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Standard horseshoe cross section geometry
Agricultural Water Management 71 (2005) 61–70 www.elsevier.com/locate/agwat
Standard horseshoe cross section geometry Gary P. Merkley* Biological and Irrigation Engineering Department, Utah State University, Logan, UT 84322-4105, USA Accepted 1 July 2004
Abstract The standard horseshoe cross section is found in many tunnels for open-channel and pressurized conduit water conveyance. The exact geometric relationships for this cross section are needed for hydraulic analysis and mathematical modeling purposes because look-up tables of numerically approximated values are typically as slow, or slower, when implemented in computer programs. Implicit solutions of the unsteady hydraulic equations require exact, analytical partial derivatives of some of the relationships, both for accuracy and numerical stability. The equations for cross-sectional area, wetted perimeter, top width, and depth to area centroid are presented as functions of the depth of water in standard horseshoe sections, in addition to the partial derivatives of wetted perimeter with respect to water depth, and depth to area centroid with respect to water depth. Sample unsteady-flow simulation results for a horseshoe cross section are compared to those of a comparably sized circular cross section. # 2004 Elsevier B.V. All rights reserved. Keywords: Horseshoe cross section; Open-channel flow; Hydraulic modeling; Hydraulic equations; Tunnels
1. Introduction Hydraulic modeling of irrigation conveyance systems is becoming more common as water management becomes more important, gate automation is considered, and infrastructure modifications are planned. Dozens of irrigation canals have tunnels with horseshoe cross sections, and the capability for their hydraulic analysis is an important part * Tel.: +1 435 797 1139; fax: +1 435 797 1248. E-mail address: [email protected]. 0378-3774/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.agwat.2004.07.004
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of complete canal system modeling. Davis and Sorensen (1969) presented tables of approximate nondimensional values for some of the geometric relationships of standard horseshoe cross sections, but without the corresponding equations. Hu (1973, 1980) presented nondimensional equations for cross-sectional area, top width, and wetted perimeter for standard horseshoe cross sections, without giving the depth to area centroid, or the partial derivative of centroid depth with respect to water depth. Equations presented by Hu (ibid.) require the application of two trigonometric functions for evaluations of geometric terms, while those presented below require only one such transcendental function, permitting greater computational speed, especially when repetitively applied in unsteady-flow hydraulic simulations. Standard horseshoe cross sections have been designed and built by the US Bureau of Reclamation and others, and are found in many tunnels for water conveyance in the USA and several other countries. The horseshoe cross section can be considered a variation of a circular section whereby the lower half of the tunnel height is widened to facilitate the passage of machinery and equipment during tunnel construction, and later for inspection and maintenance purposes. The standard horseshoe cross section is defined by the intersection of four circles: three offset circles of radius H, and one of radius H/2, as shown in Fig. 1, where a shape resembling a ‘‘horseshoe’’ might be discerned. Some of the geometric relationships can be derived from the equations for the intersecting circles, while others are obtained through integration and differentiation of those equations. Various different forms of the mathematical relationships can be derived, but those presented below are the simplest of the alternatives that were considered.
2. Equation development The depth can be divided into three logical segments (Fig. 2) as defined by the intersections of the four circles, where h1 + h2 + h3 = H. Determine h1 by solving for the
Fig. 1. Geometric definition of a standard horseshoe cross section (bold curves indicate the channel cross section as bounded by four circles).
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Fig. 2. Division of the cross section into three depth segments.
