1. One-dimensional Flow 3.1 Introduction Normal shock
In real vehicle geometry, The flow will be axisymmetric
One dimensional flow

2. The variation of area A=A(x) is gradual
Neglect the Y and Z flow variation

3. 3.2 Steady One-dimensional flow equation
Assume that the dissipation occurs at the shock and the flow up stream and downstream of the shock are uniform
Translational rotational and vibrational equilibrium

4. The continuity equation
L.H.S of C.V
(Continuity eqn for steady 1-D flow)
The momentum equation

5. Remember the physics of momentum eq is the time
rate of change of momentum of a body equals to the
net force acting on it.

6. The energy equation
Physical principle of the energy equation is the energy is the energy is conserved
Energy added to the C.V
Energy taken away from the system to the surrounding

7. 3.3 Speed of sound and Mach number
Mach angle μ
Wave front called “ Mach Wave”
Always stays inside the family of circular sound waves
Always stays outside the family of circular sound waves

8. Continuity equation A sound wave, by definition, ie: weak wave1 2
A sound wave, by definition,
ie: weak wave
( Implies that the irreversible,
dissipative conduction are negligible)
Wave front
Continuity equation

9. Momentum equation No heat addition + reversible
General equation valid for all gas
Isentropic compressibility

10. For a calorically prefect gas, the isentropic relation becomes
For prefect gas, not valid for chemically resting gases or real gases
Ideal gas equation of state

11. The physical meaning of M
Form kinetic theory
a for air at standard sea level = m/s = 1117 ft/s
Mach Number
The physical meaning of M
Subsonic flow
Kinetic energy
Sonic flow
Internal energy
supersonic flow

12. 3.4 Some conveniently defined parameters
Inagine: Take this fluid element and Adiabatically slow it own (if M>1) or speed it up (if M<1) until its Mach number at A is 1.
For a given M and T at the some point A
associated with
Its values of and at the same point

13. total temperature or stagnation temperature
In the same sprint, image to slow down the fluid elements isentropically to zero velocity ,
total temperature or stagnation temperature
total pressure or stagnation pressure
Stagnation speed of sound
Total density
Note: are sensitive to the reference coordinate system
are not sensitive to the reference coordinate
(Static temperature and pressure)

14. 3.5 Alternative Forms of the 1-D energy equation
= 0(adiabatic Flow)
calorically
prefect B
If the actual flow field is nonadiabatic form A to B → A
Many practical aerodynamic flows are reasonably adiabatic

15. Total conditions - isentropic
Adiabatic flow
isentropic
Note the flowfiled is not necessary to be isentropic
If not →
If isentropic → are constant values

17. If M → ∞ or
= 1 if M=1
< 1 if M < 1
> 1 if M > 1

18. EX. 32

19. 3.6 Normal shock relations
( A discontinuity across which the flow properties suddenly change)
The shock is a very thin region ,
Shock thickness ~ 0 (a few molecular mean free paths)
~ cm for standard condition) 1
Known
adiabatic 2
To be solved
Ideal gas E.O.S
Calorically perfect
Continuity
Momentum
Energy
Variable :
5 equations

20. Prandtl relation
Note:
1.Mach number behind the normal shock is always subsonic
2.This is a general result , not just limited to a calorically perfect
gas

21. Infinitely weak normal shock . ie: sound wave or a Mach wave
Special case 1. 2.

23. Note : for a calorically perfect gas , with γ=constant
are functions of only
Real gas effects

25. Mathematically eqns of hold for
Physically , only is possible
The 2nd law of thermodynamics
Why dose entropy increase across a shock wave ?
Large ( small)
Dissapation can not be neglected
entropy

26. To ≠ const for a moving shock Note: 1 2. Ex.3.4 Ex.3.5 Ex 3.6 Ex 3.7
To is constant across a stationary normal shock wave
To ≠ const for a moving
shock
Note:
The total pressure decreases across a shock wave
Ex.3.4 Ex.3.5 Ex 3.6 Ex 3.7

27. 3.7 Hugoniot Equation

28. Hugoniot equation
It relates only thermodynamic quantities across the shock
General relation holds for a perfect gas , chemically reacting gas, real gas
Acoustic limit is isentropic flow
1st law of thermodynamic with

29. For a calorically prefect gas
In equilibrium thermodynamics , any state variable can be expressed as a function of any other two state variable
Hugoniot curve the locue of all possible p-v condition behind normal shocks of various strength for a given

30. For a specific Straight line Note Rayleigh line ∵supersonic ∴
Isentropic line down below of Rayleigh line
In acoustic limit (Δs=0) u1→a insentrop & Hugoniot have the same slope

31. For fluids Shock Hugoniot
as function (weak) shock strength for general flow
Shock Hugoniot
For fluids

32. Coefficient
For gibbs relation

33. Let For every fluid “Normal fluid “ “Compression” shock if if
“Expansion “shock p p
s=const
s=const u u

34. 3.8 1-D Flow with heat additionq A
e.q 1. friction and thermal conduction
2. combustion (Fuel + air) turbojet
ramjet engine burners.
3. laser-heated wind tunnel
4. gasdynamic and chemical
leaser
+E.O.S
Assume calorically perfect gas

35. The effect of heat addition is to directly change the total temperature of the flow
Heat addition To
Heat extraction To

38. Given: all condition in 1 and q
To facilitate the tabulation of these expression , let state 1 be a reference state at which Mach number 1 occurs.

39. Table A.3.
For γ=1.4

40. Adding heat to a
supersonic flow
M ↓

41. To gain a better concept of the effect of heat addition on M→TS diagram

43. At point A Momentum eq. Continuity eq. Rayleigl line B A 1.0

44. At point B is maximum ds=(dq/T)rev →addition of heat ds>0 lower
B(M<1)
M<1
Heating
cooling
heating
M>1
jump
lower
At point B
is maximum
A (M=1)
ds=(dq/T)rev
→addition of heat
ds>0
MB subsonic

45. M P T T0 P0 u q 1 2 (M2>M1) (P2>P1) (T2>T1) (T02>T01)
Supersonic flow
M1>1
subsonic flow
M1<1 M
(M2<M1)
(M2>M1) P
(P2>P1) T
(T2>T1) T0
(T02>T01) P0
(P02<P01) u
(U2<U1)

46. For supersonic flow Heat addition → move close to A M → 1
→ for a certain value of q , M=1 the flow is said to be “ choked ”
∵ Any further increase in q is not possible without a drastic revision of
the upstream conditions in region 1

47. For subsonic flow heat addition → more closer to A , M →1
→ for a certain value of the flow is choked
→ If q > , then a series of pressure waves will propagate
upstream , and nature will adjust the condition is region 1
to a lower subsonic M
→ decrease
E.X 3.8

48. 3.9 1-D Flow with friction Fanno line Flow In reality , all fluids
are viscous.
- Analgous to 1-D flow with heat addition.

49. Momentum equation
Good reference for f : schlicting , boundary layer theory

50. ∵ adiabatic , To = const

51. Analogous to 1-D flow with heat addition using sonic reference condition.

52. IF we define are the station where , M = 1
F: average friction coefficient
Table A.4

53. Fanno line At point P T high u low above P , M < 1
ds < 0 P
chocked
ds > 0
At point P
T high u low above P , M < 1
T low u high below P , M > 1

54. 1-D adiabatic flow with friction
Supersonic flow
M1>1
Subsonic flow
M1<1 M
(M2<M1)
(M2>M1) P
(P2>P1) T
(T2>T1) T0
unchanged P0
(P02<P01) u
(u2<u1) ρ