ARO 3111
Gasdynamics and High-Speed Aerodynamics
Lecture 1
January 20, 2026
https://eaglepubs.erau.edu/i
ntroductiontoaerospaceflight
vehicles/chapter/flying-fast/
How does
information
propagate?
What’s the
significance
of that?
Course outline (1)
1. Introduction to compressible flow
a. Historical developments
b. Definition and flow regimes
c. Review of thermodynamics
d. Isentropic flow
e. Conservation laws of fluid mechanics in integral form
f. Continuity equation
g. Momentum equation
h. Energy equation
2. One-dimensional compressible flow
a. One-dimensional flow equations
b. Speed of sound and Mach number
c. Static, stagnation and sonic conditions
d. Different forms of one-dimensional energy equation
e. Isentropic flow relations
f. Normal shock relations
g. Rankine-Hugoniot relations
Course outline (2)
3. Oblique shock and expansion waves
a. Oblique waves
b. Oblique shock relations
c. Supersonic flow over wedges and cones
d. Shock polar
e. Wave interactions
f. Intersection of shocks of opposite families
g. Intersection of shocks of the same family
h. Mach reflection
i. Bow shock wave of a blunt body
j. Prandtl-Meyer expansion waves
k. Shock-expansion theory
l. Introduction to 1D unsteady problems and x-t diagrams
4. Quasi-one-dimensional flow
a. Governing equations
b. Area-velocity relation
c. Isentropic flow through variable-area ducts
d. Flow through diffusers
e. Supersonic wind tunnels
f. Wave reflection from a free boundary
Course outline (3)
5. One-dimensional flow with heat addition (Rayleigh flow)
a. Subsonic flows with heat transfer
b. Supersonic flows with heat transfer
c. Choked flow due to heat transfer and consequences
6. One-dimensional flow with friction (Fanno flow)
a. Subsonic flows with friction
b. Supersonic flows with friction
c. Choked flow due to friction and consequences
7. Conical flow theory (supersonic flow past un-yawed cone)
a. Physical aspects of conical flow
b. Derivation of the Taylor-Maccoll equation
c. Numerical solution procedure
d. Physical aspects of conical flow
e. Comparison with 2-D wedge solution
Course outline (4)
8. Introduction to high-speed flows
a. Pressure coefficient for compressible flow
b. Solution for weak oblique shocks
c. Solution for gradually turning supersonic flow
d. Maximum turning angle for supersonic expansion
9. Governing laws and theorems of high-speed flows in differential form
a. Conservation of mass - continuity equation
b. Conservation of linear momentum - Euler's equation
c. Conservation of energy - energy equation
d. Conservation laws in non-conservative form
e. Alternative forms of energy equation
f. The entropy equation
g. Crocco's theorem
10. The velocity potential equation
a. Irrotational flows
b. Causes of fluid rotation
c. The velocity potential equation
Course outline (5)
11. Compressible subsonic aerodynamics
a. Physical aspects of linearized flow
b. Linearized velocity potential equation
c. The Prandtl-Glauert equation/rule
d. Linearized pressure coefficient
e. Linearized subsonic flow
f. Improved compressibility corrections
g. Subsonic flow past a wavy wall - analytical solution
h. D'Alembert's paradox for subsonic flow
i. Calculation of critical Mach number & Drag divergence Mach number
12. Two-dimensional supersonic aerodynamics
a. Linearized supersonic flow past an airfoil - Ackeret's theory
b. Pressure coefficient & c. Lift coefficient
d. Wave drag coefficient
e. Pitching moment coefficient & f. Location of aerodynamic center
g. Busemann's theory
h. Supersonic flow past a wavy wall
i. Differences in physical aspects of sub- and supersonic flows
j. The characteristic equations and the Prandtl-Meyer flow
k. Linearized theory vs. exact shock-expansion theory
Course outline (6)
13. Wings of finite span in supersonic flow
a. Domain of dependence and region of influence
b. Classification of wing edges
c. Conical flow method
d. Rectangular wings in supersonic flight
e. Aerodynamics of sweptback wing of infinite span
f. Singularity-distribution method
g. Elementary singularity solutions in supersonic flow
h. Thickness problem
i. Lifting problem
14. Transonic aerodynamics
a. Transonic flow past un-swept airfoils
b. Transonic Lift and drag coefficients
c. Supercritical airfoils
d. Effect of sweepback on wing aerodynamics
e. Whitcomb's transonic area rule
f. Forward swept wings
g. Transonic aircraft
Course outline (7)
15. Hypersonic aerodynamics
a. Real gas effects - hypersonic aerothermodynamics
b. Thin shock layers
c. Blunt-body problem
d. Hypersonic aerodynamics
e. Newtonian approximation and improvements
f. Mach number independence
g. Hypersonic viscous interactions
16. Atmospheric entry aerodynamics
a. Aerodynamic braking
b. Shock wave-boundary layer interaction
c. Aerodynamic design for atmospheric entry
d. Thermal protection systems
17. High-speed aircraft design considerations
Textbooks (required, hardcopy)
1. Anderson, J.D., Jr. (2021), Modern Compressible Flow with Historical
Perspective, 4th ed., McGraw-Hill, New York, N.Y.
2. Bertin, J.J. and Cummings, R.M. (2014), Aerodynamics for Engineers, 6th ed.,
Pearson Prentice Hall, Upper Saddle River, New Jersey.
3. NACA Report 1135 (1953), "Equations, Tables and Charts for Compressible
Flow.”
References:
1. Anderson, J.D., Jr. (2015?), Fundamentals of Aerodynamics, 3rd ed., McGrawHill, New York, N.Y.
2. Hodge, B.K. and Koenig, K. (1995), Compressible Fluid Dynamics, Prentice Hall,
Englewood Cliffs, N.J.
3. Shapiro, A.H. (1953), The Dynamics and Thermodynamics of Compressible
Fluid Flow, volumes I and II, Ronald Press, New York, N.Y.
4. Liepmann, H.W. and Roshko, A. (1957), Elements of Gas Dynamics, John
Wiley & Sons, New York, N.Y.
5. McCormick, B.W. (1995), Aerodynamics, Aeronautics, and Flight Mechanics,
2nd ed., John Wiley & Sons, Inc., New York
Textbooks and material: comments
• Anderson’s book much resembles Munson’s (detailed derivations and examples).
Focus on compressible fluid dynamics, not aerodynamics per se.
• Bertin & Cummings is more of a survey/reference. Covers applied aerodynamics
across the Mach number range. Resembles McCormick in that.
• McCormick Chapter 5 summarizes the second half of our course, especially for
wing planforms: wing drag, lift curve slope and CL_max, and pitching moment
• Bertin & Cummings also cover, maybe at “more advanced level”, ARO 3011
material… like Stokes stream function, Fourier expansion for non-elliptic loading,
numerical lifting surface (as opposed to lifting line) theory, Falkner-Skan BL,…
• Anderson really likes non-ideal gasses, so he often mentions that, but nearly all of
our work assumes an ideal gas, P = ρRT.
• As in ARO 3011, expect sometimes detailed derivations, but we arrive at
equations-in-boxes which are generally how we do HW and exams.
•
Important difference from ARO 3011: tabulated results supplant turning-the-crank in
algebraic equations (easy but tedious)
Course administration
Grading: Course grade will be based on a weighted average of the following:







