Hydraulic Loss Coefficients for Culverts (2012)

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50 6.1 Summary Quantifying hydraulic roughness coefficients is commonly required in order to calculate flow rate in open channel and closed conduit applications. Much of the theory of resistance on open channel flow is derived from studies on pressurized circular pipe, which features the Darcy-Weisbach roughness coefficient, f, which is dependent upon Re, Rh, and/or k. Rela- tive to full-pipe flow, however, the behavior of open channel flow resistance is more complicated because of the presence of a free surface and because the flow area does not remain constant. A primary objective behind the development of Manning’s Equation was to create a simple open channel flow equation with a roughness coefficient (n) that was solely dependent upon the boundary roughness characteristic (e.g., rough- ness height, k). Currently, hydraulic engineering handbooks publish singular representative n values (or a small range to account for variations in material surface finish) per bound- ary material type (e.g., concrete, cast iron, clay, etc.). More recent studies, however, have suggested that Rh, k, Se, and Fr can influence n. The behavior of f and n as a function of Re, Rh, k, Se, and Fr for open channel flow was evaluated for four different boundary roughness materials, ranging from smooth to rela- tively rough, by conducting stage-discharge tests in a rectan- gular tilting flume. The test results showed that when plotting f or n versus Re, a family of curves resulted, with each curve corresponding to a specific channel slope (So). For a given So, both f and n decrease with increasing Re. The So-specific family of f curves converges to a bounding curve, unique to each boundary roughness material tested, with increasing Re, which represents a quasi-smooth flow boundary condition. For the n data, the quasi-smooth flow condition caused the n values to converge to a constant n value at larger Re values. A quasi-smooth flow boundary condition describes a condi- tion where a boundary layer develops adjacent to the channel boundary that consists of a layer of flow eddies. The bound- ary layer thickness exceeds the material roughness height, reducing the influence of the boundary roughness elements of flow resistance. With increasing Rh, f and n also decrease, with n eventu- ally approaching a constant value. The constant n assump- tion (n is independent of Re and Rh) is most appropriate for smoother boundary materials or rough boundary materi- als where a quasi-smooth flow boundary condition exists. Where a quasi-smooth condition does not exist, the con- stant n assumption is less appropriate for rougher boundary roughness materials. 6.2 Introduction Quantifying hydraulic roughness coefficients is commonly required for discharge calculations for both closed conduit and open channel flow applications. Common open channel discharge equations include the Darcy-Weisbach Equation, Equation 6-1, and Manning’s Equation, Equation 6-2, which include the friction factor (f ) and Manning’s n, respectively, as hydraulic roughness coefficients. V g f R Sh e= 8 6 1( )- V K n R Sn h e= 2 3 1 2 6 2( )- In Equations 6-1 and 6-2, V is the mean velocity, g is accel- eration due to gravity, Kn is 1.0 (International System of Units) and 1.486 (English System of Units), Rh is the hydrau- lic radius [the cross-sectional area (A) divided by the wet- ted perimeter (P), Rh = D/4 for a pipe of diameter D], and Se is the energy grade line or friction slope. Under uniform flow conditions in open channel flow, Se is equal to the channel slope (So). C h a p t e r 6 The Behavior of Hydraulic Roughness Coefficients in Open Channel Flow

51 Turbulence level and the relative roughness of the pipe or channel can influence the flow resistance or hydraulic rough- ness. Much of the current theory regarding resistance is based on knowledge gained from the study of commercial pipes flowing full. Turbulence effects are commonly quantified using f and its relationship with the Reynolds number (Re), where Re = VD/v (v represents the fluid kinematic viscosity), and relative roughness, quantified as k/D, where k is a repre- sentative value for the boundary material roughness height. For open channel flow, these parameters are represented as Re = V4Rh/v, and Rh/k, referred to as relative submergence. The behavior of the hydraulic roughness coefficients in open channel flow is not nearly as well understood as with full-pipe flow. Open channel flow resistance theory is com- monly compared qualitatively with full-pipe flow resistance, but the complexities of flow resistance behavior associated with the free surface, variable flow area, and the wider variety of boundary roughness types found in open channel situa- tions significantly complicate the behavior of f. Manning’s n is often applied to open channel