Hydraulic Loss Coefficients for Culverts (2012)

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60 7.1 Summary Manning’s Equation, which is used to estimate the head- discharge relationships in open channel flow applications, states that the mean channel flow velocity is inversely propor- tional to the Manning’s n hydraulic roughness coefficient and proportional to the hydraulic radius raised to an exponent (x′) of ²⁄³ (i.e., Rh 2/3). n and x′ represent empirical coefficients used to correlate Manning’s Equation with experimental data. In developing Manning’s Equation, Manning evaluated the stage-discharge characteristics of a range of boundary rough- ness materials ranging from smooth cement to course gravels and reported unique values of n and x′ for each boundary type. The x′ values ranged from approximately 0.65 (smooth- est boundary tested) to 0.84 (roughest boundary tested). Man- ning chose x′ = ²⁄³ as representative, compared it with field data, and suggested that it was sufficiently accurate. He also offered the caveat, however, that the use of Manning’s Equa- tion should be limited to cases where its accuracy has been validated. Chapter 6 showed that Manning’s n is not constant for all boundary materials and all stage-discharge conditions. This chapter evaluates the behavior of x′ with respect to constant n assumptions for the four boundary roughness materi- als discussed in Chapter 6 (smooth acrylic sheeting, metal lath, trapezoidal corrugations, and blocks) and the boundary roughness materials analyzed by Manning (1889). Consis- tent with the results reported by Manning (1889), this study found that the x′ = ²⁄³ assumption is appropriate for smooth boundaries (e.g., acrylic and pure cement) and for rougher boundary materials when a quasi-smooth boundary condi- tion exists. The quasi-smooth boundary condition describes a condition where the voids between the boundary rough- ness elements are filled with stable eddies, which effectively reduces the influence of the boundary roughness elements on flow resistance. For rougher boundary materials not in the quasi-smooth boundary flow condition, applying the constant Manning’s n assumption results in x′ values in excess of ²⁄³. If the constant x′ = ²⁄³ assumption is applied, then n must vary, as discussed in Chapter 6, in order to accurately predict the stage-discharge relationship using Manning’s Equation. 7.2 Introduction Uniform-flow head-discharge relationships for open chan- nel applications correlate flow rate (Q) or mean channel veloc- ity (V) to an energy gradient, taking into account the flow resistance associated with the channel cross-sectional shape and boundary roughness. Most open channel head-discharge or uniform-flow equations are in the form of Equation 7-1 (Chow, 1959). V CR Shx e y = ′ ′ ( )7 1- In Equation 7-1, Rh is the hydraulic radius [the flow area (A) divided by the wetted perimeter (P)], Se is the energy grade line or friction slope (equal to the channel slope So for uniform flow conditions), C is a flow resistance coefficient, and x′ and y′ are exponents. The Chezy Equation (Equation 7-2), Darcy- Weisbach Equation (Equation 7-3), and Manning Equation (Equation 7-4) represent three common open channel flow head-discharge relationships derived from Equation 7-1: V C R Sc h e= ( )7 2- V g f R Sh e= 8 7 3( )- V K n R Sn h e= 2 3 1 2 7( )-4 In Equations 7-2 through 7-4, g is the acceleration due to gravity; Kn = 1 for International System of Units and Kn = C h a p t e r 7 Open Channel Flow Resistance: the Hydraulic Radius Dependence of Manning’s Equation and Manning’s n

61 3.281(1-x′) for English Standard Units; and Cc, f, and n are equation-specific hydraulic roughness coefficients. Equa- tions 7-2 and 7-3 are equivalent, with the same x′ and y′ values and alternate definitions of the flow resistance coef- ficient, with the exception of the way the hydraulic rough- ness for flow resistance is quantified. Manning’s Equation is different from the other two, with x′ = ²⁄³ instead of ½. Manning (1889) made this change with the hope of devel- oping a simplified open channel equation where the rough- ness coefficient (n) would be constant for a given channel lining material (i.e., independent of stage and discharge). Equation 7-4 is commonly applied in practice with the assumption that n remains constant