Download chapter (iii) fluid flow

CHAPTER (III)
KINEMATICS OF FLUID FLOW
3.1: Types of Fluid Flow.
3.1.1: Real - or - Ideal fluid.
3.1.2: Laminar - or - Turbulent Flows.
3.1.3: Steady - or - Unsteady flows.
3.1.4: Uniform - or - Non-uniform Flows.
3.1.5: One, Two - or - Three Dimensional Flows.
3.1.6: Rational - or - Irrational Flows.
3.2: Circulation - or - Vorticity.
3.3: Stream Lines, Flow Field and Stream Tube.
3.4: Velocity and Acceleration in Flow Field.
3.5: Continuity Equation for One Dimensional Steady Flow.
1
Fluid Flow Kinematics
• Fluid Kinematics deals with the motion of
fluids without considering the forces and
moments which create the motion.
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
Types of fluid Flow
1. Real and Ideal Flow:
If the fluid is considered frictionless with zero viscosity it is called ideal.
In real fluids the viscosity is considered and shear stresses occur causing
conversion of mechanical energy into thermal energy
Ideal
Friction = 0
Ideal Flow ( μ =0)
Energy loss =0
Real
Friction = o
Real Flow ( μ ≠0)
Energy loss = 0
2. Steady and Unsteady Flow
Steady flow occurs when conditions of a point in a flow field don’t change
with respect to time ( v, p, H…..changes w.r.t. time
 
0
 t 
 
0
 t 
H=constant
steady
unsteady
H ≠ constant
V=constant
Steady Flow with respect to time
•Velocity is constant at certain
position w.r.t. time
V ≠ constant
Unsteady Flow with respect to time
•Velocity changes at certain position
w.r.t. time
3. Uniform and Non uniform Flow
Y
Y
x
x
Uniform Flow means that the
velocity is constant at certain time in
different positions (doesn’t depend
on any dimension x or y or z(
 
0
x 
 
0
x 
Non- uniform Flow means velocity
changes at certain time in different
positions ( depends on dimension
x or y or z(
uniform
Non-uniform
4. One , Two and three Dimensional Flow :
y
x
One dimensional flow means that
the flow velocity is function of one
coordinate
V = f( X or Y or Z )
Two dimensional flow means that
the flow velocity is function
of two coordinates
V = f( X,Y or X,Z or Y,Z )
Three dimensional flow means that
the flow velocity is function
of there coordinates
V = f( X,Y,Z)
4. One , Two and three Dimensional Flow (cont.)
A flow field is best characterized by its velocity
distribution.
•
A flow is said to be one-, two-, or threedimensional if the flow velocity varies in one, two,
or three dimensions, respectively.
•
However, the variation of velocity in certain
directions can be small relative to the variation in
other directions and can be ignored.
•
The development of the velocity profile in a circular pipe. V = V(r, z) and thus the
flow is two-dimensional in the entrance region, and becomes one-dimensional
downstream when the velocity profile fully develops and remains unchanged in
8
the flow direction, V = V(r).
5. Laminar and Turbulent Flow:
In Laminar Flow:
•Fluid flows in separate layers
•No mass mixing between fluid layers
•Friction mainly between fluid layers
•Reynolds’ Number (RN ) < 2000
•Vmax.= 2Vmean
Vmean
Vmax
In Turbulent Flow:
•No separate layers
•Continuous mass mixing
•Friction mainly between fluid and pipe
walls
•Reynolds’ Number (RN ) > 4000
•Vmax.= 1.2 Vmean
Vmean
Vmax
5. Laminar and Turbulent Flow (cont.):
Rotational and irrotational flows:
The rotation is the average value 
ofr rotation of two
lines in the flow.
(i) If this average = 0 then there is no rotation and the flow is called irrotational flow
6. Streamline:
A Streamline is a curve that is everywhere tangent to it at any instant
represents the instantaneous local velocity vector.
tan  
dy
v

