1. CH.12: COMPRESSIBLE FLOW It required an unhesitating boldness to undertake such a venture …. an almost exuberant enthusiasm…but most of all a completely unprejudiced imagination in departing so drastically from the known way. J. Van Lonkhuyzen, 1951, discussing designing Bell XS-1

2. Compressible flow is a fun subject. John Anderson

3. Ch.12 - WHAT CAUSES FLUID PROPERTIES TO CHANGE IN A 1-D COMPRESSIBLE FLOW FLOW? (note if isentropic stagnation properties do not change)

4. Ch.12 - COMPRESSIBLE FLOW Flow can be affected by: area change, shock, friction, heat transfer shock Q friction Q Area2 shock Area1

5. One-Dimensional Compressible Flow Rx P1 P2 dQ/dt Surface force from friction and pressure heat/cool (+ s1, h1,V1,…) (+ s2, h2, V2, …) What can affect fluid properties? Changing area, normal shock, heating, cooling, friction. - need 7 equations -

6. Affect of Area Change Assumptions = ?

7. ASSUMPTIONS • ALWAYS ~ • Steady Flow • Ideal Gas • Ignore Body Forces • only pressure work • (no shaft, shear or other work) • Constant specific heats • “One – Dimensional” • MOSTLY ~ • ISENTROPIC

8. “quasi-one-dimensional” V(s) V(x,y)

9. “quasi-one-dimensional” Flow properties are uniform across any given cross section of area A(x), and that they represent values that are some kind of mean of the actual flow properties distributed over the cross section. NOTE – equations that we start with are exact representation of conservation laws that are applied to an approximate physical model

10. One-Dimensional Compressible Flow f(x1) f(x2) + s1, h1 +s, h2 7 Variables = T(x), p(x), (x), A(x), v(x), s(x), h(x) 7 Equations = mass, x-momentum, 1st and 2nd Laws of Thermodynamics, Equations of State (3 relationships)

11. One-Dimensional Compressible Flow Variables = T(x), p(x), (x), A(x), v(x), s(x), h(x) Equations = mass, momentum, 1st and 2nd Laws of Thermodynamics, Equation of State (3 relationships) Cons. of mass (steady / 1-D) Cons. of momentum (& no FB) Cons. of energy (& only pressure work) 2nd Law of Thermodynamics . Ideal gas Ideal gas & constant cv, cp Ideal gas & constant cv, cp (steady) Eqs. of State

12. Want to find qualitative relationships for: dT, dV, dA, d ISENTROPIC

13. Affect of Change in Velocity on Temperature Isentropic, steady, cp&cv constant, no body forces, quasi-one-dimensional, ideal gas, only pressure work

14. For isentropic flow if the fluid accelerates what happens to the temperature?

15. For isentropic flow if the fluid accelerates what happens to the temperature? if V2>V1 then h2<h1

16. For isentropic flow if the fluid accelerates what happens to the temperature? if V2>V1 then h2<h1 if h2<h1 then T2<T1 Temperature Decreases !!!

17. Affect of Change in Velocity on Temperature Velocity Increases, then Temperature Decreases Velocity Decreases, then Temperature Increases NOT DEPENDENT ON MACH NUMBER Isentropic, steady, cp&cv constant, no body forces, quasi-one-dimensional, ideal gas, only pressure work

18. Affect of Change in Velocity on Pressure Isentropic, steady, no body forces, quasi-one-dimensional

19. Affect of Change in Velocity on Pressure

21. FSx = (p + dp/2)dA + pA – (p+dp)(A + dA) = pdA + dAdp/2 + pA – pA – pdA– dpA - dpdA (dm/dt)(Vx+dVx) - (dm/dt)Vx = = (Vx + d Vx) {VxA}- Vx{VxA} = Vx VxA + dVx VxA – VxVxA -dpA = VxdVxA ordp/+d{Vx2/2}=0

22. Affect of Change in Velocity on Pressure dp/+d{Vx2/2}=0 Velocity Increases, then Pressure Decreases Velocity Decreases, then Pressure Increases NOT DEPENDENT ON MACH NUMBER Isentropic, steady, no body forces, quasi-one-dimensional

