1. science
2. physics

CE 374 FLUID MECHANICS
Prepared by: Dr. Mete Köken
1st Revision by: Dr. Elçin Kentel
2nd Revision by: Dr. M. Ali Kökpınar
CHAPTER 3
KINEMATICS
Oymapınar Dam
3.1. Definition of Kinematics
Kinematics is the description of motion of fluid.
Kinematics does not deal with the forces which cause the
motion.
In fluid mechanics, we can describe the fluid motion in
two ways:
1. Mathematical description
2. Geometrical description
3.1. Definition of Kinematics
1. Mathematical description of motion is the definition of
velocity, accelaration & deformation mathematically.
We can describe the flow mathematically in two approaches:
•
Lagrangian approach
•
Eularian approach
2. Geometrical description :
In fluid flow it is difficult to visualize the flow. Therefore,
to understand & visualize the flow, we try to describe it
geometrically by introducing concepts of pathline, streamline
& streakline.
Kinematics (Graphically)
Mathematical Description of Fluid Motion
Lagrangian approach
Eulerian approach
Position vector
Velocity vector
Acceleration vector
Geometrical Description
Pathline
Streamline
Streakline
Rigid body translation
Linear deformation
Rigid body rotation
Angular deformation
Flow Visualization
Experimental
Computational
3.2 Mathematical description
a. Lagrangian Approach
Lagrangian approach of fluid flow tracks the position &
velocity of individual particles.
For example the velocity & position of a particle can be
expressed as scalar functions of time & its position at
time t=0.
Difficult to use for practical flow analysis.
Hard to follow each particle!
However, useful for specific applications such as:
Sprays, particles, bubble dynamics
Named after Italian mathematician Joseph Louis
Lagrange (1736-1813).
Lagrangian approach
Velocity of a fluid particle is:
V
z
r 0  x0 i  y0 j  z0 k
at t=t0
x=x0
y=y0
z=z0
r 0 (t0 )
r  xi  y j  z k
dx
u
dt
r (r 0 , t )
dr
dt

r0
v
dy
dt
w
dz
dt
y
Acceleration of a fluid particle is:
x
( x0 , y0 , z0 )  Locates the starting point of each particle
Parametric equations of the
path of the fluid particle ‘0’
r
t
x  x(x 0 , y0 , z 0 , t)
y  y( x 0 , y 0 , z 0 , t )
z  z( x 0 , y 0 , z 0 , t )
a
V
t

r0
dV
dt
b. Eulerian approach
Named after Swiss mathematician Leonhard Euler (1707-1783)
Eulerian approach of fluid flow  In a deformable system
such as a fluid, there are an infinite number of particles
whose motions are to be described, which makes Lagrangian
approach difficult to manage. So we employ spatial
coordinates to help identify particles in a flow. We fix a
space in the flow, & describe the motion as the fluid
particles pass thru this region.
We define field variables which are functions of space and
time:
Pressure field
Velocity field
Acceleration field
 P = P(x,y,z,t)
 V = V(x,y,z,t)
 a = a(x,y,z,t).
The region of flow that is being considered  flow field.
Eulerian Approach & Velocity Field
V  u i  v j  wk
u  u ( x, y , z , t )
v  v ( x, y , z , t )
w  w( x, y, z, t )
z
a  ax i  a y j  az k
(x,y,z)
a x  a x ( x, y , z , t )
y
a y  a y ( x, y , z , t )
a z  a z ( x, y , z , t )
x
p  p ( x, y , z , t )
3.3 Derivatives, Acceleration of a Fluid Particle
Derivatives
Let f = f ( x, y, z, t ) represents an arbitrary flow
property, where x, y and z are fixed coordinates, and t
is time.
The total change in f is given by:
f
f
f
f
df  dt  dx  dy  dz
t
x
y
z
Total (Material) Derivative
Particle A at time t+t
Particle A at time t
f (x+dx, y+dy, z+dz, t+dt) = ft+t
f (x,y,z,t) = ft
df A 
f
f
f
f
dt  dxA  dyA  dz A
t
x
y
z
dxA  udt, dyA  vdt, dzA  wdt
df A Df f
f
f
f


u v  w
dt
Dt t
x
y
z
Substitute into above
equation and divide by dt:
Total (Material) Derivative
Df f
f
f
f

u v  w
Dt t
x
y
z
Total
Local
derivative derivative
Convective
derivative
In terms of Del operator:
Df f

