Boundary Layers

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Boundary Layers
As a fluid flows over a body, the no-slip
condition ensures that the fluid next to the
boundary is subject to large shear. A pipe is
enclosed, so the fluid is fully bounded, but in
an open flow at what distance away from the
boundary can we begin to ignore this
shear? There are three main definitions of
boundary layer thickness 1. 99 thickness 2.
Displacement thickness 3. Momentum thickness
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99 Thickness
U
U is the free-stream velocity
?(x) is the boundary layer thickness when u(y)
0.99U
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Displacement thickness
There is a reduction in the flow rate due to the
presence of the boundary layer
U
This is equivalent to having a theoretical
boundary layer with zero flow
?d
U
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Displacement thickness
The areas under each curve are defined as being
equal
and
Equating these gives the equation for the
displacement thickness
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Momentum thickness
In the boundary layer, the fluid loses momentum,
so imagining an equivalent layer of lost
momentum
and
Equating these gives the equation for the
momentum thickness
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Laminar boundary layer growth
? d?
?(x)
dy
?
Boundary layer gt Inertia is of the same
magnitude as Viscosity a) Inertia Force a
particle entering the b.l. will be slowed from a
velocity U to near zero in time, t. giving force
FI ? ?U/t. But ux/t gt t ? l/U where U is the
characteristic velocity and l the characteristic
length in the x direction. Hence FI ? ?U2/l b)
Viscous force F? ? ??/?y ? ??2u/?y2 ?
?U/?2 since U is the characteristic velocity and
? the characteristic length in the y direction
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Laminar boundary layer growth
Comparing these gives
?U2/l ? ?U/?2
So the boundary layer grows according to
Alternatively, dividing through by l, the
non-dimensionalised boundary layer growth is
given by
Note the new Reynolds number characteristic
velocity and characteristic length
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Boundary layer growth
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Length Reynolds Number
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Flow at a pipe entry
d
U
d
l
If the b.l. meet while the flow is still laminar
the flow in the pipe will be laminar If the b.l.
goes turbulent before they meet, then the flow in
the pipe will be turbulent
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Length Reynolds number and Pipe Reynolds number
The critical Reynolds number for flow along a
surface is Rl3.2105
In a pipe, the Reynolds number is given by
Considering a pipe as two boundary layers
meeting, d2a2? and from above
The mean velocity in the pipe, um, is comparable
to the free-stream velocity, U
If Rl3.2105 then Re5657
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Boundary layer equations for laminar flow
These may be derived by solving the Navier-Stokes
equations in 2d.
Momentum
Continuity
U
Assume 1. The b.l. is very thin compared to the
length 2. Steady state
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Boundary layer equations for laminar flow
This gives Prandtls b.l. equation
rate of change of u with x is small compared to y
Blasius produced a perfect solution of these
equations valid for 0ltxlt3.2105, and demonstrated
the shape of the boundary layer profile
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Blasius Solution
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Laminar skin friction
The shear stress at the surface can be found by
evaluating the velocity gradient at the surface
The friction drag force along the surface is then
found by integrating over the length
where b is the breadth of the surface
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Laminar skin friction
From the Balsius solution, the gradient of the
velocity profile at y0 yields the result
The shear force can be obtained by integration
along the surface
The frictional drag coefficient can then be
calculated
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Force and momentum in fluid mechanics - refresher

• Newtons laws still apply. Consider a stream tube

u2,A2 q2u2A2
u1,A1 q1u1A1
mass entering in time, dt, is
?u1A1dt momentum entering in time, dt, is m1
(?u1A1dt)u1 momentum leaving in time, dt, is
m2 (?u2A2dt)u2 Impulse momentum change, F
(m2 m1)/ dt ?(u22A2-u12A1)
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The von Karman Integral Equation (VKI)
C
U
?2 - ?1
Boundary Layer
B
?1
?2
u2(y)
u1(y)
A
D
?x
Flow enters on AB and BC, and leaves on CD
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VKI
The momentum change between entering and leaving
the control volume is equal to the shear force
on the surface
Force on fluid
(BC)
(CD)
(AB)
By conservation of fluid mass, any fluid entering
the control volume must also leave, therefore
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VKI
As ?x ?0, the two integrals on the right become
closer and the equation may be written as a
differential
The integral is the definition of the momentum
thickness, so
if U(x)
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Turbulent boundary layers
The assumption is made that the flat plate
approximates to the behaviour in a pipe. The
free stream velocity, U, corresponds to the
velocity at the centre, and the boundary layer
thickness, ?, corresponds to the radius, R.
1/7 Power Law
From experiments, one possibility for the shape
of the boundary layer profile is
and measurements of the shear profile give
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Turbulent boundary layers
Putting the expression for the 1/7 power law into
the equations for displacement and momentum
thickness
?99
?d
?m
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Turbulent boundary layers
becomes
Equating this to the experimental value of shear
stress
Integrating gives
The turbulent boundary grows as x4/5, faster than
the laminar boundary layer.
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Turbulent boundary layers
Momentum thickness
To find the total force, first find the shear
stress
then integrate over the plate length
For a plate of length, l, and width b,
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Logarithmic boundary layer
From the mixing length hypothesis it can be shown
that the profile is logarithmic, but the
experimental values are different from those in a
pipe
and the friction coefficient
(A is a correction constant if part of the b.l.
is laminar)
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Quadratic approximation to the laminar boundary
layer
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Quadratic approximation to the laminar boundary
layer
Remember - boundary layer theory is only
applicable inside the boundary layer.
This is sometimes written with ?y/? and F(?)u/U
as
It provides a good approximation to the shape of
the laminar boundary layer and to the shear
stress at the surface
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Turbulent Boundary Layer
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Laminar Sub-Layer