for a given boundary roughness material. Chow (1959) stated that if the boundary roughness in a channel is uniform (i.e., the roughness is the same for the entire wetted perimeter over the length of the channel sec- tion) and the slope of the channel bottom is also uniform, then there is a possibility that Manning’s n could remain con- stant for all flow stages. More recently, Yen (2002) suggested that the constant n assumption is appropriate under certain conditions and makes Equation 7-4 more convenient to use than Equations 7-2 and 7-3. Data have also shown, how- ever, that n is not always constant with stage and discharge (Bathurst et al., 1981; Jarrett, 1984; and Ugarte and Madrid, 1994). These studies were performed on “steep” mountain streams with relatively rough natural channel boundaries [in some cases, the height of the roughness elements exceeded the flow depth (y)]. The results of Bathurst et al. (1981), Jarrett (1984), and Ugarte and Madrid (1994) and the statements of Yen (2002) and Chow (1959) are all somewhat supported by the discussion in Chapter 6, which documents both variable and constant n flow regimes in a rectangular channel with a uniform roughness boundary. In this chapter, the appropriateness of the constant n assumption, relative to the behavior of the other empirically determined fitting parameter in Manning’s Equation, x′, is evaluated by applying a similar analysis method to that used by Manning (1889) in the development of Equation 7-4 to the data sets used by Manning (1889) and the Manning’s n data presented in Chapter 6 (acrylic, metal lath, trape zoidal corrugation, and the block channel boundary roughness materials). 7.3 Background Chezy Equation and Darcy-Weisbach Equation The Chezy Equation (Equation 7-2) was developed circa 1769 for uniform open channel flow. Two basic assumptions contributed to its derivation: (1) the force resisting the flow per unit area of the streambed is proportional to the square of the velocity and (2) the flow gravitational force is equal and opposite to the flow resistance force (Chow, 1959). The Darcy-Weisbach Equation was developed for pressur- ized pipe flow via dimensional analysis. Values for f, which vary with k/D (k is defined as an equivalent roughness height and D is the pipe diameter) and Re, are presented for smooth-walled or non-profiled-wall pipe in the Moody Diagram, which can be found in most hydraulic handbooks. Chow (1959) stated that if Se represents the head loss per unit length of pipe or channel and if D were replaced by 4Rh, then Equation 7-3 could be applied to open channel flow. The relationship between Cc and f is shown in Equation 7-5: C C g f c= = 8 7( )-5 Manning’s Equation Aware of the variable nature of hydraulic roughness coef- ficient behavior with most open channel flow equations, including Equations 7-2 and 7-3, Manning (1889) presented an alternate open channel flow head-discharge relationship (Equation 7-4) intended to produce constant hydraulic rough- ness coefficients for given channel boundary materials (i.e., the roughness coefficient is independent of flow conditions). Manning assumed this equation would take the form of Equa- tion 7-1 with y′ = ½ as shown in Equation 7-6. V CR Shx e= ' ( ) 1 2 7-6 The empirical basis for Equation 7-4 came from experi- mental data published by Bazin (1865), who hydraulically tested four different flow boundary materials [pure cement and 2-to-1-ratio mixes of cement and fine sand, cement and small gravel (particle sizes ranging from 0.36 to 0.84 in.), and cement and large gravel (particle diameters ranged from 1.2 to 1.6 in.)]. After determining boundary-roughness-specific constant values for C in Equation 7-6, Manning reported that the boundary-roughness-specific average exponent x′ values ranged from 0.6499 to 0.8395, with x′ generally increasing with increasing boundary roughness. Manning assumed x′ = ²⁄³ (a value most consistent with smoother boundary roughness materials) to be representative and considered the resulting equation, Equation 7-4, to be “sufficiently accurate” after applying the equation to numerous experiments. Rec- ognizing the potential limitations of Equation 7-4, Manning (1889) suggested that due to its empirical nature the applica- tion of Equation 7-4 should be limited to situations where it has been tested and proven.

