Optimum finess ratio for given internal volume

[miket]

What is the optimum finess ratio for a given internal volume. I know the optimum fineness ratio for a given height what what about for a given volume? would a shorter and longer shape be better than a taller and less long shape?

[RobertSmalls]

Maximum internal volume isn't the same thing as maximum utility, but consider this simplified 2D case anyway:



Give each car the same length "c=1". Each of these sections has the same packing factor, so internal area is proportional to "t" alone. So, in the above plot, you want to maximize the value of the x-axis divided by the y-axis. It's gonna be really long and skinny.

But maybe you can do better by taking section 5, slicing it at its widest point, and extending it with flat sides in the middle?

Buses, trailers, and oil tankers are vehicles with large ratios of internal volume to aerodynamic drag.

Also, a vehicle can improve its ratio of volume to drag simply by being scaled up.

[Patrick]

Originally Posted by RobertSmalls

Maximum internal volume isn't the same thing as maximum utility, but consider this simplified 2D case anyway:



Give each car the same length "c=1". Each of these sections has the same packing factor, so internal area is proportional to "t" alone. So, in the above plot, you want to maximize the value of the x-axis divided by the y-axis. It's gonna be really long and skinny.

I'm not sure I follow your reasoning on this. Section 7 will have more internal volume than Section 5 for the same length.

Section 7 is 0.4025/0.2726 = 1.47 times as thick as Section 5.

Looking at the drag curves, Section 5 has a Cd of about .06 and Section 7 has a Cd of about .086. .086/.06 = 1.43, so Section 7 has only 1.43 times the drag, but 1.47 times the thickness.

Also, I think the greater curvature of the thicker section will include a proportionately larger area (volume if 3d) than the maximum thickness numbers would suggest.

[RobertSmalls]

Hmm, I took another look at it. I'll make it 3d by making these sections into prisms of height h=1. They're all of length c=1, and width t, which equals t/c from the table. The quantity you want to minimize is CdA/V= (Cd*h*t)/(h*t*c)=Cd/c=Cd.

Therefore, section 5 is optimal.

Or at least, I think so. It is late and I am tired. Good night.

[miket]

My guess is that submarines are close the the optimum shape.

[RobertSmalls]

Submarines can go faster with good, low-drag design, but they may have other constraints on their design.

Airships, though, should be very narrowly focused on having high displacement and low drag. You can find examples of airships with fineness ratios from 4.4:1 to 7.3:1.

[aerohead]

Originally Posted by miket

What is the optimum finess ratio for a given internal volume. I know the optimum fineness ratio for a given height what what about for a given volume? would a shorter and longer shape be better than a taller and less long shape?

For volume I believe the sphere represents the structure of minimum surface area to contain a given volume.
If I'm reading into your question,which fineness ratio for a given volume produces the least drag?,then we need to qualify the question.
For automobiles,the ultimate fineness ratio is one which produces an 'effective' 'free-air' fineness ratio of around 2.27:1.
A body length/height=5 is a relatively good metric to shoot for.
If you go below,you invite separation and high pressure drag.
If you go over,you invite additional skin friction with attendant drag increase.
If any part of the aft-body exceeds a tangent angle of 22-degrees you can expect separation.
The aft-body of the Aerodynamic Streamlining Template' respects all the criteria required for minimum drag for a road vehicle.
Dr. Hucho says the body ahead of the point of maximum cross-section "has only minor influence on the total drag.The main contributions to the drag force originate from the rear part of the body......it is very important to design a rear body surface which brings the divided streamlines smoothly together.Optimum shapes are 'streamlined' bodies having a very slender rear part.......... Lower drag can only be achieved by extending the length of the vehicle's body...........The drag coefficient for ...cars may be plotted against vehicle length.....(For) correlation...between greater body lengths and lower aerodynamic drag.....if the evaluation is limited to vehicles developed for the lowest possible drag coefficient,this expected trend is in fact confirmed."
The 2.27:1 streamlined body has the same Cd 0.04 as the 2.5:1 streamlined body used for the 'Template'.
The aft-body of the 2.27:1 streamlined body violates Mair's 22-degree angle at useful lengths of the 'Template'.
The 2.5:1 body chosen for the 'Template' is the shortest body,which can produce the lowest drag coefficient in ground-effect ( Cd0.08 ),along with respecting tangent angle limitations as spelled out by Dr.Mair for the elimination of flow separation.
NOTE: There are structures which have been shown to demonstrate lower Cd's,but they are purely 'laminar' structures,and once accelerated to critical Reynolds number,demonstrate no advantage over the 2.5:1 streamline body form.

[miket]

I thought that the 2.51 ratio is for a given height/maximum cross section. A shorter/narower but longer vehicle could have the same given volume and a smaller frontal area/cross section.

[aerohead]

Originally Posted by miket

I thought that the 2.51 ratio is for a given height/maximum cross section. A shorter/narower but longer vehicle could have the same given volume and a smaller frontal area.

miket,the size of the streamline body doesn't matter,unless we're talking golf ball size.
The Cd will always be the same as long as it is at or above critical Reynolds number.
And as Dr.Hucho says,if it's shorter,like the New Beetle, or longer,like the Hindenburg, the drag will be higher.
What you're 'packaging' can influence the shape for sure.
If you build something like the VW 1-liter car,gen-1,or gen-2 it will have higher drag than the 'template' form.Cd 0.15/0.195 vs Cd 0.10.
I guess the deal is,define your needs,then shrink-wrap the smallest sleekest envelope over it.Remember safety.Don't scrimp on crumple-zone and such and wind up at the trauma center if you get popped by Barbie & Ken in their Romancemobile.

[miket]

Im not talking about the Cd as much as the CdA. A longer vehicle will have more internal volume with a smaller cross section. Will the skin friction offset the smaller cross section?