intersection of two of the larger circles, yielding x = y (in Cartesian coordinates, where y is the vertical axis), and then substituting y for x in the equation of the uppermost circle, as follows: H2 (1) 4 Eq. (1) is quadratic with the following two (there are two intersection points, as shown in Fig. 1) solutions: pffiffiffi 1 7 y¼H (2) 4 H 2 ¼ 2y2 Hy þ
which gives the vertical distance from the center of the cross section ceiling to intersection 1 (Fig. 1). Finally, to obtain h1, subtract y from H: pffiffiffi 1þ 7 h1 ¼ H 1 (3) 4 Referring to Fig. 2, it follows that h2 ¼
H h1 2
(4)
and H (5) 2 Separate definitions for each geometric relationship are given below for each of the three depth segments, where depth, h, is measured from the center of the lower curve. h3 ¼
2.1. Top width The top width is that of the water surface, which is a function of depth, h. The top width is zero at h = 0, and reaches a maximum of T = H when h = H/2. For 0 < h h1: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ 4hð2H hÞ (6)
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or sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2 T ¼ 2H 1 1 H
(7)
Eqs. (6) and (7) are alternate expressions for the length of a chord in a circle of radius H (Beyer, 1981). For h1 < h H/2: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 2 2 T ¼2 H h H (8) 2 Eq. (8) is derived from the x-coordinate of a circle of radius H, horizontally offset from the origin by H/2 (Fig. 1). For H/2 < h < H: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2h 2 T ¼H 1 1 (9) H which is the equation for the length of a chord in a circle of radius H/2. 2.2. Cross-sectional area The cross-sectional area is simply the area occupied by water, in a plane orthogonal to the main direction of flow, based on depth and radius in the following equations. For 0 h h1:
A ¼ ðh HÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hH p hð2H hÞ þ H 2 sin1 þ H 2
(10)
For h1 < h H/2: A ¼ H2
0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2h H H H @H H 2 h A C2 þ sin1 h 2H 2 2
þ A1
(11)
where A1 is the area corresponding to h = h1 (Eq. (10)). The constant C2 is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 C21 A C1 @ 1 C1 C2 ¼ 1 1 sin (12) 2 4 2 where pffiffiffi 1þ 7 C1 ¼ 1 2
(13)
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Thus, C1 0.82288 and C2 0.38759. Eqs. (10)–(13) are based on integrations of a circular segment with respect to depth, h. Similarly, for H/2 < h H: H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 2 1 2h H hðH hÞ þ sin (14) A¼ h þ A2 2 H 4 where A2 is the area corresponding to h = H/2, which is a simplified version of Eq. (11): A2 ¼ H 2 C2 þ A1
(15)
2.3. Depth to area centroid ¯ is measured from the bottom of the channel section up The depth to the area centroid, h, to the location of the centroid (Fig. 2). The depth from the water surface down to the area centroid is sometimes required (e.g. application of the momentum function at a hydraulic ¯ The vertical distance to the area centroid can be jump), and can be defined as hc ¼ h h. defined as the first moment of area about the x-axis divided by the cross-sectional area (Stein, 1973). For 0 h h1, the first moment of area about the x-axis is Z Ah 2 Mx ¼ h dA ¼ ½hð2H hÞ 3=2 (16) 3 0 where Ah is the cross-sectional area at depth h. The depth to the area centroid is Mx h¯ ¼ þH Ax
(17)
where Ax is as calculated by Eq. (10). For h1 < h H/2, the moment of area with respect to x is " #3=2 H H 2 2 2 H 2 3 h Mx ¼ H C3 H h (18) 2 2 3 2 in which 3=2 C 21 2 C 21 þ 1 C3 ¼ 8 3 4
(19)
where C3 is a constant; and C1 is as defined in Eq. (13). The value of Mx will be negative because it is calculated based on coordinate origins at h = H/2, so the depth to the area centroid for a given water depth, h, must be shifted upward by the amount H/2: H Mx h¯ x ¼ þ 2 Ax
(20)
which will be a positive value, where Ax is the cross-sectional area from h1 up to some depth h: 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2h H H H 2A 2 1 2 @ (21) Ax ¼ H C2 þ sin h H H h 2H 2 2
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which is Eq. (11) less the A1 term, and C2 is as defined in Eq. (12). The composite value of h¯ must account for the calculations up to h = h1, so for depths from h1 to H/2, the following area-weighted relationship is used to obtain the depth to the area centroid: Ax ½ðH=2Þ þ ðMx =Ax Þ þ A1 h¯ 1 h¯ ¼ Ax þ A 1 where A1 and h¯ 1 are the values corresponding to h = h1 (Eqs. (10) and (17)). For H/2 < h H, the moment of area with respect to x is H3 2 ½hðH hÞ 3=2 12 3 The cross-sectional area from h = H/2 up to some depth h is H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 2 1 2h H sin hðH hÞ þ Ax ¼ h 2 H 4 Mx ¼
(22)
(23)
(24)
which is Eq. (14) less the A2 term. The composite value of h¯ must account for the calculations up to h = H/2, so for depths from H/2 to H, the following area-weighted relationship is used to obtain the exact depth to the area centroid: Ax ½ðH=2Þ þ ðMx =Ax Þ þ A2 h¯ 2 h¯ ¼ Ax þ A2 where A2 and h¯ 2 are the values corresponding to h = H/2 (Eqs. (11) and (22)).