62 Equations for Variable Roughness Coefficients Bathurst (2002) stated that some researchers have found success in using empirical formulas based on a power law relationship in the form of Equation 7-7 to describe hydraulic roughness coefficient variations: V V a R k h b  =   ( )7-7 In Equation (7-7), V (shear velocity) = (gRhSe)1/2, a and b are empirical coefficients, and k is the equivalent roughness height, which Chow (1959) suggests is not necessarily equal to the height or even the average height of the roughness ele- ments. The effect of the roughness elements on the hydraulic roughness coefficient is characterized by k; however, it has limited physical meaning and its definition can vary by user. It is therefore another empirical coefficient, and its physi- cal meaning depends on how it is defined for a particular equation. For example, in equations involving gravel beds, k is often defined as a representative Dr (the representative particle diameter of the channel boundary where r indicates the percentage of particles that are smaller than Dr). V/V is related to the standard hydraulic roughness coefficients as shown in Equation 7-8: V V f C g k R n g c n h  = = = 8 7 1 6 ( )-8 Bray (1979) and Griffiths (1981) published power law rela- tionships consistent with Equation 7-7 for rigid-boundary, gravel-bed rivers. Bathurst (2002) observed that, even though mountain streams may be characterized as gravel-bed riv- ers, these equations were relatively inaccurate when applied. Mountain streams are characterized by steep slopes and rela- tively low Rh/k values. According to Bathurst, one reason for these inaccuracies is that the relationships were developed by compiling data from many different river sites, fitting one curve to all the data, and then extrapolating these relation- ships to predict behaviors outside of the experimental data set. By gathering data for different flow conditions from the same river section and methodically grouping the data from similar sites, Bathurst (2002) showed that for the same type of channel (mountain streams), the data were best described by two significantly different relationships, suggesting that a and b are fairly site-specific parameters and are not solely dependent on a single channel type. Bathurst concluded that the differences between the coefficients in mountain streams were primarily related to variations in channel slope. Table 7-1 presents the coefficients for Equation 7-7 published in the referenced studies. If Equation 7-7 is simplified and solved for V, as shown in Equation 7-9, the equation takes on the form of Equation 7-6 (x′ = b + ½ and C = ag1/2/kb), which suggests a constant expo- nent, x′, and a constant roughness coefficient, C, for a given boundary roughness, provided that a and b are constant. V a g k R S b h b e=     +( )1 2 1 2 7( )-9 Applying the coefficients from Table 7-1 to Equation 7-9 shows that the x′ = ²⁄³ assumption made by Manning (1889) is not necessarily “sufficiently accurate” for all open channel flow conditions since the value of x′ can be boundary rough- ness specific, as illustrated by the data in Table 7-1. This study investigates the variation in x′ related to different boundary roughness types in a laboratory setting, where parameters are more easily controlled, to gain a better understanding of the appropriateness of the constant n assumption applied to Manning’s equation. 7.4 Experimental Method The behavior of Manning’s n for four different boundary roughness materials was investigated by conducting flow tests in a 4-ft-wide by 3-ft-deep by 48-ft-long adjustable-slope, rectangular laboratory flume. The four boundary roughness materials tested included acrylic sheeting (see Figure 6-2); a low-profile, commercially available, expanded metal lath adhered to the acrylic flume walls and floor (see Figure 6-3); regularly spaced wooden blocks (see Figures 6-4 and 6-5); and trapezoidal corrugations oriented normal to the flow direc- tion (see Figures 6-6 and 6-7). The wooden blocks measured 4 in. wide (normal to flow direction) by 3.5 in. long by 1.5 in. tall and the top edges were rounded (1-in. radius round-over). The blocks featured a painted exterior and were assembled in a closely spaced, uniform pattern. The wooden trapezoidal cor- rugation elements were 1.5 in. tall, had a top width of 1.5 in., had a base width of 4.5 in., and were spaced 1.5 in. apart. The blocks and trapezoidal corrugation elements were attached to sheets of painted marine grade plywood that were attached to the flume floor and walls. Water was supplied to the flume from a reservoir located adjacent to the laboratory and was metered using calibrated Study a b x' k Bray (1979) 5.03 0.268 0.768 D90 Griffiths (1981) 3.54 0.287 0.787 D50 Bathurst (2002) 3.84 0.547 1.047 D84 Bathurst (2002) 3.10 0.93 1.430 D84 Table 7-1. Published coefficients for the power law equation (Equation 7-7).

