The corrugation type, corrugation size, discharge, depth,

diameter and culvert slope were extracted from the [3],

[5], and [7] to compliment the data collected at the HRTF.

This data was then used to calculate the Manning’s n

value for the culvert at that depth and discharge. It was

found that there was a significant increase in Manning’s

n as the depth within the culvert decreased, which can be

seen in Figure 1, where nf is used to denote Manning’s n

for full pipe flow as determined by the Corrugated Steel

Pipe Institute [9].

Figure 1 shows that the largest variation in Manning’s

n occurs at relative depths below 0.5. Therefore a set of

equations were developed, using a least squares regres-

sion for the variation in Manning’s n, which is shown

below as

0.175

0.878for 0.5

for 0.5

f

f

nhD hD

nnhD

(2)

By dividing the equation into two portions a slightly

better fit was achieved than simply applying a power

regression to the entire relationship, improving the R2

from 0.58 to 0.62. The proportionality plot in Figure 2

shows that Manning’s Equation using a constant value

for n gives good results over the entire range. However,

using a varying Manning’s n slightly increases the corre-

lation between the observed and predicted relative depths

(Figure 3). This is better represented by Figure 4, which

shows the percent difference between the predicted and

observed data. There is significant improvement for rela-

tive depths below 0.2. Figure 5 shows the predicted re-

sults using the equation developed by Manning, 2010.

As seen it does not do a good job of predicting the

flow depth below a relative depth of a 0.2. This led to the

development of a new equation, using the same dimen-

sionless parameters that were developed by Manning [8]

using dimensional analysis. The dimensionless parame-

5

QgDters S and were then plotted against the rela-

tive depth to determine the exponential relationship be-

tween them. The exponential relationship for Ks/D could

not be determined in a similar fashion since the tests

were not conducted at the same slopes between each re-

port. Therefore the exponential relationship for Ks/D

Figure 1. Normalized Manning’s n versus relative depth for data collected at the HRTF.

Copyright © 2012 SciRes. JWARP

J. S. TOEWS, S. P. CLARK

840

Figure 2. Proportionality plot of predicted vs observed h/D

using a constant Manning’s n.

Figure 3. Proportionality plot of predicted vs observed h/D

using a varying Manning’s n.

Figure 4. Comparison of predicted and measured relative depth for Ma nning’s Equation using a constant and varying n.

Figure 5. Proportionality plot for Manning’s Equation.

and trend multiplier were then optimized to minimize the

total error between predicted and observed relative

depths. The final form of the equation is

0.463

5

0.641

hQ

S

D

gD

0.169

0.247 s

K

D

(3)

As seen in Figures 6 and 7, this equation accurately

predicts relative depths below 0.5. However the model

error tends to increase as the relative depth increases past

this threshold. This was determined not to be an issue

Figure 6. Proportionality plot for Equation (3).

since Manning’s Equation is accurate in this region.

5. Conclusions

Results indicate that Manning’s Equation using a

constant n value does a reasonable job of predicting the

uniform depth within CMP culverts over a wide range of

water depths despite the fact that the roughness

coefficient has been shown to increase with decreasing

h/D. At very low water depths (h/D < 0.15) Equation (3)

performs better than Manning’s Equation, and may be a

Copyright © 2012 SciRes. JWARP

J. S. TOEWS, S. P. CLARK 841

Figure 7. Comparison of predicted and measured relative depth for Manning’s Equation and Equation (3).

more accurate means of estimating minimum flow depths

for fish passage at low discharge. At these very shallow

flows the resistance from the corrugations seem to act as

form drag, rather than merely a uniform skin friction.

The difficulty with Equation (3) is that its accuracy has

not been verified over a significant range of corrugation

heights and culvert diameters. The advantage of using

this equation is that a varying roughness coefficient does

not need to be developed. It also has been shown to be

accurate up to the mid-depth of the culvert, which corres-

ponds to the upper depth limit in the Manitoba Stream

Crossing Guidelines for culverts. Some recommended

future works include testing other corrugation sizes at

similar slopes to determine the variation of Manning’s n

for different CMP culvert types. This would also allow

for a better determination of the relative depths reliance

on the relative roughness.

6. Acknowledgements

The authors wish to thank the NSERC, Manitoba Hydro,

Manitoba Infrastructure and Transportation, and the Cor-

rugated Steel Pipe Institute of Canada for their support.

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Copyright © 2012 SciRes. JWARP