Estudio del flujo espacialmente variado en canales de recogida de pluviales

PROJECTE O TESINA D’ESPECIALITAT Títol Estudio del flujo espacialmente variado en canales de recogida de pluviales Autor/a Elena Pons Minguillón Tutor/a Manuel Gó mez Valentín Departament Enginyeria h idráulica , marítima i ambiental. Intensificació Hidráulica i Hidrologia. Data 28/06/2013 INDEX FIGURES INDEX ........................................................................................................ 5 TABLE INDEX ............................................................................................................ 7 ABSTRACT ................................................................................................................. 9 RESUMÉ .................................................................................................................... 10 I. INTRODUCTION ............................................................................................... 11 II. OBJECTIVES .................................................................................................. 12 III. HYDRAULIC ENGINEERING APPLICATIONS ........................................... 13 Increasing Spatially Varied Flow ............................................................................. 13 Side Overflow collecting canal in dams spillways ................................................. 13 Road and Street Drainage Canals ........................................................................ 14 Roof Rain Gutters ................................................................................................ 15 Decreasing Spatially Varied Flow ............................................................................ 15 Side weirs ............................................................................................................. 15 Bottom outlet canal reach .................................................................................... 16 IV. MATHEMATICAL DESCRIPTION OF THE SPATIALLY VARIED FLOW 17 One-Dimensional Method of Flow analysis .............................................................. 17 Mean velocity ....................................................................................................... 17 Boussinesq Coefficient β....................................................................................... 18 Coriolis Coefficient α............................................................................................ 18 Continuity Equation ................................................................................................ 18 Energy Equation ...................................................................................................... 19 Momentum Equation ............................................................................................... 19 Friction Losses......................................................................................................... 19 Governing Equations for Inflow and Outflow .......................................................... 20 Flow with Increasing Discharge ............................................................................ 20 Flow with Decreasing Discharge ........................................................................... 21 General Expression .............................................................................................. 22 V. NUMERICAL APPROACH OF THE SPATIALLY VARIED FLOW ............. 23 Control Point .......................................................................................................... 23 Critical Depth Line .............................................................................................. 25 Transitional Profile .............................................................................................. 26 Control Point for a rectangular canal ................................................................... 27 Integration by 4 th Order Runge-Kutta ..................................................................... 28 VI. VALIDATION TEST ....................................................................................... 30 Test Case 1 ............................................................................................................. 31 Test Case 1.A.: .................................................................................................... 31 Test Case 1.B.:..................................................................................................... 33 Test Case 2. ............................................................................................................ 34 Test Case 3. ............................................................................................................ 36 Discussion ................................................................................................................ 39 VII. WATER PROFILE: ONE SLOPE CASES ...................................................... 40 Small bed slope canal .............................................................................................. 40 Flat slope canal ....................................................................................................... 41 “U” Shaped canal .................................................................................................... 43 VIII. WATER PROFILE: TWO SLOPES CASES ................................................ 46 Summary Table ....................................................................................................... 47 Initial Case .......................................................................................................... 48 Case 1 .................................................................................................................. 49 Case 2 .................................................................................................................. 51 Case 3.1. .............................................................................................................. 52 Case 3.2. .............................................................................................................. 53 Case 4 .................................................................................................................. 54 Case 5.1. .............................................................................................................. 55 Case 5.2. .............................................................................................................. 58 Case 6 .................................................................................................................. 60 Case 7 .................................................................................................................. 62 Case 8 .................................................................................................................. 64 Impossible Cases .................................................................................................. 65 IX. CONCLUSIONS AND FUTURE DEVELOPMENT ........................................ 