Friction in Uniform Flow Experiment/Gradually Varied Flow Experiment

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Introduction

Flows along a river through alluvial channels such as irrigation channels, and often encounter hydraulic resistance. This hydraulic resistance is a function of the fluid properties, physical characteristics of sediments, and geometry of the reach and cross-section. It is also proportional to the roughness of the boundary walls. Hydraulic resistance is categorized into four: surface resistance, form resistance, wave resistance, and resistance caused by the unsteadiness of the flow. Surface resistance is caused by viscous action on the boundary of the channel and is dependent on the relative roughness that represents the boundary’s physical characteristics and the flow’s Reynolds number. Form Resistance occurs in the alluvial channels. It occurs as a result of attached obstacles within the channel boundary such as bedforms. Wave Resistance occurs as a result of free surface distortions that generate a net pressure difference between existing cross-sections. The magnitude of resistance, irrespective of whether it follows energy or momentum concept, is often expressed in terms of resistance coefficient. For a steady uniform flow, there are three common resistance coefficients: manning coefficient, n, Chezy resistance coefficient, C, and Welsbach resistance coefficient, f. There are, therefore, three well-known formulas associated with these three formulas:

For Weisbach f: V= QUOTE RS/f

f= 8gRS/ v2

For Chezy C: V= C QUOTE

C= V/ QUOTE

For manning, n: V = QUOTE R0.66 S1/2

For manning, n, = (1.49 xR0.66 x S0.5)/v

Where v-cross-sectional average velocity (A), R-hydraulic radius and defined as ratio of cross-sectional Area to Wetted perimeter (P), f, C, and n are resistance coefficients in Daircy-Weisbach, Chezy and manning formula and S-hydraulic slope.

The objective of this experiment was to identify conditions under which a uniform can occur and determining resistance coefficients n (manning’s coefficient), C (Chezy Coefficient and f (Darcy friction factor).

Method and Experimental Procedure

The channel was first adjusted to a subcritical flow condition (mild slope) with the use of a jack controller. While ensuring the valve at the end of the downstream end of the flume is fully opened, the pump was opened. The pump was then started up while maintaining is speed at 800 rpm. This was followed by reading the value of the flow on the flow meter. To measure the flow depth where the uniform flow was established, a straight edged ruler was placed on the outside walls of the flume to help in reading the depth. The speed of the pump was then slowly reduced by 50 rpm by adjusting the slope of the flume to establish the uniform flow while recording the flow rate and the depths. The experiment was repeated up to ten different flowrates with five of these measurements being under subcritical flow condition and the rest under supercritical flow condition.

Results and Calculation

Table 01: Results for the Experiment

Profile type: m1

Flowrate (I/S): 35

Slope (m/m): 9 mm/m Profile type: S1

Flowrate (I/S): 37

Slope (m/m): 16mm/m

Distance along the flume (cm) Normal depth (Y) (mm) Distance along the flume (m) Normal depth (Y)

0 182 0 50

0.5 187 0.25 50

1 192 0.5 47

1.5 197 0.75 45

2 207 1 45

2.5 210 1.25 44

3 210 1.5 43

4 221 2 45

4.5 226 2.25 45

5 230 2.5 47

5.5 239 2.75 11

6 250 3 12

6.5 258 3.25 13

7 260 3.5 14.5

7.5 265 3.75 15.5

8 270 4 16.3

9 280 4.5 17.1

9.5 285 4.75 18.3

10 290 5 18.5

10.5 300 5.25 19.5

Question 1

Table 01:

Table 01, for v=35, and S= 9mm/m, Rh= 0.2

f= f= 8gRS/ v2

= (8×9.98x9x0.2)/35= 0.11

C= V/ QUOTE = 35/ QUOTE = 35/1.34= 26.12

For manning, n, = (1.49 xR0.66 x S0.5)/v= (1.49x R0.66 x 3)/35

n= (1.49 x0.20.66x 3)/35= 0.044

For Table 01, for v=37, and S= 16mm/m, Rh= 0.15

f= f= 8gRS/ v2

= (8×9.98x16x0.15)/37= 0.13

C= V/ QUOTE = 35/ QUOTE = 37/1.34= 23.88

For manning, n, = (1.49 xR0.66 x S0.5)/v= (1.49x R0.66 x 3)/35

n= (1.49 x0.150.66x 4)/37= 0.046

Question 2

From the above values,

V = Q/A = 40.2/6 ft/sec = 6.7 ft/sec and Reynold’s number can now be calculated as Re = ρVRh/μ= (1.94)(6.7)(0.8571)/( 2.730 x 10-5) = 4.08 x 105

The relative roughness can be found by k/Rh = (0.0015×10-3) / (0.02 m) = 0.000075

From the moody diagram, the estimated friction factor can is found to be approx 0.011.This shows that the calculated friction factors (0.11 and 0.13>0.011).

Question 4

V R0.66 S0.5

35 1.037 1.34

37 1.14 1.55

Question 4

Question 5

From the velocity, V versus R2/3S1/2 graph, the slope of the graph =1.055. From manning, n: V = QUOTE R0.66 S1/2 QUOTE represent the slope of the graph. Thus, QUOTE = 1.055. This means that n= ka/1.055

Question 6

From V= C QUOTE , C is the slope of the graph. Thus, the average value of C=slope = 1.053

Given that the Reynolds number Re= 105