Compressible Flow Intro_L8_1BACKGROUND 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 ther
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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
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What do we need from THERMODYNAMICS ?
For a reversible process,
Perfect Gas Law pV = mRT
Mass of gas
MR = 8314.4 J kg-1K-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 1st & 2nd law of thermodynamics
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Few more terms & relations (thermo)
cp → specific heat at constant pressure
cv → specific heat at constant volume
p
p
T
h
∂
∂
=
v
v
T
u
∂
∂
= ratio of sp heats, k = cp / cv
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)
s
p c
ρ
∂
∂
=
In gas, sonic velocity, Ideal gas, p kp kRT
s
= ρ
=
ρ
∂
Static pressure (or stream pressure), p
Stagnation pressure (p0)- pressure when gas brought to rest isoentropically
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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
s / m
/
* /
kP
c = ρ = 1 4 101000 1 2 = 343
at sea level, c = 340 m/s
at 11km altitude, c = 295 m/s
In water sonic speed, c = K /ρ = 2.14E09/103 =1463m/s
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
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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
c(2 ∆ t) c(3 ∆ t)
WAVE PROPAGATION (pressure disturbance)
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
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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)
Out
side
con
e
unaw
are
of s
ound
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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
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Conservation Equations for 1-D Isentropic Flow
Conservation of Mass(Continuity) : d(ρVxA) = 0 or ρVxA = m = constant
Conservation of Momentum : 0
2
2
=
+ ρ
x
V d dp
Conservation of Energy :
V h or
V h
2
0
2
2nd Law of Thermo: s = constant
Equations of State: h = h(s,p)
ρ = ρ(s,p) Isentropic Process of Ideal Gas: pk =constant
ρ (derivation available)
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1-D Analysis:
Starts with Stream tube (stream lines)
1
CV
x
y
Flow
Useful Relations:
k k
p t tan cons p
0
0 ρ
=
= ρ
Other Variables: (k )/k (k )
p
p T
ρ
ρ
=
= (from ideal gas relation)
How can we get local Isentropic variables in terms of Mach #
Pressure: p0/p = [1 + 0.5(k-1)Ma2]k/(k-1)
Temperature: T0/T = [1 + 0.5(k-1)Ma2]
Density: ρ0/ρ = [1 + 0.5(k-1)Ma2]1/(k-1)
At Sonic condition (Ma = 1)
(V* = c*)
*
*
k
k c
1
2
+
=
=
) air for ( )
k (
p
) air for ( ) k (
T
T
) air for (
) k (
p
) k /(
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Effects of Area Variaion on Properties of Isentropic Flow
2
2
=
+ ρ
x
V d
dp
x
x
dV V
dp
−
=
From Continuity:
ρ
ρ
−
−
V
dV A
dA
x x
Conbine the 2 equations:
ρ
ρ
ρ ρ
− ρ
dp
V V
dp V
dp A
x x
2 2
2
2
2
ρ
=
ρ
− ρ
=
x
x
dp d
/ dp
V V
dp A
dA
(1 Ma2)
x
x
V
dV A
dA
Above relation illustrates how area can be affected by Ma
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Effects of Area Variaion on Properties of Isentropic Flow
(1 Ma2)
x
x
V
dV A
dA
Nozzle – Diffuser Sonic velocity reached where area is minimum (throat)
Comes from the principle dA = 0 → Ma = 1
Flow Flow
Flow Regime
Sub sonic
Ma < 1
Super sonic
Ma > 1
dp<0
dVx >0
dp>0
dVx <0