27 COMPRESSIBLE FLOW CH EN 374 Fluid Mechanics

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)

• Remember: density changes when pressure changes (treating as incompressible for guesstimate)

MACH NUMBER

Mach Number • Ratio of speed of air flow to the speed of sound: • Example of dimensional analysis! • Why use speed of sound to nondimensionalize?

Speed of Sound • Sound is waves of fluctuating pressure and density • Speed of sound: • Depends on atoms, interactions

Speed of Sound

Speed of Sound • For an ideal gas:

THE ENERGY BALANCE AND THERMODYNAMICS

Compressible Energy Balances Compressibility work— energy that goes into changing density • The energy of compression/expansion, and the relationship to pressure, depends on thermodynamics of the flow • Fluid temperature might also change! • Will depend on assumptions: • Constant entropy? Constant temperature? Adiabatic (heat flow)?

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

Conservation of Mass

Flow through short nozzles • Assume: constant entropy, adiabatic, no friction loss… Mass Balance Energy Balance Speed of Sound

• Subsonic flow (Ma<1): • Sonic flow (Ma=1): • Supersonic flow (Ma>1):

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

So… • Why do rocket thrusters look like this?

Choked Flow • “Choked” flow – velocity can’t increase any more • If nozzle is long enough, exit velocity must be Mach 1!

Converging-Diverging Nozzle

SHOCK WAVES

Shock Waves • 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: • 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 •