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4.5 FLOW RESISTANCE CALCULATIONS IN ENGINEERING PRACTICE .1 Introduction
4.5.2 Selection of a flow resistance formula
Flow resistance calculations in open channels must be performed in term of the Darcy friction factor. First the type of flow regime (laminar or turbulent) must be determined. Then the fric-
tion factor is estimated using the classical results (Section 4.3.1).
In turbulent flows, the choice of the boundary equivalent roughness height is important.
Hydraulic handbooks (e.g. Idelchik, 1969, 1986) provide a selection of appropriate roughness heights for standard materials.
4.5 Flow resistance calculations in engineering practice 83
Fig. 4.12 (c) Flood plain with trees of the South Alligator River, Kakadu National Park, Northern Territory, Australia in March 1998 (Courtesy of Dr R. Rankin), (d) Flood of the Mississippi River at Prairie du Chien, Wl on 23 April 1969 (Courtesy of Dr Lou Maher) -The high bridge (on left) carries Highway 18 to Iowa.
The main limitations of the Darcy equation for turbulent flows are:
• the friction factor can be estimated for relative roughness kJD^i less than 0.05,
• classical correlations for/were validated for uniform-size roughness and regular roughness patterns.
In simple words, the Darcy equation cannot be applied to complex roughness patterns: e.g. vege- tation and trees (in flood plains), shallow waters over rough channels.
For complex channel bed roughness, practising engineers might estimate the flow resistance by combining the Chezy equation (4.23) with an 'appropriate' Gauckler-Manning or Chezy coefficient. Such an approximation is valid only for fiilly rough turbulent flows.
Discussion
Great care must be taken when using the Gauckler-Manning equation (or Strickler equation) as the values of the coefficient are empirical. It is well known to river engineers that the estimate of the Gauckler-Manning (or Strickler) coefficient is a most difficult choice. Furthermore, it must be emphasized that the Gauckler-Manning formula is valid only for fully rough turbulent flows of water.
It should not be applied to smooth (or transition) turbulent flows. It is not valid for fluids other than water. The Gauckler-Manning equation was developed and 'validated' for clear-water flows only.
In practice, it is recommended to calculate the flow resistance using the Darcy fiiction factor.
Empirical correlations such as the Bazin, Gauckler-Manning and/or Strickler formula could be used to check the result. If there is substantial discrepancy between the sets of results, experimental investigations must be considered.
Note
Several computer models of river flows using the unsteady flow equations are based on the Gauckler- Manning equation. Professionals (engineers, designers and managers) must not put too much con- fidence in the results of these models as long as the resistance coefficients have not been checked and verified with experimental measurements.
Applications
1. In a rectangular open channel (boundary roughness: PVC), the uniform equilibrium flow depth equals 0.5 m. The channel width is 10 m and the channel slope is 0.002°. Compute the discharge.
The fluid is water at 20°C.
Solution
The problem must be solved by iterations. First we will assume that the flow is turbulent (we will need to check this assumption later) and we assume kg = 0.01 mm (PVC).
An initial velocity (e.g. 0.1 m/s) is assumed to estimate the Reynolds number and hence the Darcy fiiction factor at the first iteration. The mean flow velocity is calculated using the momentum equa- tion (or the Chezy equation). The full set of calculations are summarized in following table. The total flow rate equals 1.54m^/s. The results indicate that the flow is subcritical (Fr = 0.139) and turbulent (Re = 5.6 X 10^). The shear Reynolds number 7?e* equals 0.12: i.e. the flow is smooth turbulent.
Note: As the flow is not fiilly rough turbulent, the Gauckler-Manning equation must not be used.
Iteration K (initialization) (m/s) / Qh^zy (ni^^^/s) F(m/s) (equation (4.18)) (equation (4.23)) (equation (4.23)) 0.0160 69.8 0.28 0.0134 76.6 0.305 0.0131 77.2 0.31 0.0131 77.3 0.31 2. Considering an uniform equilibrium flow in a rectangular concrete channel (B = 1 m), the total discharge is lOm^/s. The channel slope is 0.02°. Compute the flow depth. The fluid is water at 20°C.
Solution
The problem must be solved again by iterations. We will assume a turbulent flow and k^= \ mm (concrete).
At the first iteration, we need to assume a flow depth d (e.g. d = 0.5 m) to estimate the relative roughness kJDi^, the mean velocity (by continuity) and the Reynolds number. We deduce then the
1 2 3 4
0.1 0.28 0.305 0.31
4.5 Flow resistance calculations in engineering practice 85 friction factor, the Chezy coefficient and the mean flow velocity (Chezy equation). The new flow depth is deduced from the continuity equation.
The iterative process is repeated until convergence.
Iteration 1 2 3 4 5 6
d (initialization) (m)
0.5 7.1 4.0 4.2 4.15 4.16
/
(equation (4.18)) 0.018
0.0151 0.0153 0.0152 0.0152 0.0152
Qhezy (in
(equation 65.4 72.1 71.7 71.7 71.7 71.7
1/2/s)
(4.23))
r(m/s) (equation (4.23)) 0.705
1.26 1.20 1.20 1.20 1.20
^(m) 7.09 4.0 4.18 4.15 4.16 4.16 The calculations indicate that the uniform equilibrium flow depth equals 4.16 m. The flow is subcritical {Fr = 0.19) and turbulent {Re = 3.8 X 10^). The shear Reynolds number equals 52.1: i.e. the flow is at transition between smooth turbulent and frilly-rough-turbulent.
Note: as the flow is not frilly rough turbulent, the Gauckler-Manning equation must not be used.
3. Considering a rectangular concrete channel {B = 12 m), the flow rate is 23m^/s. The channel slope is 1°. Estimate the uniform equilibrium flow depth using both the Darcy friction factor and the Gauckler-Manning coefficient (if the flow is frilly rough turbulent). Compare the results. Investi- gate the sensitivity of the results upon the choice of the roughness height and Gauckler-Manning coefficient.
Solution
The calculations are performed in a similar manner as the previous example. We will detail the effects of roughness height and Gauckler-Manning coefficient upon the flow depth calculation.
For concrete, the equivalent roughness height varies from 0.3 to 3 mm for finish concrete and from 3 to 10 mm for rough concrete. For damaged concrete the equivalent roughness height might be greater than 10 mm. The Gauckler-Manning coefficient for concrete can be between 0.012 (fin- ished concrete) and 0.014 s/m^^^ (unfinished concrete). The results of the calculation are summar- ized in the following table.
Calculation Darcy friction factor
Gauckler-Manning formula
Surface Finished Unfinished Finished Unfinished
^s(mm) 0.3 1 3 10
/ 0.0143 0.0182 0.0223 0.032
'^Manning
0.012 0.014
(s/m^/3) CchezyCm^'Vs)
74.1 65.7 58.0 49.5 69.6 74.8
^(m) 0.343 0.373 0.406 0.452 0.359 0.341 First, the Reynolds number is typically within the range 7 X 10^ — 7.2 X 10^ (i.e. turbulent flow). The shear Reynolds number is between 70 and 2700. That is, the flow is frilly rough turbulent and within the validity range of the Gauckler-Manning formula. Secondly, the flow depth increases with increas- ing roughness height. The increase in flow depth results from a decrease of flow velocity with increas- ing flow resistance. Thirdly, let us observe the discrepancy of results for unfinished concrete between Darcy fiiction factor calculations and the Gauckler-Manning formula. Furthermore, note that the calculations (using the Darcy fiiction factor) are sensitive upon the choice of roughness height.