Compressible Flow Intro_L8_1 BACKGROUND INFO What are we dealing with: (gas – air) high speed flow – Mach Number > 0.3 Perfect gas law applied – expression for pressure field Need some thermodynamic concepts, eg, enthalpy, entropy, etc Pressure always expressed as absolute pressure likewise Temperature will be in absolute unit, eg Kelvin Flow Classification: - Subsonic: Ma < 1 - Supersonic: Ma > 1 - Hypersonic: Ma > 5 Density becomes a flow variable and Temperature is a variable too Application: turbomachinery – turbine, compressor, airfoil, missile Mach Number: Ma = fluid velocity / sonic velocity Sonic velocity , c,speed of sound in fluid medium Compressible Flow Intro_L8_2 What do we need from THERMODYNAMICS ? For a reversible process, Perfect Gas Law pV = mRT Mass of gas MR = 8314.4 J kg -1 K -1 Mol wt of gas Another relation: p = ρRT Few properties : P, T, u (internal energy), h (enthalpy), s (entropy) Few processes : adiabatic, isoentropic, reversible Adiabatic process: system insulated from surrounding – no heat exchange Isoentropic process: constant entropy, no change in entropy Reversible process: ideal process (most efficient); return to original state How does all these correlate? ∫ =∆ T dQ s For an irreversible process, ∫ >∆ T dQ s Adiabatic ∆s = 0 ∆s > 0 Note: the terms, u, h & s comes from 1 st & 2 nd law of thermodynamics Compressible Flow Intro_L8_3 Few more terms & relations (thermo) c p → specific heat at constant pressure c v → specific heat at constant volume p p T h c       ∂ ∂ = v v T u c       ∂ ∂ = ratio of sp heats, k = c p / c v k air ≈ 1.4 c p = c v + R h = u + p / ρ Speed of Sound (sonic velocity) sound is a measure of pressure disturbance defined as propagation of infinitesimal pressure disturbance In fluid medium process is assumed isoentropic (reversible) s p c         ρ∂ ∂ = In gas, sonic velocity, Ideal gas, kRT p k p s = ρ =         ρ∂ ∂ 0 Static pressure (or stream pressure), p Stagnation pressure (p 0 )- pressure when gas brought to rest isoentropically Compressible Flow Intro_L8_4 Note: more on stagnation pressure few slides later Sonic velocity in liquid or solid medium: ρ= /Kc where K is bulk modulus Air at moderate pressure is assumed to behave as ideal gas Speed of sound in air at a pressure 101lPa becomes s/m./*./kPc 3432110100041 ==ρ= at sea level, c = 340 m/s at 11km altitude, c = 295 m/s In water sonic speed, s/m/E./Kc 14631009142 3 ==ρ= Mach Number, Ma = U / c Inertial force compressible force U → local fluid speed Note: local sonic speed, u is defined as propagation of a infinitesimal pressure disturbance when fluid is at rest Compressible Flow Intro_L8_5 Shock Waves Finite pressure disturbances can cause sound propagation greater than local sonic speed Examples: bursting of a paper bag or a tire; disturbances caused by high velocity bullets, jet aircraft & rockets (a) Stationary fluid U = 0 c∆t c(2∆t) c(3∆t) WAVE PROPAGATION (pressure disturbance) S U < c c∆t c(2∆t) c(3∆t) U∆t U(2∆t) U(3∆t) S (b) Moving fluid Pressure disturbance occuring at an interval of every ∆t S is the disturbance source Doppler shift Ma < 1 subsonic radial propagation only radial + axial propagation