Advanced fluid mechanics (II)
Course content:
1.Compressible Fluid Mechanics
Textbook: Modern Compressible Flow, 2nd ed. , by John D
Anderson, Jr.
Reference Book
1. Gas Dynamics, 2nd ed., by James. E. A. John
2. Compressible – Fluid Dynamics by Philip A. Thompson
3. Elements of Gasdynamics by H. W. Liepmann and A.
Roshko
4. Compressible Fluid Flow. , 2nd ed. , by Michel A. Saad
Grading: 1. Homework 60%
2. Final Project 40%
Chapter I
1. Introduction and Review of Thermodynamics
What is Compressible Flow?
1.   Const
2. Energy transformation and temperature change are important
considerations → Importance of Thermodynamics
e.q Flow of standard sea level conditions,
M  2
Specific internal energy
8314
.300
RT
28
i  CvT 

 2.2 105 J / kg
 1
0.4
Specific kinetic energy
2

U
k

2
22 

U   2a
 RT
2

i  0(k )
2
 2 1.4 
8314
 300  2.5 105 J / kg
28
Chapter I
1.1 Definition of Compressible Flow
Incompressible flow → compressibility effect can be
ignored.
ν is the specific volume &   1/ v
Compressibility of the fluid
 
1 dv
v dp
Note: dp(+) → dv(-)
Physical meaning: the fractional change in volume of the fluid
element per unit change in pressure
Chapter I
1 v
)T ......
v p
…. Isothermal compressibility
1 v
) S ......
v p
…..isentropic compressibility (speed of sound)
T   (
S   (
Compressibility is a property of the fluid
Liquids have very low values of
e.g 
T
for water =
5 1010 m2 / N
at 1atm
Gases have high 
e.g  T
for air =10-5 m2/N at 1 atm,
Alternate form of 
v
1


1 d
 dp

d    dp
Chapter I
For most practical problem
d

 5% compressible
General speaking
Ma >0.3
→ Compressible effect can not be ignored
Ma < 0.3 → Incompressible flow
Chapter I
1.2. Regimes of compressible flow
Subsonic flow
Flow is
forewarned of
the presence
of the body
Streamline deflected far upstream of the body
Chapter I
Transonic flow
Mis less than 1 , but high enough to produce a pocket of
locally supersonic slow
If M  is increased to slightly above 1 , the λ shock will move to the
trailing edge of the airfoil , and bow shock appears upstream of the
leading edge.
Loosely
Defined as
the
“ Transonic
regime”
(Highly unstable)
Chapter I
Supersonic Flow
M  1 Everywhere
(We will mostly
focus on this
regimes)
T , p,  
Behind the
shock
+ Parallel the free stream flow is not forewarned of presence of the
body until the shock is encountered
+ Both flow of upstream of the shock and downstream of the shock
are supersonic
+ Dramatic physical and mathematical difference between
subsonic and supersonic flows.
Chapter I
Hypersonic Flow
T , p,   High enough to excite the internal modes of energy
dissociate or even ionize the gas.
Real gas effect !!! Chemistry comes in
Chapter I
Incompressible flow is a special case of subsonic flow limiting case
M  0
V  0
V
M 
a
Trivial , no flow
a  
a  ,  0
For incompressibility
Viscous
Flows
Flow can be also be classified as
inviscid
Viscous flow:
+ Dissipative effects : Viscosity, thermal conduction, mass
diffusion….
+ Important in regions of large gradients of V, T and Ci
e.g. Boundary layer
Chapter I
Inviscid flows: - ignore dissipative effects outside of B.L
(We will treat this kind of flow )
Also consider the gas to be “ Continuum ”
Mean free path
Kn 

L
 1
1.3 A Review of Thermodynamics
1.3.1 Ideal gas – intermolecular force are negligible
p   RT
pv  RT
p  nkT
R*
R - specific gas constant 
M
8314 (J/kg.mole.k)
Molecular weights
Boltzmann constant = 1.38 1023 J / K
For air at standard conditions
R 8314
R

 287 J /(kg.K )  1716( ft.lb) /( slug. 0 R)
M 28.9
L
d
L > 10d , for most compressible flows
Isothermal compressibility
1 v
(
)T
v p
RT
v
RT
v
1 v
1
v
;(
) 2 
 T   (
)T 
p
p
p
p
v p
p
T  
Chapter I
1.3.2. Internal Energy and Enthalpy
-Translational
-Rotational No of collisions > 5 → equilibrium
-Vibration : No of collisions > 0 (100 ) → equilibrium
Add one more time scale or length scale
-Electronic excitation + nuclear
Statistical
Thermodynamics
+
Quantum
mechanics
If the particles of the gas (called the system) are rattling about their
state of “maximum disorder”, the system of particle will be in
equilibrium.
Chapter I
Return to macroscopic view continuum
Let e be specific internal energy
Let h be specific enthalpy
h  e  pv
For both a real gas and a chemically reacting mixture of perfect
gases.
e  e(T , v)
h  h(T , p)
Thermally perfect gas
e  eT 
h  hT 
de  Cv dt
dh  C p dt
Cv (T ), C p (T )
Chapter I
Calorically perfect gas
e  CvT
h  C pT
Will be assumed in the
discussion of this class
Ratio of specific heat ,
Cp
C p , Cv are const →  
 cons tan t γ =1.4 for a diatomic gas
Cv
γ =5/3 for a monatoinic gas
Air, T<1000 K – Calorically perfect gas
O2, N2, 1000<T<2500 – Thermally perfect gas
Vibrational excited
O2 dissociate 2500<T<4000 K
N2 dissociate T>4000K
Consider caloriacally perfect gas + thermally perfect gas
e  cvT
p  RT
h  c pT
Cp  Cv
Note:
he pv

