Download 2-2 Approach

Chapter 2
Integral Forms of the Conservation Equations
for Inviscid Flow
2-1 Problem Solving
•
1. Physics  2. Approach methods
 3. Tools (mathematics)  4. Solve the problem
Approach methods :
•
Classical mechanics , no relativity： 1.Mass is conserved
2.Force = ma
3.Energy is conserved
2-2 Approach
A. Finite Control Volume Approach (macroscopic , Integral Form )
FIG. 2-1 Finite control volume approach
B. Infinitesimal Fluid Element Approach (Macroscopic , Differential Form)
Eulerian coordinate
Lagrangian coordinate
FIG.2-2 Infinitesimal fluid element approach
C. Molecular Approach (Microscopic)
Statistical averaging → Boltzmann equation from kinetic theory
Ref.
Hirschfelder , Curtiss and Bird
“Molecular Theory of Gases and Liquids”
2-3 Continuity Equation
Physical principle : Mass can be neither created nor destroyed.

mdenotes the mass flow
u
through dS
u 
are local velocity and
density at B
 
m   (u cos )ds  un ds  u  ds
FIG.2-3 Fixed control volume
The net mass flow into the control volume through the entire
control surface S
 
   u  dS
Note :
s
 
u  dS
(- )  inflow
(+)  outflow
The total mass inside the control volume
  dV
V
The time rate of change of this mass inside the C.V.

  dV
t V
Mass is conserved
  
  u  dS   dV
t V
s
Continuity equation
(Integral Form)
Applies to all flows , compressible
or incompressible , viscous or
inviscid
2-4 Momentum Equation
Physical principle : The time rate of change of momentum of a
m  const.
d
( mu )  F
dt
body equals the net force exerted on it .
F  ma
1. Body forces on V , eg. Gravitational and
E.M forces

Forces on the
control volume
Total body force =

 fdV
V


f
Body force
/ mass
2. Surface forces on S , eg. P and τ

  pdS
Total surface force
due to pressure
=
s




The total force F
F   fdV   pdS

acting on the C.V. is
v
s
We consider
inviscid flows
only
The total time rate of change of momentum of the fluid as it
flows through a fixed control volume
= the net rate of flow of momentum
across the surface S
+ the change in momentum in V due to
unsteady fluctuations in the local flow
  
  ( u  dS )u
s


  udV
t V
s
 
  ( u )dV
t
V
  
  ( u  dS )u
(For fixed C.V.)
d

 ( mu )
dt
Therefore :
Momentum Equation
  ( u  dS )u 
s

V
( u )
   fdV   pdS
t
V
s

 Fviscous
2-5 Energy Equation
Physical principle : Energy can be neither created nor destroyed ; It can only
change in form. ------- the 1st law of thermodynamics
(δQ+δW=dE)
B1= rate of heat added to the fluid inside the C.V. from the
surroundings
B2= rate of work done on the fluid inside the C.V.
B3= rate of change of the energy of the fluid as it flows through the
C.V.
B1 + B 2 = B3
B1 : volumetric heating of the fluid in V due to
a. Radiation
b. Thermal conduction & diffusion (viscous effect)
q
: the rate of heat added to C.V./mass
B1   qdV
v
B2 = the rate of doing work on a moving body =
= rate of work done on the fluid
+
inside V due to pressure forces
on S
=
F u
rate of work done on the fluid
inside V due to body forces
 
 
  Pu  dS    ( f  u )dV
s
V
B3 = net rate of flow of energy
across the C.V.
+
time rate of change of energy inside V
due to transient variations of the flow
field variables
=
u2

u2
 (e  )dV
s ( u  dS )(e  2 )  t 
2
V
Therefore ,
 
 
 qdV   pu  d S    ( f  u )dV
V
s
V
Energy Equation

u2 
u 2   (Integral Form , Inviscid)
    (e  ) dV    (e  )u  dS
t 
2 
2
V
s
More general form
 


Q  W shaft  W viscous   pu  dS    ( f  u )dV
s
V
2
2



u
u  
    (e  )  dV    (e  )u  dS
t 
2 
2
V
s