Compressible Flow Intro_L8_6 Shock Wave propagation: Case (c): U = c Ma = 1 sonic 1 2 3 1 2 3 S all wavefronts touch source S Case (d): U > c 1 2 3 S c(2∆t) c(3∆t) U∆t U(2∆t) U(3∆t) α Inside cone aware of sound The Mach Cone Ma > 1 Mach (Cone) angle: α = Sin -1 (1/Ma) O u t s i d e c o n e u n a w a r e o f s o u n d Compressible Flow Intro_L8_7 Isentropic Flow (1-D): Local Isentropic Stagnation properties Integration of differential equations result in an integration constant To evaluate this constant, a reference location is required This reference location is zero velocity where Ma = 0 Stagnation point is thus when fluid is brough to stagnant state (eg, reservoir) Stagnation properties can be obtained at any point in a flow field if the fluid at that point were decelerated from local conditions to zero velocity following an isentropic (frictionless, adiabatic) process Notation: pressure : p 0 Temperature : T 0 Density : ρ 0 Fluid chosen in most cases will be air or superheated steam which can be treated as perfect gas Compressible Flow Intro_L8_8 Conservation Equations for 1-D Isentropic Flow Conservation of Mass(Continuity) : d(ρV x A) = 0 or ρV x A = m = constant Conservation of Momentum : 0 2 2 =         + ρ x V d dp Conservation of Energy : (1 st Law of Thermo) ttanconsh V hor V hd xx ==+=         + 0 22 2 0 2 2 nd Law of Thermo: s = constant Equations of State: h = h(s,p) ρ = ρ(s,p) Isentropic Process of Ideal Gas: ttancons p k = ρ (derivation available) Compressible Flow Intro_L8_9 1-D Analysis: Starts with Stream tube (stream lines) 1 CV x y Flow Useful Relations: kk p ttancons p 0 0 ρ == ρ Other Variables: )k(k/)k( p p T T 1 0 1 00 −−         ρ ρ =         = (from ideal gas relation) How can we get local Isentropic variables in terms of Mach # Pressure: p 0 /p = [1 + 0.5(k-1)Ma 2 ] k/(k-1) Temperature: T 0 /T = [1 + 0.5(k-1)Ma 2 ] Density: ρ 0 /ρ = [1 + 0.5(k-1)Ma 2 ] 1/(k-1) At Sonic condition (Ma = 1) (V * = c * ) *** RT k k cV 0 1 2 + == )airfor(.)k(. p p )k/(k * 891150 1 0 =+= − )airfor(.)k(. T T * 21150 0 =+= )airfor(.)k(. p )k/( * 5771150 11 0 =+= ρ − Compressible Flow Intro_L8_10 Effects of Area Variaion on Properties of Isentropic Flow Momentum eqn: 0 2 2 =         + ρ x V d dp x x x V dV V dp −= ρ 2 From Continuity: ρ ρ −−= d V dV A dA x x Conbine the 2 equations: ρ ρ ρ ρ − ρ = d dp V V dp V dp A dA x xx 2 22 ( ) 2 2 2 2 11 Ma − ρ =         ρ − ρ = x x x V dp d/dp V V dp A dA ( ) 2 1 Ma - −= x x V dV A dA Above relation illustrates how area can be affected by Ma [...].. .Compressible Flow Intro_L8_11 Effects of Area Variaion on Properties of Isentropic Flow ( Nozzle – Diffuser - dVx dA = 1 − Ma 2 A Vx Sonic velocity reached where area is minimum (throat) Comes from the principle dA = 0 → Ma = 1 Nozzle Flow Regime Sub sonic Ma < 1 Super sonic Ma > 1 ) dp0 dp>0 Diffuser dVx . 0 1 2 + == )airfor(.)k(. p p )k/(k * 8 911 50 1 0 =+= − )airfor(.)k(. T T * 211 50 0 =+= )airfor(.)k(. p )k/( * 57 711 50 11 0 =+= ρ − Compressible Flow Intro_ L8_ 10 . pressure 10 1lPa becomes s/m./*./kPc 343 211 010 00 41 ==ρ= at sea level, c = 340 m/s at 11 km altitude, c = 295 m/s In water sonic speed, s/m/E./Kc 14 6 310 0 914 2 3