1.060 Engineering Mechanics II
Spring 2006
Recitation 8 - Problems
April 20th and 21st
Problem 1
Water flows in a long straight channel whose trapezoidal cross section shown in Figure 1.
The horizontal bottom is finished concrete and the sides are weedy. The bottom slope is
S0 = 0.001. The water depth over the horizontal bottom is 1 m.
a) Determine the discharge, Q.
b) Determine the corresponding mean velocity, V .
c) Determine the Froude number, Fr, of the flow.
d) Is the flow super- or subcritical?
Figure 1: Trapezoidal channel in Problem 1.
Problem 2
A long glass-walled laboratory flume has a rectangular cross-section of width 1 m, a slope
of S0 = 10−4 , and it discharges water at a flowrate Q = 0.5 m3 /s.
a) Find the critical depth, hc .
b) Find the normal depth, hn , corresponding to steady uniform flow.
c) Does the normal depth correspond to super- or subcritical flow?
Recitation 8-1
Values of the Manning Coefficient, n (Ref 5)
n
Wetted Perimeter
Natural Channels
Clean and straight
Sluggish with deep pools
Major rivers
Floodplains
Pasture, Farmland
Light brush
Heavy brush
Trees
0.030
0.040
0.035
0.035
0.050
0.075
0.15
Excavated Earth Channels
Clean
Gravelly
Weedy
Stony, Cobbles
0.022
0.025
0.030
0.035
Artifically lined channels
Glass
Brass
Steel, Smooth
Steel, Painted
Steel, Riveted
Cast iron
Concrete, Finished
Concrete, Unfinished
Planed Wood
Clay tile
Brickwork
Asphalt
Corrugated metal
Rubble masonry
0.010
0.011
0.012
0.014
0.015
0.013
0.012
0.014
0.012
0.014
0.015
0.016
0.022
0.025
Table by MIT OCW.
Recitation 8-2
Values of (VD') for water at 60o F (velocity, ft/s, diameter.in)
0.1
0.2
0.4
0.6 0.8 1
2
4
6
8 10
20
40
60
80 100
200
400
o
Values of (VD') for atmospheric air at 60
0.10
2
0.09
4
Laminar
Flow
0.08
6
8 10
Critical
Zone
20
40
60
100
200
Transition
Zone
400
600 800 1000
2000
4000
600 800 1000
8000
6000 10000
2000
20,000
4000
40,000
8000
6000 10,000
80,000
60,000 100,000
Complete turbulence rough pipes
0.05
0.07
f. 64
Re
0.03
inar
h
L V2
D 2g
Friction factor f -
Flow
0.05
0.02
0.015
0.04
0.01
0.008
0.03
0.006
Re
0.004
0.025
0.002
0.02
Relative roughness ε = ε/(4Rh)
D
0.04
Lam
0.06
0.001
0.0008
0.0006
0.0004
Smooth Pipes
0.015
0.0002
0.0001
ε 0.000.005
=
D
0.01
0.009
0.000.05
0.000.01
0.008
3
2(10 )
103
4A
D = ρ = 4Rh
3 4
5 6
4
8
2(10 )
104
3
4 5 6
5
2(10 ) 3
8
4
5 6
105
6
2(10 ) 3
8
106
VD
Reynolds Number Re =
υ
4
5 6
7
2(10 ) 3
8
4
5 6
8
107
ε 0.000.001
=
D
Re =
VD υ(4Rh)
=
υ
υ
The Moody chart for pipe friction with smooth and rough walls. This chart is identical to Eq. (6.64) for turbulent flow.
Graph by MIT OCW.
Recitation 8-2