Marine Boundary Layers

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Marine Boundary Layers Shear Stress Velocity
Profiles in the Boundary Layer Laminar
Flow/Turbulent Flow Law of the Wall Rough
and smooth boundary conditions
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Shear Stress
In cgs units Force is in dynes g cm / s2
Shear stress is in
dynes/cm2 (N/m2 in MKS)
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Z
Y
X
Each plane has three components i.e., for the x
plane For three dimensions nine
components What are the key components in the
marine boundary layer?
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XX, YY, ZZ component is the pressure force,
doesnt act to move particles XZ, YZ component
the flow is not shearing in the z-direction (in
the mean) XY, YX component assume uniform flow
(flow not rotating in the mean) End up with two
components , shear on the z-plane in x
and y directions As we get close to the seabed
and rotate into flow tb
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Simplest boundary layer case Laminar Flow
smooth boundary no turbulence
generated layers of fluid slipping past each
other In this case
Z
F
h
X
No-slip condition
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Deformation of fluid layers is at same rate for
shearing force ? linear velocity
profile Integrating Boundary
conditions Description of velocity profile
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What force (or shear stress) was needed to pull
plate A and create this velocity profile?
Molecular viscosity of the fluid (resistance of
the fluid to deformation)
Provides transfer of momentum between adjacent
fluid layers
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Another way to think about shear
stress Transfer of momentum perpendicular to
the surface on which stress is applied.
kinematic viscosity
Velocity gradient Fluid momentum
gradient Diffusion of momentum
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Turbulent Flows A random (statistically
irregular) component added to the mean flow
Dyer, 1986
Define u instantaneous velocity u random
turbulent velocity u mean velocity
u u u
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NOTE! Beware of averaging time scale. Turbulent
fluctuations follow a Gaussian distribution
Turbulence intensity can be described by the RMS
fluctuation
Frequency of occurrence
Average of u0
u
Turbulent eddies transfer momentum, much the same
way as molecular diffusion, but at appreciably
greater rates.
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Van Dyke, An Album of Fluid Motions, 1982
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Transfer of momentum can be described by eddy
viscosity - Az transfer of momentum in
z-direction (note ?? in Wright, 1995
chapter) Az gtgt ?
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Eddy fluctuations and momentum transfer u, v,
w - responsible for the transfer of momentum
Middleton Southard, 1984
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• Parcel has lower momentum at z2 by ??u
• flux of momentum
• w(??u)
• As z2 and z1 approach each other,
• u2 - u1 ?u u
• flux of momentum
• w(?u) or ?uw

This rate of change of momentum represents the
resistance to motion, or the shear stress, and
averaged over time
Reynolds Stress
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Since turbulent fluctuations difficult to
characterize, simplifying assumptions can be
made u ? u turbulent fluctuations are
proportional to the mean flow u, v, w are of
similar magnitude
Quadratic Stress Law
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• Summarize Three ways to describe shear stress
  in the turbulent bottom boundary layer.
• Eddy Viscosity
• Reynolds Stress
• Quadratic Stress Law