(25)
2.4. Wetted perimeter For 0 h h1: Wp ¼ 2Hcos
1
h 1 H
For h1 < h H/2: H 2h C1 Wp ¼ 2H cos1 cos1 þ Wp1 2H 2
(26)
(27)
where C1 is as defined in Eq. (13); and Wp1 is the wetted perimeter corresponding to h = h1 (Eq. (26)). The ‘‘cos1(C1/2)’’ term is a constant. For H/2 < h H: 2h p (28) þ Wp2 Wp ¼ H cos1 1 H 2 where Wp2 is the wetted perimeter corresponding to h = H/2 (Eq. (27)). Fig. 3 gives nondimensional curves of top width, wetted perimeter, depth to centroid, and cross-sectional area as functions of depth, all based on the above equations. For a full standard horseshoe cross section, the increase in area over a simple circle of radius H/2 is only about 5.6%, and the increase in wetted perimeter is about 4.0%, giving an increase in hydraulic radius of only 1.5%. The depth to the area centroid is less than H/2 for a full
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Fig. 3. Nondimensional curves of standard horseshoe geometric parameters with respect to normalized depth of water.
section because the lower half of the cross section represents slightly more than half of the entire cross-sectional area.
3. Partial derivatives Exact partial derivatives of wetted perimeter and area centroid with respect to depth are needed for accuracy and numerical stability when implementing implicit solutions to unsteady open-channel flow hydraulic equations. These are given below for the same three depth segments, as defined above. 3.1. Wetted perimeter with respect to depth For 0 < h h1: @Wp 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h 1 ½1 ðh=HÞ 2
(29)
For h1 < h H/2: @Wp 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h 1 ½ð1=2Þ ðh=HÞ 2
(30)
It is noted that there is a discontinuity in @Wp/@h at h = h1, between Eqs. (29) and (30).
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For H/2 < h H: @Wp 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h 1 ½1 ð2h=HÞ 2
(31)
3.2. Area centroid with respect to depth For 0 < h h1: @h¯ @ Mx Ax ð@Mx =@hÞ Mx ð@Ax =@hÞ ¼ ¼ @h @h Ax A2x
(32)
where Ax and Mx are as defined in Eqs. (10) and (16), and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @Mx ¼ 2ðh HÞ hð2H hÞ @h 0 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi @Ax H ðh HÞ2 B C ¼ hð2H hÞ þ @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h hð2H hÞ 1 ½ðh HÞ=H 2
(33)
(34)
For h1 < h H/2: @h¯ ¼ @h
Ax ½h ðH=2Þ þ A1 ðh h¯ 1 Þ Mx ðAx þ A1 Þ2
! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 H 2 ½h ðH=2Þ 2 HÞ
(35)
where Mx and Ax are as defined in Eqs. (18) and (21); A1 is from Eq. (10) with h = h1; and h¯ 1 is from Eq. (17), also for h = h1. For H/2 < h H: ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h Ax ½h H=2 þ A2 ðh h¯ 2 Þ Mx ¼ (36) ð2 ðH 2 =4Þ ½h ðH=2Þ 2 Þ 2 @h ðAx þ A2 Þ where Mx and Ax are as defined in Eqs. (23) and (24); A2 is from Eq. (15); and h¯ 2 is from Eq. (22) for h = H/2.
