Chapter 11, Compressible Flow
1. Thermodynamics of ideal Gases
p = RT
dh = cpdT
du = cvdT
s2 - s1 = cp ln (T2/T1) - R ln (p2/p1)
s2 - s1 = cv ln (T2/T1) + R ln (1/2)
cp - cv = R
cp/cv = k
cp / R = k/(k-1)
cv / R = 1/(k-1)
2. Mach Number and Speed of Sound
Mach number,
Ma = V/C
Speed of sound, C = [(p/)s]1/2
Since p = constat k
for isentropic process
(p/)s = (p/k)k k-1
= (p/)k = kRT
C = [kRT]1/2
The bulk modulus of elasticity
Ev = dp/(d/) =  (p/)s = C2
or,
C = [Ev/]1/2
3. Categories of Compressible Flow
Incompressible flow
Sub-sonic flow
Transonic flow
Super-sonic flow
Hypersonic flow
Ma  0.3
0.3 < Ma < 0.9
0.9 < Ma < 1.1
1.1 < Ma < 5.0
Ma  5.0
CD increases as Mach number increases, Fig. 11.2
4. Isentropic Flow for Ideal Gas
i) Effect of flow cross-section area variations
Continuity
m = VA = constant
ln  + ln V + ln A = constant
d/ + dV/V + dA/A = 0
-dV/V = d/ + dA/A
Newton’s second law of motion
dp + (1/2)dV2 + dz = 0
for ideal gas dz = 0
dp/(V2) = -dV/V
Combine continuity and Newton’s law
dp/(V2) = d/ + dA/A
dp/(V2) [1 - V2/(dp/d)] = dA/A
for isentropic flow,
Speed of sound, C = [(p/)s]1/2 = [dp/d]1/2
dp/(V2) [1 - (V/C)2] = dp/(V2) [1 - Ma2] = dA/A
dV/V = -[1/(1 - Ma2)]dA/A
d/ = [Ma/(1 - Ma2)]dA/A
for a converging nozzle, dA < 0
Ma < 1
Ma > 1
dV > 0
d < 0
dV < 0
d > 0
for a diverging nozzle, dA > 0
Ma < 1
Ma > 1
dV < 0
d > 0
dV > 0
d < 0
Since
dA/dV = -(A/V)(1 - Ma2)
for Ma = 1, dA/dV = 0
A converging-diverging nozzle is necessary for flow to
accelerate from sub-sonic, to sonic, and super-sonic flow
ii) Converging-diverging duct flow
For incompressible flow
stagnation pressure,
po = p + (1/2)V2
For compressible flow
po/p is a function of Mach number
for isentropic flow of ideal gas
p/k = constant = po/ok
Newton’s law
dp/ + d(V2/2) = 0
 = o(p/po ) 1/k
[dp/p 1/k ](po1/k/ o) + d(V2/2) = 0
integrate from p to po (from V to 0)
[1/(1- 1/k)](p1-1/k)(po1/k/ o) | p po= -V2/2 |V0
[k/(k-1)][po1-1/k - p1-1/k] (po1/k/ o)= V2/2
since
po1/k /o = p1/k /
[k/(k-1)][po/o - p/] = V2/2
for ideal gas, p = RT
[kR/(k-1)][To - T] = V2/2
ho - (h + V2/2) = 0
[kR/(k-1)][1 - T/To] = V2/(2 To)
T/To = 1 - [V2/(2 To)][(k-1)/(kR)]
= 1 - [V2/(kRT)](T/To)[(k-1)/2]
= 1 - Ma2 (T/To)[(k-1)/2]
T/To {1 + Ma2 [(k-1)/2]} = 1
T/To = 1/{1 + [(k-1)/2] Ma2}
Since
Table E.1 (k = 1.4)
p/(T) = po/(oTo)
T/ To = (p/po)(/o)
but
(/o) = (p/po) 1/k for isentropic flow
(p/po) = (T/To) k/(k-1)
(p/po) =  1/{1 + [(k-1)/2] Ma2} k/(k-1)
(/o) =  1/{1 + [(k-1)/2] Ma2} 1/(k-1)
At Ma = 1, Critical state ( )*
p*/po = [2/(k+1)]k/(k-1)
*/o = [2/(k+1)]1/(k-1)
T*/To = 2/(k+1)
Ex. 11-5 thru 11-7
For choked flow in a converging-diverging nozzle, the
mass flow rate can not be increased by increasing Mach
number; or m is maximum at Ma = 1.0
from Conservation of Mass, the maximum mass flow rate
AV = *V*A*
where * designates the Mach number of unity at the throat
A/A* = (*/)(V*/V)
V* = C* = [kRT*]1/2
V = Ma [kRT]1/2
A/A* = (*/o)(o/)(T*/To)(To/T)(1/ Ma )
= (1/ Ma ){1 + [(k-1)/2] Ma2}/{1 + (k-1)/2}
(k+1)/[2(k-1)]
5. Normal Shock Waves (Non-Isentropic Flow)
Thin shock layer,
Ax = A y = A
Continuity
xVx = yVy
Momentum
pxA - pyA = m(Vy - Vx)
Energy
hx +Vx2/2 = hy +Vy2/2
Second Law of Thermodynamics
sy - sx = cp ln (Ty/Tx) - R ln (py/px)
State (ideal gas)
p = RT
hy - hx = cp(Ty - Tx )
Across a normal shock, the stagnation temperature
remains constantant (Tox = Toy)
Ty/Tx = (Ty/Toy) (Toy/Tox) (Tox/Tx)
= {1 + [(k-1)/2] Max2}/{1 + [(k-1)/2] May2}
Table E.4 (k = 1.4)
Ex. 11-18 thru 11-20
Flow is non-isentropic across a shock wave, but isentropic
up stream and down stream of the shock wave.