application because, according to Streeter and Wylie (1979), n is thought to be an absolute roughness coefficient, i.e., dependent upon sur- face roughness only. Representative Manning’s n values for common channel lining materials are typically presented in hydraulic handbooks as singular values or as a high, aver- age, and low value to account for surface finish variations. Streeter and Wylie (1979) also state that n actually depends upon the size and shape of the channel cross-section in some unknown manner. This dependency and others have been described by researchers with equations where n = F(Rh, k, Fr, and Se) (Limerinos, 1970; Jarrett, 1984; Bathurst, 2002; Ugarte and Madrid, 1994). It is useful to know, however, that for many of the test conditions evaluated by Bathurst (2002), the height of the roughness elements making up the bound- ary exceeded the flow depth (y) (i.e., small Rh/k values) in some cases. Other research has suggested adjusting other open channel head-discharge relationship flow parameters in lieu of adjusting hydraulic roughness coefficients to better match physical conditions. For example, Christensen (1992) proposed an alternative definition of Rh based on the idea that the shear stress values are not constant along the wetted perimeter; Blench (1939) proposed a change to the exponent of the Rh term. The current study presents similarities and differences between full-pipe flow and open channel flow resistance coef- ficients by evaluating the behavior of f and n with respect to Re, Rh, k, So, and Froude number (Fr). Special attention was given to the validity of the assumption of a constant n value and how it might relate to f. The analysis was based on open channel flow testing conducted in a rectangular tilting flume featuring boundary roughness materials ranging from smooth to relatively rough. 6.3 Background Darcy-Weisbach f The Darcy-Weisbach Equation (Equation 6-1) dates back to the mid 1800s (Rouse and Ince, 1957). Nikuradse (1933) performed tests on turbulent flow in artificially roughened pipes (pipes roughened with uniformly sized sand grains) flowing full to investigate the behavior of f. Nikuradse made two important conclusions. At low Re for pipe with rela- tively small sand grains (high Rh/k values), the values of f were similar to smooth pipe values [f = F(Re) only and the flow condition is known as smooth turbulence or smooth- walled pipe flow]. At relatively low Rh/k values and high Re values, f is solely a function of Rh/k, and the flow condition is known as fully rough turbulence. A transitional turbulence Re range also exists where f is a function of both Re and Rh/k. Colebrook (1939), using commercial pipe data, developed an empirical equation that describes the dependencies of f on Rh/k and Re. From Colebrook’s Equation, the Moody Dia- gram was developed, and it has become a common source for assigning a value to f for full-pipe flow under turbulent conditions. Chow (1959) compiled data from various open channel flow tests performed in rough channels with turbulent flow. Some of the data compiled by Chow show that at relatively high Re, f becomes independent of Re and is solely depen- dent on Rh and k. Chow also observed, for some data, that f decreased with increasing Re, with the minimum f values bounded by an equation in the form of Equation 6-3, where f is a function of Re and the coefficients a and b are boundary roughness specific (k): 1 6 3 f a Re f b = log ( )- In Equation 6-3, a and b are empirical coefficients specific for a given channel shape and boundary roughness. Prandtl developed an equation (commonly referred to as the Prandtl- von Kármán Equation) of the form of Equation 6-3, which reasonably describes f data for smooth-walled pipe, with a and b equal to 2 and 2.51, respectively (Crowe et. al, 2001). The open channel flow stage-discharge data presented by Chow (1959) suggest that a and b will vary with boundary rough- ness type, i.e., f values increase with increasing boundary roughness or increasing k values. Chow (1959) also suggests that when the behavior of f for a given boundary roughness material can be described by Equation 6-3 with a constant set of empirical coefficients (a and b), a quasi-smooth flow condi- tion exists. The idea of a quasi-smooth boundary flow condi- tion was introduced by Morris (1955) and describes a flow state where the areas between the roughness elements are filled with stable eddies, creating a pseudo wall flow boundary

