Transcript AE 301 Aerodynamics I

Vortex Lattice Method
•
The lifting line method was very useful to understand
the basic concepts of span loading and the elliptical
distribution.
•
However, lifting line theory is limited to high aspect
ratio, unswept wings – and even then is approximate.
•
To do analysis for more generalized geometries we need
to go to a numerical approach like that of the Vortex
Lattice Method.
•
To keep our numerical method simple lets assume a
planar wing – I.e. the entire wing lies on the z=0 plane.
•
We can still model twist and/or camber, but we will
apply the slope boundary condition at z=0 just like we
did in Thin Airfoil Theory.
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Vortex Lattice Method [2]
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Our approach is to panel the wing planform and place a
horseshoe vortex at the c/4 of each panel.
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In addition, we place a flow tangency boundary
condition point at the mid-span and 3c/4 point of each
panel.
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For 6 spanwise and 2 chordwise panels, the result is:
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Surface Paneling
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How to space the panels?
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The most obvious is to use even spacing in both
directions.
•
However, on most wings, the greatest change in span
loading occurs at the wing tips .
•
To better resolve these variations, a spacing which
clusters panels near the tips can be used.
•
Thus, if we have nj spanwise panels and ni chordwise
panels, the sides of the panels may be given by:
 nj  j 
b
y j 1,nj 1  cos
 
2  nj

•
 i 1 
xi 1,ni 1  xle  c

 ni 
Note: there is one more side then there are panels!
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Surface Paneling [2]
•
Note that we really are not interested in the panels
themselves – just the vortex corner points and the flow
tangency points.
•
To find the vortex corner points directly use:
 nj  j 
b

yc (i, j )  cos

 n

2
j


xc (i, j )  xle, j
 i  .75 

 c j 
 ni 
i  1, ni  1
j  1, nj  1
•
This gives one extra set of corner points behind the
wing, but is useful for the next step.
•
Now, average the corner points to find the tangency
point in the center:
x(i, j)  14 xc (i, j)  xc (i  1, j)  xc (i  1, j  1)  xc (i, j  1)
y(i, j )  14  yc (i, j )  yc (i  1, j)  yc (i  1, j  1)  yc (i, j  1)
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Surface Paneling [3]
•
•
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For each span station, the local chord and leading edge
values can be found from the planform:
2 yc ( j,1)
2 yc ( j,1)
xle, j 
tan  le
ct  cr 
c j  cr 
b
b
Where the wing parameters can be determined from:
b
2c
ct  cr
c
cr 
AR
1 

cr  ct 
tan  le  tan  c / 4 
2b
All this set takes a bit of work – don’t be surprised if
half your program is just setting up the geometry.
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Induced Downwash
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We have already developed an equation for the
downwash due to a horseshoe vortex based upon the
Biot-Savart Law.
•
However, the equation, shown below, is only good for
when the bound vortex is perpindicular to the trailing
ones.

cos 1  cos  2 
w
4d
 



1  cos  1  
4l1 
2

  
 

 cos  2    1
4l2  
2 
AE 401 Advanced Aerodynamics
l2
d
l1
1
2
z
y
x
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Induced Downwash [2]
•
For the more general case of a skewed vortex
arrangement, the following equation applies:
( xk , yk )
2
 1
cos 1  cos  2 
w
y2  yb  yk

4  d
( xb , yb ) d  3
1
1  cos 0 

y1
y1  yk  ya
1

1
cos 3   1

0
y2

( xa , ya )
• From this relation, we can determine the downwash
induced at any flow tangency point due to any panel
horseshoe vortex.
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Induced Downwash [3]
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To aid in evaluating the angles used in the previous
equation, it is helpful to set our x-y axis at one corner
point – and then rotate it to align with the bound
vortex.
( xk , yk )
( xb , yb )
2
2
l  xb  xa    yb  ya 
y y
  xb  xa  
l
  atan 

x


y

y
a 
 b
 x
xk  xk  xa cos   yk  ya sin 
( xa , ya )
yk  xk  xa sin    yk  ya cos
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Induced Downwash [4]
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In terms of the coordinates of this rotated system, the
distance to the point and the angles are then:
d  xk
 xk 
1  atan 
 yk 
 xk 

 2  atan 
 y k  l 
0 

2
   1
3  2 
•

2
( xk , yk )
2
( xb , yb )
y
d
3
1
0
x
 x
( xa , ya )

Now we are ready to set up our solution method.
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Solution Method
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Our problem is that we have a total of nk unknown
vortex strengths to find, one for each panel.
nk  ni  nj
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When programming, it will be easiest to loop over all
the chordwise and spanwise panels using the counter:
k  j  nji 1
•
The velocity induced at each flow tangency point, k,
due to all the panel vortices is:
nk
wk   kk wk ,kk
kk 1
•
Where, from our earlier relation:

1  1
1
1
cos 1  cos  2   1  cos 0  cos 3   1
wk ,kk 

4  d
y1
y2

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Solution Method [2]
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At the same time, for flow tangency, the downwash
velocity must equal:
dz
wk  V
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Where, for a simple untwisted, uncambered wing:
dz
•
dx
dx
  sin 
From this, we see that our matrix solution formulation is
going to be:
[wk ,kk ]kk / V    sin 
•
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Once this equation is solved for the individual vortex
strengths, the lift and drag must then be computed.
The lift is easy - just sum the
L


V

individual lift contributions:
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nk
  y 
k 1
k
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k
Solution Method [3]
•
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Or, in terms of the lift coefficient:
2 nk
k yk 
CL 

SV k 1
The drag coefficient is found by from the induced
downwash velocity due to the trailing vortices:
2
CD 
SV2
wt k ,kk
•
nk


 k yk  wtk ,kk kk 

k 1 
kk 1

nk

1  1
1
1  cos0   cos3   1


4  y1
y2

A little computation could be saved if the above wake
downwash term was saved from when it was computed
as part of the earlier downwash term.
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Solution Refinements
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The preceding analysis method will give good results for
a wide range of problems.
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Wing twist and camber can easily be added by including
the appropriate surface slope at each tangency point.
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To model winglets or wings with extreme dihedral
angles, the method must be reformulated in 3-D.
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One advantage of doing this is the ability to model
ground effect via a mirror wing below the surface.
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To find surface pressure distributions for boundary layer
calculations, thickness must also be modeled.
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This is usually done by having both upper and lower
surface panels.
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