49swarm, and you have the Mach number.
Here again we have some magic from
the thermodynamics. This time, while
we still want to know the temperature,
pressure and density within the swarm,
these properties then depend on the Mach
number alone. Amazing!
Review what has been said. We started
off with the dynamics of motion within
the swarm. Then as the swarm started to
move, we had two forms of motion: internal
and external. The faster the speed, the less
the dependence on the thermal internal
motion within the swarm, to dependence
on the motion of the swarm itself.Transonic and Supersonic
Around about Mach 0.8, shock-waves
attached to the wing (the swarm has re-
shaped into a wing: swarms do that!) start
to grow from near the high point upper
surface on the section. With more speed,
the upper surface shocks continue to grow
while new ones appear near the lower
surface trailing edge.
At about Mach 1.2, these shocks move
off the wing altogether.
We have now defi ned two regions of
speed. Subsonic below Mach 0.8 and
transonic between 0.8 and 1.2. When all
the local fl ow over the wing is more than
Mach 1.2, we have the supersonic region.
Viscosity has not gone away. The
eff ect of viscosity is small, but diffi cult
to quantify. There is now no equivalent
magic numbers like the Reynolds and
Mach numbers. So let us go faster still.
At fi ve times the speed of sound, Mach 5,
we defi ne a new region, the “hypersonic”region. Here the text books have titles like
“Hypersonic inviscid fl ow”.Newtonsonic
With hypersonic fl ow we ignore the internal
thermal motion of the air. The speed is so
great that the swarm behaves like a spray
of machine gun bullets. The shock-waves
are angled so far back that they are nearly
on the surface of the hypersonic body.
The bullets don't collide with each other,
they all go in the same direction, and they
then behave as for the Newtonian model of
a perfect gas!
Shock Horror! Newtons model for low
speed was that the lift depended on the
square of the sine of the angle of attack.
This didn't work. But it works great for
hypersonic fl ow, so well that it is all
anybody uses.Clearly, this fl ow regime should
have been called “Newtonsonic” or
“Isaacsonic”, although “hypersonic” does
have some cachet to recommend it.Where Have We Been?
When you try to un-muddle the text books,
try to place the context in which the air
has internal motion (like the swarm) and
external motion (like the swarm trying to
catch and kill you), up to the point where
the internal motion can be neglected, and
you are simply being machine-gunned.
On that latter point, I once observed a
demonstration of a Vickers .303 machine
gun fi ring tracer across a valley. The
stream of bullets impressed me greatly.
But when I was told that between each
round of tracer there were 5 rounds of
slugs I just felt sick. ■Diagram taken from the 1949 publication “Gas Turbines for Aircraft” by Godsey and Young. This very
important concept, to me as a propeller designer, alerted me to the effects of compressibility. The
equation is also called the Prandtl-Glauert rule: it recognises the “geometric similitude” which permits
scaling of an aerofoil section at one Mach number to that at another. The concept of “similitude” is
a diffi cult thing to express in words while avoiding the word “similar”, which can quickly reduce your
expression to a tautology. The best account I have found is on page 40, “Hypersonic Inviscid Flow” by
Hayes and Probstein. This text also gives Newton another good airing!These three remarkably
simple equations refer to
compressibility at the leading
edge stagnation point of an
aerofoil section. The pressure
(Ps), temperature (Ts) and
density (rhos) are expressed
as a ratio at a remote point
(P,T,rho)in the free-stream. M
of course is the free-stream
Mach number. Again, in
propeller design, the pressure
at the leading edge stagnation
point determines the lift of
the associated blade element,
so that these equations are
vital at M = 0.3 and above,
i.e. where the change from
incompressible fl ow to
compressible fl ow becomes
signifi cant.Nice, instructive shot of a shock-wave formed over an aerofoil section.
Taken from the 1957 text “Fluid Mechanics for Engineers, by P. P. Barna.
Two lessons may be drawn. Firstly, the airfl ow behind the shock has caused
the boundary layer to break away, a source of drag. Secondly, note that the
section is at a small angle of attack. It is not enough to make the section
thin, as the growth of the shock also depends on the angle of attack.
Anything that increases the fl ow speed over the section increases shock
strength. Here endeth the Sermon.ARMCHAIR PART 29.indd 49 26/04/2018 14:31