52 similar to a smooth wall (See Figure 6-1). The results from this study confirm that Equation 6-3 is a relative limiting boundary to f and also show that this limiting boundary has relevance to the assumption of a constant n. Manning’s n Equation 6-4 relates the Manning’s n roughness coefficient and f. V V f K R n g n h  = = 8 6 4 1 6 ( )- In Equation 6-4, V is the shear velocity [V = (gRhSe)1/2]. Manning (1889) developed Equation 6-2 with the expressed intent of providing a simplified open channel flow equation where, contrary to existing equations, the empirical coeffi- cients (including the roughness coefficient) would remain constant for a given channel boundary type, independent of Q and Rh variations. Manning applied Equation 6-2 with river-reach-specific constant n values to more than 100 data points taken from various rivers and concluded that it was “sufficiently accurate.” Chow (1959) states that if the bed and banks of a chan- nel are equal in roughness and the slope is uniform, then n is usually assumed to be constant for all flow depths (y). Chow (1959) presents Manning’s n data (constant values) and photo graphs for a number of different channel types as a reference for designers. More recent studies, however, have shown that n is not necessarily a constant even under the con- ditions described by Chow (1959). A number of relationships have been developed based on the results of these studies in order to predict the behavior of n. For example, Limerinos (1970), Bray (1979), Griffiths (1981), and Bathurst (2002) have presented relationships suggesting that n is a function of Rh/k. Jarrett (1984) suggested that n is dependent upon Se and Rh. Ugarte and Madrid (1994) proposed relationships for n involving Rh, k, Se, and Fr. These relationships were developed based on studies where Manning’s Equation was applied to a specific type of channel. The Limerinos, Bray, and Griffith relationships were developed for rivers with gravel beds; the Bathurst, Ugarte and Madrid, and Jarrett relationships were specific to “mountain streams” characterized as steep with relatively small Rh/k values. Yen (2002) maintains, however, that for a given boundary roughness, n should be relatively constant, independent of Re, and Rh, provided that the equiv- alent f value per Equation 6-4 is in the fully turbulent range [i.e., f = F(Rh and k)]. Froude Number Effects Flow state is commonly characterized by the value of the Froude number (Fr) (see Equation 6-5), which represents the ratio of inertial to gravitational forces. In Equation 6-5, T is the channel top width. When Fr < 1, gravitational forces are dominant, flow velocities are low, and the flow condition is referred to as subcritical. When Fr > 1, the inertial forces are dominant, the velocity is high, and the flow condition is referred to as supercritical. Fr V gA T= ( )1 2 6 5( )- Chow (1959) states that when Fr < 3, the influence of Fr on open channel roughness coefficients is negligible. Chow con- cedes, however, that as more data become available, the influ- ence of Fr on open channel roughness coefficients may need to be reconsidered. Ugarte and Madrid (1994) concluded that n has Fr dependencies; however, it is important to note that their study was generally limited to relatively small Rh/k val- ues. Bathurst et al. (1981) also found that Fr was a factor in quantifying the n; however, instead of using the traditional Fr definition, Rh was substituted for A/T in Equation 6-5. Objectives The objectives of this study were to investigate the rela- tionships of the roughness coefficients f and n with Re, Rh, k, and Fr in open channel flow in an effort to better understand the appropriateness of the constant n value assumption for a given boundary roughness. Comparisons are made for four different roughness materials ranging from smooth (acrylic sheeting) to relatively rough (block and trapezoidal corrugated roughness elements). 6.4 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 channel boundary materials tested include acrylic sheeting (see Figure 6-2); a low- profile, commercially available expanded metal lath adhered to the acrylic walls and floor of the flume (see Figure 6-3); regularly spaced wooden blocks (see Figures 6-4 and 6-5); and Figure 6-1. Illustration of the quasi-smooth flow boundary theory.

53 trapezoidal corrugations oriented normal to the flow direc- tion (see Figures 6-6 and 6-7). The wooden blocks, measur- ing 4 in. wide (normal to flow direction) by 3.5 in. long by 1.5 in. tall, with the top edges rounded (1-in. radius round- over), 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. and a base 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, which were attached to the flume floor and walls. Assigning a k value to various types of roughness material is not an exact process. For gravel-lined channels, the mean grain size diameter is often used. In this study, all roughness materials, save the acrylic sheeting, have more than one geo- metric dimension that influences the hydraulic roughness Figure 6-2. Acrylic boundary roughness material. Figure 6-3. Metal lath boundary roughness material. Figure 6-4. Block boundary roughness material. Figure 6-5. Schematic of block boundary roughness material (dimensions shown in inches). Figure 6-6. Trapezoidal corrugation boundary roughness material.

