Wings and Bodies in Compressible Flows

2. Slender wing theory

3. Slender body theory

Prandtl-Glauert-Goethert Transformation

Potential equation:

If we choose and

Laplace eqn.

The transformation has stretched the x co-ordinate by

Values of at corresponding are identical.

z co-ordinates remain the same, therefore at corresponding points are also the same.

Thus the equations (1) and (2) may be solved by solving equation (4) for a wing of greater sweep, smaller aspect ratio and same section shape.

Leading edge sweep angles are related by;

Similarly,

Therefore,

Also,
Since ,

Section Lift

Note:

When , the equation reduces to the Laplace eqn.

Source:

For , h is real inside the Mach cone and imaginary outside.

Inside the Mach cone, velocity components are .

Thus, the supersonic flow about a slender, non-lifting body can be analyzed by superposing on the main flow the perturbation velocities of a line of supersonic sources of strengths c = c(x), whose Mach cones intersect the body surface upstream of any given surface point.

For 2-D flows, and derivatives are constant at every point on a given Mach line; but the perturbation velocities of supersonic decrease with distance from the x-axis within the Mach cone.

Velocity potential for constant source distribution and uniform flow at a point P

where f(x )dx = 2p time source strength along dx .

Problem:

Determine source density distributions f(x ) such that the body surface is a streamline.

boundary condition:

Neglecting quadratic terms.

Von Karman showed that where , the rate of change of dS area with x of the body.

Assuming that can be represented by a Fourier series,

Then wave drag is and

Computation

Given M find m

Cut fuselage using planes inclined at m (what matter is the area in this plane) There are infinite orientations for this plane within the Mach cone.

Then

Find at n points; solve for ans.,

Then

Total wave drag coefficient is
Average over all possible cutting plane orientations.

Lift-dependant drag is calculated using vortex panel method.

At given M

Slender Wing Theory: Results

For a more complete discussion on slender wing theory, please see the graduate course on Unsteady Aerodynamics:
1. Slender Wing Theory
2. Selected Results from Slender Body Theory
3. Unsteady Slender Wing Theory

In the Low Speed Aerodynamics course, we saw that the lift and induced drag of a straight, tapered finite wing of high aspect ratio could be analyzed using Prandtl's Lifting Line Theory. The general idea there is that it is the spanwise distribution of lift (or circulation) that matters; the entire chordwise distribution could be ignored and replaced by a single vortex along the quarter-chord line.

That is the "high aspect ratio limit" of the general theory called Lifting Surface Theory. Now consider what happens at the other limit: a wing whose chord is a lot longer than its span; a wing whose aspect ratio is very small, of the order of 1. This is called a "slender wing". In this limit, we will, not surprisingly, see that the spanwise distribution can be represented by something very simple: it turns out that in the slender wing limit, the spanwise circulation distribution is elliptic.

An interesting idea is used to simplify the theory: that idea is very similar to that used in the boundary layer theory. We say: "look, as you move along the chord, the rate at which properties change is pretty gradual, when you compare it to the sharp changes as you move along the span.. So, its the flow pattern in the cross-section plane that really matters. "

We started with the general lifting surface theory for incompressible flow. Then we specialized it to the limit of high aspect ratio, where lifting-line theory gave useful results. Now we seek useful simplifications at the other extreme of aspect ratio: the very slender wing. Assume:
The terminology and derivation below follow the treatment given in the textbook for this course: Katz and Plotkin, "Low Speed Aerodynamics".

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. It means that the vertical velocity component at the surface depends on the slope of the surface, which depends on the angle of attack and the camber.

Slender wing simplifications: Distance scales along x are much larger than those along y or z. .

; , 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.

Transverse Forces and Moments on a Slender Body

(see Ashley and Landahl, Eq. 6-106)

Here the subscript B refers to the base section.

For a Wing alone, , so that .

So, lift coefficient referred to base area is simply

If the body is pointed at the rear, this says that lift should be zero. There is only a pitching moment, and it is destabilizing. In reality, viscous forces will cause a small positive lift.

From the wing-alone result,

, with

;

, where S is the wing planform area, we see that

, so that the lift coefficient referred to the planform area is:

. These equations hold only for wings with monotonically increasing span from pointed apex to the base. If the span decreases anywhere along the chord, a wake is formed. Thus, the lift on usual wings is dependent on the forward sections.