Transcript Slide 1
Fluid Mechanics
Lecture 6
The boundary-layer equations
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The need for the boundary-layer model
• While the flow past a streamlined body may be
well described by the inviscid flow (and even the
potential flow) equations over almost all the flow
region, those equations do not satisfy the fact that
– because of finite viscosity of real fluids – the
flow velocity at the wall itself must vanish.
• So, we need a flow model that uses the simplest
possible form of the Navier Stokes equation but
which does enable the no-slip condition to be
satisfied.
• Such a model was first developed by Ludwig
Prandtl in 1904.
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Objectives of this lecture
• To explore the simplification of the Navier
Stokes equations to obtain the boundary
layer equations for steady 2D laminar flow.
• To understand the assumptions used in
deriving these equations.
• To understand the conditions in which the
boundary-layer equations can be used
reliably.
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The governing equations
• Navier-Stokes equations:
Continuity:
U
x
y-momentum:
y
0
2
2U
U
U
V
2
2
x
y
x
x
y
2
2V
V
U
V
2
2
x
y
y
x
y
U
x-momentum:
V
V
U
V
P
P
• We seek to simplify these equations by
neglecting terms which are less important
under particular circumstances.
• Key assumptions: the thickness of the region
where viscous effects are significant,δ, is very
thin , i.e. d << L and ReL >>1.
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Non-dimensionalized form of N-S Equations
• Non-dimensionalize equations using
V, a constant
(approach) velocity,
L ,an overall
dimension i.e.
• U*= U/ V;
V*=V/V; x*=x/L;
y*=y/L
• P*=???
( A question for you)
U
x
V
y
0
2U
2
U
P
1
U
U
U
V
2
2
R e L x
x
y
x
y
2V
2
V
V
V
P
1
U
V
2
2
R e L x
x
y
y
y
L
L
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Non-dimensionalized N-S equations
• Since 0 x L ,
0 x*
x
L
1.
x has a magnitude
comparable to L
x* has an order of
magnitude of 1.
Hence we write x O (1)
y O (d / L ) O (d )
u O (1)
O (1)
L
p O (1)
O (1)
k O (1)
L
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Non-dimensionalized N-S equations
• Since
0 yd
and
d L
,
0 y*
y
L
Hence we write
y*= O(d).
d
. 1
L
y* is at least an order of
magnitude smaller than 1.
x O ( x / L ) O (1)
• Also we have
y O (d / L ) O (d )
U
O (1)
L
P O (1)
O (1)
k
O (1)
C p O (1)
L
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Continuity equation
U
x
V
y
x O (1)
0
y O (d )
U
O (1)
O (1)
[V ]
O (d )
O (1)
0
O (1)
[V ]
O (d )
0
[V*] has to be of order O(d) to satisfy continuity, i.e..
*
V O (d )
No term can be omitted hence the continuity equation
remains as it is, i.e.
U
x
V
y
0
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x-momentum equation
O ((dx /) L )
xy O
U
U
O (1
O (1),
U
x
U
x
V
O (1)
y
x
2
U
2
O (1)
P
1
O (1),
V
O (1)
U
P
x
V
2 U 2U
2
2
R e L x
y
x
y
U
O (d )
O (1)
O (1)
O (d )
u
y
O (1),
x
*
Vy OO((dd /)L )
U
O (1)
P
O (1)
O
O (1),
O (d )
P Ox(1) O
O (d )
p
O (1)
O (1)
O (1)
O (1)
k O (1)
C p O (1)
O (1)
O (1)
2
U
y
2
O (1)
O (1)
O (1)
2
O (d )
O (1) O (
1
d
2
)
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x-momentum equation
U
U
x
V
2 U 2U
2
2
R e L x
y
x
y
U
O (1) O (1) O (1)
P
1
Re
L
1
1
O (1) O ( d 2 )
To make the above equation valid, we must have:
Re
L
O(
1
d
2
)
ReL has to be large and x-gradients in the viscous term can be dropped
in comparison with y-gradients. The dimensional form of the equation
thus becomes:
U
V
x
V
U
y
P
x
2
U
y
2
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y-momentum equation
U
V
x
V
2V 2V
2
2
R e L x
y
y
y
V
P
1
y O (d )
V
To do an order of magnitude analysis for each
P
term and estimate the order of magnitude for
y
*
O (d )
[Re] O (
1
d
2
)
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y-momentum equation
U
V
x
V
2V 2V
2
2
R e L x
y
y
y
V
P
1
y O (d )
V
*
O (d )
[Re] O (
1
d
2
)
O (d )
O (d )
O [1]
O [d ]
2
1
O [1]
O [d ]
O
(1)
y
O (d )
O(
)
O (d )
O (d )
P
1
d
P
2
1
O (d ) O (d ) O [
] O (d ) O (d ) O ( )
d
y
Hence
P
y
O [d ]
2
at most
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y-momentum equation
P
y
O (d )
P
y
0
The pressure can be assumed to be constant
across the boundary layer over a flat plate. Hence
the pressure only varies in the x-direction and the
pressure at the wall is equal to that at the edge of the
layer, i.e. P(x,y)=P(x).
L
L
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Two qualifiers
• If the surface has substantial longitudinal
curvature (d/R >0.1)
it may not be adequate to
assume constant pressure across boundary layer.
Then one needs to apply radial equilibrium to
compute P (see Slide 16)
• In 3D boundary layers (not covered in this course
but very important in the industrial world) one
needs to be able to work out the presssure
variations in the y-z plane (normal to the mean
flow) to compute the secondary velocities .
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Summary of assumptions
• Basic assumption:
• Derived results
– V is small, i.e.
– Re must be large:
d L
V O (d )
[Re] O (
1
d
2
)
and then only velocity gradients normal to the wall are
significant in the viscous term
– The pressure is constant across the boundary layer
(for 2D nearly straight) flows, i.e.
P
y
0
P(x,y)=P(x).
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Boundary layer equations
Continuity
x-momentum
U
x
U
V
y
0
U
x
V
U
y
dP
dx
2
U
y
2
2
U
• Since x 2 disappears, the equations become of
parabolic type which can be solved by knowing
only the inlet and boundary conditions... i.e. no
feedback from downstream back upstream.
• Unknowns: U and V; (P may be assumed known)
• Boundary conditions:
y = 0; U = V = 0
At wall :
Free stream: y = d; U = V U
U(x0,y), V(x0,y)
Inlet:
L
L
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Boundary layer over a curved surface
Pressure gradient
across boundary layer:
P
y
U
2
R
Assume a linear velocity distribution, i.e.U
integrating from y=0 to d gives
2
P (d ) P (0)
U d
3R
P
U y
P (d ) P (0)
2
U
d
d
3R
Hence pressure variations
across the boundary layer
are negligible when d R
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Limitations
• Large Reynolds number, typically Re >1000
• Boundary-layer approximations inaccurate
beyond the point of separation.
• The flow becomes turbulent when Re > 500,000.
In that case the averaged equations may be
describable by an adapted for of momentum
equation – to be treated later.
• Applies to boundary layers over surfaces with
large radius of curvature.
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