4. Sample application of the equations All of the above equations were compared to various numerical approximations of the geometric variables in a spreadsheet application, validating each of the equations given herein. The equations were also applied in a mathematical model of unsteady open-channel flow for a standard horseshoe cross section of H = 3 m, a length of 800 m, a longitudinal slope of 0.00005 m/m, and a Manning roughness of n = 0.012. The inflow was constant at 0.1 m3/s, and the downstream boundary was a rectangular weir with a crest length of 1 m, a coefficient of 1.8, an exponent of 1.5, and a fixed crest height of 0.2 m. One simulation was performed for the above parameters, and another with a 3-m diameter circular section,
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Fig. 4. Simulation results for a standard horseshoe cross section with H = 3 m, and a circular section with a 3-m inside diameter.
which is the same as the standard horseshoe section in the upper half. All other parameters were held constant for the two simulations. Each simulation started with a dry channel whereby the 800-m reach was partially filled, and then attained a steady-state hydraulic condition within 3 h, using a time step of 1 min. Fig. 4 shows the results in terms of water depth at the upstream and downstream boundaries of the channel during the 3 h simulation period. It is seen that the depth differences are significant for this small flow rate. However, for flow rates near the channel capacity, where the water depth is a significant fraction of the 3-m channel depth, the hydraulic differences between the horseshoe and circular sections are much less.
5. Conclusions The equations presented above can be applied in a spreadsheet computer program to generate tables of geometric relationships for standard horseshoe cross sections, when appropriate, or can be directly applied, as in mathematical models of steady (uniform and gradually varied) or unsteady flow, both for open channels and pressurized conduits with this cross-sectional shape. Most of the equation forms are simpler and easier to apply than previously available equations, and they include previously unpublished partial derivatives, and depth to area centroid relationships. The hydraulic characteristics of a standard horseshoe cross section are significantly different from those of a circular section with an inside diameter of ‘‘H’’ for relatively small flows and depths.
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Acknowledgment The author would like to thank the support from the Utah Agricultural Experiment Station, Project 788.
References Beyer, W.H. (Ed.), 1981. Standard Mathematical Tables, 26th ed. CRC Press. Boca Raton, FL. Davis, C.V., Sorensen, K.E. (Eds.), 1969. Handbook of Applied Hydraulics. McGraw-Hill Book Company, New York, NY. Hu, W.W., 1973. Hydraulic elements for USBR standard horseshoe tunnel. J. Transportation Eng. Div., ASCE 99 (4), 973–980. Hu, W.W., 1980. Water surface profile for horseshoe tunnel. Transportation Eng. J., ASCE 106 (2), 133–139. Stein, S.K., 1973. Calculus and Analytical Geometry. McGraw-Hill Book Company, New York, NY.
Standard horseshoe cross section geometry Gary P. Merkley* Biological and Irrigation Engineering Department, Utah State University, Logan, UT 84322-4105, USA Accepted 1 July 2004
Abstract The standard horseshoe cross section is found in many tunnels for open-channel and pressurized conduit water conveyance. The exact geometric relationships for this cross section are needed for hydraulic analysis and mathematical modeling purposes because look-up tables of numerically approximated values are typically as slow, or slower, when implemented in computer programs. Implicit solutions of the unsteady hydraulic equations require exact, analytical partial derivatives of some of the relationships, both for accuracy and numerical stability. The equations for cross-sectional area, wetted perimeter, top width, and depth to area centroid are presented as functions of the depth of water in standard horseshoe sections, in addition to the partial derivatives of wetted perimeter with respect to water depth, and depth to area centroid with respect to water depth. Sample unsteady-flow simulation results for a horseshoe cross section are compared to those of a comparably sized circular cross section. # 2004 Elsevier B.V. All rights reserved. Keywords: Horseshoe cross section; Open-channel flow; Hydraulic modeling; Hydraulic equations; Tunnels
1. Introduction Hydraulic modeling of irrigation conveyance systems is becoming more common as water management becomes more important, gate automation is considered, and infrastructure modifications are planned. Dozens of irrigation canals have tunnels with horseshoe cross sections, and the capability for their hydraulic analysis is an important part * Tel.: +1 435 797 1139; fax: +1 435 797 1248. E-mail address: [email protected]. 0378-3774/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.agwat.2004.07.004
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of complete canal system modeling. Davis and Sorensen (1969) presented tables of approximate nondimensional values for some of the geometric relationships of standard horseshoe cross sections, but without the corresponding equations. Hu (1973, 1980) presented nondimensional equations for cross-sectional area, top width, and wetted perimeter for standard horseshoe cross sections, without giving the depth to area centroid, or the partial derivative of centroid depth with respect to water depth. Equations presented by Hu (ibid.) require the application of two trigonometric functions for evaluations of geometric terms, while those presented below require only one such transcendental function, permitting greater computational speed, especially when repetitively applied in unsteady-flow hydraulic simulations. Standard horseshoe cross sections have been designed and built by the US Bureau of Reclamation and others, and are found in many tunnels for water conveyance in the USA and several other countries. The horseshoe cross section can be considered a variation of a circular section whereby the lower half of the tunnel height is widened to facilitate the passage of machinery and equipment during tunnel construction, and later for inspection and maintenance purposes. The standard horseshoe cross section is defined by the intersection of four circles: three offset circles of radius H, and one of radius H/2, as shown in Fig. 1, where a shape resembling a ‘‘horseshoe’’ might be discerned. Some of the geometric relationships can be derived from the equations for the intersecting circles, while others are obtained through integration and differentiation of those equations. Various different forms of the mathematical relationships can be derived, but those presented below are the simplest of the alternatives that were considered.
