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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 equals V sub * times (a plus b times the logarithm of (R divided by k sub s) (C.1)

where,

V= mean channel velocity, m/s (ft/s)
V*= shear velocity which is square root of ( g times R times S sub f, 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:

n equals alpha times R to the one-sixth power divided by the square root of g divided by (a plus b times the logarithm of (R divided by k sub s) (C.2)

where,

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.

square root of (one over f) equals a plus b times the logarithm of (d divided by k) (C.3)

where,

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
V*/V*crit a b
1.0 0.15 1.85
2.5 0.29 3.50

Critical shear velocity is estimated as the minimum value computed from the following two equations:

V sub *crit equals alpha sub 1 plus alpha sub 2 times MEI squared (C.4a)
V sub *crit equals alpha sub 3 times MEI to the 0.106 power (C.4b)

where,

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).

k divided by h equals 0.14 times the quantity ( the quantity (MEI divided by tau sub o) to the one-fourth power divided by h) to the eight-fifths power (C.5)

where,

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 Density-stiffness, MEI
Retardance Class m ft N·m2 lb·ft2
A 0.91 3.0 300 725
B 0.61 2.0 20 50
C 0.20 0.66 0.5 1.2
D 0.10 0.33 0.05 0.12
E 0.04 0.13 0.005 0.012

Eastgate (1966) showed a relationship between a fall-board test (Appendix E) and the MEI property as follows:

MEI equals alpha times h sub b to the 2.82 power (C.6)

where,

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.

MEI equals C sub s times h to the 2.82 power (C.7)

where,

Cs= grass density-stiffness coefficient

Combining Equations C.5 and C.7 gives:

k equals 0.14 C sub s to the 0.4 power times h to the 0.528 power times 1 over tau sub o to the 0.4 power (C.8)

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.

n equals alpha times C sub l times k (C.9)

where,

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:

C sub l equals 2.5 times C sub s to the -0.3 power (C.10)

Substituting Equations C.10 and C.8 in Equation C.9 yields the following:

n equals alpha times 0.35 times C sub s to the 0.1 power times h to the 0.528 power times (1 over tau sub o) to the 0.4 powe (C.11)

Defining a grass roughness coefficient, Cn as,

C sub n equals 0.35 times C sub s to the 0.10 power times h to the 0.528 power (C.12)

yields the following relationship for Manning's n:

n equals alpha times C sub n divided by tau sub o to the 0.4 power (C.13)

data show a linear relationship of n as a function of k. n equals 0.5028 times k
Figure C.1a. Relative Roughness Relationships for Excellent Vegetated Conditions


data show a linear relationship of n as a function of k. n equals 0.686 times k
Figure C.1b. Relative Roughness Relationships for Good Vegetated Conditions


data show a linear relationship of n as a function of k. n equals 1.3158 times k
Figure C.1c. Relative Roughness Relationships for Poor Vegetated Conditions


graph showing C sub l equals 2.5 times C sub s to the -0.3 power
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 divided by V sub * equals alpha times d sub a to the one-sixth power divided by n divided by square root of g equals function of Fr times function of REG times function of CG (C.14)

where,

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.

n equals alpha times d sub a to the one-sixth power divided by square root of g divided by function of Fr divided by function of REG divided by function of CG (C.15)

The functions of Froude number, roughness element geometry, and channel geometry are given by the following equations:

function of Fr equals (0.28 times Fr divided by b) to the power of (logarithm of (0.755 divided by b)) (C.16)
function of REG equals 13.434 times (T divided by Y sub 50) to the 0.492 power times b to the (1.025 times (T divided by Y sub 50) to the 0.118 power) (C.17)
function of CG equals (T divided by d sub a) to the minus b power (C.18)

where,

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:

b equals a times (d sub a divided by S sub 50) to the c power (C.19)

where,

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:

c equals 0.64 times sigma to the -0.134 power (C.20)

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:

a equals (1.175 times (Y sub 50 divided by T) to the 0.557 power) raised to the c power (C.21)

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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Updated: 04/07/2011
 

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