23. Affect of Change in Area on Pressure and Velocity Isentropic, steady, no body forces, quasi-one-dimensional

24. EQ. 11.19b EQ.12.1a {d(AV) + dA(V) +dV(A)}/{AV} = 0 steady isentropic, steady

25. isentropic, steady

26. EQ. 12.5 EQ.12.6 isentropic, steady

27. Affect of Change in Area on Velocity dV/V = - (dA/A)/[1-M2] M <1 Velocity and Area change oppositely M > 1 Velocity and Area change the same Isentropic, steady, no body forces, quasi-one-dimensional

28. Affect of Change in Area on Pressure dp/(V2) = (dA/A)/[1-M2] M <1 Pressure and Area change the same M > 1 Pressure and Area change oppositely Isentropic, steady, no body forces, quasi-one-dimensional

29. isentropic, steady, ~1-D M<1 dVx > 0 or dVx < 0 dp > 0 or dp < 0

30. isentropic, steady, ~1-D M<1 dVx > 0 or dVx < 0 dp > 0 or dp < 0

31. isentropic, steady, ~1-D M >1 dp > 0 or dp < 0 dVx > 0 or dVx < 0

32. isentropic, steady, ~1-D M >1 dp > 0 or dp < 0 dVx > 0 or dVx < 0

33. isentropic, steady,~1-D M<1 dp > 0 or dp < 0 dVx > 0 or dVx < 0

34. isentropic, steady,~1-D M<1 dp > 0 or dp < 0 dVx > 0 or dVx < 0

35. isentropic, steady,~1-D M>1 dp > 0 or dp < 0 dVx > 0 or dVx < 0

36. isentropic, steady,~1-D M>1 dp > 0 or dp < 0 dVx > 0 or dVx < 0

37. isentropic, steady, ~1-D Subsonic Nozzle Subsonic Diffuser (dp and dV are opposite sign) M<1 If M < 1 then [ 1 – M2] is +, then dA and dP are same sign; and dA and dV are opposite sign qualitatively like incompressible flow

38. isentropic, steady, ~1-D Supersonic Nozzle Supersonic Diffuser (dp and dV are opposite sign) M>1 If M > 1 then [ 1 – M2] is -, then dA and dP are opposite sign; and dA and dV are the same sign qualitatively not like incompressible flow

39. And this is the reason!!!!!!! If  is constant then dA and dV must be opposite signs, but for compressible flows all bets are off, e.g. for M>1 both dV and dA can have the same sign

40. Affect of Change in Area on Density Isentropic, steady, no body forces, quasi-one-dimensional

41. Affect of Change in Velocity on Pressure

42. Affect of Change in Velocity on Pressure -(dA/A)/(1-M2) = -dA/A - d/ d/ = (dA/A)[1/(1-M2) - 1] d/ = (dA/A)[M2/(1-M2)]

43. d =? d/ = (dA/A)[M2/(1-M2)]

44. d/ = (dA/A)[M2/(1-M2)] d/ > 0 d/ < 0 d/ > 0 d/ < 0

45. WHAT HAPPENS AT M = 1 ? ? dp/(V2) = (dA/A)/(1-M2); dV/V = -(dA/A)/(1-M2); d/ = (dA/A)[M2/(1-M2)]

46. If M = 1 have a problem, Eqs. blow up! Only if dA  0 as M  1 can avoid singularity. Hence for isentropic flows sonic conditions can only occur where the area is constant. dp/(V2) = (dA/A)/(1-M2); dV/V = -(dA/A)/(1-M2); d/ = (dA/A)[M2/(1-M2)]

47. What happens to dp and d across the throat in a supersonic nozzle with steady, isentropic flow?

48. What happens to dp and d across the throat in a supersonic nozzle with steady, isentropic flow? dp/ = - d{Vx2/2} Vx continues to increase so p continues to decrease d/ = (dA/A)[M2/(1-M2)] (dA/A)[M2/(1-M2)] is always negative so  continues to decrease

49. What happens to dp and d across the throat in a supersonic diffuser steady, isentropic flow?

50. What happens to dp and d across the throat in a supersonic diffuser with steady, isentropic flow? dp/ = - d{Vx2/2} Vx continues to decrease so p continues to increase d/ = (dA/A)[M2/(1-M2)] (dA/A)[M2/(1-M2)] is always positive so  continues to increase