Area Rule - graphic method

[[user-BG]]

Hello friends,
my first post here.

So let me introduce you a Area Rule's graphic method.
There is no double & triple integrals or some complex mathematics with no shape representation in the end.

Any comments welcome

Regards

 

[VTOLicious]

Welcome to the forum!

Nice drawing! However, what is the method? Can you elaborate?

 

[HoHun]

Hi,

So let me introduce you a Area Rule's graphic method.
There is no double & triple integrals or some complex mathematics with no shape representation in the end.

Any comments welcome

Really cool! I would love to see this style of diagram for the first-generation of jets. Are you using a dedicated software for the diagrams?

Regards,

Henning (HoHun)

 

[Sundog]

Note that what he is showing is only for Mach=1.

 

[icyplanetnhc (Steve)]

Yup. For supersonic area rule, the cross-sections areas are measured via Mach plane cuts, with Amax biased to the rear based on your desired optimal Mach number.

 

[sienar]

OpenVSP makes this very easy;

It can import quite a few 3d model formats.

 

[[user-BG]]

OpenVSP makes this very easy;
View attachment 719272
It can import quite a few 3d model formats.

Hello & thanks for reply.
When I introduced this method I have't idea of OpenVSP.
Please pay attention to the slopes of your curves and fact you have 2 peaks. This design is far away from perfect.
Also decreasing/increasing one section you'll have wavy surfaces, adding details just to fill gaps in the curve is also not the greatest idea ...
I passed through all this.

I'm achieving perfect smooth one peak curve from first try following one simple rule.

Regards

 

[[user-BG]]

Hello guys,
thanks for replies to everybody

Another first try design with one peak smooth sections distribution curve.

 

[[user-BG]]

Welcome to the forum!

Nice drawing! However, what is the method? Can you elaborate?

Hello and thanks for you kindness!

Area Rule from Wikipedia (in History part there is a some objective information): https://en.wikipedia.org/wiki/Area_rule

Regards

 

[[user-BG]]

Here I decide to broke some of the rules of my method, just to see what will happens (model is built from the drawing found in the web).
Please note I performing exactly Area Rule just like it is done back in 40s, not something else!
I don't know what some software is doing, how it doing, how much it depend on a mesh errors ... etc. This is not subject of my research at all!

This is the result for today (first one is transparent background png):







Regards

 

[Richard N]

You might change the vertical grid to red or green to separate it from the background.

 

[[user-BG]]

You might change the vertical grid to red or green to separate it from the background.

Hello and thanks for reply, to be honest I thought the same ... I'll! Job is not done yet ...

 

[[user-BG]]

... I'll improve graphic representation, but in scientific point of view curve is not that bad.

 

[gwiz]

Takes me back to the start of my career. We had a little program to get the wave drag from the area distribution, but the drawings were still done by hand and I spent a lot of time with a planimeter actually getting those area distributions to input. Things soon got easier when the wing geometry was digital and I only had to measure the fuselage, but it was quite a time before the whole aircraft was digital.

 

[[user-BG]]

Takes me back to the start of my career. We had a little program to get the wave drag from the area distribution, but the drawings were still done by hand and I spent a lot of time with a planimeter actually getting those area distributions to input. Things soon got easier when the wing geometry was digital and I only had to measure the fuselage, but it was quite a time before the whole aircraft was digital.

Hello gwiz,
thank you so much for your comment. I'm glad somebody that spent time on Area Rule can appreciate/understand my idea. The point with software is that you never know what/how exactly it calculate. Also I prefer to have full control over 3d model, parametric packages are not my type, I'm into old school nurbs modeling.

Best regards.

 

[Scott Kenny]

Takes me back to the start of my career. We had a little program to get the wave drag from the area distribution, but the drawings were still done by hand and I spent a lot of time with a planimeter actually getting those area distributions to input. Things soon got easier when the wing geometry was digital and I only had to measure the fuselage, but it was quite a time before the whole aircraft was digital.

Question for you, since I don't have enough of an engineering background to understand much of the area rule other than "it's needed."

@[user-BG]'s method puts the area change into a nice smooth curve if the area ruling is done correctly. Is there some optimum shape to this curve that you can express as a math formula? (basically working backwards from this formula to make the plane shape)

 

[overscan (PaulMM)]

Yes, look up Sears-Haack body.