 (V  ) f
Dt t



 i
j k
x y
z
V 
is the transport operator
Acceleration
Df f

 (V  ) f
Dt t
let f = V
DV V

 (V  )V  a
Dt
t
DV
a
Dt
Total
Acceleration
=
ax 
Du u
u
u
u

u v  w
Dt t
x
y
z
ay 
Dv v
v
v
v
 u v  w
Dt t
x
y
z
az 
Dw w
w
w
w

u
v
w
Dt
t
x
y
z
V
t
+
Local
Acceleration
V
V
V
u
v
w
x
y
z
Convective
Acceleration
3.4 Geometric Description of Flow with Flow
Lines
Three lines help us in describing a flow field:
Pathline
 is a result of Lagrangian Approach,
Streamline
 is a result of Eulerian Approach,
Streakline
 is a result of Experimental Studies.
Pathline – line traced out by a given particle as it
flows from one point to another.
Streamline – line (or more precisely, curve) which is
everywhere tangent to the velocity fields.
Streakline – curve joining all particles in a flow that
have previously passed through a given point.
Pathline
Pathline is the curve traversed by any one particle in the
flow, & corresponds to the trajectory. The Lagrangian
description leads the definition of pathline. The relation
between the velocity & pathline is given by:
dx
dy
dz
u
v
w
dt
dt
dt
For a given velocity, the integration of
velocity equations will give the pathline
in parametric form as:
x(t )   udt
z
y(t )   vdt
r  r (r0 , t )
r 0 (t0 )
y
x
z (t )   wdt
Subject to initial condition: at t=0, (x,y,z)=(x0,y0,z0)
Streamline
A streamline is a line such that the velocity vector of each particle
occupying a point on a streamline is tangent to the streamline.
Let dr be a displacement vector on a streamline and V be the
velocity vector as shown in figure: V // dr
If two vectors are parallel, then their cross product must be zero.
V
y
ds
V
dr
dx
v
V ×dr = 0
this equation will give us the
equation of streamlines:
dy
dx dy dz


u
v
w
VV
u
x
STREAMLINE
IN
A
FLOW
FIELD
The trajectories of velocity field are called streamlines.
Streamlines are the instantaneous pictures of flow.
V ×dr = 0
Streamlines
NASCAR surface pressure
contours and streamlines
Airplane surface pressure
contours, volume streamlines, and
surface streamlines
Streamlines in hurricanes
Streakline
A line connecting all particles that have
passed successively a given point in space is
called streakline.
For a streakline to occur,
1.
a certain duration must pass, and
2. a fixed point is needed.
Streakline
Let us draw a streakline. Consider a chimney as a fixed point in space.
Let us draw the pathline of particles leaving out of chimney at different
times. During a period of 5 seconds, four different particle has left the
chimney. At time t=5 sec they occupy the positions shown by a number 5.
5
5
5
4
4
4
3 3
5
4
3
2
2
1
0
Pathline of particle who
left chimney at t=0
Pathline of particle who
left chimney at t=1
Pathline of particle who
left chimney at t=2
Pathline of particle who
left chimney at t=3
Streakline at t=5 sec
Flowlines generated by smoke ejected
from a chimney
Flowlines generated by smoke ejected
from a chimney
At time =to
B(t=t0)
A(t=t0)
wind
tt0
Pathlines
Streamlines
Streaklines
Flowlines generated by smoke ejected
from a chimney
C(t=t1)
wind
t0<t<t1
B(t=t1)
A(t=t1)
Pathlines
Streamlines
Streaklines
Flowlines generated by smoke ejected
from a chimney
C(t=t2)
D(t=t2)
wind
A(t=t2)
B(t=t2)
t1<tt2
Pathlines
Streamlines
Streaklines
For steady flow, pathlines, streamlines, & streaklines
are identical.
For unsteady flow, they can be very different.
Streamlines are an instantaneous picture of the flow
field.
Pathlines & streaklines are flow patterns that have a
time history associated with them.
3.5 Deformation of Fluid Elements
In general, deformation is change of position & shape
during a motion.
We can identify four different types of deformation:
1.
2.
3.
4.
Rigid body translation,
Linear deformation,
Rigid body rotation,
Angular deformation
=
General
motion
+
Rigid body
translation
+
Linear
deformation
+
Rigid body
rotation
Angular
deformation
Rigid Body Translation
In rigid body translation, the shape and volume remain the
same, only the coordinates of the centroid of the body will
change  all points in the element have the same velocity.
y
ut
v
O
t=t’
t=0
O1
vt
u
Translation during time interval t in the
absence of velocity gradients
t=t’’
x
Linear Deformation
Let us consider an elementary volume which has sides x,
y, and z. If velocity gradients are present then the body
will undergo a volumetric deformation during time interval
t  difference in velocity will cause a stretching of the
volume element
B
y
u
x
C1
u
u
A
u
x
x
 u 
(x )   x t
 x 
x
xyz
u
O
C
A1
u
x
x
Change in 
during t
The rate of
change of  per
unit volume
u
u
 =  x t zy    t
 x 
 x 
1
()
1 d () u
lim


