Example 1: Calculate the theoretical velocity distribution u(z) using ...
Example 1: Calculate the theoretical velocity distribution u(z) using ...
Example 1: Calculate the theoretical velocity distribution u(z) using ...
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Inhaltsverzeichnis 19<br />
<strong>Example</strong> 1:<br />
<strong>Calculate</strong> <strong>the</strong> <strong>the</strong>oretical <strong>velocity</strong> <strong>distribution</strong> u(z) <strong>using</strong> <strong>the</strong> measured <strong>velocity</strong> <strong>distribution</strong><br />
u e (z). For this purpose <strong>the</strong> general logarithmic law can be considered as a two-parameter<br />
analytic model, whereby <strong>the</strong> values of <strong>the</strong> parameters local shear-stress <strong>velocity</strong> and referent<br />
distance from <strong>the</strong> riverbed are determined by <strong>the</strong> measured <strong>velocity</strong> profile. The<br />
approximation is based on <strong>the</strong> following equation, which is represented by a straight line in<br />
<strong>the</strong> diagram.<br />
⎛z⎞<br />
⎛ h ⎞<br />
u( z)<br />
= 5,75 ⋅u* ⋅ log⎜<br />
⎟+ 5,75 ⋅u*<br />
⋅log⎜ ⎟<br />
⎝h⎠ ⎝z0<br />
⎠<br />
(0.19)<br />
measured <strong>the</strong>oretical<br />
water depth relative depth <strong>velocity</strong> ue <strong>velocity</strong> u(z)<br />
z [m]<br />
z/h [-]<br />
[m/s]<br />
[m/s]<br />
difference<br />
ue<br />
− u<br />
⋅ 100 [%]<br />
u<br />
0,00 0,00 0,00 0,000 0,00<br />
0,10 0,029 1,080 1,072 0,74<br />
0,20 0,057 1,163 1,138 2,15<br />
0,40 0,114 1,212 1,204 0,66<br />
0,80 0,229 1,270 1,270 0,00<br />
1,20 0,343 1,316 1,309 0,53<br />
1,60 0,457 1,332 1,337 0,38<br />
2,00 0,571 1,360 1,358 0,15<br />
2,40 0,686 1,370 1,375 0,36<br />
2,80 0,800 1,370 1,390 1,46<br />
3,20 0,914 1,392 1,403 0,79<br />
3,50 1,000 1,392 1,412 1,44<br />
e<br />
Note that <strong>the</strong> relative depth in this diagram is in <strong>the</strong> logarithmic scale. The best-fit line is<br />
drawn for a region close to <strong>the</strong> bed: z/h < 0.
20 Inhaltsverzeichnis
Inhaltsverzeichnis 21<br />
Solution 1:<br />
Two characteristic values u(z/h) are determined:<br />
⎛z<br />
⎞<br />
⎛ h ⎞<br />
u⎜<br />
= 1⎟<br />
= 0,0+ 5,75⋅u* ⋅ log⎜<br />
⎟ = u1<br />
= 1,41m/s<br />
⎝h<br />
⎠ ⎝z0<br />
⎠<br />
⎛z<br />
⎞<br />
⎛ h ⎞<br />
u⎜<br />
= 0,1⎟<br />
= −5,75 ⋅ u* + 5,75 ⋅u* ⋅ log⎜<br />
⎟ = u2<br />
= 1,19 m / s<br />
⎝h<br />
⎠ ⎝z0<br />
⎠<br />
By <strong>the</strong> subtraction of <strong>the</strong> second equation from <strong>the</strong> first one, <strong>the</strong> expression in order to<br />
calculate <strong>the</strong> value of <strong>the</strong> local shear <strong>velocity</strong> is derived:<br />
u<br />
*<br />
u1 − u2<br />
= = 0,038 m/s<br />
5,75<br />
The calculation of <strong>the</strong> distance from <strong>the</strong> referent distance yields to:<br />
u<br />
− 1<br />
5,75⋅u<br />
−6<br />
z = h⋅ 10<br />
*<br />
= 1,4⋅10 m<br />
0<br />
The calculated <strong>velocity</strong> <strong>distribution</strong> is graphically represented<br />
The relative differences between <strong>the</strong> <strong>the</strong>oretical and empirical <strong>velocity</strong> <strong>distribution</strong>s are given<br />
in <strong>the</strong> last column of Table 1. The best match appears near <strong>the</strong> bed, whereas errors increase<br />
towards <strong>the</strong> free-surface, as expected.