# Frisbee-circular plate aerodynamics

Discussion in 'Hydrodynamics and Aerodynamics' started by Mikko Brummer, Apr 18, 2012.

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### Mikko BrummerSenior Member

I finally took the time analyse, on Leo's suggestion, the frisbee aerodynamics.

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• ###### Frisbee Aerodynamics AIAA-2002-3150.pdf
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### Leo LazauskasSenior Member

The lip of the frisbee makes it a very tough problem.
That's why I suggested you try a simple flat circular plate first.

My contribution (with E.O Tuck) was to try to find cambered circular wings
with zero pitching moment. The preprint of the paper which appeared in J. Ship Research can be found at:

Good luck!
Leo.

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### Mikko BrummerSenior Member

I started with the circular plate, easier to model in CAD. I ran 2 studies with different turbulence modeling: RANS (Reynolds averaged Navier-Stokes) and LES (large eddy simulation), and compared the results against the Potter & Crowther windtunnel experiments as well as Leo's LSP-lifting surface program results.

The results are good: The LES simulation (orange circles) is spot on the Potter & Crowther presented experimental curve (in black) all the way up to 30 deg angle of attack. The simpler RANS-simulation (green squares) also does well up until CL about 0,8. Above that it falls between Leo's LSP curves for 0% (blue) and 100% leading edge suction (red). Considering the LSP is inviscid and the flow over the plate is almost totally separated, the 100% LE suction model does a great job, too.

#### Attached Files:

• ###### Circular plate CD vs CL.jpg
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### Mikko BrummerSenior Member

Flow patterns from the RANS-study, at 10, 20 and 30 deg

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• ###### Circular 30 deg_2.jpg
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### Mikko BrummerSenior Member

I only chose the plate because it needed no modelling... It would be nice to do the 2 frisbee models tested, to see if they rate similarly against each other in CFD as in the experiment. Now that you mention it, should have done the frisbee first, don't know when I will have the time for that.

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### daiquiriEngineering and Design

Looks nice. However, the flow in the pictures looks symmetric, which in case of a frisbee imo shouldn't be so. Did you include a disc angular velocity in the simulation?

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### Mikko BrummerSenior Member

No spin here... the force & surface pressure plots are extensively covered in the AIAA paper for the non rotating discs. While there are some results for the spinning frisbee (fig 8 in the paper), which one is it? Not likely the circular plate. So I took the easy way out. If I do the real frisbee, I could try it with a spin, too.

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### Mikko BrummerSenior Member

Animated flow patterns for the LES-simulation. The LES-simulation is transient, time dependent, so in the video you can observe how the flow develops with time, speeded instantly from rest to 20 m/s. The LES also captures more accurately the real behaviour of the turbulence: with LES, the turbulence is resolved directly up to a chosen level, and only the smallest eddies (within the boundary layer) are modeled, while in RANS the turbulence is averaged and modeled all the way. The accuracy comes at a cost, the LES takes about 24 hours for 1 angle of attack, while the RANS runs in 20 minutes.

In the LES simulation, you can see the smaller vortex cores spinning around the larger, initial trailing vortices, starting from frame 5 or so, while they are absent in RANS. Observe also how the starting vortex for the circular wing is - circular. The animation is about 30 times in slow motion, the whole thing happens in reality in 0,15 sec, the time the air travels through the 3 m long "windtunnel". The trailing vortices are coloured with vorticity, the plate with velocity.

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### Leo LazauskasSenior Member

Splendid work, Mikko!

Just one minor point - my code is NOT a Vortex Lattice Method (VLM).
VLM produces rubbish for lifting surfaces with curved leading and/or
trailing edges because the straight edges of the horseshoe vortices
"intersect" or come close to the the collocation points.
That is one reason I like the circular wing as a test case. The other is that
there is an exact solution for the Lifting Surface Integral Equation.

I could go into the messy numerical details but it would send you and everybody to sleep!

Leo.

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### DCockeySenior Member

I've seen "Vortex Lattice Method" applied to methods with skewed horseshoe vortices which alleviate the problems cited above. A typical application is a tapered and/or swept wing. It's also worth noting that vortex latice methods are equivalent to a panel method which uses constant strength doublet distribution over each panel.

