Transcript Lecture 4

Marine Boundary Layers
Shear Stress
Velocity Profiles in the Boundary Layer
Laminar Flow/Turbulent Flow
“Law of the Wall”
Rough and smooth boundary conditions
Suspended Load
U z   b   critical 
threshold
of motion
Bed Load
Shear Stress
Force ML 1
M
 
 2  2  2
Area
T
L
T L
In cgs units: Force is in dynes = g * cm / s2
Shear stress is in dynes/cm2
(N/m2 in MKS)
Z
 xz
Y
 xy
 xx
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?
XX, YY, ZZ component – is the pressure force, doesn’t 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:

zy
,

shear on the z-plane in x and y directions
zx
As we get close to the seabed and rotate into flow:
τb
Simplest boundary layer case:
Laminar Flow – smooth boundary
no turbulence generated
Z
layers of fluid slipping past each other
F
h
X
“No-slip” condition
In this case:
 zx
F

 constant
A
Deformation of fluid layers is at same rate for shearing
force
 linear velocity profile
du
K
dz
Integrating:
Boundary conditions:
Description of velocity profile:
What force (or shear stress) was needed to pull plate A and
create this velocity profile?
 zx
u

z

Molecular viscosity of the fluid (resistance of the fluid
to deformation)
  f T , Sal , P

Provides transfer of momentum between adjacent fluid layers
Another way to think about shear stress:
Transfer of momentum perpendicular to the surface on
which stress is applied.
u  

 zx  

z  

 zx
  u 
 

z

u  momentum
kinematic viscosity
Velocity gradient
Fluid momentum gradient
Diffusion of momentum
Turbulent Flows
A random (statistically irregular) component added to
the mean flow
Dyer, 1986
Define u = instantaneous velocity
u’ = random turbulent velocity
ū = mean velocity
u = ū + u’
NOTE! Beware of averaging time scale.
Turbulent fluctuations follow a Gaussian distribution:
Frequency of occurrence
Average of u’==0
u’
Turbulence intensity can be described by the RMS fluctuation
 
sqrt u'2
Turbulent eddies transfer momentum, much the same way as
molecular diffusion, but at appreciably greater rates.
Van Dyke, “An Album of Fluid Motions”, 1982
Transfer of momentum can be described by:
“eddy” viscosity - Az – transfer of momentum in zdirection
(note:  in Wright, 1995 chapter)
du
 zx  (   Az )
dz
Az >> 
du
 zx  Az
dz
Eddy fluctuations and momentum transfer:
u’, v’, w’ - responsible for the transfer of momentum
Middleton & Southard, 1984
• Parcel has lower momentum
at z2 by ρΔu
Z
• flux of momentum:
w’•(ρΔu)
• As z2 and z1 approach each
other,
u2 - u1 = Δu u’
ū
• flux of momentum:
w’•(ρu’) or u’w’
This rate of change of momentum represents the resistance to
motion, or the shear stress, and averaged over time:
 zx    u' w'
Reynolds Stress
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
 zx  u
 zx  Cd u
2
2
Quadratic Stress Law
Summarize: Three ways to describe shear stress in
the turbulent bottom boundary layer.
• Eddy Viscosity
du
 zx  Az
dz
•Reynolds Stress
 zx    u' w'
•Quadratic Stress Law
 zx  Cd u
2