Fluid Flow Concepts
p and
Reynolds Transport Theorem
Fluid
Kinematics
®
Descriptions of:
®
®
Fluid Dynamics
®
®
A l i Approaches
Analysis
A
h
®
®
fluid motion
fluid flows
temporal and spatial classifications
Lagrangian vs. Eulerian
Moving from a system to a control volume
®
Re nolds Transport Theorem
Reynolds
Descriptions of Fluid Motion
®
streamline
®
®
®
®
®
®
®
Defined instantaneously
V2, b2
has the direction of the velocity vector at each point
no flow across the streamline
steady flow streamlines are fixed in space
unsteady flow streamlines move
pathline
®
Streamlines
Defined as particle moves (over time)
V1, b1
path of a particle
same as streamline for steady flow
streakline
®
®
tracer injected continuously into a flow
same as pathline and streamline for steady flow
Tom Hsu’s numerical simulation
Ideal flow machine
Descriptors of Fluid Flows
®
Laminar flow
®
®
®
Temporal/Spatial Classifications
®
fluid moves along smooth paths
viscosity damps any tendency to swirl or mix
Steady - unsteady
®
Turbulent flow
®
®
®
fluid moves in very irregular paths
efficient mixing
velocity
y at a point
p
fluctuates
®
Uniform - nonuniform
®
Analysis Approaches
®
Lagrangian (system approach)
® Describes
D
ib a ddefined
fi d _____
iti velocity,
l it
mass ((position,
®
®
Changing in time
acceleration, pressure, temperature, etc.) as functions
of time
Track the location of a migrating bird
®
field (velocity, acceleration,
Describes the flow ______
pressure, temperature, etc.) as functions of position
i
andd time
Count the birds passing a particular location
If you were going to study water flowing in a pipeline, which
approach would you use? ____________
Eulerian
Changing in space
The Dilemma
®
The laws of physics in their simplest forms
d
describe
ib systems
t
(the
(th Lagrangian
L
i approach)
h)
®
®
Eulerian
®
Can turbulent flow
b steady?
be
d _______
If
________________
averaged over a
________________
suitable time
®
®
Conservation of Mass, Momentum, Energy
It is impossible to keep track of the system
y fluids pproblems
in many
The laws of physics must still hold in a
Eulerian world!
We need some tools to bridge the gap
Control Volume Conservation
Equation
Reynolds Transport Theorem
®
®
®
A moving system flows through the fixed
control
t l volume.
l
The movingg system
y
transports
p
extensive
properties across the control volume
surfaces.
We need a bookkeeping method to keep
track of the properties that are being
transported into and out of the control
volume
B =__________________________
Total amount of some property in the system
b = Amount of the property ___________
per unit mass
DBsys
Dt
Rate of increase
of the property
in the system
Control Volume Conservation
Equation
®
®
Dt


  bdV  cs  bV  nˆ dA
t cv
0 = -11 + ((-00 + 1)
0 = 1 + (-1 + 0)
0 = 0 + (-0 + 0)
=
Rate of increase
of the property
in the control
volume
+
Rate of efflux of
the property across
the control volume
boundary
Application
pp
of Reynolds
y
Transport Theorem
®
DBsys


  bdV  cs  bV  nˆ dA
t cv
Conservation of mass (for all species)
Newton’s 2nd law of motion (momentum)
F = ma
_______
First law of thermodynamics (energy)
Summary
®
Reynolds Transport Theorem can be applied
to a control volume of finite size
®
®
®
We don
don’tt need to know the flow details within
the control volume!
We do need to know what is happening at the
control surfaces.
W will
We
ill use Reynolds
R
ld Transport
T
t Theorem
Th
to solve many practical fluids problems
Mt St
Mt.
St. Helens