Leo, a brief description of the method your code would be appreciated by me, though most others may find it a bit esoteric.

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### PetrosSenior Member

Very interesting, great work!

looking at the Cl/Cd polar it appears L/D is only about 4:1? A best L/D is only around 7;1 or so? Seems kind of low from my casual observations. The spinning I think might be a component too large to ignore, greatly affects Rn, which has a very large effect on low speed objects.

Anyone ever measure the L/D of a real Frisbee? and compare it with these model codes?

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### Mikko BrummerSenior Member

Look at the AAIA paper above - they have tested two types of real frisbees and the flat plate I am referring to. Frisbees have a higher lift, but also more drag because they are "thick".

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### Leo LazauskasSenior Member

There are VLM's that use skewed horsehoe vortices (i.e. a slanted bound
vortex with parallel trailing vortices) but they are not guaranteed to work
with curved leading and trailing edge panels. The problem is that a
collocation point can lie on the (infinite) line through the slanted vortex or
close to it. This creates very severe numerical problems when the
influence matrix is inverted.

The method I use is due to E.O. Tuck and further developed by D.W.F.
Standingford in his PhD thesis. See the LSP pdfs in previous threads for biblios.

Essentially, the method solves the once (x-wise) integrated form of the
Lifting Surface Integral Equation. The method was used by Tuck to solve
many different types of Cauchy singular integral equations.

I have also used several numerical tricks we developed over a few years.
For example, using a Chebyshev grid improves convergence with the
number of panels in the x-wise and y-wise directions.

A nice trick is to assume square root behavior at the trailing edge (i.e. the
Kutta condition) and to build this into the matrix of influence coefficients.
That increases convergence with respect to the number of panels in the
x-direction.

VLM's have trouble capturing the singular behaviour at the leading edge of
rectangular and curved leading edges. There is often a "kink" in the
methods to ameliorate that kink, but the best I have seen is due to
Standingford. His technique builds into the influence matrix an analytical
correction that eliminates the kink for rectangular and straight leading
edges.

Despite these tricks and techniques, there are still some funny wobbles
lift coefficient is accurate, but there are still difficulties with the leading
edge suction. That is a much more difficult quantity to capture accurately.
CL is pretty easy. Once the leading edge suction is known it can be used
to calculate the induced drag directly from a balance of the forces on the
wing. Other (e.g Trefftz plane) methods also work, but I prefer the "direct"
method for flat wings.

As I said, I like the circular planform case because it has an "exact"
solution which makes it an excellent benchmark. The lift slope is
1.79002303. Peter Jordan used Gegenbauer polynomials to get that value.
Boersma and others have verified it. Touvia Miloh got close with a
brilliantly simple approximation, but his solution was for a wing with tiny
twists near the wing tips, so it is actually the solution for a non-planar
wing. See the paper I wrote with Tuck for a potted history.

I could go on and on describing the delights of this wonderful,
controversial, difficult, classical aerodynamic problem, but it's 5am and I
must go to bed to dream of singularities and solutions.

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### Leo LazauskasSenior Member

Also, LSP is really only valid for small angles of attack (AoA), typically a
maximum of about 5 to 10 degrees. The suction analogy I use makes it
reasonable over a wider range of AoA

LSP gives zero CL and zero drag at AoA=0.0. An additional component of
profile drag could be added to give non-zero drag at AoA=0 which might
improve the correlations at those low AoA.

Did you use an finite thickness plate in your simulations?
If so, what is the shape of the leading edge and trailing edge?

I have seen various results for flat wings with rounded, bevelled, and
squared-off leading edges. That can make a difference to predictions, but
it would probably be a pain to grid accurately.

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### Leo LazauskasSenior Member

There are some advantages for low aspect ratio wings of circular (or near
circular) planform because of their characteristics at high angles of attack.

A strange aeroplane dubbed "The Flying Pancake" was investigated
towards the end of WWII because it could take off from a short runway.
I think developments in VTOL killed it off. See:
http://en.wikipedia.org/wiki/Vought_V-173

These planforms have recently made a come back with small UAV's where,
as you point out, Reynolds number effects can be very important.

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