Chapter four fluid mechanics

Introduction
 Fluid Kinematics deals with the motion of fluids
without considering the forces and moments which
create the motion.
 Items discussed in this Chapter.
 Introduction
 Velocity Field
 Acceleration Field
 Control Volume and System Representation
 Reynolds Transport Theorem
 Examples

Introduction
• If a Fluids is subjected to shear force then it tends to flow
• Fluids that are subjected to pressure imbalance resulted in
fluid flow
• In kinematics we are not concerned with the force, but the
motion, thus, we are interested in flow visualization.
• We can learn a lot about flows from watching.

Velocity field-Continuum Hypothesis
 Continuum Hypothesis: The flow is made of tightly packed fluid
particles that interact with each other. Each particle consists of
numerous molecules, and we can describe the field variables
velocity, acceleration, pressure, and density of these particles at a
given time.

Velocity field
 Different particles in fluid flow, which move at different
velocities and may be subjected to different accelerations.
 The velocity and acceleration of a fluid particle may change
both with respect to time and space.
 In the study of fluid flow it is necessary to observe the motion
of the fluid particles at various points in space and at a
successive instants of time.

Velocity field - Lagarangian Vs Eulerian method
 Generally there are two methods by which the motion of a fluid
may be described, Lagarangian and Eulerian method.
 In the Lagarangian method any individual fluid particle is selected ,
and observation is made about the behavior of this particle during
its course of motion through space (Control mass approach).
 In the Eulerian method any point in the space occupied by the fluid
is selected and observation is made of whatever changes of
velocity, density and pressure which take place at that point (control
volume approach).

Lagrangian Description
 Lagrangian description of fluid flow tracks the position and velocity
of individual particles.
 Based upon Newton's laws of motion, it is difficult to use Lagrangian
description for practical flow analysis as.
 Fluids are composed of billions of molecules.
 Interaction between molecules hard to describe/model.
 However, useful for specialized applications in
 Sprays, particles, bubble dynamics, rarefied gases.
 Coupled Eulerian-Lagrangian methods.
 Named after Italian mathematician Joseph Louis Lagrange (1736-
1813).

Eulerian Description
 In a Eulerian description of fluid flow a flow domain or control volume is
defined by which fluid flows in and out.
 We define field variables which are functions of space and time.
 Pressure field, P = P(x, y, z, t)
 Velocity field,
 Acceleration field,
 These (and other) field variables define the flow field.
 Well suited for formulation of initial boundary-value problems (PDE's).
 Named after Swiss mathematician Leonhard Euler (1707-1783).
( ) ( ) ( ), , , , , , , , ,V u x y z t i v x y z t j w x y z t k= + +
  
( ) ( ) ( ), , , , , , , , ,x y za a x y z t i a x y z t j a x y z t k= + +
 
( ), , ,a a x y z t=
 
( ), , ,V V x y z t=
 

Lagarangian Vs Eulerian method
Lagrangian
Measurement of fluid temperature
Eulerian
Eulerian methods are commonly
used in fluid experiments or
analysis—a probe placed in a
flow.

Example: Coupled Eulerian-Lagrangian Method
 Simulation of micron-scale
airborne probes. The probe
positions are tracked using a
Lagrangian particle model
embedded within a flow field
computed using an Eulerian
CFD code.
http://www.ensco.com/products/atmospheric/gem/gem_ovr.htm

Velocity field- Uniform Vs Non-Uniform
 Uniform flow: If the flow velocity is the same magnitude and direction
at every point in the fluid at the given instant of time, it is said to be
uniform.
 Non-uniform: If at a given instant, the velocity is not the same at
every point the flow is non-uniform. (In practice, fluid that flows near
a solid boundary will be non-uniform – as the fluid at the boundary
must take the speed of the boundary, usually zero.
0=
∂
∂
s
V
0≠
∂
∂
s
V

Velocity field- Steady Vs Unsteady
 Steady Flow: The velocity at a given point in space does not vary
with time.
 Very often, we assume steady flow conditions for cases where there
is only a slight time dependence, since the analysis is “easier”
 Unsteady Flow: The velocity at a given point in space does vary with
time.
 Almost all flows have some unsteadiness. In addition, there are
periodic flows, non-periodic flows, and completely random flows.

