Boundary Layer Velocity Profile

z

ū

zU

Viscous sublayer

Buffer zone

Logarithmic

turbulent zone

Ekman Layer, or

Outer region

(velocity defect layer)

But first.. a definition:

2*ub

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,

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 **

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.

At l,

Make assumption of isotropic turbulence:

|u’| ~ |v’| ~ |w’|

Therefore, |u’| ~ |w’| ~

Through the Reynolds Stress formulation,

dzudu

l'

~

dz

udl

dz

udl

22~

''

dz

udl

wu

zx

zx

Prandtl Mixing Length Formulation

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,

dz

udKzu

dz

udzKzx

*

222

in terms of u*

Solving for the velocity profile:

ln z

ū

Intercept, b, depends on roughness of the bed - f (R*)

Rename b, based on boundary condition:

z = zo at ū = 0

Karmen-Prandtl Eq.

or Law of the Wall

o

z

z

z

Ku

uln

1

*

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

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

sku

R **

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

2. Hydraulically Rough Flow (HRF) 70**

Sku

R

** no viscous sublayerzo is a function of the roughness elements

Nikaradze pipe flow experiments:

Karmen-Prandtl equation becomes:

For turbulent flow over a hydraulically rough boundary with no bedforms, no stratification, etc.

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)

4.1

20

H

Hzo

3. Hydraulically Transitional Flow (HTF) 705 **

Sku

R

zo is both fraction of the viscous sublayer thickness and a function of bed roughness.Karmen-Prandtl equation is defined as:

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*

o

z

z

z

Ku

uln

1

*ln z

ūz

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

*