54 (e.g., the block height, width, length, and spacing). Chow (1959) explains that while k represents a measure of a bound- ary’s roughness, it is an empirical parameter that doesn’t nec- essarily correspond to a specific geometric dimension of the roughness element that can be measured using a linear scale and that k is influenced by many factors such as roughness element shape, orientation, and distribution. In this study, k was assumed to be equal to the physical height of the rough- ness elements, for lack of a more appropriate alternative. The acrylic sheeting k value was selected to be consistent with published values (k = 0.00006 in.). Water was supplied to the flume from a reservoir located adjacent to the laboratory and was metered using calibrated orifice flow meters. Flow depths were measured using a preci- sion 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 6-2 when uniform flow exists in the channel and y and Q are known. Due to the limited length of the laboratory flume, uniform flow depth could not be achieved for all test con- ditions. In laboratory practice, a tailgate is often used to help establish uniform depth in a flume by increasing the downstream flow depth and truncating part of the gradually varied flow (GVF) profile. According to Yen (2003), this method does not guarantee the presence of a uniform flow condition. In addition to a constant flow depth, the velocity distribution, pressure, and turbulence characteristics must also be uniform for uniform flow to exist. Yen (2003) states that even though a constant depth may be forced in a short channel with the use of a tailgate, the effects of the channel inlet and tailgate may affect the characteristics of the flow, resulting in a flow condition that is not “uniform.” In the current study, all tests featured a free-overfall downstream boundary condition. For flow conditions that did not achieve normal depth naturally, Manning’s n values were determined using a computational GVF profiling technique. For each steady state flow condition, the GVF profile was determined by measuring flow depths (ymeasured) at 33 different locations along the length of the flume. The Manning’s n coefficient was determined for each flow condition by adjusting the Manning’s n value in a GVF computer program until the computed water surface profile best matched the measured profile. To determine the “best fit” of the data, a coefficient of determination (r2), Equation 6-6, was maximized. r y y y y measured calculated measured av 2 2 1= − −( ) − ∑ erage( )∑ 2 6 6( )- In Equation 6-6, ycalculated is the flow depth calculated by the GVF computer program and yaverage is the average of ymeasured. The data collection proceeded as follows. For each slope and discharge, the water surface was measured in relation to the flume floor at 2-ft intervals over the upstream half of the flume and at 1-ft intervals over the downstream half. Due to the nature of the block and trapezoidal corrugation roughness materials, no single channel invert datum was present. Conse- quently, a representative datum was determined by calculating the total volume of the roughness elements (blocks or trap- ezoidal corrugations) divided by the total flume floor area and adding the resulting height to the elevation of the plywood floor upon which the roughness elements were installed. Using this GVF method, a separate Manning’s n value was determined for each flow condition. Early in the data collection process, however, it became apparent that for the relatively rough boundary materials (blocks and trape- zoidal corrugations), Manning’s n exhibited variability with flow depth for a common flow rate. Figure 6-8, for example, shows Manning’s n data for a number of flow conditions in the block-lined channel. With steeper channel slopes, where uniform flow conditions were more prevalent, n values were determined using the measured normal depth (yn), Q, and Equation 6-2. For milder sloping channels, where uniform flow profiles were less common, n values were determined using the GVF profile method. A comparison of the block- lined Manning’s n values determined using both techniques is presented in Figure 6-8, which plots n versus the average channel profile flow depth (yaverage). The uniform flow depth data in Figure 6-8 show that for the block roughness, n var- ies (0.087 ≥ n ≥ 0.038) with changes in uniform 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 vari- able with depth throughout a GVF profile. Based on the vari- able nature of n with flow depth in GVF profiles, it may be Figure 6-7. Schematic of trapezoidal corrugation boundary roughness material (dimensions shown in inches).

