AAE 3710
Fundamentals of Aerodynamics
Lecture 7 Example problems for
fluid kinematics
02/06/2006
Reynolds Transport Theorem
It shifts from the governing laws expressed using system concepts (consider a given
mass of the fluid) to those expressed using control volume concepts (consider a given
volume)
Acceleration Vector
Time derivative term
Convective terms
Because of convective terms, even the fluid is steady it still possibly has accelerations.
Streamline
definition
Finite Control Volume Analysis
Governing eqns based on
system method
(Normally differential form)
Reynolds Transport Theorem
Control volume formulas
(Normally integral form)
Physical laws that will be considered:
1. Conservation of Mass
1.1 fixed, nondeforming C.V.
1.2 moving, nondeforming C.V.
1.3 deforming C.V
2. Newton’s 2nd law
linear momentum equation
3. First law of thermodynamics
Apply conservation of mass to
control volume
Mass conservation law:
time rate of change of the system mass = 0
DMsys/Dt = 0
Resort to
Reynolds
This is applied to a system of
mass. We are interested not in
a specific fluid but how fluid
behaves in a field or control
volume.
Transpor
t Theorem
How to proceed?
-Extensive property of the system: Bsys (=M mass)
-Intensive property of the system: b (=1; M=M*b)
-Bsys=
Apply conservation of mass to control volume
Sign convention
zero
time rate of change of
the mass of the contents of the control
volume
net rate of mass flow through
the control surface
Continuity eqn with a
fixed, nondeforming
control volume
Outflow: “+”
Mass flow rate calculation:
--If density and velocity are both uniform through a section of control surface (most common expression)
Normal component of velocity
For general nounifo replaced by average value
Inflow: “-”
Apply conservation of mass to control volume
Volume flow rate
Apply conservation of mass to control volume
Continuity equation
There are 3 control surface, i.e. inlet, outlet,
and the pipe wall. Only first two are valid
contributions to the integral while the pipe wall
is parallel to the flowing direction so there is
no flow penetrating it.
Apply conservation of mass to control volume
Different forms of continuity equations:
Steady flows
Steady, incompressible flows
Multiple inlets
and outlets