From: David Lednicer <firstname.lastname@example.org>
Subject: Area rule - wassat all about then?
Date: Mon, 21 Feb 2000 11:55:07 +0000
The Area Rule states that two configurations with the same
cross-sectional area distribution will have the same wave drag, to first
order. The Sears-Haack body is regarded as that having minimum wave
drag, so designers tailor the cross-sectional area distribution of
transonic aircraft to be similar to that of the Sears-Haack body. The
XF-102, when first flown, had big spikes in the cross-sectional area
distribution where the wing grew out of the fuselage, etc. This
resulted in high wave drag and it couldn't go supersonic. By carving
away at the fuselage, the cross-sectional area distribution more closely
represented the optimum, the wave drag was lower and the aircraft was
now capable of going supersonic.
Modern fighter aircraft have more power available, so designers worry
less about the area rule. However, it still does come into play.
An interesting example, involving an airliner, is the Boeing 747. The
cross-sectional area distribution of the 747-200 isn't too bad, but when
they stretched the upper deck, resulting in the 747-300, the
distribution got better. This allows the 747-300 to cruise at a
slightly higher Mach number. A comparison of the wind tunnel data for
these two variants is shown in Kuethe & Chow's Foundations of
There is also a way of applying the area rule to a local area. John
Kutney of GE has written quite a bit about this. His work in this
regard helped improve the cruise drag of the Convair 990 and was also
used on the S-3 Viking and Airbus A300. The Airbus A340 had a problem
inboard of the outboard pylon that was fixed by use of the local area
David Lednicer | "Applied Computational Fluid Dynamics"
Analytical Methods, Inc. | email: email@example.com
2133 152nd Ave NE | tel: (206) 643-9090
Redmond, WA 98052 USA | fax: (206) 746-1299
From: firstname.lastname@example.org (Mark Drela)
Subject: Re: Subsonic Area Rule?
Date: 16 Sep 1995 09:57:00 -0500
In article <CMM.0.90.4.811172853.rdd@netcom12>,
>I am familiar with the area rule concept as it applies to supersonic
>flow, but applying it to a subsonic case is not as clear to me. My
>collection of text books have not been very enlightening. I am
>starting to wonder if the term subsonic area rule is perhaps a bit of a
>misnomer. Any way, here are a few of the burning questions I have:
> - Does the same principle of a smooth area distribution (a Sears-Hack
> body) apply and just the term sqrt (1-M^2) substituted for
> - Or, is it something more subtle, like locally altering the surface
> contours to maintain some favorable pressure gradient or mach
The subsonic area rule is somewhat qualitative in nature. It's like saying
that to avoid local Cp spikes, an airfoil's surface must be smooth.
The reason that the _total_ X-sectional area at a given streamwise x location
is important is that near M=1 the streamtubes going past the body cannot
change in area very much. For any given streamtube, the mass flow
m = rho V A is a constant. At low speeds, the density (rho) is nearly
constant, and increase in velocity V will cause the streamtube area A to
contract proportionately, so that it can "squeeze" past the body.
At higher speeds, there is a significant decrease in pressure and hence
a decrease in rho which offsets some of this area decrease. At M=1,
the cancellation between rho and V changes is perfect, and A cannot change.
Quantitatively, the fractional area change is related to the fractional
velocity change by
dA 2 dV
-- = (M - 1) --
Hence dA = 0 if M = 1. Whitcomb called flow near M=1 "pipefitter's flow".
The streamtubes have to get past the body by being redistributed, not by
The upshot of this is that any "bulge" in the aircraft near M = 1 can
strongly communicate its presence to all other points at that same
x location, since the sideways shove it gives to a surface streamtube
doesn't die off with distance. The streamtube moves sideways without
contracting, shoving the next streamtube by the same amount. and so
the signal propagates sideways without attenuation. So the effect of
all such bulges at any x location simply adds up, and so the total
area variation must be smooth in order to disturb the overall flow
by the least amount. If the flow IS disturbed a great deal, then
local high-velocity regions will occur, along with shock waves and
The Sears-Haack body you mention was derived assuming linearized supersonic
flow, which is NOT valid near M = 1. This is the transonic regime, which
is a rather different animal. There is no reason to believe that the
Sears-Haack shape, which is optimal for supersonic flow, represents
the best X-sectional area near M = 1.
> - Has this principle been applied to any lower speed aircraft, i.e.; GA
> aircraft, transports, etc?
> - Have (or are) area distribution plots been widely used in the design of
> subsonic aircraft?
At lower speeds, below M = 0.7 say, the effect of all bulges doesn't
pile up like it does near M = 1.0, since the streamtubes can contract
significantly and the lateral signals attenuate. Hence, it's not too
important to have a smooth total X-sectional area.
Area ruling IS a consideration in typical M = 0.80 or 0.85 jets,
but it's not a major design driver.
Mark Drela First Law of Aviation:
MIT Aero & Astro "Takeoff is optional, landing is compulsory"