## Jet Research Organization – Los Angeles California

Welcome to our blog. This blog is a place for enthusiasts to discuss the ongoing improvements to jet aircraft, various theories, strategies for improvements for the members in the greater Los Angeles California area.

First a bit of background on Jet aircraft & their history:

The first type of “jet” was the turbojet and was created by both Frank Whittle as well as Hans Von Ohain. This led to the 1st turbojet aircraft being developed which was the Heinkel He 178 prototype which was piloted by Erich Warsitz in the German Luftwaffe (Air Force) on August 27, 1939. The first jet flight to gain notoriety was made by Major Mario De Bernadi in the Italian Caproni Campini N.1 motorjet.

The first Japanese jet was the Imperial Japanese Navy’s Nakajima J9Y Kikka. These new plane designs were quite extreme for their day. Other jet type aircraft for Japan during this time includes the canard design Shinden, as well as the rocket-propelled Mitsubishi J8M. The bulk of the designs were derivative works from the Germans with the Kikka being based on the Messerschmitt Me 262 and the J8M on the Messerschmitt Me 163). The Japanese only had drawings to work from, so the designers, engineers and manufacturers played major roles in the final design specifications. Even though significant progress was made, the advances that Japan made came too late to have an impact during W.W. II. (The Kikka only flew twice before the end of World War 2 – so it was hardly battle ready.)

Now that we’ve covered a bit of the history, lets move on to more modern day theory:

Most planes are designed with the shape that is known as the Sears-Haack body. This shape provides the lowest theoretical wave drag for a given shape and it provides nearly the same cross sectional area at each point along its length. As most persons are aware from the Whitecomb’s area rule, the derivative of cross sectional area yields wave drag.

(ATTENTION: Math area ahead. Expect Calculus, and Geometric calculations!) The cross sectional area can be calculated as follows:

A(x) = \frac {16V}{3L\pi}[4x-4x^2]^{3/2} = \pi R_{max}^2[4x-4x^2]^{3/2}

The volume of a Sears–Haack Body is:

V = \frac {3\pi^2R_{max}^2L}{16}

The radius of a Sears-Haack Body is:

r(x) = Rmax(4x − 4×2)3 / 4

The derivative (slope) is:

r’(x) = 3Rmax(4x − 4×2) − 1 / 4(1 − 2x)

The 2nd derivative is:

r”(x) = − 3Rmax[(4x − 4x2) − 5 / 4(1 − 2x)2 + 2((4x − 4x2) − 1 / 4)]

where:

x ranges from 0 to 1 (distance)

r is the radius

Rmax is the radius at its maximum (occurs at center of the shape)

V is the volume

L is the length

ρ is the density

U is the velocity

From Slender-body theory:

D_{wave} = – \frac {1}{4 \pi} \rho V^2 \int_0^\ell \int_0^\ell S”(x_1) S”(x_2) \ln |x_1-x_2| \mathrm{d}x_1 \mathrm{d}x_2

alternatively:

D_{wave} = – \frac {1}{2 \pi} \rho V^2 \int_0^\ell S”(x) \mathrm{d}x \int_0^x S”(x_1) \ln (x-x_1) \mathrm{d}x_1

These formulas may be combined to get the following:

D_{wave} = \frac{64 V^2}{\pi L^3}

CD_{wave} = \frac {24V} {L^3}