55 expected that the predicted normal depths associated with the variable n values should also vary. Consequently, based on the good correlation between the uniform flow and GVF n data presented in Figure 6-8, yaverage was selected as the repre- sentative flow depth parameter for calculating Rh, Re, V, etc., rather than a predicted normal depth, for flow conditions where a uniform flow was not present. Manning’s n data for the acrylic boundary were collected at three different slopes (i.e., So = 0.0002, 0.0003, and 0.0022) with the number of flow conditions at each slope ranging from 6 to 17. The metal lath boundary was tested at four different slopes (i.e., So = 0.0066, 0.0118, 0.0179, and 0.022) with 4 to 29 flow conditions tested at each slope. The block and trapezoidal corrugation boundaries were each tested at five slopes (i.e., So = 0.0004, 0.0018, 0.0095, 0.0237, and 0.05) with seven different flow conditions per slope. The channel discharges ranged from 0.24 to 23 cfs. 6.5 Discussion and Analysis f Relationships Figure 6-9 plots the Darcy-Weisbach f versus Re data for each of the roughness materials in a uniformly lined channel. The data from the acrylic-lined channel generally follow the Prandtl-von Kármán smooth-wall pipe flow curve. Although they are not necessarily discernable in Figure 6-9 due to the scale of the y-axis, the acrylic experimental f values exceed the Prandtl-von Kármán curve values at higher Re values. At a given Re value, f increases with increasing boundary rough- ness (i.e., f of the blocks is greater than the metal lath, which is greater than the acrylic). At first glance, there appears to be considerable scatter in the data for the two larger roughness materials (block and trapezoidal roughness materials) in Figure 6-9; however, a closer look reveals families of curves segregated by So. The data show that for a prismatic channel where So is held constant, f decreases as Re increases. For a constant Re, f increases with increasing So. As Re increases, the roughness-element-specific, slope-dependent family of curves converges to a single curve. There is no single Re value, however, at which the individual curves converge. The Re value at which a slope-specific curve converges to the bounding curve for an individual roughness material increases with increasing So. The bounding curve to which the acrylic, metal lath, and trapezoidal corrugation data converge is consistent with Equation 6-3, which, as described by Chow (1959), becomes a limiting boundary to the decreasing effect of the bound- ary roughness on the total resistance to the flow. Figure 6-9 shows that the block slope-specific data curves do not fully converge to a single curve within the range of Re tested; how- ever, the trend lines appear to be converging toward a single bounding curve with increasing Re. The convergence of the metal lath and the trapezoidal corrugation roughness data to a single bounding f versus Re curve indicates that the condi- tions in the channel have reached a quasi-smooth boundary flow condition consistent, in theory, with the illustration in Figure 6-1. Figure 6-8. Comparison of Manning’s n values determined using the uniform flow depth and the GVF technique (non-uniform flow conditions) versus yaverage of the measured water surface (data from block-lined channel).

56 n Relationships If n were constant (as is often assumed) and solely depen- dent on k, four horizontal lines, one for each roughness material tested, should result when plotting n versus Re. The results in Figure 6-10 show relatively constant n values for the smooth acrylic data and over most of the Re data range for the metal lath. There is a small range of relatively small Re values over which n for the metal lath varies. For the two rougher materi- als (block and corrugation roughness), n varies significantly over the range of Re tested. The data for these roughness materials show trends similar to the f data presented in Figure 6-9: there is a family of curves segregated by So, n decreases with increasing Re, n increases with an increasing slope (at a constant Re value), and the So-specific curves converge as Re increases. An inspection of the data in Figures 6-9 and 6-10 reveals a subtle but important difference between the behavior of n and f with Re. In Figure 6-10, the slope-dependent Man- ning’s n data curves converge to a constant (minimum) value as Re increases, indicating that n is solely dependent upon k at higher Re values. In contrast, the So-specific f curves in Figure 6-9 converge to a bounding curve in the form of Equation 6-3 as Re increases. Although the slopes of the bounding curves become relatively small at higher Re values, the bounding curves do not reach a zero slope, indicating that f remains a function of Re and k over the range of Re numbers tested. The data in Figure 6-10 also suggest that the appropriate- ness of a constant n value assumption increases as the relative smoothness of the channel boundary increases. The n val- ues for the acrylic and metal lath channels are constant over the majority of the Re range tested. As the relative roughness increases (e.g., the blocks and trapezoidal corrugations), the range of Re over which n is constant diminishes. Based on the data presented in Figure 6-10, the constant n assumption, commonly used when applying Manning’s Equation (Equa- tion 6-2), is appropriate for smooth-wall channel lining Figure 6-9. f versus Re data for acrylic, metal lath, trapezoidal corrugation (A), and block (B) roughness materials.

