3D Aerodynamics and Finite Wings

AA200

Lecture 11

March 6, 2011

AA200 - Applied Aerodynamics 1

AA200 - Applied Aerodynamics Lecture 11

Basic 3D Theory

We start our discussion of three dimensional aerodynamics by consideringthe simple case of irrotational, inviscid flow. When compressibility can alsobe neglected, we consider solutions to Laplaces equation in 3D:

2 = 0

Just as in the 2-D case, since this equation is linear, we might constructsolutions by superimposing known solutions. In 2-D we used sources andvortices extensively to construct the flow over airfoils. In 3-D we also usesources and vortices to model the flow over wings and bodies.

This section shows how this is done, starting with some results for somefundamental 3-D singularities.

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AA200 - Applied Aerodynamics Lecture 11

Fundamental Singularities in 3D Potential Flow

One may derive fundamental solutions to Laplaces equation in 3-D, justas we did in 2-D (although complex variables are not quite so useful).

3-D Source

It was easily discovered that the potential: = kr satisfied Laplacesequation in 3-D. Since V = , the velocity associated with this solutionis directed radially with a magnitude: V = k

r2. It is easily shown that the

constant k is related to the volume flow rate, S by: k = S4pi, so:

V =S

4pir2

. The velocity distribution associated with this 3-D source dies off as r2

rather than r as in the 2-D case.

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AA200 - Applied Aerodynamics Lecture 11

Point Doublet

Another basic solution, that has been used with some success insupersonic aerodynamics programs is the point doublet, obtained by movinga point source and sink together while keeping the product of their strength,S, and separation, L, constant. With = SL, the velocity associated withthe point doublet is:

~V = cos 2pir3

r + sin 4pir3

Vortex Filament

One of the most useful fundamental solutions to the 3-D Laplaceequation is that of a vortex filament. A vortex filament may be visualized asa thin tube in which the flow has vorticity, . In the limit as the diameterof the tube is made small, but the circulation, , is held fixed, this region ofvorticity is called a vortex filament. We often represent regions of vorticityas vortex filaments or discrete vortex elements.

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AA200 - Applied Aerodynamics Lecture 11

Helmholtz Vortex Theorems

Helmholtz summarized some of the properties of vortex filaments, orvortices, in 1858 with his vortex theorems. These three theorems governthe behavior of inviscid three-dimensional vortices:

1. Vortex strength is constant

2. Vortices are forever (end on boundaries or form a closed path)

3. Vortices move with the flow

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AA200 - Applied Aerodynamics Lecture 11

Vortex strength is constant: A vortex line in a fluid has constantcirculation.

This can be proved by imagining a closed 3-D loop around the vortexline as shown:

The integral around the closed loop from a to b to c to d to a cutsthrough no vorticity so from Stokes theorem the integral is zero. But asthe slit is made very small the integral approaches the sum of the integralfrom b to c and the integral from d to a. These are the local circulationsaround the vortex line and so, the circulations must be constant along theline.

Since the vortex strength is constant along the vortex line the strength

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AA200 - Applied Aerodynamics Lecture 11

cannot suddenly go to zero. Thus, a vortex cannot end in the fluid. It canonly end on a boundary or extend to infinity. Of course in an real, viscousfluid, the vorticity is diffused through the action of viscosity and the widthof the vortex line can become large until it is hardly recognized as a vortexline. A tornado is an interesting example. One end of the twister is on aboundary; but at the other end, the vortex diffuses over a large area withvorticity.

As discussed in the section on sources and vortices, singularities suchas vortices in the flow move along with the local flow velocity. Here,interactions of the vortices in the trailing wake, cause them to curve aroundeach other and to form the nonplanar wake shown below.

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AA200 - Applied Aerodynamics Lecture 11

Image from Head 1982 in van Dyke, An Album of Fluid Motion, usedwith permission.

