One-dimensional Flow

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. 1 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. Form kinetic theory a for air at standard sea level = 340.9 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 Adiabaticallyslow 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. Note: are sensitive to the reference coordinate system are not sensitive to the reference coordinate 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 (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. = 1 if M=1 < 1 if M < 1 > 1 if M > 1 If M → ∞ or

18. EX. 32

19. 1 Known 2 To be solved adiabatic 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) 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 is constant across a stationary normal shock wave To ≠ const for a moving shock Note: 1 2. 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 Rayleigh line Note ∵supersonic ∴ Isentropic line down below of Rayleigh line In acoustic limit (Δs=0) u1→a insentrop & Hugoniot have the same slope

31. 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. q A 3.8 1-D Flow with heat addition 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 B A 1.0 Momentum eq. Continuity eq. Rayleigl line ∴ At point A , M=1

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

45. q 1 2

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