Chapter 4: Fluid Kinematics
Eric G. Paterson
Department of Mechanical and Nuclear Engineering
The Pennsylvania State University
Spring 2005
Note to Instructors
These slides were developed1 during the spring semester 2005, as a teaching aid
for the undergraduate Fluid Mechanics course (ME33: Fluid Flow) in the Department of
Mechanical and Nuclear Engineering at Penn State University. This course had two
sections, one taught by myself and one taught by Prof. John Cimbala. While we gave
common homework and exams, we independently developed lecture notes. This was
also the first semester that Fluid Mechanics: Fundamentals and Applications was
used at PSU. My section had 93 students and was held in a classroom with a computer,
projector, and blackboard. While slides have been developed for each chapter of Fluid
Mechanics: Fundamentals and Applications, I used a combination of blackboard and
electronic presentation. In the student evaluations of my course, there were both positive
and negative comments on the use of electronic presentation. Therefore, these slides
should only be integrated into your lectures with careful consideration of your teaching
style and course objectives.
Eric Paterson
Penn State, University Park
August 2005
1 These
slides were originally prepared using the LaTeX typesetting system (http://www.tug.org/)
and the beamer class (http://latex-beamer.sourceforge.net/), but were translated to PowerPoint for
wider dissemination by McGraw-Hill.
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Chapter 4: Fluid Kinematics
Overview
Fluid Kinematics deals with the motion of fluids
without considering the forces and moments
which create the motion.
Items discussed in this Chapter.
Material derivative and its relationship to Lagrangian
and Eulerian descriptions of fluid flow.
Flow visualization.
Plotting flow data.
Fundamental kinematic properties of fluid motion and
deformation.
Reynolds Transport Theorem
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Chapter 4: Fluid Kinematics
Lagrangian Description
Lagrangian description of fluid flow tracks the
position and velocity of individual particles.
Based upon Newton's laws of motion.
Difficult to use for practical flow analysis.
Fluids are composed of billions of molecules.
Interaction between molecules hard to
describe/model.
However, useful for specialized applications
Sprays, particles, bubble dynamics, rarefied gases.
Coupled Eulerian-Lagrangian methods.
Named after Italian mathematician Joseph Louis
Lagrange (1736-1813).
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Chapter 4: Fluid Kinematics
Eulerian Description
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, V  V  x, y, z , t 
V  u  x, y, z, t  i  v  x, y, z, t  j  w  x, y, z, t  k
Acceleration field, a  a  x, y, z, t 
a  ax  x, y, z, t  i  a y  x, y, z, t  j  az  x, y, z , t  k
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).
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Chapter 4: Fluid Kinematics
Example: Coupled Eulerian-Lagrangian
Method
Global Environmental
MEMS Sensors (GEMS)
Simulation of micronscale 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
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Chapter 4: Fluid Kinematics
Example: Coupled Eulerian-Lagrangian
Method
Forensic analysis of Columbia accident: simulation of
shuttle debris trajectory using Eulerian CFD for flow field
and Lagrangian method for the debris.
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Chapter 4: Fluid Kinematics
Acceleration Field
Consider a fluid particle and Newton's second law,
Fparticle  m particle a particle
The acceleration of the particle is the time derivative of
the particle's velocity.
dVparticle
a particle 
dt
However, particle velocity at a point is the same as the
fluid velocity, V particle  V  x particle  t  , y particle  t  , z particle  t  
To take the time derivative of, chain rule must be used.
V dt V dx particle V dy particle V dz particle
a particle 



t dt x dt
y dt
z dt
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Chapter 4: Fluid Kinematics
Acceleration Field
Since
dx particle
dt
 u,
a particle
dy particle
dt
 v,
dz particle
dt
w
V
V
V
V

u
v
w
t
x
y
z
In vector form, the acceleration can be written as
dV V
a  x, y , z , t  

