Finite Wing- Vortex Lattice Methods

8/11/2019 Finite Wing- Vortex Lattice Methods

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Session 03

Introduction to Vortex Lattice Methods

Session delivered by:Session delivered by:

. . .. . .

103 M.S. Ramaiah School of Advanced Studies, Bengaluru



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Session Objectives

-- At the end of this session the dele ate would have

understood

• The vortex lattice methods

• Application of BCs in the vortex lattice methods

• Vortex dynamics, Helmholtz theorems

• Application of Biot-Savart Law




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ess on op cs

1. Introduction to Vortex Lattice Method

2. Application of BCs in VLM

3. Linearisation of the BC

4. Transfer of the BC to the Mean Surface. ,

6. Vortex Dynamics, Helmholtz Theorems

7. Biot-Savart law

8. Application of Biot-Savart Law to horseshoe vortex




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-

There is another method that is very close to the Panel

e o s we s u e an g ves ns g n o e w ng

aerodynamics: The Vortex Lattice Methods (VLM). Thesewere first formulated in the 1930’s.

The Vortex Lattice Methods are also based on the solution

In these methods also, singularities are placed on the

sur aces an non-pene ra on angency con on s sa s e .

Formulation of a system of linear algebraic equations and


their solution determine the strengths of the singularities.



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The similarity of VLMs with the Panel Methods is mentioned in

e as pane . ow e erences:

VLMs are specially meant for lifting surfaces and ignore

thickness. Hence they are restricted to thin surfaces. Recall that

the panel methods are not restricted in this fashion. (Q: Why?)

The boundary conditions are applied on a mean surface, not

the actual surface. There is no singularity for upper and lower

sur aces separa e y. ence we o no ge p upper an p lower

separately but only Δ C p .


Singularities are not distributed over the entire surface.



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The boundary conditions are applied in an approximate manner on

a mean sur ace ra er an on e p ys ca sur aces. e

following 2-D example of an airfoil makes the idea clear.

The boundary condition required here is V • n = 0.




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The airfoil surface is given by the equation

F ( x, y) = 0 = y – f ( x, y).The unit normal vector n is:

∞

presence of the body by a disturbance velocity q( x,y) = u + iv:




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Then the total velocity components are:

And the boundary conditions reduce to

The BC is specified in terms of normal component but we need it


for u & v. Applying this on the wing surface ( x, y) = 0 leads to



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Or in terms of f we get on y = f ( x):

This is an exact relation for v but cannot be used since u is also an


unknown. (Note that boundary condition is V n = 0, not v = 0.)



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The exact BC

will be shifted to the mean surface so that it becomes solvable.

First, we assume that α is small and hence

Substituting this in the BC we get




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Linearisation of the BC (2)

The body is assumed to be a thin surface. This, along with small α

∂ F/ ∂ x << ∂ F/ ∂ y , u << V ∞ & v << V ∞ .

ence we can neg ect u ∞

compare to un ty ea ng to t e

linearised BC

large or if the flow details near the leading edge are important. But

this form of the BC can be applied though it is inconvenient since it


is not known on coordinate lines. Hence we simplify it further.



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The linearised BC

is now applied on the chord instead of on the wing surfaceassuming

This can be justified using the Taylor series of v about a point on

-

Note that here Δ y = f ( x) and also (∂ v/ ∂ y) is very small. (Q:

Justify. Use continuity eqn.). Also note that we are changing


geometry when we replace a thin airfoil by a flat plate for BC.



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Now the interesting part. The airfoil geometry (camber &

c ness , we now, s very cruc a ; u we wan o c u a e

information on the x-axis, i.e. y = 0. This is done cleverly bymaking a distinction between the upper and lower surfaces and

applying the corresponding BCs for the transverse velocity v:

See that it accounts for camber and also upper and lower surface

geometry (thickness). These approximations turn out to be good,


surprisingly even in transonic and supersonic flow.