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AA200 - Applied Aerodynamics Lecture 11

Biot-Savart Law

The Biot-Savart law relates the velocity induced by a vortex filamentto its strength and orientation. The expression, used frequently inelectromagnetic theory, can be derived from the basic equations for the3D potential. The result is:

In the simple case of an infinite vortex we obtain the 2-D result:

||V || = 2pir

In the case of a horseshoe vortex, the two trailing legs contribute:

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AA200 - Applied Aerodynamics Lecture 11

In the web notes you will find a simple subroutine to compute thevelocity components due to a vortex filament of length Gx, Gy, Gz with thestart of the vortex rx, ry, rz from the point of interest.

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AA200 - Applied Aerodynamics Lecture 11

Finite Wing Models

One may apply the results of 3-D potential theory in several ways. Wefirst consider the theory of finite wings.

We might start out by saying that each section of a finite wing behavesas described by our 2-D analysis. If this were true then we would stillfind that the lift curve slope was 2pi per radian, that the drag was0, and the distribution of lift would vary as the distribution of chord.Unfortunately, things do not work this way. There are several reasons for this:

One explanation is that the high pressure on the lower surface of thewing and the low pressure on the upper surface causes the air to leak aroundthe tips, causing a reduction in the pressure difference in the tip regions.

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AA200 - Applied Aerodynamics Lecture 11

In fact, the lift must go to zero at the tips because of this effect. We willnext see how and why we must model the 3-D wing differently from 2-D.

If we were to take the naive view that the 2-D model would work in3-D, we might have the picture shown on the right. If each section had thedistribution of vorticity along its chord that it had in 2-D, the lift would beproportional to the chord, and would not drop off at the tips as we know itmust.

This sort of model does not conform to our physical picture of whathappens at the wing tips. And indeed, it does not satisfy the equations of

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AA200 - Applied Aerodynamics Lecture 11

3-D fluid flow. The reason that this does not work is that in this case thestreamlines are not confined to a plane. They move in 3-D and the flowpattern is quite different.

We could go back to the governing equations and startsimply with the linear Laplace equation. By superimposingknown solutions we could obtain a simple model of a 3-D wing.We might start by superimposing vortices on the wing itself:

But this is no more than the strip theory model that did not work.The reason that this model (which seems just to be a superposition ofknown solutions) is not adequate is that it violates the governing equationsin certain regions. The model does not satisfy the Helmholtz laws sincevorticity ends in the flow near the tips. Some additional requirements must

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AA200 - Applied Aerodynamics Lecture 11

be imposed on the model. The requirements for such a model are just theHelmholtz vortex theorems, discussed previously. Our simple 3-D modelabove may be modified as shown below to satisfy the first of the Helmholtztheorems.

In fact, as can be seen from the picture here, this vortex model is nottoo far from reality.

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AA200 - Applied Aerodynamics Lecture 11

The downwash field and the existence of trailing vorticesare not just some strange mathematical result. They arenecessary for the conservation of mass in a 3-D flow.

Air is pushed downward behind the wing, but this downward velocitydoes not persist far from the wing. Instead it must move outward. Theoutward-moving air is then squeezed upward outboard of the wing and theflow pattern shown above develops.

The trailing vortex is visualized by NASA engineers byflying an agricultural airplane through a sheet of smoke.

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AA200 - Applied Aerodynamics Lecture 11

The main effect of this vortex wake is to produce a downwash field onthe wing.

This downwash field has several very significant effects:

1. It changes the effective angle of attack of the airfoil section. Thischanges the lift curve slope and has many implications.

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AA200 - Applied Aerodynamics Lecture 11

2. Induced drag: Lift acts normal to flow in 2D. This accounts for about40% of the fuel used in a commercial airplane, and as much as 80

3. It produces interference effects that are important in the analysis ofstability and control.

The magnitude of the downwash can be estimated using the Biot-Savartlaw, discussed previously.

When applied to our simple model with two discrete trailing vortices,the equation predicts infinite downwash at the wing tips, a result that isclearly wrong. In fact, the induced downwash is not even very large.

The failure of this simple model led Prandtl to develop a slightly moresophisticated one in 1918. Rather than representing the wing with justone horseshoe-shaped vortex, the wing is represented by several of them:

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AA200 - Applied Aerodynamics Lecture 11

In this way the circulation on the wing can vary from the root to the tip.The strength of the trailing vortex filaments is rel