 R
T
T
 h 
Cp  


T

p
 e 
Cv   
 T v
specific heat at constant pressure
specific heat at constant volume
Chapter I
Perfect gas
Cv , C p ,   cons tan t
Ideal gas
e  e(T )
h  h(T )
p   RT
Cp
R
R
Cp  Cv  R;    Cp 
, Cv 
Cv
 1
 1
1.3.3. First law of the thermodynamics
-Conservation of Energy
Consider a system, which is a fixed mass of gas separated from the
surroundings by a flexible boundary. For the time being, assume the
system is stationary, i.e., it has no directed kinetic energy
 q   w  de
An incremental
amount of heat
added to the
system across the
boundary
e is state variable, de is an exact differential
depends only on the initial and final states
of the system
The work done on the system by the
surrondings
Chapter I
For a given de , there are in general an infinite different ways
(processes) of
q  w
We will be primarily concerned with 3 types of processes:
1. Adiabatic process  q  0
2. Reversible process – no dissipative phenomena occur, i.e,.
Where the effects of viscosity, thermal conductivity, and
mass diffusion are absent
 w   pdv (see any text on thermodynamic)
3. Isentropic process - both adiabatic & reversible
2nd law of thermodynamic
Chapter I
1.3.4 Entropy and the Second Law of Thermodynamic
Define a new state variable, the entropy,
ds 
qrev or
T
ds 
q
T
 siirev
The actual heat added/T,
siirev  0
A contribution from the
irreversible dissipative
phenomena of viscosity
thermal conductivity, and
mass diffusion occurring
within the system
These dissipative phenomena “ always” increase
the entropy
For a reversible process
ds 
q
T
If the process is adiabatic,  q  0
2nd law
 ds  sirrev  0
In summary, the concept of entropy in combination with the 2nd law allow us to
predict the direction that nature takes.
Chapter I
Assume the heat is reversible,
 q   qrev   qrev  Tds
 q  pdv  de  Tds  de  pdv
1st law becomes
h  e  pv dh  de  pdv  vdp
Tds  dh  vdp
For a thermally perfect gas,
ds  C p
dh  C p dT
dT vdp

T
T
If the gas also obey the ideal gas equation of state p  RT
ds  C p
dT
dp
R
T
p
T2
dT
P2
 R ln
Note
Integrate s2  s1   Cp
T
P
1
T1
C p  C p (T )
Chapter I
If we further assume a calorically perfect gas,
T2
P2
s2  s1  Cp ln  R ln
T1
P1
T2
v2
 Cv ln  R ln
T1
v1
C p  Cv  R
Tds  de  pdv
1.3.5. Isentropic realtions
For an adiabatic process q  0 and for a reversible process dsirrev  0
Hence, from eq ds  q  dsirrev  0
,i.e.,
T
the entropy is constant. s2  s1
Chapter I

T
P
P
T C /R T
0  C p ln 2  R ln 2  2  ( 2 ) p  ( 2 )  1
T1
P1
P1
T1
T1
1
T2
v2
v2
T2  Cv / R T2   1
0  Cv ln  R ln   ( )
( )
T1
v1
v1
T1
T1
 2 T2  1
( )
1
T1
1

P2

T
 ( 2 )  ( 2 )  1
P1
1
T1
Outside B.L-Isentropic relations prevail
e.g.
T=1350K
P=?
T=2500 K
P=15atm
M=12, Cp=4157 J/kg.K

P2
T
 ( 2 )  1  0.0248
P1
T1

Cp
Cv
  1.2
; Cv  C p  R  4157 
8314
 3464 J / kg.k
12
Chapter I
1.3.6. Aerodynamic forces on a Body
Main concerns : Lift & drag
Forces on a body of airfoil
-Surface forces: pressure
shear stress
-Body forces : gravity ;
electric-magnetic
Sources of aerodynamic force, resultant force and its resolution
into lift and drag
Chapter I
Let n & m be unit vectors perpendicular and parallel, respectively
to the element ds,



dF   pnds  mds




F   dF    pds   mds
Lift L is the component of F perpendicular to the relative
wind V
Drag D is the component of F parallel
In our plot. L// y, D//

 

L    pds 
y
x
 



mds    pds (resonable )
y
inviscid
y
Chapter I

 

D    pds 
x
Pressure drag ->
wave drag, e.g
slender supersonic
shapes with shock
waves

mds

x
Skin friction drag
-We consider only inviscid flows and both pressure and skinfriction drags are important
-In the most cases, we can not predict the drag accurately
For blunt bodies, Dp dominates
For streamlined bodies, Dskin dominates
with shock wave, Dwave drag dominate and Dskin can be
neglected
D can be predicted reasonably