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We get insight by decomposing the BCs in terms of combination of

ang e o a ac , cam er an c ness. c ema ca y




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Hence

Notice that only the thickness term sign changes for the upper and

ower s es.The Laplace equation is to be solved subject to this BC. Further,

superposition of individual solutions is possible.




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We start with the exact relation for C p

and then use the approximations of thin airfoil introduced earlier

here:

This gives




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Since α, (u / V ∞

), (v / V ∞

) << 1, the exact relation for C p

reduces to:

This is the thin airfoil or linearised pressure distribution formula.




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DeltaC

Due to Airfoil Camber and Al ha (1)

We will calculate now the net load on the wing

= - p p lower p upper

starting from the formula

Using superposition, the pressure can be written as a combination

from the wing thickness, camber and angle of attack α

C p lower = C pt + C pc + C pαC p upper = C pt - C pc - C pα

This gives C p = 2 (C pc + pα

In the resent linear model disturbance velocit and ressure


distribution are not influenced by thickness.



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Line Source or Vortex

The same expression describes a "point" source or vortex in 2-D

- .

When K is real the expression describes a source with radiallydirected induced velocity vectors; imaginary values lead to vortex

flows with induced velocities in the tangential direction. Further

discussion of these flows is given in the next section.




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ne ource or or ex on

=

Φ = (- Γ / 2 π) ln R




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filament in a viscous

fluid. At t = 0, uθ

= Γ /

π r . as e nes

correspond to the case ofrigid body rotation

corresponding roughly to

core radii proportional to

.

From: Kuethe & Chow




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This 1/r behavior of the vortex induced velocity is not just a

.

equilibrium. We can easily see that the velocity must vary as 1/r

for the pressure gradients to balance the centrifugal force acting


on e u . e er va on s s own a er.



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ree or ces

Singularities that are free to move in the flow do not behave in

response to F = m a (what is m?). Rather they move with the

local flow velocity. Thus, vortices and sources are convected

.

produce complex motions due to their mutual induced

velocities.




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A pair of counter-rotating

vortices moves downward

Co-rotating vortices orbit

each other under the

because of their mutual

induced velocities.

influence of their mutual

induced velocities.




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Fundamental Sin ularities in 3D Potential FlowOne may derive fundamental solutions to Laplace's equation in

3-D, just as we did in 2-D (although complex variables are not

.

3-D Source

t was eas y scovere t at t e potent a : φ = - r sat s e

Laplace's equation in 3-D.

Since V = grad φ, the velocity associated with this solution isdirected radially with a magnitude:

V = k/r 2. It is easily shown that the constant k is related to the

= , ,

V = S / 4π r 2.

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


as r rat er t an r as n t e - case.



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Another basic solution, that has been used with some success in

supersonic aerodynamics programs is the point doublet,

obtained by moving a point source and sink together while

kee in the roduct of their stren th S and se aration L

constant. With μ = SL, the velocity associated with the point

doublet is:




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e m o z or ex eoremsHelmholtz summarized some of the properties of vortex filaments,

or vortices, in 1858 with his vortex theorems. These three theorems

govern the behavior of inviscid three-dimensional vortices:

. .

2. Vortices are forever (They cannot end abruptly. They should

extend to infinity, or end on boundaries or form a closed path).. or ces move w e ow n n a y rro a ona , nv sc

flow remains irrotational).

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




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-vortex line as shown:

The integral around the closed loop from a to b to c to d to a

cuts through no vorticity so from Stokes theorem the integral is

.

the sum of the integral from b to c and the integral from d to a.

These are the local circulations around the vortex line and so,

the circulations must be constant along the line.


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strength cannot suddenly go to zero. Thus, a vortex cannot end

in the fluid. It can only end on a boundary or extend to infinity.

Of course in an real, viscous fluid, the vorticity is diffused

through the action of viscosity and the width of the vortex linecan become lar e until it is hardl reco nized as a vortex line. A

tornado is an interesting example. One end of the twister is on a

boundary; but at the other end, the vortex diffuses over a large

.