Velocity field- Steady Vs Unsteady
 Periodic flow: “fuel injectors” creating a periodic swirling in the
combustion chamber. Effect occurs time after time.
 Random flow: “Turbulent”, gusts of wind, splashing of water in the
sink
 Steady or Unsteady only pertains to fixed measurements, i.e.
exhaust temperature from a tail pipe is relatively constant “steady”;
however, if we followed individual particles of exhaust they cool!

Velocity field- Steady Vs Uniform
 Combining the above we can classify any flow into one of four type:
 Steady uniform flow. Conditions do not change with position in the
stream and with time at a point.
 An example is the flow of water in a pipe of constant diameter
at constant velocity.
 Steady non-uniform flow. Conditions change from point to point in the
stream but do not change with time at a point.
 An example is flow in a tapering pipe with constant velocity at
the inlet - velocity will change as you move along the length of
the pipe toward the exit.

Velocity field- Steady Vs Uniform
 Unsteady uniform flow. At a given instant in time the conditions at
every point are the same, but will change with time.
 Example : An example is a pipe of constant diameter connected
to a pump pumping at a constant rate which is then switched off.
 4. Unsteady non-uniform flow. Every condition of the flow may
change from point to point and with time at every point.
 For example waves in a channel.

Velocity Field- 1D, 2D, and 3D Flows
 Most fluid flows are complex three dimensional, time-dependent
phenomenon, however we can make simplifying assumptions allowing
an easier analysis or understanding without sacrificing accuracy. In
many cases we can treat the flow as 1D or 2D flow.
 Three-Dimensional Flow: All three velocity components are
important and of equal magnitude. Flow past a wing is complex 3D
flow, and simplifying by eliminating any of the three velocities would
lead to severe errors.

Velocity Field- 1D, 2D, and 3D Flows
 Two-Dimensional Flow : In many situations one of the velocity
components may be small relative to the other two, thus it is
reasonable in this case to assume 2D flow.
 One-Dimensional Flow: In some situations two of the velocity
components may be small relative to the other one, thus it is
reasonable in this case to assume 1D flow. There are very few
flows that are truly 1D, but there are a number where it is a
reasonable approximation.

Flow Visualization
 Flow visualization is the visual examination of flow-field
features.
 Important for both physical experiments and numerical (CFD)
solutions.
 Numerous methods
 Streamlines and streamtubes
 Pathlines
 Streaklines
 Timelines
 Refractive techniques
 Surface flow techniques

Velocity field - Streamlines
 A Streamline is an imaginary curve
drawn through the flowing fluid in such a
way that the tangent to it at any point
gives the direction of the velocity at that
point.
 Because the fluid is moving in the
same direction as the streamlines,
fluid can not cross a streamline.
 Streamlines can not cross each other.
If they were to cross this would
indicate two different velocities at the
same point. This is not physically
possible.
 The above point implies that any
particles of fluid starting on one
streamline will stay on that same
streamline throughout the fluid.

Streamlines
 A Streamline is a curve that is
everywhere tangent to the
instantaneous local velocity vector.
 Consider an arc length
 must be parallel to the local
velocity vector
 Geometric arguments results in the
equation for a streamline
dr dxi dyj dzk= + +
 
dr

V ui vj wk= + +
  
dr dx dy dz
V u v w
= = =

Velocity field - Streamlines
NASCAR surface pressure contours
and streamlines
Airplane surface pressure contours,
volume streamlines, and surface
streamlines

Velocity field - Pathlines
( ) ( ) ( )( ), ,particle particle particlex t y t z t
 A Pathline is the actual path
traveled by an individual fluid
particle over some time period.
 Same as the fluid particle's
material position vector
 Particle location at time t:
 Particle Image Velocimetry (PIV) is
a modern experimental technique
to measure velocity field over a
plane in the flow field.
start
t
start
t
x x Vdt= + ∫
 

Velocity field - Streakline
 A Streakline is the locus of
fluid particles that have passed
sequentially through a
prescribed point in the flow.
 Easy to generate in
experiments: dye in a water
flow, or smoke in an airflow.

Comparisons
 For steady flow, streamlines, pathlines, and streaklines are
identical.
 For unsteady flow, they can be very different.
 Streamlines are an instantaneous picture of the flow field
 Pathlines and Streaklines are flow patterns that have a time
history associated with them.
 Streakline: instantaneous snapshot of a time-integrated flow
pattern.
 Pathline: time-exposed flow path of an individual particle.