57 materials (e.g., smooth acrylic sheeting) or for “rougher” boundary materials when a quasi-smooth boundary condi- tion is present (e.g., metal lath and trapezoidal corrugation roughness material n versus Re data become constant). Under these conditions, n is a function of k and is no longer depen- dent on Re, So, or Rh/k. The behavior the block roughness n data in Figure 6-10 is similar to that of the f data in that the n data do not fully con- verge to a constant value (a bounding curve for the f data) due to the limited range of experimental Re values. It is assumed, however, that similar to the trapezoidal corrugations, the block data will converge to a constant n value at higher Re values. Figure 6-11 presents n versus Rh/k for the block and trap- ezoidal corrugations. The block data show a strong depen- dence on Rh/k (n decreases with increasing Rh/k) and are relatively independent of So, as the data essentially collapse to a single curve. The fact that the n versus Rh/k data are essentially independent of So means that for the rectangular flume used in this study n was solely a function of flow depth. This means that n will be the same for two different channel slopes, provided that flow depths are the same, independent of the differences in Q, V, and Re for the two slope conditions. As a result, when correlating n versus Rh/k, n is essentially independent of V and Re. The trapezoidal corrugation data in Figure 6-11 also show a strong dependence on Rh/k; however, a slight data segregation (family of curves) associated with So exists (more than with the block data). The reason for the variation in the behavior of n versus Rh/k between the block and trapezoidal corrugation materials isn’t clear, but the variation may be related to the nature of the flow paths near the boundaries. With the blocks, flow passes over and around the individual roughness elements. With the trapezoidal corrugations, the flow only passes over the rough- ness elements, making the velocity profile near the boundary primarily two dimensional rather than three dimensional like the blocks. The disparity between the So-specific n versus Rh/k Figure 6-10. n versus Re data for acrylic, metal lath, trapezoidal corrugation (A), and block (B) roughness materials.

58 curves in Figure 6-11, however, is significantly reduced rela- tive to the n versus Re data in Figure 6-10. It is also interesting to note that despite the fact that the block and trapezoidal corrugation roughness elements are the same height, the n versus Rh/k data trend differently in Figure 6-11. At smaller flow depths (e.g., Rh/k = 1.0), the flow resistance of the blocks is larger (larger n) than that of the trapezoidal cor- rugations. For the trapezoidal corrugations, n decreases more rapidly with increasing Rh/k than for the blocks, and the point at which n becomes constant occurs at a lower Rh/k value. This suggests that equating k to the height of the roughness ele- ment does not adequately characterize the influence of the roughness elements on flow resistance. Although perhaps not a general conclusion, it is interesting to note that the trape- zoidal corrugations and the block, which were approximately the same height, both approach approximately the same con- stant n value at high Rh/k values (n~0.033). More research is recommended to investigate the characteristic differences between the flow resistance behavior of two-dimensional and three-dimensional boundary roughness element types. With respect to the data presented in Figure 6-11, the quasi- smooth flow boundary condition occurs for rougher bound- ary materials when a sufficiently high Rh/k condition, referred to as relative submergence or the boundary roughness ele- ments, is reached and n becomes constant. For Rh/k values below the quasi-smooth flow limit, the constant n assumption is not appropriate. According to the data presented in Figure 6-11, the level of relative submergence required to produce a quasi-smooth flow condition varies with the boundary rough- ness characteristics, which are partially described by k and Rh/k. Manning (1889) reported relatively constant n values for numerous river channel sections. The river channel sec- tions most likely featured sufficiently high Rh/k values to vali- date a constant n assumption. Subcritical versus Supercritical Flow Nineteen of 35 data points taken from a metal-lath-lined flume featured supercritical flow conditions and were dispersed over the range of Re tested. Three of seven flow conditions corresponding to the steepest channel slope for the block and trapezoidal roughness produced supercritical flow. Although the data are not specifically identified as subcritical or super- critical flow in Figure 6-10, the consistent trends in the data sets indicate that n is relatively independent of Fr over the range of Fr values tested. For the entire data set (all four boundary roughness data sets) Fr ranged from 0.33 to ~1.54. These results concur with Chow (1959), who stated that for small Fr (Fr < 3), the effect of gravity on flow resistance is negligible. 6.6 Conclusions Quantifying hydraulic roughness coefficients is commonly required to calculate flow rate in open channel and closed conduit applications. Much of the theory of resistance on open channel flow is derived from studies on pressurized cir- cular pipes and features the Darcy-Weisbach roughness coef- ficient, f, and its relationship with Re, Rh, and k. Figure 6-11. n versus Rh/k for block and trapezoidal corrugation roughness materials.