 

[Scott Kenny]

Yes, look up Sears-Haack body.

It's not sounding like that's what I meant.

My question was if there was a mathematical description of the change in cross-sectional area that was considered ideal, not the body shape directly. Also, apparently the Sears-Haack body doesn't apply at transonic speeds, because the Prandtl-Glauert Equation is invalid there.

 

[gwiz]

@[user-BG]'s method puts the area change into a nice smooth curve if the area ruling is done correctly. Is there some optimum shape to this curve that you can express as a math formula? (basically working backwards from this formula to make the plane shape)

The method was applied to the cross-sectional area distribution using cutting planes tangential to the Mach cone and averaged over roll angle around the cone. If there was a difference between area at the nose and tail, eg a blunt base like a rocket, the Von Karman ogive between nose and tail was subtracted leaving a distribution pointed at both ends. The optimum shape for the residual area distribution was the Sears-Haack one, but the actual wave drag was calculated from the formula, the double integral one, in the Wikipedia entry for Sears-Haack. As the formula uses the second derivative of the area distribution, it is highly sensitive to the smoothness of the distribution.

In the transonic region, the equations describing the flow are a lot more complicated than in pure subsonic or supersonic flow and you need a full CFD method to get good predictions.

 

[afcivileng]

Can you import an F102 and then an F106 to highlight the effect on essentially the same design

 

[Scott Kenny]

The method was applied to the cross-sectional area distribution using cutting planes tangential to the Mach cone and averaged over roll angle around the cone. If there was a difference between area at the nose and tail, eg a blunt base like a rocket, the Von Karman ogive between nose and tail was subtracted leaving a distribution pointed at both ends. The optimum shape for the residual area distribution was the Sears-Haack one, but the actual wave drag was calculated from the formula, the double integral one, in the Wikipedia entry for Sears-Haack. As the formula uses the second derivative of the area distribution, it is highly sensitive to the smoothness of the distribution.

Ugh. Way over my math paygrade (I didn't do so well in Calculus, though I didn't find it all that hard. Just didn't do anywhere near enough problems to internalize the formulas.)

In the transonic region, the equations describing the flow are a lot more complicated than in pure subsonic or supersonic flow and you need a full CFD method to get good predictions.

Bugger.

 

[Scott Kenny]

Can you import an F102 and then an F106 to highlight the effect on essentially the same design

Second this!

 

[[user-BG]]

Question for you, since I don't have enough of an engineering background to understand much of the area rule other than "it's needed."

@[user-BG]'s method puts the area change into a nice smooth curve if the area ruling is done correctly. Is there some optimum shape to this curve that you can express as a math formula? (basically working backwards from this formula to make the plane shape)

Hello and thanks for the interest on my topic.
If I understand you right (since English is not my native language) you are asking for mathematical prescription of a optimum section distribution curve.

It is called "Normal Distribution" or "Gaus-Laplace distribution" or "Laplace distribution" in English.
Specially for you I attaching a picture and pdf with method formulas.

The method I introduced is simple and effective and can be converted into software package.

Regards

 

[[user-BG]]

Can you import an F102 and then an F106 to highlight the effect on essentially the same design

Hello, thanks for reply.
Please pay attention to the tubes of both planes - sections where wings are decreased just because of Area Rule.
However, I think a lot of things can be improved in those planes also on a lot of existing planes build in hundreds.

Regards

 

[VTOLicious]

Hello, thanks for reply.
Please pay attention to the tubes of both planes - sections where wings are decreased just because of Area Rule.
However, I think a lot of things can be improved in those planes also on a lot of existing planes build in hundreds.

Regards

 

[[user-BG]]

Hello VTOLicious & other guys, thanks for replies to everybody.
Great examples for F-102 modifications based on Area Rule.
What we can see is that even modifications result is 2 peaks curve.

Without being into American airplane designs a lot, I think Convair XF-92 is also representation of Area Rule.

Plane look (95%) like this:



Regards

 

[[user-BG]]

... and Area Rule for the Convair XF-92, two peaks again - at section 04 and even more in section 09. Please note linear segments representing section's area are decreased 100 times.





... all comments welcome.

Regards