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
Flow Classification: - Subsonic: Ma < 1
- Supersonic: Ma > 1
- Hypersonic: Ma > 5
Pressure always expressed as absolute pressure
likewise Temperature will be in absolute unit, eg Kelvin
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
Perfect Gas Law pV = mRT MR = 8314.4 J kg-1K-1
Mass of gas
Mol wt of gas
Another relation: p = ρRT
What do we need from THERMODYNAMICS ?
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? Adiabatic
dQ
For a reversible process, ∆s = ∫ ∆s = 0
T
dQ ∆s > 0
For an irreversible process, ∆s > ∫
T
Note: the terms, u, h & s comes from 1st & 2nd law of thermodynamics
Compressible Flow Intro_L8_3
Few more terms & relations (thermo)
cp → specific heat at constant pressure
cv → specific heat at constant volume
∂h ∂u ratio of sp heats, k = cp / cv
cp = cv =
∂T p ∂T v
kair ≈ 1.4
cp = cv + 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)
∂p ∂p p
In gas, sonic velocity, c = Ideal gas, = k 0 = kRT
∂ρ
∂ρ s ρ
s
Static pressure (or stream pressure), p
Stagnation pressure (p0)- 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: c = K /ρ
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
c = kP / ρ = 1.4 * 101000 / 1.2 = 343m / s
at sea level, c = 340 m/s
at 11km altitude, c = 295 m/s
In water sonic speed, c = K / ρ = 2.14E09 / 10 3 = 1463m / s
Inertial force
Mach Number, Ma = U / c
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
WAVE PROPAGATION (pressure disturbance)
Pressure disturbance occuring at an interval of every ∆t
S is the disturbance source c∆t
c(2∆t)
c(3∆t) c(2∆t) c(3∆t)
c∆t S
U=0 U Compressible Flow Intro_L8_6
Shock Wave propagation:
Case (c): U = c Inside cone aware
c(2∆t) of sound
nd
α c(3∆t)
ou
eo e
1
a r co n
2
fs
3
aw e
S S
un tsid
3 2 1 3 2 1
Ou
U∆t
U(2∆t)
Ma = 1 sonic U(3∆t)
The Mach Cone
all wavefronts touch Ma > 1
source S Case (d): U > c
Mach (Cone) angle: α = Sin-1(1/Ma)
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 : p0
Temperature : T0
Density : ρ0
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(ρVxA) = 0 or ρVxA = m = constant
dp Vx2
Conservation of Momentum : +d
2
=0
ρ
Conservation of Energy : Vx2 Vx2
d h +
=0 or h+ = h 0 = cons tan t
(1st Law of Thermo) 2 2
2nd Law of Thermo: s = constant
Equations of State: h = h(s,p)
ρ = ρ(s,p)
p
Isentropic Process of Ideal Gas: = cons tan t (derivation available)
ρ k
Compressible Flow Intro_L8_9
1-D Analysis: CV
y
Starts with Stream tube (stream lines) Flow
x
Useful Relations: p p0
= cons tan t = k 1
ρk ρ0
( k −1) / k ( k −1)
Other Variables: T0 p 0 ρ
= = 0 (from ideal gas relation)
T p
ρ
How can we get local Isentropic variables in terms of Mach #
Pressure: p0/p = [1 + 0.5(k-1)Ma2]k/(k-1) At Sonic condition (Ma = 1)
2k (V* = c*)
Temperature: T0/T = [1 + 0.5(k-1)Ma2] V =c =
* * *
RT0
k +1
Density: ρ 0/ρ = [1 + 0.5(k-1)Ma2]1/(k-1) p0
*
= 0.5(k + 1)k /( k −1) = 1.89 ( for air )
p
ρ0 T0
= 0.5(k + 1)1 /( k −1) = 1.577 ( for air ) = 0.5(k + 1) = 1.2 ( for air )
p* T*
Compressible Flow Intro_L8_10
Effects of Area Variaion on Properties of Isentropic Flow
dp Vx2 dp dV
Momentum eqn: +d =0 =− x
ρ 2 ρVx2 Vx
dA dV dρ
From Continuity: =− x −
A Vx ρ
dA dp dp ρVx2 dρ
Conbine the 2 equations: = −
A ρVx2 ρVx2 dp ρ
Vx2
dA
=
dp
1 −
dp / dρ =
dp
(1 − Ma 2 )
A ρVx2 ρVx2
dA
A
=
- dVx
Vx
(
1 − Ma 2 )
Above relation illustrates how area can be affected by Ma
Compressible Flow Intro_L8_11
Effects of Area Variaion on Properties of Isentropic Flow
dA
A
=
- dVx
Vx
(
1 − Ma 2 )
Nozzle – Diffuser Sonic velocity reached where area is minimum (throat)
Comes from the principle dA = 0 → Ma = 1
Nozzle dp>0 Diffuser
dp0 dVx 1
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