M E 320
Professor John M. Cimbala
Lecture 41
Today, we will:
•
•
Discuss isentropic, compressible, adiabatic flow in ducts
Discuss converging-diverging ducts, and introduce choking and shock waves
2. One-Dimensional Isentropic Adiabatic Flow in Ducts
a. Setup and equations
Pressurized
reservoir
Duct with non-constant cross-sectional area
Stagnation
conditions:
V≈0
T = T0
P = P0
h = h0
Goal: Predict P, T, r, V, Ma,
etc. at some downstream
location in the duct
For simplicity, we approximate the flow as isentropic (negligible friction and other
irreversibilities) and adiabatic (no heat transfer from the air to the surroundings – insulated
duct walls).
Equations for isentropic, compressible, adiabatic flow of an ideal gas:
k
1
T
k −1
ρ  k − 1 2  k −1 P0  k − 1 2  k −1
Ma 2 0= 1 +
Ma 
= 1 +
Ma 
• For any ideal gas: 0 = 1 +
ρ 
2
P 
2
T
2


2.5 P
3.5
ρ
T
0
= (1 + 0.2Ma 2 )
• For air (k = 1.4): 0 = 1 + 0.2Ma 2 0= (1 + 0.2Ma 2 )
ρ
T
P
ρ T 
• Or, for air in terms of temperature, 0 =  0 
ρ T 
2.5
P0  T0 
= 
P T 
3.5
1/ 2
c T 
also 0 =  0 
c T 
Subsonic water nozzle
Supersonic rocket nozzle
2
T0 k + 1
A
1 (1 + 0.2Ma )
= 1.2 from which we get
For air: * =
and=
T*
2
A Ma
1.728
3
2.5
3.5
1/ 2
ρ0  T0 
P0  T0 
c0  T0 
2.5
3.5
1/ 2
=
=
=
=
=
=
1.2
1.577
1.2
1.893
=
1.2
=
1.095
 *
 *
 *=

*
*
*
ρ T 
P T 
c T 
But these are usually listed in terms of their inverses, i.e.,
T*
2
c*
P*
ρ*
= = 0.8333
= 0.5283
= 0.9129 = sonic or critical values
= 0.6339
T0 k + 1
c0
ρ0
P0
We also define a critical Mach number, Ma*, as
V V c Ma ⋅ c Ma ⋅ kRT
T
=
=
=
= Ma *
*
*
*
c
cc
c
T
kRT *
We plot all these as functions of Mach number in Appendix A-13 (for air):
*
Ma=
Should be r/r0
Consider a stationary normal shock wave (as in a supersonic wind tunnel)
Properties that increase
across the shock:
• P2 > P1
• T2 > T1, thus:
o c 2 > c1
Properties that decrease
o h2 > h1
across the shock:
• r2 > r1
• Ma2 < Ma1
• s2 > s 1
• P02 < P01
• A2* > A1*
• r02 > r02
• V2 < V1
Properties that stay the same across the shock:
• T02 = T01
• h02 = h01
Consider instead a moving normal shock wave (as in a blast wave from an explosion)
1
2
V2
Vs
V1 = 0
•
•
•
•
The shock is moving into quiescent air (region 1)
In this frame of reference we define Ma1 = Vs/c1
The shock wave travels into region 1 at supersonic speed (Ma1 > 1)
The air behind the shock (region 2) follows at a slower speed
The “dime analogy” (model the moving shock as rows of dimes that pile up when
pushed by a rod or “piston” as sketched; three sequential times):
Time: t = 0
2
1
Time: t = t1
1
V2
V1 = 0
Time: t = t2
2
1
V2
Vs
V1 = 0
2
V2
V2
Vs
V1 = 0
V2
Comments:
• The vertical red line is analogous to a shock wave: V1 = 0, Vs > V2, r2 > r1 (there is
sudden increase in density, and the “wave front” of dimes moves faster than the piston).
• The dimes in region 1 don’t “know” anything is happening until the shock hits them.
• Similarly in a shock wave in air, the air in region 1 does not “know” anything is
happening until the shock wave hits it.