 t  0 t
 dt
x
v
w
Including
y and z equivalent terms in y and z directions for 3D
case
1 d() u v w

 
   V  div V
 dt
x y z
Volumetric
dilatation rate
Angular Deformation
u
v
and
Consider the velocity gradients,
y
x
u
B
y
(v / x)xt v
  tan( ) 
 t
x
x
u
y
y
(u / y )yt u
  tan( ) 
 t
y
y
B1

v
O u
x

A1 v  v x
x
A
    
d (  )
 v u
 xy 
 lim
 

t

0
dt
t x y
Rate of Angular Deformation
Similarly,
u
u
y
y
B

  xz
∂
w ∂
u


∂
x ∂
z
  yz
∂
v ∂
w


∂
z ∂
y
B1
y
A1
v
O
u
x

A
v
v  x
x
Rigid Body Rotation
The rotation, ωz, of the element about the z axis is defined as the
average of angular velocities ωA-A and ωB-B of the two mutually
perpendicular lines A-A and B-B.
u
B
u
y
y

(v / x )xt v
  tan() 
 t
x
x
( u / y )yt
u
  tan( ) 
  t
y
y
A

A
y
x

v
u
B
x
v
v  x
x
a
+
 v

t 0 t
x
 A A  lim

u

t 0 t
y
 B  B  lim
1
1  v u 
 z  (  A A   B  B )    
2
2  x y 
y
Similarly,
1  w v 
x  
 
2  y z 
and
Angular velocity vector can be
written as:
1  u w 
y   

2  z x 
  x i  y j  z k
i
1
1
1 
  curlV    V 
2
2
2 x
u
j

y
v
k

z
w
Vorticity
In fluid mechanics, twice of angular velocity is called
vorticity, 
  2   V  curlV
A flow for which  = 0 is called irrotational, otherwise
if   0, it is called rotational.
3.5. Classification of Fluid Flow:
One-, two and three-dimensional flows
Dimensionality of a flow field can be determined as
the minimum number of space coordinates required to
express the velocity vector.
Parallel plates
y
Diverging plates
y
x
x
V=V(y) 1-D Flow
V=V(x,y) 2-D Flow
Flow over a sphere
z
V=V(x,y,z) 3-D Flow
y
x
Steady and Unsteady Flows
A flow field is said to be steady if all fluid and flow
properties are independent of time (/t=0).
Conversely, a flow field is classified as unsteady if
any flow or fluid property is time dependent (/t0).
Uniform and non-uniform flows
A flow is classified as uniform when identical velocity
profiles are observed at different sections of the
flow field. Mathematically
Flow is uniform when (V)V = 0 (eg. parallel plates)
Flow is non-uniform when (V)V  0 (eg. diverging plates)
Compressible and incompressible flows
Fluid flows in which fluid particle density remain
constant (i.e. fluid elements deform without volumetric
change) are called incompressible.
d d


0


 V  0
Rotational and irrotational flows
   V
Irrotational if  = 0.
Rotational if   0.
Viscous and inviscid flows
Flow fields in which the shearing stresses are assumed
to be negligible are said to be inviscid, nonviscous or
frictionless.
Inviscid if m=0
=0
Viscous if m0
0
Laminar and turbulent flows
UL
Re 
m
Laminar
if Re < Re-critical
Turbulent if Re  Re-critical
Reynolds Tranport 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.
Reynolds Tranport Theorem (RTT)
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).
Reynolds Tranport 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).
Reynolds Tranport 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

  bd   bV  ndA
cs
Dt
t CV
DBsys
RTT – Special Cases
For moving and/or deforming control volumes,


( b)d   bVr  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.
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