Homework (~15)
Computer assignment
Aircraft Project (PPT + Report)
Class Journal
Quizzes (6)
Midterm Exams (2)
Final Exam
10%
10%
5%
5%
25% - will drop the lowest quiz
25%
20%
Quizzes will be closed-book, closed-notes, in essay form. Exams will be openbook, open-notes, emphasizing quantitative- and derivation-type problems, like
the homework problems.
Journals… free-form, in your own words, collected at mid-semester and final
HW: you are encouraged to work together, but your written answers must be
written by yourselves, in your own words
Course Schedule (1)
Course Schedule (2)
Course Schedule (3)
Our first assignment
• Reading:
• J. D. Anderson: chapters 1 and 2
• PS - 1
•
•
Chapter 1: Problems 3, 4 and 5
Derivation of Chapter 1’s equations (1.37)
• Due January 27, 2026 at 11:59 PM (Tuesday).
T
v
s2  s1  cv ln 2  R ln 2
T1
v1
Today’s topics
• What is gas dynamics (and compressible flow)?
•
•
•
•
•
Waves and information-flow in compressible media
Significance of compressibility
History of gas dynamics
Definition of speed of sound
Flow regimes based on Mach number
• A brief review of thermodynamics, Part 1
•
•
•
•
First law of thermodynamics
Definition of enthalpy
Ratio of specific heats
Perfect gas and calorically perfect gas
Why are we here?
• Gas Dynamics is the study of the dynamic properties of gases and their
thermal effects (temperature, pressure and density changes).
What is Compressible flow?
• The branch of fluid mechanics where fluid density changes significantly due to
the flowfield (gases, mostly, display such behavior.)
• Relevant to high-speed aircraft, jet engines, rocket motors, high-speed entry
into a planetary atmosphere, gas pipelines, commercial applications such as
abrasive blasting, and many other fields. Also astronomy, plasma physics,
lithotripsy, and even highway traffic flow (the “traffic equation”).
From fluid dynamics to gas dynamics
• Gases are highly compressible because their molecules are far apart and not
directly bonded to each other
• Compression <==> statistical mean spacing between molecules is smaller
• Based on concept of kinetic theory of gases… origin of thermodynamics and
justification for continuum assumption (at small Knudsen number)
Kn 