dx
u

u
v

dx
dy
in
general
for 3  D
u
v
w Stream line equation


dx
dy
dz
Where :
u velocity component in -X- direction
z
v velocity component in-Y- direction
w velocity component in -Z- direction
w
V
v
y
u
x
V 
u 2  v2
 w2
velocity vector can written as:
V  ui  vj  wk
Where :
i, j, k are the unit vectors in +ve x, y, z directions
Acceleration Field
•
From Newton's second law,
Fparticle  m particle a particle
•
The acceleration of the particle is the time derivative of the particle's velocity.
•
However, particle velocity at a point is the same as the fluid velocity,
a particle 
dVparticle
dt
V  f
n.
( x, y , z , t )
Mathematically the total derivative equals the sum of the partial derivatives
u
u
u
u
dx 
dy 
dz 
dt
x
y
z
t
du
u dx
u dy
u dz
u





dt
x dt
y dt
z dt
t
u
u
u
u

u
v
w
x
y
z
t
u
u
u
u
u
v
w

x
y
z
t
du 
ax
ax
ax
Convective component
Local component
Similarly :
ay  u
az  u
a
v
v
v
v
v
w

x
y
z
t
w
w
w
w
v
w

x
y
z
t
a2x  a2 y  a2z
NASCAR surface pressure contours
and streamlines
Airplane surface pressure contours,
volume streamlines, and surface
streamlines
7. Streamtube:
• Is a bundle of streamlines
• fluid within a streamtube remain constant
and cannot cross the boundary of the streamtube.
(mass in = mass out)
Types of motion or deformation of
fluid element
Linear translation
Rotational translation
Linear deformation
angular deformation
8- Rotational Flow & Irrotational Flow:
The rate of rotation can be expressed or equal to the angular velocity vector(
):

1  w
v 
1  u
w 
  
  i  


2  y z 
2  z x 
Note:
x 

y
1  w
v 




2  y
z 

1  u
w 




2  z
x 
z 
1  v
u


2
y
 x




j
1  v u 
 
 k
2  x y 
The flow is side to be rotational if :
x
or  y
or  z

0
The fluid elements are rotating in space (see Fig. 4-44 )
The flow is side to be irrotational if :
x

y

z 
0
The fluid elements don’t rotating in space (see Fig. 4-44 )
rotational flow
Irrotational flow
9- Vorticity ( ξ ):
Vorticity is a measure of rotation of a fluid particale
Vorticity is twice the angular velocity of a fluid particle
x
 w
v 




y

z


u
w 
y  



x 
 z
 v
u 
z  



x

y


10- Circulation ( Г ):
The circulation ( Г ) is a measure of rotaion and is defined as the line integral
of the tangential component of the velocity taken around a closed curve in
the flow field.
+
θ
. cos θ
NOTE:
The flow is irrotational if
ω=0,
ξ=0,
Г=0
For 2-D Cartesian Coordinates
Y
u
v
dy
u
dy
y
+
dx
u


d  udx  (v 
v
u
dx)dy  (u 
dy )dx  vdy
x
y
v u
(

) dxdy
x
y
  z . area
Г = ξ . area
v
v
dx
x
x
Conservation of Mass ( Continuity Equation
( Mass can neither be created nor destroyed ) )
The general equation of continuity for three dimensional steady flow
( w 
w
dz ).dxdy
z
v.dx.dz
z
( u 
dz
u.dy.dz
dx
x
dy
y
( v 
v
dy ).dx.dz
y
w.dxdy
u
dx).dy.dz
x
Net mass in x-direction=
v.dx.dz-
Net mass in y-direction=
Net mass in z-direction=
( u 
u.dy.dz-

u
dx.dy.dz
x
v
( v 
dy ).dx.dz
=
y

v
dx.dy.dz
y
w
dz ).dx.dy =
z

w
dx.dy.dz
z
( w 
w.dx.dy-
u
dx).dy.dz
=
x
Σ net mass = mass storage rate

u
dx.dy.dz
x

u
x



v
dx.dy.dz
y
v
y

w
z

w
dx.dy.dz
z
=

u
w
v



t
x
z
y

t
=
0
=

( dx.dy.dz )
t
 u v w



0
t
x
y
z
General equation fof 3-D , unsteady and compressible fluid
Special cases:
1- For steady compressible fluid
 )
2- For incompressible fluid ( ρ= constant
0
t

u
v
w



0
t
x
y
z
u
v
w


 0
x
y
z
Note : The above eqn. can be used for steady & unsteady for
incompressible fluid
3- For 2-D :
u
v

0
x
y
u
w

0
x
z
v
w

0
y
z