Plots of Data
 A Profile plot indicates how the value of a scalar property varies
along some desired direction in the flow field.
 A Vector plot is an array of arrows indicating the magnitude and
direction of a vector property at an instant in time.
 A Contour plot shows curves of constant values of a scalar property
for magnitude of a vector property at an instant in time.

Acceleration Field
Lagrangian Frame:
Eulerian Frame: we describe the acceleration in terms of position and time
without following an individual particle. This is analogous to describing the
velocity field in terms of space and time.
A fluid particle can accelerate due to a change in velocity in time (“unsteady”)
or in space (moving to a place with a greater velocity).

Acceleration Field: Material (Substantial) Derivative
time dependence
spatial dependence
We note:
Then, substituting:
The above is good for any fluid particle, so we drop “A”:

Writing out these terms in vector components:
x-direction:
y-direction:
z-direction:
Writing these results in “short-hand”:
where,
k
z
j
y
i
x
ˆˆˆ()
∂
∂
+
∂
∂
+
∂
∂
=∇ ,
Fluid flows experience fairly
large accelerations or
decelerations, especially
approaching stagnation
points.

Acceleration field
 The time dependant term in the acceleration field is called the local
acceleration and is nonzero only for unsteady flows.
 The spacial dependant term in the acceleration field is called the
advective acceleration and accounts for the effect of the fluid
particle moving to a new location in the flow, where the velocity is
different.
 The total derivative operator d/dt is called the material derivative
and is often given special notation, D/Dt.
 Advective acceleration is nonlinear: source of many phenomenon
and primary challenge in solving fluid flow problems.
 Advective acceleration provides ``transformation'' between
Lagrangian and Eulerian frames.
 Other names for the material derivative include: total, particle,
Lagrangian, Eulerian, and substantial derivative.

Applied to the Temperature Field in a Flow:
The material derivative of any variable is the rate at which that variable changes
with time for a given particle (as seen by one moving along with the fluid—
Lagrangian description).

Acceleration Field: Unsteady Effects
If the flow is unsteady, its paramater values at any location may change with
time (velocity, temperature, density, etc.)
The local derivative represents the unsteady portion of the flow:
If we are talking about velocity, then the above term is local acceleration.
In steady flow, the above term goes to zero.
If we are talking about temperature, and V = 0, we still have heat transfer
because of the following term:
0 0 0
=

Acceleration Field: Unsteady Effects
Consider flow in a constant diameter pipe, where the flow is assumed to be
spatially uniform:
0
0 00 0

Acceleration Field: Convective Effects
The portion of the material derivative represented by the spatial derivatives is
termed the convective term or convective accleration:
It represents the fact the flow property associated with a fluid particle may
vary due to the motion of the particle from one point in space to another.
Convective effects may exist whether the flow is steady or unsteady.
Example 1:
Example 2:
Acceleration = Deceleration

Control Volume and System Representations
Systems of Fluid: a specific identifiable quantity of matter that may consist of
a relatively large amount of mass (the earth’s atmosphere) or a single fluid
particle. They are always the same fluid particles which may interact with their
surroundings.
Control Volume: is a volume or space through which the fluid may flow,
usually associated with the geometry.
Example: following a system the fluid passing through a compressor
We can apply the equations of motion to the fluid mass to describe their
behavior, but in practice it is very difficult to follow a specific quantity of matter.
When we are most interested in determining the forces put on a fan, airplane,
or automobile by the air flow past the object rather than following the fluid as it
flows along past the object.
Identify the specific volume in space and analyze the fluid flow within,
through, or around that volume.

Control Volume and System Representations
Fixed Control Volume:
Fixed or Moving
Control Volume:
Deforming Control
Volume:
Surface of the Pipe
Surface of the Fluid
Volume Around The
Engine
Inflow
Outflow
Outflow Deforming Volume

Reynolds Transport Theorem: Preliminary Concepts
All the laws of governing the motion of a fluid are stated in their basic form in
terms of a system approach, and not in terms of a control volume.
The Reynolds Transport Theorem allows us to shift from the system
approach to the control volume approach, and back.
General Concepts:
B represents any of the fluid properties, m represent the mass, and b
represents the amount of the parameter per unit volume.
Examples:
Mass b = 1
Kinetic Energy b = V2/2
Momentum b = V (vector)
B is termed an extensive property, and b is an intensive property. B is
directly proportional to mass, and b is independent of mass.