59 In developing Equation 6-2, Manning’s (1889) primary objective was a simple open channel flow equation with a roughness coefficient (n) that was solely dependent upon k. Currently, hydraulic engineering handbooks publish singu- lar representative n values (or a small range to account for variations in material surface finish) per boundary material type (e.g., concrete, cast iron, clay, etc.). Manning concluded that the constant n assumption was “sufficiently accurate” after applying Equation 6-2 to numerous data taken from rivers; however, later studies (Limerinos, 1970; Jarrett, 1984; Bathurst et al., 1981; Ugarte and Madrid, 1994) suggest that n can be influenced by Rh, k, Se, and Fr. The behavior of f and n as a function of Re, Rh, k, So (rather than Se), and Fr in open channel flow was evaluated in a rect- angular tilting flume for four different boundary roughness materials ranging from smooth to relatively rough. Based on the results of this study, the following is concluded: 1. In relation to Re, the f and n data from this study have similar characteristics to the data presented by Chow (1959). At a constant So, both f and n decrease with increasing Re. The Re-dependent f data were bound by a material-roughness-specific limiting curve consistent with Equation 6-3; the corresponding n data were bound by a limiting constant n value. Chow (1959) suggested that the f-data bounding curves are consistent with a smooth surface condition, analogous to the Prandtl-von Kármán smooth pipe wall boundary condition, or a quasi-smooth boundary flow condition, which describes a condition where the voids between boundary roughness elements are filled with stable eddies, reducing the influence of the boundary roughness elements on flow resistance. The constant n assumption is appropriate for smooth and quasi-smooth flow conditions. For rougher boundary materials, n can vary considerably for non-quasi-smooth flow conditions, which if not appropriately accounted for, could significantly increase the level of uncertainty associ- ated with open channel flow stage-discharge calculations. 2. For a single boundary roughness material (characterized by k), flow resistance testing over a range of channel slopes produced a family of curves dependent on channel slope (So) (see Figures 6-9 and 6-10). The families of f and n data curves in Figures 6-9 and 6-10 do not necessarily con- firm So as a significant parameter influencing flow resis- tance behavior; it is more likely that these families are an indicator that there are additional system parameters that influence open channel flow resistance that are not appro- priately accounted for in an f or n versus Re analysis. The differences between the So-dependent curves for a single boundary roughness material increased as the material roughness height, k, increased (i.e., the metal lath family of curves is more closely spaced than the curves for the block or trapezoidal corrugation boundary materials in Figures 6-9 and 6-10). Figures 6-9 and 6-10 also show that f and n increase with increasing So for a given boundary roughness material. 3. Figure 6-11 shows that the So-dependent family of curves collapse relatively well to a single curve where n is plotted with respect to Rh/k, indicating that n is the same for two different channel slopes, provided that flow depths are the same. The So-dependent family of curves is also indepen- dent of the differences in Q, V, and Re for the two slope conditions. The trapezoidal corrugations show a greater scatter in the data than do the blocks when plotted with respect to Rh/k; however, the Rh/k relationship is a great improvement to the Re relationship in collapsing the data to a single curve. More research is needed to fully explain the scatter shown in the data. 4. According to Figures 6-10 and 6-11, the appropriateness of the constant Manning’s n assumption or the existence of a quasi-smooth flow condition is dependent upon the boundary roughness and a specific value of Rh/k. There exists a minimum Rh/k value for each boundary roughness material tested above, which n is essentially constant. The constant n minimum values of Rh/k decrease as k decreases (as the boundary becomes more smooth). It is interesting to note that despite the fact that the trapezoidal corrugation and block elements had similar height dimensions (1.5 in.) used to quantify their k values, the constant n minimum Rh/k values differed appreciably (as shown in Figure 6-11). This suggests that simply using the vertical dimension or height of a boundary roughness element, particularly for relatively rough boundary materials, does not sufficiently characterize their equivalent roughness height (k). Height, width, length, spacing, uniformity, and surface texture, etc. will all influence the behavior of n with variations in Rh/k. It is also interesting to note that despite the fact that the block and trapezoidal corrugations reach the constant n condition at differing values of Rh/k, the block and trap- ezoidal corrugation boundary roughness materials con- verge to approximately the same constant n values. 5. Consistent with the findings of Chow (1959), n was found to be independent of Fr for Fr < 3 (all test data from this study were less than Fr = 3). Ugarte and Madrid (1994) reported that n was Fr dependent, but their test conditions were limited to relatively small values of Rh/k (large rough- ness elements and/or shallow flow depths) relative to the current study. The appropriateness of assuming material-specific con- stant Manning’s n values for all stage and discharge conditions is limited to smooth (physically smooth or quasi-smooth) boundary flow conditions. Additional research is needed to provide engineers with more comprehensive Manning’s n data that better characterize the flow resistance behavior of common channel lining materials for design purposes.