1. 27. Compressible Flow
CH EN 374: Fluid Mechanics

2. Compressibility
The equations we have used in this class have assumed incompressibility
Without that assumption, math becomes more complicated
I am more interested in y’all learning concepts
Will present equations, but mostly to help illustrate

3. Review: When is flow compressible?
Technically always
But usually just a little
We consider flow compressible for:
Gasses
When pressure change is very large
This happens
At high speeds (e.g. around airplanes)
In very long gas pipelines (think natural gas)

4. 𝑃 𝜌 + 1 2 𝑣 2 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Δ 𝑣 2 =𝐵𝐼𝐺 Δ𝑃=𝐵𝐼𝐺
Remember: density changes when pressure changes
𝑃 𝜌 𝑣 2 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
(treating as incompressible for guesstimate)
𝒗 𝟐 =𝒗𝒆𝒓𝒚 𝒇𝒂𝒔𝒕
Δ 𝑣 2 =𝐵𝐼𝐺
Δ𝑃=𝐵𝐼𝐺
𝒗 𝟏 =𝟎

5. Δ 𝑃 𝑙𝑜𝑠𝑠 =2𝜌𝑓 𝐿 𝐷 𝑣 2 𝐿=𝐵𝐼𝐺 Δ𝑃=𝐵𝐼𝐺
Remember: density changes when pressure changes
Δ 𝑃 𝑙𝑜𝑠𝑠 =2𝜌𝑓 𝐿 𝐷 𝑣 2
(treating as incompressible for guesstimate) 𝑳
𝐿=𝐵𝐼𝐺
Δ𝑃=𝐵𝐼𝐺

6. Mach Number

7. Mach Number 𝑀𝑎= 𝑣 𝑐 𝑀𝑎>0.3→𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒
Ratio of speed of air flow to the speed of sound:
𝑀𝑎= 𝑣 𝑐
𝑀𝑎>0.3→𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒
Example of dimensional analysis!
Why use speed of sound to nondimensionalize?

8. Speed of Sound ! Sound is waves of fluctuating pressure and density
𝑐= 𝜕𝑃 𝜕𝜌 𝑆
Depends on atoms, interactions

9. Speed of Sound 𝑐= 𝜕𝑃 𝜕𝜌 𝑆 Δ𝑃→𝑏𝑖𝑔 Δ𝜌 Δ𝑃→𝑠𝑚𝑎𝑙𝑙 Δ𝜌 Is c big or small?
𝑐= 𝜕𝑃 𝜕𝜌 𝑆
Δ𝑃→𝑏𝑖𝑔 Δ𝜌
Is c big or small?
Will flow be compressible at higher or lower speeds? (𝑀𝑎= 𝑣 𝑐 )
Δ𝑃→𝑠𝑚𝑎𝑙𝑙 Δ𝜌

10. Speed of Sound 𝑐= 𝛾𝑅𝑇 𝑀𝑊 For an ideal gas: 𝛾:𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 ℎ𝑒𝑎𝑡 𝑟𝑎𝑡𝑖𝑜
𝑐= 𝛾𝑅𝑇 𝑀𝑊
𝛾:𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 ℎ𝑒𝑎𝑡 𝑟𝑎𝑡𝑖𝑜
For air at 27°𝐶, 𝛾=1.40

11. The energy balance and thermodynamics

12. Compressible Energy Balances
Δ𝑃 𝜌 𝛼Δ 𝑣 2 +𝑔Δ𝑧= 𝑤 𝑆 − 𝑤 𝐿 + 𝑤 𝐶
Compressibility work—energy that goes into changing density
The energy of compression/expansion, and the relationship to pressure, depends on the thermodynamics of the flow
Fluid temperature might also change!
Will depend on assumptions:
Constant entropy? Constant temperature? Adiabatic (heat flow)?

13. 𝒗 𝟏 𝑨 𝟏 = 𝒗 𝟐 𝑨 𝟐 No More

14. A water rocket has a thruster like this:
A fueled rocket has a thruster like this:
Why?

15. Conservation of Mass 𝑚 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝜌 𝑉 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝜌𝑣𝐴=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑑 𝜌𝑣𝐴 =0
𝑣𝐴𝑑𝜌+𝜌𝐴𝑑𝑣+𝜌𝑣𝑑𝐴=0
d𝜌 𝜌 + 𝑑𝑣 𝑣 + 𝑑𝐴 𝐴 =0

16. Conservation of Mass
d𝜌 𝜌 + 𝑑𝑣 𝑣 + 𝑑𝐴 𝐴 =0
𝑨 𝟐
𝑨 𝟏
𝑨 𝟏
𝑨 𝟑

17. Flow through short nozzles
Assume: constant entropy, adiabatic, no friction loss…
Mass Balance
d𝜌 𝜌 + 𝑑𝑣 𝑣 + 𝑑𝐴 𝐴 =0
𝑑𝑃 𝜌 +𝑣𝑑𝑣=0
Energy Balance
Speed of Sound
𝑐= 𝜕𝑃 𝜕𝜌

18. 𝑑𝐴 𝑑𝑣 =− 𝐴 𝑣 1−𝑀 𝑎 2 𝑑𝐴 𝑑𝑣 = Subsonic flow (Ma<1):
𝑑𝐴 𝑑𝑣 =− 𝐴 𝑣 1−𝑀 𝑎 2
𝑑𝐴 𝑑𝑣 =
Subsonic flow (Ma<1):
Sonic flow (Ma=1):
Supersonic flow (Ma>1):
𝑑𝐴 𝑑𝑣 =
𝑑𝐴 𝑑𝑣 =

19. Nozzles and Diffusers Nozzles speed flow, diffusers slow flow
Supersonic Nozzle
Subsonic Nozzle
Subsonic Diffuser
Supersonic Difuser

20. So…
Why do rocket thrusters look like this?

21. Choked Flow “Choked” flow – velocity can’t increase any more
𝑀𝑎=1
𝑀𝑎=1
𝑀𝑎=0.9
𝑀𝑎=0.8
“Choked” flow – velocity can’t increase any more
If nozzle is long enough, exit velocity must be Mach 1!

22. Converging-Diverging Nozzle
𝑀𝑎=1
𝑀𝑎<1
𝑀𝑎>1

23. Shock waves

24. Shock Waves Sonic boom: https://www.youtube.com/watch?v=fW2Y8DLwvxs
Supersonic speeds → sound waves “bunch up” → large pressure differentials
Subsonic
Supersonic
Some people think you hear at sonic boom at the moment when the plane reaches the speed of sound. Why is that incorrect?
Shocks can also happen in pipes with choked flow!

25. Pipe Flow

26. Pipe Flow Assume: Math is complicated!
𝑃 1 , 𝜌 1 , 𝑣 1 , 𝑇 1
𝑃(𝑥),𝜌(𝑥),𝑣(𝑥),𝑇(𝑥)
Assume:
Adiabatic (flow too fast for heat transfer)
Significant friction loss
Temperature and entropy non-constant
Math is complicated!
Derivation in your book, p
Final equations p. 343

27. Pipe Flow
𝑃 1 , 𝜌 1 , 𝑣 1 , 𝑇 1
𝑃(𝑥),𝜌(𝑥),𝑣(𝑥),𝑇(𝑥)
Combining mass balance, energy balance, momentum balance we learn:
Subsonic flow will speed up to 𝑀𝑎=1
Supersonic flow will slow down to 𝑀𝑎=1
𝑀𝑎=1 flow is choked