2. Equation development The depth can be divided into three logical segments (Fig. 2) as defined by the intersections of the four circles, where h1 + h2 + h3 = H. Determine h1 by solving for the
Fig. 1. Geometric definition of a standard horseshoe cross section (bold curves indicate the channel cross section as bounded by four circles).
G.P. Merkley / Agricultural Water Management 71 (2005) 61–70
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Fig. 2. Division of the cross section into three depth segments.
intersection of two of the larger circles, yielding x = y (in Cartesian coordinates, where y is the vertical axis), and then substituting y for x in the equation of the uppermost circle, as follows: H2 (1) 4 Eq. (1) is quadratic with the following two (there are two intersection points, as shown in Fig. 1) solutions: pffiffiffi 1 7 y¼H (2) 4 H 2 ¼ 2y2 Hy þ
which gives the vertical distance from the center of the cross section ceiling to intersection 1 (Fig. 1). Finally, to obtain h1, subtract y from H: pffiffiffi 1þ 7 h1 ¼ H 1 (3) 4 Referring to Fig. 2, it follows that h2 ¼
H h1 2
(4)
and H (5) 2 Separate definitions for each geometric relationship are given below for each of the three depth segments, where depth, h, is measured from the center of the lower curve. h3 ¼
2.1. Top width The top width is that of the water surface, which is a function of depth, h. The top width is zero at h = 0, and reaches a maximum of T = H when h = H/2. For 0 < h h1: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ 4hð2H hÞ (6)
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or sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2 T ¼ 2H 1 1 H
(7)
Eqs. (6) and (7) are alternate expressions for the length of a chord in a circle of radius H (Beyer, 1981). For h1 < h H/2: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 2 2 T ¼2 H h H (8) 2 Eq. (8) is derived from the x-coordinate of a circle of radius H, horizontally offset from the origin by H/2 (Fig. 1). For H/2 < h < H: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2h 2 T ¼H 1 1 (9) H which is the equation for the length of a chord in a circle of radius H/2. 2.2. Cross-sectional area The cross-sectional area is simply the area occupied by water, in a plane orthogonal to the main direction of flow, based on depth and radius in the following equations. For 0 h h1:
A ¼ ðh HÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hH p hð2H hÞ þ H 2 sin1 þ H 2
(10)
For h1 < h H/2: A ¼ H2
0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2h H H H @H H 2 h A C2 þ sin1 h 2H 2 2
þ A1
(11)
where A1 is the area corresponding to h = h1 (Eq. (10)). The constant C2 is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 C21 A C1 @ 1 C1 C2 ¼ 1 1 sin (12) 2 4 2 where pffiffiffi 1þ 7 C1 ¼ 1 2
(13)
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Thus, C1 0.82288 and C2 0.38759. Eqs. (10)–(13) are based on integrations of a circular segment with respect to depth, h. Similarly, for H/2 < h H: H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 2 1 2h H hðH hÞ þ sin (14) A¼ h þ A2 2 H 4 where A2 is the area corresponding to h = H/2, which is a simplified version of Eq. (11): A2 ¼ H 2 C2 þ A1
(15)
2.3. Depth to area centroid ¯ is measured from the bottom of the channel section up The depth to the area centroid, h, to the location of the centroid (Fig. 2). The depth from the water surface down to the area centroid is sometimes required (e.g. application of the momentum function at a hydraulic ¯ The vertical distance to the area centroid can be jump), and can be defined as hc ¼ h h. defined as the first moment of area about the x-axis divided by the cross-sectional area (Stein, 1973). For 0 h h1, the first moment of area about the x-axis is Z Ah 2 Mx ¼ h dA ¼ ½hð2H hÞ 3=2 (16) 3 0 where Ah is the cross-sectional area at depth h. The depth to the area centroid is Mx h¯ ¼ þH Ax
