Fundamentals of Fluid Mechanics
Chapter 4: Fluid Kinematics
Department of Hydraulic Engineering - School of Civil Engineering - Shandong University - 2007
Overview
Fluid Kinematics deals with the motion of fluids
without necessarily 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
Two ways to describe motion are Lagrangian and
Eulerian description
Lagrangian description of fluid flow tracks the position
and velocity of individual particles. (eg. Brilliard ball on a
pooltable.)
Motion is described based upon Newton's laws.
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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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
EXAMPLEL A: A Steady Two-Dimensional
Velocity Field
A steady, incompressible, twodimensional velocity field is
given by
A stagnation point is defined
as a point in the flow field
where the velocity is identically
zero. (a) Determine if there are
any stagnation points in this
flow field and, if so, where? (b)
Sketch velocity vectors at
several locations in the domain
between x = - 2 m to 2 m and y
= 0 m to 5 m; qualitatively
describe the flow field.
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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 at any instant in time
t is the same as the fluid velocity,
V particle  V  x particle  t  , y particle  t  , z particle  t  ,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
Where  is the partial derivative operator and d is the total
derivative operator.
Since
dx particle
dt
a particle
 u,
dy particle
dt
 v,
dz particle
w
dt
V
V
V
V

u
v
w
t
x
y
z
In vector form, the acceleration can be written as
a  x, y , z , t  
dV V

 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
EXAMPLE: Acceleration of a Fluid
Particle through a Nozzle
Nadeen is washing her car,
Howusing
to apply
this equation
to the
a nozzle.
The nozzle
is problem,
V
ft)
V long,
V with
Van
3.90
in (0.325
a particle 
u
v
w
inlet diameter
t
of
x 0.420
y in z
(0.0350 ft) and an outlet
diameter of 0.182 in. The
volume flow rate through the
garden hose (and through the
nozzle) is 0.841 gal/min
(0.00187 ft3/s), and the flow
is steady. Estimate the
magnitude of the acceleration
of a fluid particle moving
down the centerline of the
nozzle.
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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
EXAMPLE B: Material Acceleration of a
Steady Velocity Field
Consider the same velocity
field of Example A. (a)
Calculate the material
acceleration at the point (x = 2
m, y = 3 m). (b) Sketch the
material acceleration vectors at
the same array of x- and y
values as in Example A.
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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
While quantitative study of fluid
dynamics requires advanced
mathematics, much can be
learned from flow visualization
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
EXAMPLE C: Streamlines in the xy
Plane—An Analytical Solution
For the same velocity field of Example
A, plot several streamlines in the right
half of the flow (x > 0) and compare to
the velocity vectors.
where C is a constant of
integration that can be set to
various values in order to plot
the streamlines.
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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
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Streamtube
A streamtube consists of a
bundle of streamlines (Both are
instantaneous quantities).
 Fluid within a streamtube must
remain there and cannot cross the
boundary of the streamtube.
In an unsteady flow, the
streamline pattern may change
significantly with time. the mass
flow rate passing through any
cross-sectional slice of a given
streamtube must remain the same.
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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
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Pathlines
A modern experimental technique called particle
image velocimetry (PIV) utilizes (tracer) particle
pathlines to measure the velocity field over an entire
plane in a flow (Adrian, 1991).
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Pathlines
Flow over a cylinder
Top View
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Side View
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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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Streaklines
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Streaklines
Karman Vortex street
Cylinder
x/D
A smoke wire with mineral oil was heated to generate a rake of Streaklines
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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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Comparisons
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Timelines
A Timeline is a set of
adjacent fluid
particles that were
marked at the same
(earlier) instant in
time.
Timelines can be
generated using a
hydrogen bubble wire.
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Timelines
Timelines produced by a hydrogen bubble wire are used to
visualize the boundary layer velocity profile shape.
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Refractive Flow Visualization Techniques
Based on the refractive property of light waves in fluids with different
index of refraction, one can visualize the flow field: shadowgraph
technique and schlieren technique.
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Plots of Flow Data
Flow data are the presentation of the flow
properties varying in time and/or space.
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 the magnitude of a vector
property at an instant in time.
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Profile plot
Profile plots of the horizontal component of velocity as a
function of vertical distance; flow in the boundary layer
growing along a horizontal flat plate.
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Vector plot
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Contour plot
Contour plots of the pressure field due to flow impinging
on a block.
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