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1. General Slender Wing Theory: That's Where We Are Now!

2. Selected Results from Slender Body Theory

3. Unsteady Slender Wing Theory

SLENDER WING THEORY

Assume:

Small angle of attack: ;

Thin wing: .

No spanwise camber (this is just for convenience in the derivation) .

The wall boundary condition is:

. This can be applied to the upper surface as well as the lower surface.

Slender wing simplifications: Distance scales along x are much larger than those along y or z. (Covering 50% of the distance between the east and west coasts or the Florida Keys, or going from the highest to the lowest point of the Florida Keys is much easier than covering 50% of the distance between the north and south ends of the Floriday Keys).

; , so that ; . (If you move a 100 yards across the Florida Keys, you change from water's edge to mid-continent. If you move 100 yards along the main road, well, you've moved 100 yards out of a few hundred miles: nothing has changed.

Thus, . The Laplace equation then reduces to: .

In other words, the cross-flow (flow in the y-z plane) is the dominant feature of the problem to be solved. Of course there is a high flow velocity along the x-direction, but its effects vary slowly along x, compared to the sharp changes occurring along y and z.

At any x-station, a local 2-D cross-flow solution is sufficient. This also implies that Mach number dependence is lost in small-disturbance compressible flow over a slender wing, and the solutions that we obtain may be applicable (with some care and thought) to supersonic flow as well. This is a very interesting aspect. It is used to split up the supersonic flow over a slender wing or body at small angle of attack into two parts: an external (far-field) solution which is for supersonic flow, plus a near-surface solution which is for the local incompressible cross-flow. There must of course be some careful matching at the boundaries between these solution domains.

Local angle of attack:

. Since

;

; we can simplify the general integral equation for a lifting surface as follows:

Using a doublet distribution to create antisymmetry (pressure jump) in the z direction,

Since ; ; for comparable levels of changes to occur, the kernel becomes:

Thus, portions of the wing ahead of x will have influence on the conditions at x, but the influence of downstream stations is negligible: So wake effects are negligible!

Thus the equation now reduces to:

SLENDER WING ANALYSIS OF R.T.JONES

R.T. Jones approached the steady-flow solution over a slender wing from an unsteady-flow point of view.

Consider a thin flat-plate, pointed, slender wing moving through an imaginary plane in the fluid, at the freestream speed, and the appropriate angle of attack.

Premise

The flow pattern in the y-z plane cutting the wing at distance x from the nose is the 2-D flow caused by a flat plate of normal velocity , moving down! The width of the plate, and hence the scale of the flow, continually increase.

Local lift per unit chord l = (downward velocity)*(rate of increase of apparent mass per unit chord)

where .

Apparent mass per unit chord where is the local span in the cross-flow plane.

Therefore, the local lift coefficient is:

For the 2-D potential cross-flow, the surface potential is distributed spanwise as:

where is a function of y. This gives:

Notes:

1. Pressure distribution has an infinite peak along the sloping sides of the wing.

2. Distribution along rays (lines of constant y/y1) is uniform.

3. The center of pressure coincides with the centroid of the area.

4. Maintenance of lift up to the trailing edge is associated solely with the case of zero width: does not exist for finite Aspect Ratio.

5. Sections downstream of maximum width will not generate lift. A wake exists: no infinite suction peak downstream of the location of maximum width: Kutta condition.