 V V
dt
t


First term is called the local acceleration and is nonzero only for
unsteady flows.
Second term 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.
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Chapter 4: Fluid Kinematics
Material Derivative
The total derivative operator d/dt is call the material
derivative and is often given special notation, D/Dt.
DV dV V


 V  V
Dt
dt
t


Advective acceleration is nonlinear: source of many
phenomenon and primary challenge in solving fluid flow
problems.
Provides ``transformation'' between Lagrangian and
Eulerian frames.
Other names for the material derivative include: total,
particle, Lagrangian, Eulerian, and substantial
derivative.
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Chapter 4: Fluid Kinematics
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
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Chapter 4: Fluid Kinematics
Streamlines
A Streamline is a curve that is
everywhere tangent to the
instantaneous local velocity
vector.
Consider an arc length
dr  dxi  dyj  dzk
dr must be parallel to the local
velocity vector
V  ui  vj  wk
Geometric arguments results
in the equation for a streamline
dr dx dy dz



V
u
v
w
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Chapter 4: Fluid Kinematics
Streamlines
NASCAR surface pressure contours
and streamlines
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Airplane surface pressure contours,
volume streamlines, and surface
streamlines
Chapter 4: Fluid Kinematics
Pathlines
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
x
particle
 t  , y particle  t  , z particle  t  
Particle location at time t:
t
x  xstart 
 Vdt
tstart
Particle Image Velocimetry
(PIV) is a modern experimental
technique to measure velocity
field over a plane in the flow
field.
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Chapter 4: Fluid Kinematics
Streaklines
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.
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Chapter 4: Fluid Kinematics
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 timeintegrated flow pattern.
Pathline: time-exposed flow path of an individual
particle.
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Chapter 4: Fluid Kinematics
Timelines
A Timeline is the
locus of fluid particles
that have passed
sequentially through a
prescribed point in the
flow.
Timelines can be
generated using a
hydrogen bubble wire.
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Chapter 4: Fluid Kinematics
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.
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Chapter 4: Fluid Kinematics
Kinematic Description
In fluid mechanics, an element
may undergo four fundamental
types of motion.
a)
b)
c)
d)
Translation
Rotation
Linear strain
Shear strain
Because fluids are in constant
motion, 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
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Chapter 4: Fluid Kinematics
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:
V  ui  vj  wk
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:
1  w v 
1  u w 
1  v u 
    i     j    k
2  y z 
2  z x 
2  x y 
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Chapter 4: Fluid Kinematics
Linear Strain Rate
Linear Strain Rate is defined as the rate of increase in length per unit
length.
In Cartesian coordinates
 xx 
u
v
w
,  yy  ,  zz 
x
y
z
Volumetric strain rate in Cartesian coordinates
1 DV
u v w
  xx   yy   zz 
 
V Dt
x y z
Since the volume of a fluid element is constant for an incompressible
flow, the volumetric strain rate must be zero.
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Chapter 4: Fluid Kinematics
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  u v 
1  w u 
1  v w 
 xy     ,  zx     ,  yz    
2  y x 
2  x z 
2  z y 
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Chapter 4: Fluid Kinematics
Shear Strain Rate
We can combine linear strain rate and shear strain
rate into one symmetric second-order tensor called
the strain-rate tensor.
  xx

