Design of Roadside Channels with Flexible Linings
Hydraulic Engineering Circular Number 15, Third Edition
C.1 GENERAL RELATIONSHIPS
Resistance to flow in open channels with flexible linings can be accurately described using the universal-velocity-distribution law (Chow, 1959). The form of the resulting equation is:
|V||= mean channel velocity, m/s (ft/s)|
|V*||= shear velocity which is , m/s (ft/s)
|a, b||= empirical coefficients|
|R||= hydraulic radius, m (ft)|
|kS ||= roughness element height, m (ft)|
|g||= acceleration due to gravity, m/s2 (ft/s2)|
Manning's equation and Equation C.1 can be combined to give Manning's roughness coefficient n in terms of the relative roughness. The resulting equation is:
|α||= unit conversion constant, 1.0 (SI) and 1.49 (CU)|
C.2 Grass Lining Flow Resistance
General form of the relative roughness equation (Kouwen and Unny, 1969; Kouwen and Li, 1981) is as follows.
|f||= Darcy-Weisbach friction factor|
|a, b||= parameters for relative roughness formula|
|d||= depth of flow, m (ft)|
|k||= roughness height (deflected height of grass stem), m (ft)|
Coefficients "a" and "b" are a function of shear velocity, V*, relative to critical shear velocity, V*crit, where the critical shear velocity is a function of the density-stiffness property of the grass stem. Table C.1 provides upper and lower limits of the relative roughness coefficients. Within these limits, values of the coefficients can be estimated by linear interpolation on V*/V*crit.
Table C.1. Resistance Equation Coefficients
Critical shear velocity is estimated as the minimum value computed from the following two equations:
|V*crit||= critical shear velocity, m/s (ft/s)|
|MEI||= density-stiffness parameter, N·m2 (lb·ft2)|
|α1||= unit conversion constant, 0.028 (SI) and 0.092 (CU)|
|α2||= unit conversion constant, 6.33 (SI) and 3.55 (CU)|
|α3||= unit conversion constant, 0.23 (SI) and 0.69 (CU|
Note: For MEI < 0.16 N·m2 (0.39 lb·ft2) the second equation controls, which is in the D to E retardance range.
The roughness height, k, is a function of density, M, and stiffness (EI) parameter (MEI). Stiffness is the product of modulus of elasticity (E) and second moment of stem cross sectional area (I).
|h||= grass stem height, m (ft)|
|τo||= mean boundary shear stress, N/m2, lb/ft2|
Values of h and MEI for various classifications of vegetative roughness, known as retardance classifications, are given in Table C.2.
Table C.2. Relative Roughness Parameters for Vegetation
||Average Height, h
Eastgate (1966) showed a relationship between a fall-board test (Appendix E) and the MEI property as follows:
|hb||= deflected grass stem height resulting from the fall-board test, m (ft)|
|α||= unit conversion constant, 3120 (SI) and 265 (CU)|
Kouwen collected additional data using the fall-board test (Kouwen, 1988). These data have been interpreted to have the following relationship.
|Cs||= grass density-stiffness coefficient|
Combining Equations C.5 and C.7 gives:
Over a range of shallow depths (y < 0.9 m (3 ft)), the n value is a function of roughness height as shown in Figure C.1. The linear relationships shown between roughness height, k, and Manning's n differ with vegetation condition.
|Cl||= k-n coefficient|
|α||= unit conversion constant, 1.0 (SI) and 0.213 (CU)|
As is apparent in Figure C.2, a relationship exists between Cl and Cs and is quantified in the following equation:
Substituting Equations C.10 and C.8 in Equation C.9 yields the following:
Defining a grass roughness coefficient, Cn as,
yields the following relationship for Manning's n:
Figure C.1a. Relative Roughness Relationships for Excellent Vegetated Conditions
Figure C.1b. Relative Roughness Relationships for Good Vegetated Conditions
Figure C.1c. Relative Roughness Relationships for Poor Vegetated Conditions
Figure C.2. Relationship between Cl and Cs
C.3 Bathurst Resistance Equation
Most of the flow resistance in channels with large-scale (relative to depth) roughness is derived from the form drag of the roughness elements and the distortion of the flow as it passes around roughness elements. Consequently, a flow resistance equation for these conditions has to account for skin friction and form drag. Because of the shallow depths of flow and the large size of the roughness elements, the flow resistance will vary with relative roughness area, roughness geometry, Froude number (the ratio of inertial forces to gravitational forces), and Reynolds number (the ratio of inertial forces to viscous forces).
Bathurst's experimental work quantified these relationships in a semi-empirical fashion. The work shows that for Reynolds numbers in the range of 4 x 104 to 2 x 105, resistance is likely to fall significantly as Reynolds number increases. For Reynolds numbers in excess of 2 x 105, the Reynolds effect on resistance remains constant. When roughness elements protrude through the free surface, resistance increases significantly due to Froude number effects, i.e., standing waves, hydraulic jumps, and free-surface drag. For the channel as a whole, free-surface drag decreases as the Froude number and relative submergence increase. Once the elements are submerged, Froude number effects related to free-surface drag are small, but those related to standing waves are important.
The general dimensionless form of the Bathurst equation is:
|V||= mean velocity, m/s (ft/s)|
|V*||= shear velocity = (gdS)0.5 , m/s (ft/s)|
|da||= mean flow depth, m (ft)|
|g||= acceleration due to gravity, 9.81 m/s2 (32.2 ft/s2)|
|n||= Manning's roughness coefficient|
|Fr||= Froude number|
|REG||= roughness element geometry|
|CG||= dchannel geometry|
|α||= unit conversion constant, 1.0 (SI) and 1.49 (CU)|
Equation C.14 can be rewritten in the following form to describe the relationship for n.
The functions of Froude number, roughness element geometry, and channel geometry are given by the following equations:
|T||= channel top width, m (ft)|
|Y50||= mean value of the distribution of the average of the long and median axes of a roughness element, m (ft)|
|b||= parameter describing the effective roughness concentration|
The parameter b describes the relationship between effective roughness concentration and relative submergence of the roughness bed. This relationship is given by:
|S50||= mean of the short axis lengths of the distribution of roughness elements, m (ft)|
|a, c||= constants varying with bed material properties|
The parameter, c, is a function of the roughness size distribution and varies
with respect to the bed-material gradation. σ, where:
For standard riprap gradations the log standard deviation is assumed to be constant at a value of 0.182, giving a c value of 0.814.
The parameter, a, is a function of channel width and bed material size in the cross stream direction, and is defined as:
In solving Equation C.15 for use with this manual, it is assumed that the axes of a riprap element are approximately equal for standard riprap gradations. The mean diameter, D50, is therefore substituted for Y50 and S50 parameters.
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