VELOCITY PROFILE AND SHEAR STRESSES CALCULATION IN HIGH

VOLUME RELATIVE BED ROUGHNESS FLOW

Wojciech Bartnik

Andrzej Struzynski

Krakow Agriculture UniversityKrakow Agriculture University

Flow zones – Introduction

Laboratory measurements

Bed roughness measurements

Log-law velocity distribution

Calculation of velocity and shear stresses

Conclusions

Presentation SchedulePresentation Schedule

Bed roughness and water surface acts on the flowing water

Flow zonesFlow zones

Flow zonesFlow zones

I - laminar flow II - log-law velocity distribution III - wake region IV - free surface region

Flow zonesFlow zones

flat bed

III

III

IV

Flow zonesFlow zones

rough bed

I

II

III

IV

IV

Flow zonesFlow zones

[Williams J.J., 1996]

Bed roughness and water surface acts on the shape of flowing water velocity profile.

Flow zonesFlow zones

The shape of velocity profile depend on: flow depth, av. velocity of flowing water, bed roughness, relative roughness ...

For hydraulically rough flow conditions I and IV flow zone decreases

Fr = 0.074 Fr = 1.38

4D

Flow zonesFlow zones

Laboratory measurementsLaboratory measurements

Flume dimensions: l2.0 x 0.5 x 0.6 m(glass walls)Flume rig: micro-propeller flow-meter slope measurements

Bed slope, water surface slope

Discharge: max 0.13 qm s-1 Artificial grains Ø – 4 to 8 cm

Bed roughness measurementsBed roughness measurements

homogeneous roughnessks = K (1.926 SF2 – 0.488 SF + 4.516)

N

n n Hhn

K1

2

1

1

Profile-meter AG-1

Log-law velocity distributionLog-law velocity distribution

Maximum velocity moves with relative roughness change flat bed

Log-law velocity distributionLog-law velocity distribution

Maximum velocity moves with relative roughness change rough bed

Log-law velocity distributionLog-law velocity distribution

For the same bed roughness curves are parallel flat bed

Log-law velocity distributionLog-law velocity distribution

For the same bed roughness curves are parallel grains 4M

Log-law velocity distributionLog-law velocity distribution

For the same bed roughness curves are parallel grains 4D

Log-law velocity distributionLog-law velocity distribution

For the same bed roughness curves are parallel grains 6D

Log-law velocity distributionLog-law velocity distribution

For the same bed roughness curves are parallel grains 8D

Calculation of velocity and shear stressesCalculation of velocity and shear stresses

Log-law velocity distribution for whole profile is used

U/Umax = A log (y/Y) + B

sk

yUU

30log 75,5 * Modified Prandtl equation

B becomes constant - B = 1.12 ± 3%

Calculation of velocity and shear stressesCalculation of velocity and shear stresses

U/Umax = A log (y/Y) + B

A value changes with relative depth Y/K

Calculation of velocity and shear stressesCalculation of velocity and shear stresses

U/Umax = A log (y/Y) + B

Comparison of measured to calculated A constant

835.0

38.6

K

YA

Calculation of velocity and shear stressesCalculation of velocity and shear stresses

Velocity profile reflects shear stresses

|

dy

du

Use of logarithmic equation allow calculating 0 for rough flow conditions

0 = 2.303 K UM y

U

Calculation of velocity and shear stressesCalculation of velocity and shear stresses

ConclusionsConclusions

•Near bed the velocity and velocity profile slope calculations (in logarithmic scale) are correct within the second and third flow zone. The use of equation (4) makes the bed level (zero velocity) estimation error negligible (B=1.12).•The use of mentioned method is limited to the rough flow conditions where the maximum velocity lays close to the water surface (the near surface region decreases to 20% of water depth).•The measurements of surface velocity, water depth and bed roughness can be used for calculation of water velocity profile and bed shear stresses for rough flow conditions.