Boundary Layer Velocity Profile

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Boundary Layer Velocity Profile
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But first.. a definition
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1. Viscous Sublayer - velocities are low, shear
stress controlled by molecular processes As in
the plate example, laminar flow dominates,
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 gt 70 rough
turbulent no viscous sublayer R lt 5 smooth
turbulent yes, viscous sublayer
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2. Log Layer Turbulent case, Az is NOT
constant in z Az is a property of the flow, not
just the fluid To describe the velocity profile
we need to develop a profile of Az. Mixing
Length formulation Prandtl (1925) which is a
qualitative argument discussed in more detail
Boundary Layer Analysis by Shetz, 1993 Assume
that water masses act independently over a
distance, l Within l a 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,
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. l ? z l
Kz where K is a universal turbulent momentum
exchange coefficient von Karmens constant. K
has been found to be 0.41 Near the bed,
in terms of u
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Solving for the velocity profile
ln z
u
Intercept, b, depends on roughness of the bed - f
(R)
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Rename b, based on boundary condition z zo
at u 0
Karmen-Prandtl Eq. or Law of the Wall
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Hydraulic Roughness Length, zo zo is the
vertical intercept at which uz 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
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1. Hydraulically Smooth Flow (HSF)
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)
zo is a function of the roughness
elements Nikaradze pipe flow experiments
no viscous sublayer
Karmen-Prandtl equation becomes
For turbulent flow over a hydraulically rough
boundary with no bedforms, no stratification, etc.
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Notes on zo in 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)
where H is the
ripple height and ? is the ripple
wavelength Suspended Sediment Smith (1977) zo
f (excess shear stress, and zo from ripples)
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3. Hydraulically Transitional Flow (HTF)
zo is both fraction of the viscous sublayer
thickness and a function 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), use the
formulations above. If you can avoid it.. do
so. With multiple velocity measurements, use the
Law of the Wall to get u
ln z
uz
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To determine ?b (or u ) from a velocity
profile 1. Fit line to data 2. Find slope -
3. Evaluate