Velocity Profile Analysis
The log and power function velocity distribution functions will be used to obtain a mathematical
description of the velocity profile.
(a) Logarithmic Velocity Distribution Equation
The log velocity relation is Equation 2.75:
v/V* = 1/k ln (y/y') = 2.31/k log (y/y')
Determine the value of k and y':
V* = (g yoSf )1/2 = (9.81 x 2.38 x 0.000206)1/2 = 0.069 m/sec
Table 2.6 is prepared for both log and power forms with the values shown.
Using regression techniques for the logarithmic form of the equation on the values of v/V* and ln y'
with the data in Table 2.6, obtain:
v/V* = 2.564 (ln y - ln 0.009) with a regression coefficient R = 0.996 giving the values of k = 0.39 and
y' = 0.009. The final equation for the data is
v/V* = 2.564 ln (y/0.009)
Table 2.6. Velocity Profile Calculations.
y (m)
v (m/s)
v/V*
v/V
y/y0
0.152
0.518
7.51
0.702
0.064
0.305
0.633
9.17
0.858
0.128
0.457
0.701
10.16
0.950
0.192
0.762
0.762
11.04
1.033
0.321
1.07
0.838
12.14
1.136
0.449
1.37
0.884
12.81
1.198
0.577
1.68
0.945
13.70
1.281
0.705
1.90
0.945
13.70
1.281
0.801
2.29
0.991
14.36
1.340
0.962
(b) Power Velocity Distribution Equation
The power form of the velocity distribution equation is:
v/V = a (y/yo)b.
To simplify the regression calculations take the ln transform giving:
ln v/V = ln a+ b ln (y/yo) using linear regression of ln v/V vs ln (y/yo) calculate the values of a and b.
The results are:
a
=
1.368
b
=
0.240
R
=
0.998
2.72
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