(17)
where Ax is as calculated by Eq. (10). For h1 < h H/2, the moment of area with respect to x is " #3=2 H H 2 2 2 H 2 3 h Mx ¼ H C3 H h (18) 2 2 3 2 in which 3=2 C 21 2 C 21 þ 1 C3 ¼ 8 3 4
(19)
where C3 is a constant; and C1 is as defined in Eq. (13). The value of Mx will be negative because it is calculated based on coordinate origins at h = H/2, so the depth to the area centroid for a given water depth, h, must be shifted upward by the amount H/2: H Mx h¯ x ¼ þ 2 Ax
(20)
which will be a positive value, where Ax is the cross-sectional area from h1 up to some depth h: 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2h H H H 2A 2 1 2 @ (21) Ax ¼ H C2 þ sin h H H h 2H 2 2
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which is Eq. (11) less the A1 term, and C2 is as defined in Eq. (12). The composite value of h¯ must account for the calculations up to h = h1, so for depths from h1 to H/2, the following area-weighted relationship is used to obtain the depth to the area centroid: Ax ½ðH=2Þ þ ðMx =Ax Þ þ A1 h¯ 1 h¯ ¼ Ax þ A 1 where A1 and h¯ 1 are the values corresponding to h = h1 (Eqs. (10) and (17)). For H/2 < h H, the moment of area with respect to x is H3 2 ½hðH hÞ 3=2 12 3 The cross-sectional area from h = H/2 up to some depth h is H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 2 1 2h H sin hðH hÞ þ Ax ¼ h 2 H 4 Mx ¼
(22)
(23)
(24)
which is Eq. (14) less the A2 term. The composite value of h¯ must account for the calculations up to h = H/2, so for depths from H/2 to H, the following area-weighted relationship is used to obtain the exact depth to the area centroid: Ax ½ðH=2Þ þ ðMx =Ax Þ þ A2 h¯ 2 h¯ ¼ Ax þ A2 where A2 and h¯ 2 are the values corresponding to h = H/2 (Eqs. (11) and (22)).
(25)
2.4. Wetted perimeter For 0 h h1: Wp ¼ 2Hcos
1
h 1 H
For h1 < h H/2: H 2h C1 Wp ¼ 2H cos1 cos1 þ Wp1 2H 2
(26)
(27)
where C1 is as defined in Eq. (13); and Wp1 is the wetted perimeter corresponding to h = h1 (Eq. (26)). The ‘‘cos1(C1/2)’’ term is a constant. For H/2 < h H: 2h p (28) þ Wp2 Wp ¼ H cos1 1 H 2 where Wp2 is the wetted perimeter corresponding to h = H/2 (Eq. (27)). Fig. 3 gives nondimensional curves of top width, wetted perimeter, depth to centroid, and cross-sectional area as functions of depth, all based on the above equations. For a full standard horseshoe cross section, the increase in area over a simple circle of radius H/2 is only about 5.6%, and the increase in wetted perimeter is about 4.0%, giving an increase in hydraulic radius of only 1.5%. The depth to the area centroid is less than H/2 for a full
G.P. Merkley / Agricultural Water Management 71 (2005) 61–70
67
Fig. 3. Nondimensional curves of standard horseshoe geometric parameters with respect to normalized depth of water.
section because the lower half of the cross section represents slightly more than half of the entire cross-sectional area.
3. Partial derivatives Exact partial derivatives of wetted perimeter and area centroid with respect to depth are needed for accuracy and numerical stability when implementing implicit solutions to unsteady open-channel flow hydraulic equations. These are given below for the same three depth segments, as defined above. 3.1. Wetted perimeter with respect to depth For 0 < h h1: @Wp 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h 1 ½1 ðh=HÞ 2
(29)
For h1 < h H/2: @Wp 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h 1 ½ð1=2Þ ðh=HÞ 2
(30)
It is noted that there is a discontinuity in @Wp/@h at h = h1, between Eqs. (29) and (30).