 ij    yx

 zx
 xy
 yy
 zy
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
u

x

 xz  
1  v u 

 yz      
 2  x y 

 zz  
 1  w  u 
 2  x z 

23
1  u v 
 

2  y x 
v
y
1  w v 
 

2  y z 
1  u w  
 

2  z x  
1  v w  




2  z y  

w


z

Chapter 4: Fluid Kinematics
Shear Strain Rate
Purpose of our discussion of fluid element
kinematics:
Better appreciation of the inherent complexity of fluid
dynamics
Mathematical sophistication required to fully describe
fluid motion
Strain-rate tensor is important for numerous
reasons. For example,
Develop relationships between fluid stress and strain
rate.
Feature extraction and flow visualization in CFD
simulations.
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Chapter 4: Fluid Kinematics
Shear Strain Rate
Example: Visualization of trailing-edge turbulent eddies
for a hydrofoil with a beveled trailing edge
Feature extraction method is based upon eigen-analysis of the strain-rate tensor.
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Chapter 4: Fluid Kinematics
Vorticity and Rotationality
The vorticity vector is defined as the curl of the velocity
vector z    V
Vorticity is equal to twice the angular velocity of a fluid
particle. z  2
Cartesian coordinates
 w v   u w 
 v u 
z    i     j    k
 y z   z x 
 x y 
Cylindrical coordinates
   ru  ur
 1 u z u 
 ur u z 
z 



 e  
 er  
z 
r 

 z
 r 
 r

 ez

In regions where z = 0, the flow is called irrotational.
Elsewhere, the flow is called rotational.
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Chapter 4: Fluid Kinematics
Vorticity and Rotationality
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Chapter 4: Fluid Kinematics
Comparison of Two Circular Flows
Special case: consider two flows with circular streamlines
ur  0, u   r
1    ru  ur
z  

r  r

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

1    r 
 0  ez  2ez
 ez  

r
r



28
K
r
1    ru  ur
z  

r  r

ur  0, u 


1  K 
 0  ez  0ez
 ez  
r  r


Chapter 4: Fluid Kinematics
Reynolds—Transport Theorem (RTT)
A system is a quantity of matter of fixed identity. No
mass can cross a system boundary.
A control volume is a region in space chosen for study.
Mass can cross a control surface.
The fundamental conservation laws (conservation of
mass, energy, and momentum) apply directly to systems.
However, in most fluid mechanics problems, control
volume analysis is preferred over system analysis (for
the same reason that the Eulerian description is usually
preferred over the Lagrangian description).
Therefore, we need to transform the conservation laws
from a system to a control volume. This is accomplished
with the Reynolds transport theorem (RTT).
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Chapter 4: Fluid Kinematics
Reynolds—Transport Theorem (RTT)
There is a direct analogy between the transformation from
Lagrangian to Eulerian descriptions (for differential analysis
using infinitesimally small fluid elements) and the
transformation from systems to control volumes (for integral
analysis using large, finite flow fields).
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Chapter 4: Fluid Kinematics
Reynolds—Transport Theorem (RTT)
Material derivative (differential analysis):
Db b

 V  b
Dt t


General RTT, nonfixed CV (integral analysis):


 b  dV  CS bV ndA
CV
dt
t
dBsys
Mass Momentum
Energy
Angular
momentum
H
r V
B, Extensive properties
m
mV
E
b, Intensive properties
1
V
e


In Chaps 5 and 6, we will apply RTT to conservation of mass, energy, linear
momentum, and angular momentum.
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Chapter 4: Fluid Kinematics
Reynolds—Transport Theorem (RTT)
Interpretation of the RTT:
Time rate of change of the property B of the
system is equal to (Term 1) + (Term 2)
Term 1: the time rate of change of B of the
control volume
Term 2: the net flux of B out of the control
volume by mass crossing the control surface


b  dV   bV ndA

CV t
CS
dt
dBsys
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Chapter 4: Fluid Kinematics
RTT Special Cases
For moving and/or deforming control volumes,


b  dV   bVr ndA

CV t
CS
dt
dBsys
Where the absolute velocity V in the second
term is replaced by the relative velocity
Vr = V -VCS
Vr is the fluid velocity expressed relative to a
coordinate system moving with the control
volume.
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Chapter 4: Fluid Kinematics
RTT Special Cases
For steady flow, the time derivative drops out,
0


b  dV   bVr ndA   bVr ndA

CV t
CS
CS
dt
dBsys
For control volumes with well-defined inlets and
outlets
dBsys d
  bdV  avg bavgVr ,avg A   avg bavgVr ,avg A
dt
dt CV
out
in
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Chapter 4: Fluid Kinematics