63 orifice flow meters located in the supply piping. Flow depths were measured using a precision point gage, readable to 0.008 in., attached to a movable carriage located above the flume. Manning’s n can be directly calculated via Equation 7-4 when uniform flow exists in the channel and the flow depth (y) and flow rate (Q) are known. Due to the limited length of the laboratory flume, uniform flow depths could not be achieved for all test conditions. For non-uniform flow con- ditions, a gradually varied flow (GVF) profile analysis tech- nique was used, as discussed in Section 6-4. Figure 6-8 shows a plot of the n data calculated using the uniform flow and the GVF methods versus y for the block boundary rough- ness. The plotted data show good agreement between the two methods. The uniform flow data in Figure 6-8 show that, for the block roughness, n varies (0.087 ≥ n ≥ 0.038) with changes in uni- form flow depth (0.13 ≤ y ≤ 0.9). Analysis of various truncated sections of a single GVF profile using the GVF n method also produced different predictive values for n, suggesting that n is also variable with depth (and velocity) throughout a GVF profile. Based on the variable nature of n with y in the GVF profiles, the predicted normal depths (yn) associated with the variable n values would also vary. Consequently, based on the good correlation between the uniform flow and GVF n data presented in Figure 6-8, yaverage, the average value of y in the measured GVF profile, was selected as the repre- sentative flow depth parameter in this analysis for calcu- lating Rh, Re, V, etc. For flow conditions where uniform flow developed, yaverage = yn. The four boundary roughness materials were tested over a range of channel slopes and discharges. 7.5 Discussion and Results For Manning’s n coefficient to remain constant for a given channel lining material, independent of stage and discharge, the following two conditions must be met: 1. The mean flow velocity can be represented by an equation in the form of Equation 7-6. 2. x′ will equal ²⁄³, independent of the channel lining material. If these conditions are not met, then n must vary in order to match Equation 7-4 with the actual head-discharge relation- ship. Conditions 1 and 2 were tested by plotting log(V/Se1/2) ver- sus log(Rh) using data from Bazin (1865) and the current study. To satisfy Condition 1, the data should be well represented by a linear equation of the form of Equation 7-10. In Equation 7-10, C is equal to the y-intercept on the plot, and x′ is the slope. The corresponding x′ values are presented in Table 7-2. log log ( ) V S C x R e h1 2 7     = + ′ ( ) -10 The x′ values corresponding to Bazin’s data in Table 7-2 are consistent with those calculated and reported by Man- ning (1889) for the same data sets. The r2 values [coeffi- cient of determination applied to the linear relationship of log(V/Se1/2) versus log(Rh)] in Table 7-2, which are all ≈1.0, indicate that V is relatively well represented by Equation 7-6, and Condition 1 is satisfied. However, according to the x′ data presented in Table 7-2, which vary and are boundary roughness specific, Condition 2 is not met. The smoother roughness boundary x′ values (e.g., pure cement, cement/sand mix, and acrylic) are approximately Boundary Roughness Material Description x' r2 Bazin Study (1865) Pure Cement Pure cement lining 0.676 0.998 Cement-Sand Mix 2/3 cement, 1/3 fine sand mix 0.684 0.994 Small Gravel Diameters ranging from 0.36 in. to 0.84 in. 0.721 0.997 Large Gravel Diameters ranging from 1.2 in. to 1.56 in. 0.822 0.999 Laths of Wood (Corrugations) 0.36 in. tall, 1.1 in. wide, spaced 1.92 in. apart, oriented normal to flume centerline 0.732 0.997 Current Study Acrylic Acrylic lining of flume boundary (see Figure 6-2) 0.644 0.982 Metal Lath Commercially available expanded metal lath with a thickness of 0.125 in. (see Figure 6-3) 0.795 0.989 Trapezoidal Corrugations 1.5 in. in height, top width of 1.5 in., and bottom width of 4.5.in., spaced 1.5 in. apart, oriented normal to flume centerline (see Figure 6-6) 0.968 0.968 Blocks 4.5 in. wide by 3.5 in. long by 1.5 in. tall, with the top edges rounded (1- in. radius round-over) (see Figure 6-4) 1.160 0.997 Table 7-2. Optimal x′ values.