L
Mean free path
Relevant length scale
• Mathematically, “compressible” means
D
0
Dt
Waves
consequence of flows or gases being compressible is that perturbations (small
changes in local gas density or pressure) propagate at finite speed, as waves
What is a wave? a moving disturbance that transfers energy
through matter or space
Are ducks
supersonic?
Examples of waves
Water waves
light waves
Sound waves
microwaves
Examples of waves
Radio waves
Ultraviolet waves
X-ray waves
seismic waves
Transverse vs. longitudinal waves
(electromagnetic waves, light)
(e.g., Sound Waves)
Slinky Spring
Speed of Sound – a longitudinal wave
• The idea that there is a speed of sound, and that it can be
measured, is a major human achievement.
• Motivation: thunder storm
• We see a flash and followed by a loud noise
• the difference between the two, allows us to infer the local speed of sound
Speed of Sound induced by pressure perturbation
Continuity
1u1   2u2
(   d )(a  du )  a
ad  du  0
du  a
Momentum 1u1 u1  p1   2u2 u2  p2
(   d )(a  du)(a  du )  ( p  dp)  a 2  p
dp  2  adu  a 2 d  0
dp
 a2
Isentropic process
d
dp  a 2 d ad
du 

2 a

d

Estimation of speed sound
p
a c 
 S
2
2
differentiation is with holding entropy constant
Or approximately:
a2  c2 
p
 S
p
We can find
from the gas’s bulk modulus, B

c2 
dp B

d 
c
B


As we will see later, using the isentropic
relations, we find that
And using the ideal gas law:
B
dp
d
Elastic property
Inertial Property
p
p
 const 



 S

p
c 2   RT
Typical Sound Speeds (STP)
Gas
Air
Ar
C3H6
1117
1038
1009
mi/hr
762
708
688
C3H8
810
552
247
CH4
1447
987
441
CO
CO2
1136
869
775
593
346
265
H2
4236
2888
1291
H2O
1381
941
421
He
N2
3280
1136
2236
775
1000
346
O2
1061
723
323
238
299
204
91
4340
3080
3810
6102
4390
4240
4757
4888
4990
mi/hr
2959
2100
2598
4161
2993
2891
3244
3333
3402
UF6
ft/s
Liquid
Benzene
Carbon Tetrachloride
Ethanol
Glycerin
Kerosene
Machine Oil
Mercury
Water, fresh
Water, salt
ft/s
m/s
341
316
307
m/s
1323
939
1161
1860
1338
1292
1450
1490
1521
Solid
Aluminum
Beryllium
Brass
Brick
Concrete
Copper
Cork
Glass
Gold
Iron
Hickory
Ice
Lead
Platinum
Rubber
Steel
Wood
ft/s
16896
42290
11401
13701
10600
12799
1312
12999
10630
19521
13189
10499
3799
10696
328
19554
12999
mi/hr
11520
28834
7773
9341
7228
8726
895
8863
7248
13310
8992
7158
2590
7292
224
13332
8863
m/s
5150
12890
3475
4176
3231
3901
400
3962
3240
5950
4020
3200
1158
3260
100
5960
3962
Generally, sound travels
faster in solids than liquids
and faster in liquids than
gases.
Consequences of finite speed of sound
• A finite speed of sound implies wave propagation phenomena, which has
profound effects on flow fields and gas dynamics
• Leads us to the concept of a dimensionless Mach number
V
M   Mach number
a
V  speed of a subject
a  speed of sound of the ambient fluid / gas
For most practical problems, when
d

