boundary layer velocity profile
TRANSCRIPT
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Boundary Layer Velocity Profile
z
zU
Viscous sublayer
Buffer zone
Logarithmic
turbulent zone
Ekman Layer, or
Outer region
(velocity defect layer)
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But first.. a definition:
2*ub
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1. Viscous Sublayer - velocities are low, shear stress
controlled by molecular processes
As in the plate example, laminar flow dominates,
z
ub
Put in terms of u*
integrating,
boundary conditions,
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When do we see a viscous sublayer?
v
= f (u*, , ks)
where ks== characteristic height of bed
roughness
Roughness Re:
R*> 70 rough turbulent
no viscous sublayer
R*< 5 smooth turbulent
yes, viscous sublayer
sku
R *
*
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2. Log Layer:
Turbulent case, Azis NOT constant inz
Azis a property of the flow, not just the fluid
To describe the velocity profile we need to develop a
profile ofAz.
Mixing Lengthformulation Prandtl (1925) which is
a qualitative argument discussed in more detail
Boundary Layer Analysis by Shetz, 1993
Assume that water masses act independently over adistance, l
Within la change in momentum causes a fluctuation
to adjacent fluid parcels.
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At l,
Make assumption of isotropic turbulence:
|u| ~ |v| ~ |w|
Therefore, |u| ~ |w| ~
Through the Reynolds Stress formulation,
dzud
ul
'~
dz
ud
l dz
ud
l
2
2~
''
dz
udl
wu
zx
zx
Prandtl Mixing Length
Formulation
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Von Karmen (1930) hypothesized that close to a boundary,
the turbulent exchange is related to distance from the
boundary.
lz
l=Kz
where K is a universal turbulent momentum exchangecoefficient == von Karmens constant.
Khas been found to be 0.41
Near the bed,
dz
udKzu
dz
udzK
zx
*
2
22
in terms of u*
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Solving for the velocity profile:
ln z
Intercept, b, depends on roughness of the
bed - f (R*)
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Rename b, based on boundary condition:
z=zo at = 0
Karmen-Prandtl Eq.
or Law of the Wall
o
z
z
z
Ku
uln
1
*
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Hydraulic Roughness Length,zo
zois the vertical intercept at which
z= 0
zo= f ( viscous sublayer,
grain roughness,
ripples & other bedforms,
stratification)
This leads to two forms of the Karmen-Prandtl Equation
1) with viscous sublayer HSF
2) without viscous sublayer HRF
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Can evaluate which case to use with R*
where ks== roughness length scale
in glued sand, pipe flow experiments
ks=D
in real seabeds with no bedforms,
ks=D75
in bedforms, characteristic bedform scale
ks~ height of ripples
skuR *
*
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1. Hydraulically Smooth Flow (HSF) 50 *
*
Sku
R
** boundary layer is
turbulent, but there is
a viscous sublayer
zo
is a fraction of the
viscous sublayer
thickness:
Karmen-Prandtl equation
becomes:
For turbulent flow over a
hydraulically smooth boundary
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2. Hydraulically Rough Flow (HRF) 70*
*
Sku
R
** no viscous sublayerzois a function of theroughness elements
Nikaradze pipe flow
experiments:
Karmen-Prandtl equation
becomes:
For turbulent flow over a
hydraulically rough boundary with
no bedforms, no stratification, etc.
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Notes on zoin HRF
Grain Roughness:
Nikuradze (1930s) - glued sand grains on pipe flow
zo=D/30
Kamphius (1974) - channel flow experiments
zo=D/15
Bedforms:
Wooding (1973)
whereHis the ripple height
and is the ripple wavelengthSuspended Sediment:
Smith (1977)
zo= f (excess shear stress, andzofrom ripples)
4.1
20
HHz
o
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3. Hydraulically Transitional Flow (HTF) 705 **
Sku
R
zois both fraction of the viscous sublayer thickness and afunction of bed roughness.
Karmen-Prandtl equation is
defined as:
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Bed Roughness is never well known or characterized, but fortunately
not necessary to determine u*
If you only have one velocity measurement (at a single elevation), usethe formulations above.
If you can avoid it.. do so.
With multiple velocity measurements, use the Law of the Wall to
get u*
o
z
z
z
Ku
uln
1
*ln z
z
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To determine b(or u*) from a velocity profile:
1. Fit line to data
2. Find slope -
3. Evaluate
)(
lnln
12
12
uu
zzm
mu
K
*