As discussed in the section on sources and vortices, singularities

such as vortices in the flow move along with the local flowvelocity. Here, interactions of the vortices in the trailing wake,

cause them to curve around each other and to form the


nonplanar wake shown below.

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, ,used with permission.


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The Biot-Savart law relates the velocit induced b a vortex

filament to its strength and orientation. The expression, usedfrequently in electromagnetic theory, can be derived from the

.

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


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,contribute:

A simple subroutine is provided to compute the velocitycomponents due to a vortex filament of length Gx, Gy, Gz with

the start of the vortex rx r rz from the oint of interest.


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To determine the velocit associated with a vortex line we

consider the expression for vorticity :

, ,

so we can write

where A is called the vector potential. We are free to choose A

so that it satisfies


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This is a Poisson equation for A which has the well-known

solution:

So the contribution of a length dl

of vortex filament A is:


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This expression may be integrated along the vortex line for the

velocity induced by the filament to obtain the Biot-Savart law:


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xamp e- n n n e y ong ra g or ex

The velocit induced b an infinitel lon strai ht vortex at a

point p located at a distance h from the vortex will be evaluated

using the Biot-Savart Law. In 2-D it was stated to be

We verify that it is

Consistent with the

3-D formulation.




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Refer to the notes b W.H. Mason for details. After com letin

the integration we get the correct 2-D limit

NOTES

1. Observe the radial dependence of V p in 2-D & 3-D.. at s t e rect on o p n t e - case same as n -

3. What is the direction of dV p due to the vortex element of

length dl ? Is there an axial component of velocity?

4. See that the vortex element farther away from point p willhave smaller effect which drops off like (1 / r pq

2 ). This


infinite range.

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xamp e- e em - n n e ra g or ex

In this and the next exam le we consider a vortex that has an

end point(s). This may be on the surface or we may consider a

part of the vortex where the geometry changes.

extends up to infinity.

The induced velocity is:

It gives the correct limit


.

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See the comments on the previous example.

The induced velocity V p is:

It gives the correct limit

for the infinite and

semi-infinite vortices. The velocity at a point on the axis but outside vortex is zero.


indeterminate form? This result will be needed later. (Contd)

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We express the result in another form that will be used later.

We re-write the formula:

s ormu a w e an y.

Plot V on a line arallel to the vortex a distance h awa .


Identify the location of the maximum velocity. Justify it physically.

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We have seen the downwash due to a horseshoe vortex. Here

AB is the bound vortex and we have two trailing vortices.

evaluated based on the Biot-Savart law formulas we have

developed in the previous examples.

Note that Γ has to be

same for the three legs.

Induced velocity V p


to the sum of three values.

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We sum the values from three legs obtained by the formula

given earlier:

Note that the point C need not have to be co-planar with the

vor ex. n ac , we can ex en s me o o any com na on

of vortices, since superposition is allowed.

Here Ψ (vector) and Ω (scalar) represent:


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Explicit expressions are available for the three legs of the

horseshoe vortex ( for the terms (c1mn , c2mn & c3mn ) shown

below; see, e.g. W.H. Mason) that can be used in computer. n .

By summing we write at any point m, velocity at location m

ue o e n orses oe vor ex

V mn = (c1mn + c2 mn + c3mn ) Γ n

= (C mn ) Γ n

th


mn

vortex effect at location m.

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Summary

The following topics were dealt in this session

1. Introduction to Vortex Lattice Method

2. Application of BCs in VLM

.

4. Transfer of the BC to the Mean Surface

5. Decomposition BC to Camber, Thickness & Alpha

6. Vortex Dynamics, Helmholtz Theorems7. Biot-Savart law


. pp ca on o o - avar aw o orses oe vor ex

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Finite Wing- Vortex Lattice Methods

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