64 equal to the ²⁄³ value used by Manning (see Equation 7-4). For the rougher boundaries, x′ varied significantly (up to 1.16 for the blocks). Figure 7-1 illustrates the relevance to the assumption of a constant roughness coefficient of the differ- ent x′ values associated with Equation 7-6. Figure 7-1 compares the roughness coefficients from three different versions of Equation 7-6: the Chezy Equation (Equa- tion 7-2) or Darcy-Weisbach (Equation 7-3) Equation (where x′ = ½), Manning’s Equation (Equation 7-4) (x′ = ²⁄³), and Equation 7-6 where x′ is varied per Equation 7-10 in order to maintain a constant n value (nopt). For convenience, the hydraulic roughness coefficient results in Figure 7-1 are all presented in terms of an equivalent n value (neq). This was done by replacing C in Equation 7-6 with Kn/neq and noting that Kn = 3.281(1-x′) for the individual boundary roughness materials (i.e., x′ varies with boundary roughness type). Figure 7-1(A) presents neq versus Rh for the pure cement lining data reported by Bazin (1865). The data show a variable nc; n and nopt are relatively constant and equal. The constancy of n and nopt is due to the fact that Conditions 1 and 2 are both satisfied. The Manning’s n data for the acrylic and the cement-sand mixture boundary conditions (not presented) had similar x′ values to the pure cement and behaved similarly. Figures 7-1(B) and (C) present the data for the large gravel roughness (Bazin, 1865) and the block roughness, respec- tively. These figures show examples where Condition 2 is not met and, therefore, Manning’s Equation requires a vari- able n value to match the results of the experimental data. While Manning’s Equation (Equation 7-4) improves upon the Chezy Equation (Equation 7-2)—that is, the difference between the maximum and minimum neq values decreases from 0.0124 (nc curve) to 0.008 (n curve)—the roughness coefficient is not constant unless x′ of Equation 7-6 is opti- mized for these specific boundary roughness materials, as evidenced in the nopt curve. The results clearly indicate that either x′ or the bound- ary roughness coefficient (n, f, or C) must vary to accurately describe the hydraulic behavior of the stage-discharge rela- tionship as Rh varies. Although some research has suggested correcting Manning’s Equation by changing x′ (Blench, 1939), more recent research has focused on variable roughness coef- ficient predictive techniques (Limerinos, 1970; Bray, 1979; Griffiths, 1981; Bathurst et al., 1981; Jarret, 1984; Ugarte and Madrid, 1994; and Bathurst, 2002) for use in Equations 7-2, 7-3, and/or 7-4. Equation 7-9 shows that using the power law equation to determine a variable hydraulic roughness coef- ficient is basically the equivalent of changing the x′ value of Equation 7-6 and applying a constant roughness coefficient. These power law equations are generally developed for a specific boundary roughness type with the underlying assumption that the equation applies to a range of roughness element sizes (generally characterized by k). For example, Bray (1979) and Griffiths (1981) present equations developed for channels with rigid gravel beds; Bathurst (2002) presents an equation for mountain streams. Each of these equations uses a k value defined by gravel Dr. They assume that a single x′ value may apply to a range of roughness element sizes that can be characterized by a common Dr value for a certain type of boundary roughness. The r2 value reported for the Bray (1979) and Griffiths (1981) equations are 0.355 and 0.591, respectively, suggest- ing that a considerable amount of scatter exists in the data. Griffiths (1981) attributes the scatter to inadequate descrip- tions of the channel reach and hydraulic variables, restric- tions and errors in data collection procedures, irregularities in the alignments and channel cross-sections, and the rugged bed topography. Bathurst (2002) found that if the data were divided into groupings based on channel similarities, the scatter decreased significantly (increased r2 values). Dividing the data into two groups resulted in two equations with x′ values of 1.047 and 1.43, respectively. The difference in these two x′ values was attributed to differences in the channel slope: 1.047 for Se < 0.8% and 1.43 for Se > 0.8%. The results from the current study (see Table 7-2) suggest that roughness element size may have a significant effect