4.4 Flow resistance in open channels 69
Negative surge
(as seen by an observer standing on the bank)
Gate opening
J S -
////////////////////////////////////
Fig. 4.9 Negative surge.
Discussion
Positive surge flows are solved using the quasi-steady flow analogy (Section 4.3.2). For negative surges, the flow is unsteady and no quasi-steady flow analogy exists. The complete unsteady analysis is necessary (e.g. Henderson, 1966; Liggett, 1994). This will be developed in Chapters 16 and 17.
For practising engineers, it is important to recognize between positive and negative surge cases. The table below summarizes the four possible cases:
Surge wave:
Moving
Positive surge Negative surge
downstream upstream downstream upstream Front of deep water deep water shallow water shallow water Energy balance loss of energy loss of energy gain of energy gain of energy Wave front stability stable stable unstable unstable Analysis quasi-steady quasi-steady unsteady unsteady
Notes
1. A positive surge is characterized by a steep advancing front. It is easy to recognize. Negative surges are more difficult to notice as the free-surface curvature is very shallow.
2. For a negative surge, the quasi-steady analysis is not valid (Chapters 16 and 17).
4.4 FLOW RESISTANCE IN OPEN CHANNELS
Bernoulli equations can be applied to estimate the downstream flow properties as functions of the upstream flow conditions and boundary conditions. However, the approximation of fiictionless flow is no longer valid for long channels. Considering a water supply canal extending over several kilometres, the bottom and sidewall fiiction retards the fluid, and, at equilibrium, the fiiction force counterbalances exactly the weight force component in the flow direction.
The laws of flow resistance in open channels are essentially the same as those in closed pipes (Henderson, 1966). In an open channel, the calculations of the boundary shear stress are compli- cated by the existence of the fi'ee surface and the wide variety of possible cross-sectional shapes.
Another difference is the propulsive force acting in the direction of the flow, hi closed pipes, the flow is driven by a pressure gradient along the pipe whereas, in open channel flows, the fluid is propelled by the weight of the flowing water resolved down a slope.
Head loss
For open channel flow as for pipe flow, the head loss AH over a distance A^- (along the flow direction) is given by the Darcy equation:
A^ V^
^=fir^ (4.12)
w h e r e / i s the Darcy coefficient'*, F i s the mean flow velocity and D^ is the hydraulic diameter or equivalent pipe diameter. In open channels and assuming hydrostatic pressure distribution, the energy equation can be conveniently rewritten as:
d^ cos di + ZQI + aj - ^ = (^2 cos 02 + ^o2 + ^2 + ^^ (4.13) 2g 2g
where the subscripts 1 and 2 refer to the upstream and downstream cross-section of the control volume, d is the flow depth measured normal to the channel bottom, 6 is the channel slope, ZQ is the bed elevation, Fis the mean flow velocity and a is the kinetic energy correction coefficient (i.e. Coriolis coefficient).
Notes
1. Henri P.G. Darcy (1805-1858) was a French civil engineer. He gave his name to the Darcy- Weisbach friction factor.
2. Hydraulic diameter and hydraulic radius
The hydraulic diameter and hydraulic radius are defined respectively as:
^ ^ cross-sectional area A A Dy^ = 4 = —
wetted perimeter P^
_ cross-sectional area A
7?H = = — (e.g. Henderson, 1966: p. 91) wetted perimeter P^
where the subscript H refers to the hydraulic diameter or radius, A is the cross-sectional area and P^ is the wetted perimeter.
^ Also called the Darcy-Weisbach friction factor or head loss coefficient.
4.4 Flow resistance in open channels 71 The hydraulic diameter is also called the equivalent pipe diameter. Indeed it is noticed that:
Dn = pipe diameter D for a circular pipe
The hydraulic radius was used in the early days of hydraulics as a mean flow depth. We note that:
Rn = flow depth d for an open channel flow in a wide rectangular channel
The author of the present textbook believes that it is preferable to use the hydraulic diameter rather than the hydraulic radius, as the fiiction factor calculations are done with the hydraulic diameter (and not the hydraulic radius).
Bottom shear stress and shear velocity
The average shear stress on the whetted surface or boundary shear stress equals:
To = Q i p F ^ (4.14a) v^here Q is the skin friction coefficient^ and F i s the mean flow^ velocity. In open channel flow,
it is common practice to use the Darcy friction factor/, w^hich is related to the skin friction coefficient by:
It yields:
f 7
^0= JP^ (4.14b) The shear velocity V* is defined as (e.g. Henderson, 1966: p. 95):
V. = J l l (4.5)
w^here TQ is the boundary shear stress and p is the density of the flowing fluid. The shear velocity is a measure of shear stress and velocity gradient near the boundary.
As for pipe flows, the flow regime in open channels can be either laminar or turbulent. In industrial applications, it is commonly accepted that the flow becomes turbulent for Reynolds numbers larger than 2000-3000, the Reynolds number being defined for pipe and open channel flows as:
Re = p^^ (4.16) where /JL is the dynamic viscosity of the fluid, D^ is the hydraulic diameter and V is the mean
flow velocity.
^ Also called drag coefficient or Fanning friction factor (e.g. Liggett, 1994).
Most open channel flows are turbulent. There are three types of turbulent flows: smooth, transition and fully rough. Each type of turbulent flow can be distinguished as a function of the shear Reynolds number defined as:
Re. = - ^ (4.17)
V
where k^ is the average surface roughness (e.g. Henderson, 1966: p. 95-96). For turbulent flows, the transition between smooth turbulence and fully rough turbulence is approximately defined as:
Flow situation Open channel flow Pipe flow (Reference) (Henderson, 1966) (Schlichting, 1979) Smooth turbulent Re* < 4 Re*<5 Transition 4 < i?e* < 100 5 < Re* < 75
Fully rough turbulent 100 < Re* 75 < Re*
Notes
1. The shear velocity being a measure of shear stress and velocity gradient near the boundary, a large shear velocity V* implies large shear stress and large velocity gradient. The shear velocity is commonly used in sediment-laden flows to calculate the sediment transport rate.
2. The shear velocity may be rewritten as:
where Fis the mean flow velocity.
Friction factor calcuiation
For open channel flow the effect of turbulence becomes sensible for Re > 2000-3000 (e.g.
Comolet, 1976). In most practical cases, open channel flows are turbulent and the friction factor (i.e. Darcy coefficient) may be estimated from the Colebrook-Whiteformula (Colebrook,
1939):
^ = - 2 . 0 log, -f 2-^1 \ ff • " ' \ 3 . 7 1 Z ) H Reff]
(4.18) where k^ is the equivalent sand roughness height, D H is the hydraulic diameter and Re is the Reynolds number defined as:
Re = p^^ (4.16) Equation (4.18) is a non-linear equation in which the friction factor/is present on both the left-
and right-sides. A graphical solution of the Colebrook-White formula is the Moody diagram (Moody, 1944) given in Fig. 4.10.
0.4 0.6 0.8 1
Values of (\/D") for water at 60° F (velocity In ft/s x diameter in Inches)
2 4 6 8 10 20 40 60 80 100 200 400 600 8001000 2000 4000 6000 800010,000
0.1 0.09 0.08 0.07
0.06
0.051-
.9 0.025
0.015
0.01 0.009 0.008