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Near the leading edge of a flat plate, the boundary layer is wholly laminar. For a
laminar boundary layer the velocity distribution is parabolic.
The thickness of the boundary layer (δ) increases with distance from the leading edge
x, as more and more fluid is slowed down by the viscous boundary, becomes unstable and
breaks into turbulent boundary layer over a transition region.
For a turbulent boundary layer, if the surface is smooth, the roughness projections are
covered by a very thin layer which remains laminar, called laminar sub layer. The
velocity distribution in the turbulent boundary layer is given by Log law of Prandtls one-
seventh power law.
Development of Boundary layer
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δ increases as distance x increases.
δ decreases as U increases.
δ increases as kinematic viscosity (ѵ) increases.
When U decreases in the downward direction, boundary layer growth is reduced.
When U decreases in the downward direction, flow near the boundary is further
retarded, boundary layer growth is faster and is susceptible to separation.
Ʈ0 decreases as x increases as
Boundary layer thickness (δ ):
The velocity within the boundary layer increases from zero at the surface to the velocity
of the main stream asymptotically. Therefore, the thickness of the boundary layer is
arbitrarily defined as that distance from the surface boundary in which the velocity
reaches 99 per cent of the free stream velocity.( i.e. u = 0.99U).
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V(x, y, z) in cartesian or V(r, , z) in cylindrical coordinates
r z
V(r)
velocity profile
(remains unchanged)
One-dimensional flow in a circular pipe
One, Two and Three Dimensional Flows
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DIFFERENTIAL ANALYSIS OF FLUID FLOW
Finite control volume approach is very practical and useful, since it does not generally require a detailed knowledge of the pressure and velocity variations within the control volume
Problems could be solved without a detailed knowledge of the flow field
Unfortunately, there are many situations that arise in which details of the flow are important and the finite control volume approach will not yield the desired information
How the velocity varies over the cross section of a pipe, how the pressure and shear stress vary along the surface of an airplane wing
In these circumstances we need to develop relationships that apply at a point, or at least in a very small region infinitesimal volume within a given flow field. This approach - DIFFERENTIAL ANALYSIS
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• DIFFERENTIAL ANALYSIS PROVIDES VERY DETAILED KNOWLEDGE OF A FLOW FIELD
Control volume analysis
Interior of the CV is
BLACK BOX
Differential analysis
All the details of the flow are solved at every point within the flow domain
Flow out
Flow in
Flow out
Control volume
F
Flow out
Flow out
Flow domain
F
Flow in
x
y
z
zyu
zyxx
uzyu
zxv
zyx
y
vzxv
zyx
z
wyxw
yxw
Equation of Continuity
Let
u, v and w are the velocities in x, y and z direction.
Mass flow rate entering in the x – direction
Mx =
And mass flow out of element Mx+δx =
zyx
x
uzyu
x
x
MxMx
zyu
=
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Similar the Mass flow rate entering and leaving
the element in y and z - direction are shown in
figure.
Now, balanced equation for element can be written
as….
Mass inflow – Mass outflow = Rate of change
of mass inside the element
zyxt
zyxz
wyxw
zyxy
vzxv
zyxx
uzyu
yxwzxvzyu
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zyxt
zyxz
wzyx
y
vzyx
x
uzyx
t
This is the differential equation of continuity in
Cartesian Coordinate for unsteady and
compressible fluid flow.
For Steady and compressible flow,
0
z
w
y
v
x
u
0
z
w
y
v
x
u
t
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For Steady and incompressible flow,
0
z
w
y
v
x
u
In polar coordinate, the general form of continuity
equation for compressible and unsteady flow,
0
1
z
VV
rr
V
r
V
t
zrr
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Momentum Equation Consider an element, two dimensional
control volume ( dx . dy . Unit depth) within the boundary layer region.
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Assumption
Let u = velocity of fluid flow at left hand face AB,
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Since we are concerned only with the momentum in x-direction,
momentum of fluid moving in y direction is obtain by, multiplying mass
flow rate in y direction (my) by x-direction of velocity (u).
Velocity of fluid flow at face AD =
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For study and incompressible flow in,
Blasius Exact Solution for Laminar Boundary Flows
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LINEAR DEFORMATION
Because of the presence of velocity gradients, the element will generally be deformed and rotated as it moves. For example, consider the effect of a single
velocity gradient , On a small cube having sides x
u
zandy,x
A O
B C
O
B C
A
uu x
x
uu x
x
δx
δy
u
u
ux t
x
C’
A’
δy
δx
x component of velocity of O and B = u x component of velocity of A and C = This difference in the velocity causes a “STRETCHING” of the volume element by a volume Rate at which the volume V is changing per unit volume due the gradient
xx
uu
tzyxx
u
x
u
x
u
t
tx
u
0tLim
td
Vd
V
1
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Blasius Exact Solution for Laminar Boundary Flows
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Thermal Boundary Layer
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Whenever a flow of fluid takes place past a heated or cold surface, a
temperature field is set up in the field next to the surface.
[If the surface of the plate is hotter than fluid, the temperature distribution
will be as shown in the Figure.]
The zone or this layer wherein the temperature field exists is called the
thermal boundary layer.
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The thermal boundary layer thickness (δt) is arbitrarily defined as the
distance y from the plate surface at which
[The shape of thermal boundary layer when ]
The thermal boundary layer (δt) depends on the fluid properties like
specific heat, thermal conductivity, flow velocity, and viscosity.
The relative magnitude of δ and δt affected by the thermo-physical
properties of fluid.
The governing non dimensional parameter / number is prendtl number.
δ = δt when Pr=1; δ > δt when Pr > 1 & δ < δt when Pr < 1
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Figure shows a hot fluid flowing over a cool flat plate, and development of the
thermal boundary layer.
Consider control volume (dx X dy X unit depth) in the boundary layer. The
enlarged view of this control volume is shown in Fig. (b)
Differential Energy equation
Involving principle of conservation of energy for the steady state condition,
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As the temperature gradient in x direction in very small the conduction in x direction
is negligible. …..
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