on the value of x′. The Bazin (1865) gravel data produced x′ val- ues equal to 0.721 and 0.822 for the small and large gravel tests. The block data, which are somewhat representative of a rigid gravel or small cobble bed (flow can pass over and around the projecting roughness elements), produced an x′ value equal to 1.16, suggesting that x′ increases with increasing gravel or roughness element size. The smoothest boundary materi- als (acrylic, pure cement, and cement-sand mix) produced the smallest and relatively constant x′ values of 0.644, 0.676, and 0.684, respectively. The corrugated boundary roughness materials produced increasing x′ values with increasing cor- rugation size (x′ = 0.732 for Bazin’s “laths of wood” and x′ = 0.968 for the relatively larger trapezoidal corrugations). For the roughness materials evaluated in this study, chan- nel slope was not a significant factor of the x′ value (i.e., the data in Figure 7-1(C) fall on a single curve regardless of the channel slope). Although Bathurst (2002) points out differ- ences between the channel geometries and typical boundary roughness materials used in flume studies and those found in natural mountain streams, both the Bathurst (2002) results and the current study indicate that x′ is dependent on more than simply the roughness material type or channel geom- etry. Therefore, an equation in the form of Equation 7-6, with a constant hydraulic roughness coefficient, will not accurately describe the stage-discharge relationship for a general boundary type classification such as gravel channels. x′ will vary with the size, density, spacing, and alignment of the boundary roughness elements.

Figure 7-1. Equivalent Manning’s n coefficients (nc, n, and nopt) for pure cement (data from Bazin, 1865) (A), large gravel (data from Bazin, 1865) (B), and block (C) roughness data.

66 The prospect of developing equations specific to the bound- ary roughness type as well as the size, density, and distribution of the individual roughness elements is a somewhat daunting task. Manning’s (1889) original intent was a single simple equa- tion that would produce “sufficiently accurate” results consid- ering the information available. It is interesting that Manning’s x′ = ²⁄³ and his boundary-specific, constant n assumption have withstood the test of time for so long considering the result- ing range of required x′ values determined in this and other studies required to support a constant n value. A closer look at the data provides insight on the longevity and relative reliabil- ity of Manning’s Equation. The x′ values reported in Table 7-2 represent the data with a single optimized head-discharge curve. Figure 7-2 pre- sents log(V/Se1/2) versus log(Rh) plots for the acrylic, metal lath, block, and trapezoidal corrugation channel lining material data in Figure 7-2 are better represented by two linear trend lines, each with a different slope (x′), as described by Equa- tion 7-10. Consistent with Manning’s Equation (Equation 7-4), the acrylic data correlated well with the x′ = ²⁄³ trend line slope represented on the plot by a dashed line. The metal lath and the trapezoidal corrugation data sets both exhibit vari- able dependence on Rh as shown by the two distinct trend lines of differing slope corresponding to the “higher” and “lower” Rh data ranges. x′ values for the higher Rh data ranges (metal lath and trapezoidal corrugation data) are reasonably represented by x′ = ²⁄³ (Manning’s Equation). The smaller Rh data ranges for both data sets require x′ > ²⁄³ to match the experimental data (e.g., x′ = 0.9 for the metal lath and x′ = 1.25 for the trapezoidal corrugations are required to better match the larger Rh experimental data). The block data correspond to a single linear trend line with x′ = 1.2. This result, however, may be due only to the fact that sufficiently high Rh values could not be achieved in the test facility to identify a range of Rh where the x′ = ²⁄³ is appropriate. Note that the higher Rh block data (top 7-8 data points) are begin- ning to deviate from the trend line slightly. In summary, the acrylic boundary (over the full range of Rh) and the metal lath and trapezoidal corrugation channel lining materials at larger Rh values produced an x′ = ²⁄³. For all other condi- tions, including the block channel lining material, alternate x′ values were required in order to fit the