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For H/2 < h H: @Wp 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h 1 ½1 ð2h=HÞ 2
(31)
3.2. Area centroid with respect to depth For 0 < h h1: @h¯ @ Mx Ax ð@Mx =@hÞ Mx ð@Ax =@hÞ ¼ ¼ @h @h Ax A2x
(32)
where Ax and Mx are as defined in Eqs. (10) and (16), and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @Mx ¼ 2ðh HÞ hð2H hÞ @h 0 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi @Ax H ðh HÞ2 B C ¼ hð2H hÞ þ @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h hð2H hÞ 1 ½ðh HÞ=H 2
(33)
(34)
For h1 < h H/2: @h¯ ¼ @h
Ax ½h ðH=2Þ þ A1 ðh h¯ 1 Þ Mx ðAx þ A1 Þ2
! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 H 2 ½h ðH=2Þ 2 HÞ
(35)
where Mx and Ax are as defined in Eqs. (18) and (21); A1 is from Eq. (10) with h = h1; and h¯ 1 is from Eq. (17), also for h = h1. For H/2 < h H: ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @h Ax ½h H=2 þ A2 ðh h¯ 2 Þ Mx ¼ (36) ð2 ðH 2 =4Þ ½h ðH=2Þ 2 Þ 2 @h ðAx þ A2 Þ where Mx and Ax are as defined in Eqs. (23) and (24); A2 is from Eq. (15); and h¯ 2 is from Eq. (22) for h = H/2.
4. Sample application of the equations All of the above equations were compared to various numerical approximations of the geometric variables in a spreadsheet application, validating each of the equations given herein. The equations were also applied in a mathematical model of unsteady open-channel flow for a standard horseshoe cross section of H = 3 m, a length of 800 m, a longitudinal slope of 0.00005 m/m, and a Manning roughness of n = 0.012. The inflow was constant at 0.1 m3/s, and the downstream boundary was a rectangular weir with a crest length of 1 m, a coefficient of 1.8, an exponent of 1.5, and a fixed crest height of 0.2 m. One simulation was performed for the above parameters, and another with a 3-m diameter circular section,
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Fig. 4. Simulation results for a standard horseshoe cross section with H = 3 m, and a circular section with a 3-m inside diameter.
which is the same as the standard horseshoe section in the upper half. All other parameters were held constant for the two simulations. Each simulation started with a dry channel whereby the 800-m reach was partially filled, and then attained a steady-state hydraulic condition within 3 h, using a time step of 1 min. Fig. 4 shows the results in terms of water depth at the upstream and downstream boundaries of the channel during the 3 h simulation period. It is seen that the depth differences are significant for this small flow rate. However, for flow rates near the channel capacity, where the water depth is a significant fraction of the 3-m channel depth, the hydraulic differences between the horseshoe and circular sections are much less.
5. Conclusions The equations presented above can be applied in a spreadsheet computer program to generate tables of geometric relationships for standard horseshoe cross sections, when appropriate, or can be directly applied, as in mathematical models of steady (uniform and gradually varied) or unsteady flow, both for open channels and pressurized conduits with this cross-sectional shape. Most of the equation forms are simpler and easier to apply than previously available equations, and they include previously unpublished partial derivatives, and depth to area centroid relationships. The hydraulic characteristics of a standard horseshoe cross section are significantly different from those of a circular section with an inside diameter of ‘‘H’’ for relatively small flows and depths.
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Acknowledgment The author would like to thank the support from the Utah Agricultural Experiment Station, Project 788.
References Beyer, W.H. (Ed.), 1981. Standard Mathematical Tables, 26th ed. CRC Press. Boca Raton, FL. Davis, C.V., Sorensen, K.E. (Eds.), 1969. Handbook of Applied Hydraulics. McGraw-Hill Book Company, New York, NY. Hu, W.W., 1973. Hydraulic elements for USBR standard horseshoe tunnel. J. Transportation Eng. Div., ASCE 99 (4), 973–980. Hu, W.W., 1980. Water surface profile for horseshoe tunnel. Transportation Eng. J., ASCE 106 (2), 133–139. Stein, S.K., 1973. Calculus and Analytical Geometry. McGraw-Hill Book Company, New York, NY.