 5%, the flow is considered as compressible
We can use Mach number to identify flow compressibility effects
General speaking
Mach >0.3 → Compressible effect can not be ignored
Mach < 0.3 → Incompressible flow
Mach angle and Mach number
Subsonic vs. Supersonic motion
Mach wave and Mach angle
Mach cone and shock waves
Mach
wave
How Shock Waves Form
Think of “a” as ‘information speed’ and M = V/a as
ratio of flow speed to information speed
If M < 1, information available throughout
flow field
If M > 1, information confined to some region
of flow field
no upstream influence in supersonic flow
M<1
M=1
M>1
Flow regimes vs. Mach number (1)
Flow Regime
Density
Gradient
V
M
a
Shock Waves
M  0 .3
Incompressible Negligible
None
0.3  M  0.8
Subsonic
Small
None
0.8  M  1.2
Transonic
Significant
First appear
1.2  M  5  6 Supersonic
Significant
Significant
56 M
Dominant
Dominant
Hypersonic
Flow regimes vs. Mach number (2)
Subsonic
Can still be compressible, but there
are no shock waves anywhere
Transonic (1)
Anderson, Fig. 1.3
Supersonic
Outside of leading edge
(nose) and boundary
layer, all of the flow is
strictly supersonic
Hypersonic
No bow-shock, but supersonic pockets (where Cp is
lowest) terminated by a shock
Transonic (2)
Shock angle is close to body angle, so
that the flow as it were “impinges” on
the body… and BL occupies much or
nearly all of the layer beneath the shock
and the body. Also, chemical effects, γ
may vary, ….
Bow-shock followed by extensive subsonic region, acceleration to supersonic,
then termination right at the trailing edge with a upper/lower shock
Aircraft taxonomy with
respect to Mach Number
V
M
c
Supersonic: 1.2 < M < 5
https://www.historynet.com/tu-144-sst-soviet-supersonic-transport/
Transonic: ~0.7 < M < 1.2
Most commercial jets fly
between M = 0.75 and M = 0.9
https://flyvolato.c
om/fastestprivate-jets-inthe-world/
https://upload.wikimedia.org/wikipedia/common
s/c/c6/I-DPCN_at_work_03_%28cropped%29.jpg
Hypersonic: M > 5
Subsonic: M <0.35
https://airandspac
e.si.edu/collectionmedia/NASMA19690360000NASM2018-10037
History of gas dynamics and compressible flow
• Beginning of the 19th century, interest in artillery: projectiles and bullets
• 19th century: development of thermodynamics, mostly with application to steam
engines
• 1880s: Ernest Mach illustrated the existence of shocks around a supersonic bullet.
• “shadowgraph” optical technique: resolves changes in air density spatial gradients
•
•
•
•
•
•
1890s: de Laval and his converging-diverging nozzle
1890s: Parsons steam turbine
1920s: transonic flow encountered in propeller aerodynamics
1930s: Volta conference
1930s-1940s: supersonic wind tunnels go mainstream
1945: Project Paperclip - discovering German research
•
•
•
•
•
•
Late 1940s: wellspring of supersonic flow theory and experiments
1947: Chuck Yeager and first manned supersonic flight
1950s and 1960s: supersonic aircraft and ICBMs. High-subsonic jet airliners.
1961-1969: huge effort towards Apollo program (hypersonics)
1970s onward: supersonic CFD
Modern era: "quiet" supersonic tunnels
• See discussions throughout Anderson’s books!
Mach's Shadowgraph (Schlieren is similar)
Ernst Mach (1838–1916)
Ernst Mach’s historic 1887
photograph (shadowgraph) of a
bow shockwave around a
supersonic bullet
De Laval's nozzle
In 1890 Laval developed a nozzle to increase the steam jet to supersonic
speed, working off of the kinetic energy of the steam, rather than its pressure.
The nozzle, now known as a de Laval nozzle, is used in modern rocket engine
nozzles. De Laval turbines can run at up to 30,000 rpm.
Carl Gustaf de Laval 1875
1893 Chicago World Fair Exhibit
Sonic “Barrier”
Chuck Yeager sonic flight with Bell X-1
manned flight, 14 October 1947: Captain Charles "Chuck" Yeager
Aircraft was dropped from a B-29 Superfortress bomber, then accelerated to
Mach 1.06 (700 miles per hour (1,100 km/h; 610 kn))
Thermodynamics – intro/review
Thermodynamics is the study of relation between heat, work and internal
energy of a system, and its interaction with its environment.
Example systems
• Gas in a container
• Charging & discharging a battery
• Chemical reactions
• Refrigerator
• Supersonic wind tunnel
System
Environment
Universe
•
•
•
•
Extensive property: depends on system size, usually mass
Intensive property: independent of size (mass normalized-out)
State Postulate: a “state” is fixed if we have two independent, intensive properties
• Example: p  p (T ,  )  p (T , s )
Equilibrium: where a system has a “desire” to go, and where it finally gets
• Thermal equilibrium: uniformity of temperatures
Our objective in this course: use thermodynamics to describe how
mechanical energy and internal energy (temperature) in a gas interchange
Thermodynamic Intensive properties
p, Static Pressure ( lbf/ft2 or N/m2 )
T, Static Temperature (oR or K )
, Static Density ( slug/ft3 or kg/m3 )
c, Acoustic speed ( ft/sec or m/sec )
 p 
c   
   s
2
e, Specific internal energy ( lbf-ft/slug or J/kg )

