Boundary Layer Velocity Profile

8/13/2019 Boundary Layer Velocity Profile

1/19

Boundary Layer Velocity Profile

z

zU

Viscous sublayer

Buffer zone

Logarithmic

turbulent zone

Ekman Layer, or

Outer region

(velocity defect layer)


2/19

But first.. a definition:

2*ub


3/19

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,


4/19

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 *

*


5/19

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.


6/19

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


7/19

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*


8/19

Solving for the velocity profile:

ln z

Intercept, b, depends on roughness of the

bed - f (R*)


9/19

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

*


10/19


11/19

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


12/19


13/19

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 *

*


14/19

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


15/19

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.


16/19

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


17/19

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:


18/19

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


19/19

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

*

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Boundary Layer Velocity Profile