinite distribution of angle of attack, angle of zero lift can satisfy the assigned distributionof load.

In this case we look for an appropriate distribution of angle of attack.

noto

= =

0.52

Chord from strength.Mean line (CL0) from cavitation

Lifting Surface - outline

Lifting Surface - outline

Lifting Surface - outline

Lifting Surface - outline

Lifting Surface - outline

Lifting Surface - outline