h
e
p   e   1
 e 
cv   
 T  v
 h 
cp   
 T  p
h, Specific enthalpy ( lbf-ft/slug or J/kg )
cp & cv, Specific heats ( lbf-ft/slug-oR or J/kg-K )
, Ratio of specific heats

cp
cv
h  e
p

First Law of Thermodynamics
Energy can be neither created nor destroyed
•Heat and work are equivalent (James Joule’s experiment)
• Energy is conserved in any transformation
• The change of energy of a system is independent of the path taken
e = internal energy
q = heat
w = work
P = pressure
V = volume
Conceptualizing the first law of thermodynamics
Heat input or output
System
(reference
mass)
Internal
energy, e
Surroundings
Work input or output
boundary
Internal energy e, can only change if:
1. Heat is added or removed from the system (δq, heat per unit mass)
2. Work is done to (or by) the system (δw, work done per unit mass)
de  q  w
What is work?
Work at the surface =
[force] [length] =
[pressure] [area] [displacement]
P
dA
dw  p dA x
x
W   px dA  p  x dA   pV
A
A
w   pdv
• We’re assuming a reversible process (no work irrecoverably lost)
• dv is incremental change in volume due to displacement of system boundary,
per unit mass… so it’s the reciprocal of differential of density
de  q  pdv
Enthalpy and specific heats
de  q  pdv
dh  de  d ( pv)  de  pdv  vdp  q  vdp
h  e  pv
Constant volume process
Constant pressure process
Fixed boundary
δq
Specific heat
δq
T, p, v
c
q
dT
Resulting value of dT, when
adding δq, depends on the
type of process
cv 
q
dT v
q
cp 
dT p
T, p, v
Piston moves
up and down,
to keep
pressure
constant
At constant volume
At constant pressure
Thermally perfect gas: constant specific heats
de  q  pdv
Enthalpy and specific heats
de  q  cv dT
For a constant volume process, dv = 0
e  cvT
0
For a constant pressure process, dp = 0
dh  q  vdp  c p dT
h  c pT
(h = 0 at T = 0)
For air: (T < 1000 K)
cp
cv  720 J / kg K

c p  1008 J / kg K
 1.4
cv
Integrations hold for a calorically perfect gas,
where specific heats are constant (compare to
thermally perfect gas, where specific heats are
a function of T, but not of v or p)
c pT  cvT  h  e  pv  RT  c p  cv  R
1
R
 1
R
1  
 cp
cp 
Adiabatic process
q 0
Reversible process: no friction or
dissipation (such as viscosity or diffusion)
R
 1  
cv
R
cv 
 1
Isentropic process: both
adiabatic and reversible
(idealization)
Ideal Gas Law (Equation of State) (1)
P  pressure [Pa]
V  volume [m3]
m  mass [kg]
R  gas constant [J/kg-K]
T  temperature [K]
ρ  density [kg/m3]
v  specific volume [m3/kg]
n  number of moles [mol]
Ru  universal gas constant [J/kmol-K]
MWgas  molecular weight [kg/kmol]
k = Boltzmann constant = 1.380649×10−23 J/K
Nava = Avogadro’s number = 6.02214076 × 1023 /mole
Ru = 8.314 kJ/(kmol-K) or kPa-m3/(kmol-K)
= 0.0831447 bar-m3/(kmol-K)
= 1.98588 Btu/(lbmol-R)
= 1545.37 (ft-lbf)/(lbmol-R)
PV  mRT
p  nkT
P/  Pv  RT
P  ρRT
Ru
R
M W gas
R   kJ / kg  K
 kPa  m / kg  K
3
Ideal Gas Law (Equation of State) (2)
• Relates P, T, and specific volume (v) of a substance.
• The best-known EoS for the gas phase is the ideal-gas EoS for P-v-T behavior.
•
•
•
Different gases have different gas constants.
Air molecular weight = 28.9 kg/k-mol
So air’s R =8.3144 /28.9= 0.287 kJ/(kg-K)
The one to
remember!
Summary
• We discussed the physical meaning of bulk modulus, compressibility, finite
speed of sound, information-flow and Mach angle

2
 p q 0.5 V 2
V 
~
~ 

0
.
5
  
2

B
B
c 
a
Incompressible if
0.5M 2  1
• First law of thermodynamics… work and heat and things like that.
• Refer to Anderson's books (Compressible Flow, Intro to Flight) for history
• In this course, we will almost universally assume the ideal gas law
• More thermo to come… 2nd law, and consequences of being isentropic