27. Compressible Flow CH EN 374: Fluid Mechanics

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

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)

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

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

Mach Number

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?

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

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

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

The energy balance and thermodynamics

Compressible Energy Balances Δ𝑃 𝜌 + 1 2 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)?

𝒗 𝟏 𝑨 𝟏 = 𝒗 𝟐 𝑨 𝟐 No More

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

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

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

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

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 =

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

So… Why do rocket thrusters look like this?

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!

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

Shock waves

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!

Pipe Flow

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. 341-343 Final equations p. 343

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