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Boundary Layer Velocity Profile

Ekman Layer, or Outer region (velocity defect layer). z. Logarithmic turbulent zone. Buffer zone. Viscous sublayer. ū. Boundary Layer Velocity Profile. But first.. a definition:. 1. Viscous Sublayer - velocities are low, shear stress controlled by molecular processes

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Boundary Layer Velocity Profile

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  1. Ekman Layer, or Outer region (velocity defect layer) z Logarithmic turbulent zone Buffer zone Viscous sublayer ū Boundary Layer Velocity Profile

  2. But first.. a definition:

  3. 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,

  4. 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

  5. 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.

  6. At l, Make assumption of isotropic turbulence: |u’| ~ |v’| ~ |w’| Therefore, |u’| ~ |w’| ~ Through the Reynolds Stress formulation, Prandtl Mixing Length Formulation

  7. 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 Karmen’s constant. K has been found to be 0.41 Near the bed, in terms of u*

  8. Solving for the velocity profile: ln z ū Intercept, b, depends on roughness of the bed - f (R*)

  9. Rename b, based on boundary condition: z = zo at ū = 0 Karmen-Prandtl Eq. or Law of the Wall

  10. Hydraulic Roughness Length, zo zo is the vertical intercept at which ūz= 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

  11. 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

  12. 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

  13. 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.

  14. 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)

  15. 3. Hydraulically Transitional Flow (HTF) zois both fraction of the viscous sublayer thickness and a function of bed roughness. Karmen-Prandtl equation is defined as:

  16. 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 ūz

  17. To determine b (or u*) from a velocity profile: 1. Fit line to data 2. Find slope - 3. Evaluate

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