II' Kinematics of Fluid Motion

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CEE 262A HYDRODYNAMICS
Lecture 18 Autumn 2007-2008
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Boundary layers

• Thin layers in which dynamics are different from
  rest of fluid key concept of inner and outer
  regions
• Often diffusive fluxes dominate in direction
  normal to boundary layer (momentum, heat, scalar)
• Usually determines friction or heat/mass transfer

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Cases

• Stokes layer under water waves
• Stokes 2nd problem
• Stokes layer as an example of asymptotic matching
• Ekman layer under geostrophic flow
• Boundary layer on a solid surface due to flow
  past a 2D body
• Flat plate boundary layer (zero pressure
  gradient, Blasius)
• Effects of pressure gradients

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The Stokes layer
A flow closely related to the flow of Stokes
first problem is Stokes second problem! Now we
consider the flow induced above a flat plate at
x30 that oscillates according to
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First, what happens?
If we move the plate to the right, we impart
negative vorticity to the fluid This diffuses
upwards
Now, when the plate reverses direction. A new
front begins to diffuse having opposite sign
vorticity.
Thus as the plate oscillates for some time, we
expect that the vorticity from successive fronts
eventually cancels out one-another far enough
away from the boundary. Near the boundary, we
expect to see only the current forcings front.
old front
new front
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To confirm this picture, lets look at the forced
or periodic soln.
We choose dimensionless variables using
To create a parameter free eqn. we set
Expressing the fact that we expect the dominant
length scale will be the distance vorticity can
diffuse in one period
Thus
with
Real part of complex
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We now set (and drop )
u is complex
The P.D.E. gives (same method as for the Ekman
layer)
We can eliminate one of these values of k by
insisting that
Eliminate since unbounded.
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Here
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These two components look like this
(pic in page V39)
Note the strong similarity between this soln and
that of the Ekman Layer.
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The Stokes layer under gravity waves (Phillips
The Dynamics of the Upper Ocean)
a
With the dispersion relation
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If we calculate the horizontal velocity
We see that (as expected), the inviscid wave
velocity field does not satisfy the no slip
condition on the bottom
Thus, we are lead to question the utility of the
irrotational wave flow when/where is it valid?
To address this question, we convert the
linearized momentum equations to a dimensionless
form.
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We start by choosing
Note that this scaling supposes that k d -1. We
could also consider separately the case where k
ltlt d -1.
We can find P from the inviscid linear momentum
balance
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With these definitions the full (i.e., including
the viscous stresses) linearized x momentum eqn
Drop s and divide by as2
where
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I.e. The vorticity-containing Stokes layer which
might exist at the bottom must be much smaller
than the depth.
d
Stokes layer
z
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(return )
or
In this area viscosity is as important as
pressure gradients and inertia. If we look at the
z momentum eqn, we must set
In which case we find that
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How do we solve these incorporate the fact that
the inviscid solution is good in most of the
domain?
The central idea of Prandtls boundary layer
theory is that we solve the inviscid problem the
usual waves for all of the fluid except near the
wall. There, we must include viscous effects.
These two solutions must be matched combined
appropriately such that far from the boundary
i.e.
we see the inviscid result.
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To the waves, the Stokes layer looks infinitely
thin whereas to the Stokes layer flow, the
inviscid flow looks to be infinitely deep.
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Outer solution, far from BL
Inner solution, within BL
Since pressure is constant in BL, we use the
pressure gradient from the bottom of the outer
solution as the gradient throughout the inner
solution.
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The pressure condition from the z momentum eqn
tells us that the pressure is constant within the
Stokes layer is equal to the inviscid pressure
at z 0.
We can now construct the necessary solution
Let the flow inside the Stokes layer be
Viscous flow
Inviscid flow
Since the inviscid velocity obeys the relation
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The no-slip condition implies that, at z'0,
This has the same form as the oscillating plate
solution above
Let
So
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We find that employing the bottom boundary
condition requires
And so
In dimensional terms this gives
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However, we can make a composite solution by
adding this correction to the full velocity
field. Note that for

so the BL correction is
invisible far from the wall
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A variety of conclusions can be derived from this
viscous solution

• There is a mean flow driven by the viscous
  stresses within the boundary layer (in addition
  to the Stokes drift that occurs in the absence of
  friction).
• The wave loses energy due to viscous damping
  (dissipation) in this layer

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Geostrophic Flow the Ekman Layer
We consider a geostrophic flow with flow far from
a no-slip surface that is entirely in the x1
direction such that the x2 momentum equation is
Low
Meanwhile in the x1 direction we have
High
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Suppose that the vertical scale of this flow is
H, i.e. we have flow in a layer of thickness H
Geostrophic interior
H
UG
x3
No slip
x1
How do we add viscous stresses to this and
satisfy the no slip condition?
We start by making the overall equations
nondimensional
Let
L a horizontal length scale
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With all of the above we get
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But, near the wall, where ,
viscous stresses are important, so lets focus on
this inner region
With this choice we have
Note that continuity implies that
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Thus the vertical momentum eqn. reads (same
assumptions as above)
Pressure inside Ekman layer Pressure outside
So if we let
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Our momentum equations read
These are the Ekman Layer equations
since the geostrophic p.g. is exactly balanced by
the geostrophic velocity
To cancel out geostrophic velocity
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Substitution of into
Requires
Two roots
Decays choose this one
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At
This is the Ekman spiral (again)
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Which looks like this
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Thus, there is a net flow to x2 driven by the
pressure gradient
The change in transport (from geostrophic flow)
This means a reduction near the boundary of the
x1 flow and an increase in flow in the x2
direction
Low
q2
High
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Effect on flow in atmosphere
Low P
High P
Air sinks
Air rises
Air moves towards low
Air moves away from high
High-pressure systems cool, dry air Low-pressure
systems warm, moist air
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Or, the tea cup after stirring
Interior (solid body rotation)
Sidewall boundary layer (Stewartson layer)
Ekman layer
Tea leaves