Intuition The one core ideaA hypersonic vehicle turns the air's motion energy into heat so violently that the gas
stops behaving like a simple spring-loaded gas and starts to vibrate, break apart, and glow .
Everything on the parent page is just careful bookkeeping of where that energy goes — so before
we can read that page, we must first own every symbol used to track speed, temperature, pressure,
and energy.
This page assumes you have seen none of the notation. We build each symbol one at a time, each
one leaning only on the ones before it. Read top to bottom.
V V how fast the air moves past the vehicle, in metres per second. Picture standing
on the nose of a re-entry capsule: the air rushes at you at speed V
We draw the stream as a bundle of parallel arrows, all the same length (the length = the speed).
Intuition Why speed alone is not enoughA car at 30 m/s and a bullet at 30 m/s feel different to the air only because the air's own
"signalling speed" differs. Air pushes back by sending pressure ripples — sound — ahead of the
object. What matters is your speed compared to how fast those ripples travel . That comparison is
the Mach number, next.
a a how fast a tiny pressure disturbance travels through the gas . Picture tapping the air:
the "tap" spreads outward as an expanding sphere at speed a less where the air is colder.
Why does the topic need a shock .
M M = a V how many times faster than sound you are going. M = 1 M = 5
M M < 1 M > 1 M
M a same speed V M ⇒ a ⇒ M M where ?"
M Supersonic Flow & Area-Mach Relations and is the master dial for the whole
parent topic.
T T average jiggling energy of the gas molecules moving in straight lines
(translation). Picture a swarm of balls bouncing around a box: hotter = faster average bouncing.
We use kelvin (K), which starts at absolute zero (no jiggle), so T
Why the topic needs T somewhere ,
and the first place it goes is into this jiggling — i.e. temperature shoots up.
Intuition A crucial subtlety for laterT translational (and rotational) jiggling. Molecules can also stretch and bend
(vibration) or break apart — and that hidden energy does not show up as T γ
p p push per unit area the gas exerts, from countless molecules hammering a surface.
Picture rain on an umbrella: more drops, or faster drops, means more push. p ∞ ∞ undisturbed air far upstream, before the vehicle
disturbs it.
ρ ρ how much mass sits in each cubic metre of gas. Picture packing more balls into the
same box. ρ ∞ ρ 1 ρ 2 before and
after a shock (see next).
Why the topic needs both: pressure gives the force (lift, drag, heating loads), and density
tells us how tightly the gas packs into the thin shock layer.
The parent page decorates symbols with little subscripts. Each is a place in the flow.
Definition Reading the subscripts
∞ far upstream , the calm air the vehicle flies into (V ∞ , p ∞ , ρ ∞ 0 stagnation , where the flow has been brought fully to rest (T 0 , h 0 1 just before a shock ; 2 just after a shock .
Intuition Why "stagnation" (subscript 0) is the star of the showAt the very front of a blunt body there is a single point where the air is fully stopped: the
stagnation point . All the motion energy there has become heat, so T 0 hottest the
gas gets. The whole "why does it get so hot" story is about computing T 0
Stagnation quantities are the subject of Stagnation Properties & Isentropic Relations .
h h total internal + flow energy stored in each kilogram of gas. You do not need its
deep definition yet — treat h h = c p T c p
Why the topic needs this: it is the single equation that says "the gas gets hot because it
stopped." Everything about T 0
These four describe how a particular gas stores energy . They are the bridge between temperature
and everything else.
R R links pressure, density, and temperature : p = ρR T
γ γ = c v c p single number capturing how "springy" the gas is . Cold air has γ = 1.4 γ drops because energy hides in
vibration and broken bonds.
Intuition The picture: degrees of freedomf x y z degree of
freedom f f ⇒ ⇒ c v ⇒ γ = 1 + f 2
Real-gas behaviour of these constants lives on
Real Gas Thermodynamics & Dissociation .
θ θ angle between the oncoming stream and the surface the air hits. Picture the
flow skimming a ramp: θ C p = 2 sin 2 θ θ
α α how much the whole body is tilted relative to the flow. For a flat plate, the
windward face's θ α
R n R n how rounded the front of the body is . A large R n R n 1/ R n
Shock angles and their equations come from Normal and Oblique Shock Waves .
C p C p = 2 1 ρ ∞ V ∞ 2 p − p ∞ the extra pressure a surface feels, measured in units of the flow's motion energy
density . The denominator 2 1 ρ ∞ V ∞ 2 dynamic pressure — the
"punch" the stream carries. Dividing by it makes C p
Why the topic needs C p C p = 0.134 Mach-independence principle.
stagnation temperature T0
specific heat ratio gamma
real gas and dissociation
Read it as: temperature sets the speed of sound , which with speed sets M M γ stagnation temperature ; that high temperature opens
degrees of freedom , which lower γ real-gas effects — the loop that
makes hypersonics special.
Cover the right side and test yourself. If any answer surprises you, reread that section.
What does M Your speed V a
Why can the same speed be different Mach numbers? Because a a M
What does temperature T not ? It counts translational/rotational jiggling; it does not count vibration or bond-breaking energy.
What do subscripts ∞ 0 1 2 Far upstream; stagnation (fully stopped); just before a shock; just after a shock.
State the energy equation that makes the nose hot. h + 2 1 V 2 = h 0 = const
What is γ The ratio c p / c v 1.4
How does opening more degrees of freedom f γ More f c v γ = 1 + 2/ f
What two identities convert the energy equation into T 0 / T = 1 + 2 γ − 1 M 2 c p = γ R / ( γ − 1 ) a 2 = γ R T
What is C p The pressure coefficient ( p − p ∞ ) / ( 2 1 ρ ∞ V ∞ 2 )
Why does nose radius R n Stagnation heat flux scales as
1/ R n , so blunter (large
R n ) means less heating.
Ready? Head back to Hypersonic flow — Mach 5+, high temperature effects and every symbol on that
page will now be one you have already met.