chapter (iii) kinematics of fluid flow 3.1: types of fluid flow. 3.1.1: real - or - ideal fluid....
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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.
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Fluid Flow Kinematics
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• 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 timePressure field, Velocity field
, , , , , , , , ,V u x y z t i v x y z t j w x y z t k
, , ,V V x y z t
Acceleration field,
, , , , , , , , ,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
),,,( tzyxPP
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Types of fluid Flow 1. Real and Ideal Flow:
Friction = 0Ideal Flow ( μ =0)Energy loss =0
Friction = oReal Flow ( μ ≠0)Energy loss = 0
IdealReal
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
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2 .Steady and Unsteady Flow
H=constant
V=constant
Steady Flow with respect to time•Velocity is constant at certain position w.r.t. time
Unsteady Flow with respect to time•Velocity changes at certain position w.r.t. time
H ≠ constant
V ≠ constant
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
0
t
tsteady
unsteady
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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(
3 .Uniform and Non uniform Flow
Non- uniform Flow means velocity changes at certain time in different positions ( depends on dimension x or y or z(
YY
x x
0
0
x
x
uniform
Non-uniform
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4. One , Two and three Dimensional Flow :
Two dimensional flow means that the flow velocity is function of two coordinatesV = f( X,Y or X,Z or Y,Z )
One dimensional flow means that the flow velocity is function of one coordinate V = f( X or Y or Z )
Three dimensional flow means that the flow velocity is function of there coordinatesV = f( X,Y,Z)
x
y
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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 three-dimensional 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 the flow direction, V = V(r).
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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
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 VmaxVmean
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5. Laminar and Turbulent Flow (cont.):
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Rotational and irrotational flows:
rThe rotation is the average value of rotation of two lines in the flow .)i (If this average = 0 then there is no rotation and the flow is called irrotational flow
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6. Streamline:
A Streamline is a curve that is everywhere tangent to it at any instant represents the instantaneous local velocity vector.
tan
3
dy v
dx uu v
dx dy
in general for D
u v w
dx dy dz
w
uvx
z
y
Where: u velocity component in -X- directionv velocity component in-Y- directionw velocity component in -Z- direction
Stream line equation
222 wvuV
V
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velocity vector can written as:
V ui vj wk
Where: i, j, k are the unit vectors in +ve x, y, z directions
• From Newton's second law,
• 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,
Acceleration Field
particle particle particleF m a
particleparticle
dVa
dt
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t
u
z
uw
y
uv
x
uua
t
uwz
uvy
uux
ua
t
u
dt
dz
z
u
dt
dy
y
u
dt
dx
x
u
dt
dua
dtt
udzz
udyy
udxx
udu
tzyxfV
x
x
x
n
),,,(.
Convective component
Local component
Mathematically the total derivative equals the sum of the partial derivatives
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Similarly:
t
v
z
vw
y
vv
x
vuay
t
w
z
ww
y
wv
x
wuaz
zyx aaaa 222
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NASCAR surface pressure contours and streamlines
Airplane surface pressure contours, volume streamlines, and surface streamlines
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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)
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Types of motion or deformation of fluid element
Linear translation
Rotational translation
Linear deformation
angular deformation
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8 -Rotational Flow & Irrotational Flow:The rate of rotation can be expressed or equal to the angular velocity vector )
:(
Note:
y
u
x
v
x
w
z
u
z
v
y
w
z
y
x
2
1
2
1
2
1
ky
u
x
vj
x
w
z
ui
z
v
y
w
2
1
2
1
2
1
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The flow is side to be rotational if:
0zyx oror
The flow is side to be irrotational if:
0 zyx
The fluid elements are rotating in space (see Fig. 4-44 )
The fluid elements don’t rotating in space (see Fig. 4-44 )
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Irrotational flow
rotational flow
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9 -Vorticity ( ξ ):
Vorticity is a measure of rotation of a fluid particaleVorticity is twice the angular velocity of a fluid particle
x
y
z
w v
y z
u w
z x
v u
x y
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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.
.cos θ
θ+
NOTE:The flow is irrotational if ω=0, ξ=0, Г=0
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For 2-D Cartesian Coordinates
x
Y
dy
dx
dxx
vv
dyy
uu
u
v +
vdydxdyy
uudydx
x
vvudxd
)()(
area
dxdyy
u
x
v
z .
)(
Г = ξ . area
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Conservation of Mass ( Continuity Equation ) )Mass can neither be created nor destroyed(
The general equation of continuity for three dimensional steady flow
x
y
dx
dz
dy
z
dzdydxx
uu .).(
dzdyu ..
dzdxv ..
dxdyw.dzdxdy
y
vv .).(
dxdydzz
ww ).(
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Net mass in x-direction= - =dzdyu .. dzdydxx
uu .).(
dzdydxx
u..
Net mass in y-direction= - =dzdxv .. dzdxdyy
vv .).(
dzdydxy
v..
Net mass in z-direction= - =dydxw .. dydxdzz
ww .).(
dzdydx
z
w..
Σ net mass = mass storage rate
dzdydxz
w..
dzdydxy
v..
dzdydxx
u..
= )..( dzdydxt
x
u
y
v
z
w
=t
x
u
y
v
z
w
=t
0
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General equation fof 3-D , unsteady and compressible fluidSpecial cases:
1 -For steady compressible fluid 2 -For incompressible fluid ( ρ= constant ) 0
t
0
z
w
y
v
x
u
t
0
z
w
y
v
x
u
Note : The above eqn. can be used for steady & unsteady for incompressible fluid
0
z
w
y
v
x
u
t
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3 -For 2-D: 0
y
v
x
u
0
z
w
x
u
0
z
w
y
v