Reynolds Transport Theorem: Preliminary Concepts
For a System: The amount of an extensive property can be calculated by
adding up the amount associated with each fluid particle.
Now, the time rate of change of that system:
Now, for control volume:
For the control volume, we only integrate over the control volume, this is
different integrating over the system, though there are instance when
they could be the same.

Reynolds Transport Theorem: Derivation
Consider a 1D flow through a fixed control volume between (1) and (2):
CV, and system at t1
System at t2
System at t2
Writing equation in terms of the extensive parameter:
Originally,
At time 2:
Divide by δt:

Noting,
Let,
Time rate of change of mass within the control volume:
The rate at which the extensive property flows out of the control surface:
(1) (2) (3) (4)
(1)
(2)
(4)

The rate at which the extensive property flows into the control surface:
(3)
Now, collecting the terms:
or
Restrictions for the above Equation:
1) Fixed control volume
2) One inlet and one outlet
3) Uniform properties
4) Normal velocity to section (1) and (2)

The Reynolds Transport Theorem can be derived for more general conditions.
Result:
This form is for a fixed non-deforming control volume.

Reynolds Transport Theorem: Physical Interpretation
(1) (2) (3)
(1) The time rate of change of the extensive parameter of a system, mass,
momentum, energy.
(2) The time rate of change of the extensive parameter within the control
volume.
(3) The net flow rate of the extensive parameter across the entire control
surface. “outflow across the surface”
“inflow across the surface”
“no flow across the surface”
Mass flow rate:

Reynolds Transport Theorem: Analogous to Material Derivative
Time dependant Portion Convective Portion
Steady Effects:
Unsteady Effects (inflow = outflow):

Kinematic Description
 In fluid mechanics, an element
may undergo four fundamental
types of motion.
a) Translation
b) Rotation
c) Linear strain
d) Shear strain
 Fluids motion and deformation is
best described in terms of rates
a) velocity: rate of translation
b) angular velocity: rate of rotation
c) linear strain rate: rate of linear
strain
d) shear strain rate: rate of shear
strain

Rate of Translation and Rotation
 To be useful, these rates must be expressed in terms of velocity
and derivatives of velocity
 The rate of translation vector is described as the velocity vector.
In Cartesian coordinates:
 Rate of rotation at a point is defined as the average rotation
rate of two initially perpendicular lines that intersect at that
point. The rate of rotation vector in Cartesian coordinates:
V ui vj wk= + +
  
1 1 1
2 2 2
w v u w v u
i j k
y z z x x y
ω
   ∂ ∂ ∂ ∂ ∂ ∂ 
= − + − + −    ∂ ∂ ∂ ∂ ∂ ∂    
 

Linear Strain Rate
 Linear Strain Rate is defined as the rate of increase in length per unit length.
 In Cartesian coordinates
 Volumetric strain rate in Cartesian coordinates
 Since the volume of a fluid element is constant for an incompressible flow, the
volumetric strain rate must be zero.
, ,xx yy zz
u v w
x y z
ε ε ε
∂ ∂ ∂
= = =
∂ ∂ ∂
1
xx yy zz
DV u v w
V Dt x y z
ε ε ε
∂ ∂ ∂
= + + = + +
∂ ∂ ∂

Shear Strain Rate
 Shear Strain Rate at a point is defined as half of the rate of
decrease of the angle between two initially perpendicular lines
that intersect at a point.
 Shear strain rate can be expressed in Cartesian coordinates
as:
1 1 1
, ,
2 2 2
xy zx yz
u v w u v w
y x x z z y
ε ε ε
   ∂ ∂ ∂ ∂ ∂ ∂ 
= + = + = +    ∂ ∂ ∂ ∂ ∂ ∂    

Vorticity and Rotationality
 The vorticity vector is defined as the curl of the velocity vector
 Vorticity is equal to twice the angular velocity of a fluid particle.
 Cartesian coordinates
 In regions where ζ = 0, the flow is called irrotational.
 Elsewhere, the flow is called rotational.
Vζ = ∇×
  
2ζ ω=
 
w v u w v u
i j k
y z z x x y
ζ
   ∂ ∂ ∂ ∂ ∂ ∂ 
= − + − + −    ∂ ∂ ∂ ∂ ∂ ∂    
  

Chapter four fluid mechanics

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