data for each of the roughness materials. These results suggest that Conditions 1 and 2 are met when either the roughness boundary itself is smooth (e.g., the acrylic and cement boundaries) or at higher Rh values for rougher boundaries. This finding is consistent with the quasi-smooth boundary condition theory discussed in Chap- ter 6, where stable eddies form between the roughness ele- ments of the rougher boundaries, creating a quasi-smooth flow condition above the roughness elements. The acrylic and cement boundaries represent smooth-flow boundary conditions. When x′ for the rougher boundary materials is equal to ²⁄³ (larger Rh values), the flow condition is consistent with the quasi-smooth boundary condition. When channels lined with rougher boundary materials operate outside of the quasi-smooth flow condition, then Conditions 1 and 2 are no longer met, x′ ≠ ²⁄³, and/or n cannot be considered constant. The longevity and relative reliability of the use of Manning’s Equation (Equation 7-4) with boundary-specific constant n values suggests that many of the channels used in practice have relatively smooth flow boundaries (e.g., cement-lined channels) or that they may commonly operate in the quasi- smooth flow condition. Figure 7-2. Plot of log(V/Se1/2) versus log(Rh) data for acrylic, metal lath, block, and trapezoidal corrugation boundary roughness materials.

67 7.6 Conclusions When applying Manning’s Equation, the assumption is often made that n is a constant value, independent of flow depth and discharge for a given channel lining material. An inspection of the experimental data from the current study and from Bazin’s (1865) showed that the applicability of the constant n assumption diminishes as the roughness of the boundary increases. To produce a constant n value for a given boundary roughness material at all flow conditions, the mean velocity must be well represented by an equation in the form of Equation 7-6, and the representative x′ coefficient must equal ²⁄³. This study evaluated these two conditions for a range of boundary roughness materials and produced the following conclusions: 1. The data showed that Equation 7-6 provided a relatively good overall fit to the data for each of the lining materials tested. 2. Only the smooth boundary materials (e.g., acrylic sheet- ing and pure cement) produced an x′ = ²⁄³, based on the Equation 7-6 relationship for the entire range of Rh tested. x′ was found to be a unique value for each boundary material tested, ranging from 0.644 (acrylic sheeting) to 1.16 (blocks), with the x′ value increasing with increasing boundary roughness. 3. Relative to the other hydraulic roughness coefficients (Cc and f ), Manning’s n exhibited less variability with respect to changes in Rh (see Figure 7-1). As Rh increases, n approaches becoming or becomes constant. Based on the range of flow conditions tested (in a rectangular flume), the range of Rh values over which n is constant decreases as the roughness of the boundary material increases. For very smooth boundar- ies (e.g., acrylic sheeting), n was approximately constant over the entire range of Rh tested. 4. The value of x′ that corresponds to Equation 7-6 varied with boundary roughness material type and Rh. The metal lath and trapezoidal corrugation data in Figure 7-2 show that two separate linear curves that correspond to differ- ent x′ (Equation 7-6) values are required in order to match the experimental data. This means that over the range of Rh tested, a constant n value cannot be applied to these boundary roughness materials when using Manning’s Equation (Equation 7-4) with x′ = ²⁄³. Manning’s Equa- tion with a constant n value gives a good representation of the data at larger Rh values where quasi-smooth-type flow conditions exist. The block data also showed evidence that at larger Rh values there would be a shift in the x′ value. Sufficiently high Rh data for the blocks were not obtain- able with the experimental test setup to confirm the high Rh block x′ value. 5. The results of this study show that Manning’s n will not likely be a constant value for canals, streams, and rivers with rough boundaries such as large gravels and cobbles unless the Rh is sufficiently large. The limiting Rh above which quasi-smooth flow conditions exist and n becomes constant will be specific for each boundary roughness type and must be determined by testing.