66 REFERENCES ........................................................................................................... 67 FIGURES INDEX Figure 1: Hoover Dam, between Nevada and Arizona, USA. ....................................... 13 Figure 2. Fuensanta Reservoir, Albacete, Spain. ......................................................... 14 Figure 3. Street Drainage Canals ISVF. ...................................................................... 14 Figure 4. Overflow in a street drainage canal. ............................................................. 14 Figure 5. Roof Rain Gutter ISVF. ............................................................................... 15 Figure 6. Elevation view of a Side Weir DSVF. .......................................................... 15 Figure 7. Bottom outlet DSVF. ................................................................................... 16 Figure 8. Velocity Distribution. ................................................................................... 17 Figure 9. Increasing Spatially Varied Flow. ................................................................. 18 Figure 10. Momentum Control Volume. ...................................................................... 19 Figure 11. Increasing SVF Control Volume. ................................................................ 20 Figure 12. Integration direction if Control Point located downstream. ........................ 24 Figure 13. Integration direction if Control Point Located in the canal reach. .............. 24 Figure 14. Integration direction if Control Point Located Upstream ........................... 25 Figure 15. Ill-conditioned Transitional Profile. ............................................................ 27 Figure 16. Test Case 1.B. Critical Depth and Transitional profile. .............................. 32 Figure 17. Test Case 1.A. Water Profile and Bottom Slope. ....................................... 32 Figure 18. Test Case 1.A. Froude Number. ................................................................. 33 Figure 19. Test Case 1.B., Critical Depth and Transitional profile. ............................. 34 Figure 20. Test Case 2. Critical Depth and Transitional profile. ................................. 35 Figure 21. Test Case 2. Water Profile and Bottom Slope. ........................................... 35 Figure 22. Test Case 2. Froude Number...................................................................... 36 Figure 23. Test Case 3. Critical depth and Transitional Profile. ................................. 37 Figure 24. Test Case 3, Water Profile and Bottom Slope. ........................................... 38 Figure 25.Test Case 3, Froude Number. ...................................................................... 38 Figure 26. Small Slope Case Critical depth and Transitional profile............................ 40 Figure 27. Small Bed Slope. Water Profile and Bottom Slope. .................................... 41 Figure 28. Small Bed Slope Case 4. Froude Number. .................................................. 41 Figure 29. Flat Slope Case. Water Profile. .................................................................. 42 Figure 30. Flat Slope Case. Froude Number. .............................................................. 43 Figure 31. Cross Sections used by ACO Company. ..................................................... 43 Figure 32. “U” Shaped Cross Section. ......................................................................... 44 Figure 33. U” Shaped. Water Profiles and bottom slope. ............................................ 45 Figure 34. Two Slope Cases. ....................................................................................... 47 Figure 35. Two slope Initial Case Water Profile .......................................................... 49 Figure 36. Two slope Case1 Water Profile. ................................................................. 50 Figure 37. Two slope Case 2 Water Profile. ................................................................ 52 Figure 38. Two slope Case 3.1. Water Profile. ............................................................ 53 Figure 39. Two slope Case4 Water Profile. ................................................................. 55 Figure 40. Two slope Case5.1. Independent Water Profiles. ........................................ 56 Figure 41. Two slope Case5.1. Hydraulic Jump in R2. ................................................ 57 Figure 42. Two slope Case 5.1. Hydraulic Jump in R1. ............................................... 58 Figure 43. Two slope Case 5.1. Water Profile. ............................................................ 58 Figure 44. Two slope Case 5.2 Hydraulic Jump in S1 or S2. ....................................... 59 Figure 45. Two slope Case 5.2 Final Water Profile with bottom slope. ....................... 60 Figure 46. Two slope Case 6 Independent Water Profiles............................................ 61 Figure 47. Two slope Case5.2 Hydraulic Jump in S1 ................................................... 61 Figure 48. Two slope Case5.2 Final Water Profile....................................................... 62 Figure 49. Two slope Case 6 Final Water Profile with bottom slope. .......................... 62 Figure 50. Two slope Case 7 Water Profile with bottom slope. ................................... 63 Figure 51. Two slope Case 8 Water Profile with bottom slope. ................................... 64 TABLE INDEX Table 1. Test Case 1 Data .......................................................................................... 31 Table 2. Test Case 2 ISP location Literature. ............................................................. 31 Table 3. Test Case 2 ISP location Literature. ............................................................. 31 Table 4. Test Case 1.B. ISP location Literature. ......................................................... 33 Table 5. Test Case 1.B. ISP location. .......................................................................... 33 Table 6. Test Case 2 Data. ......................................................................................... 34 Table 7. Test Case 2 ISP location Literature. ............................................................. 34 Table 8. Test Case 2 ISP location. .............................................................................. 35 Table 9. Test Case 2. Water Level Profile Comparison, Upstream. ............................ 36 Table 10. Test Case 2. Water Level Profile Comparison, Downstream. ....................... 36 Table 11. Test Case 3 Data. ........................................................................................ 37 Table 12. Test Case 3 ISP location literature. ............................................................. 37 Table 13. Test Case 3 ISP location. ............................................................................ 37 Table 14. Test Case 3. Water Level Profile Comparison, Upstream. ........................... 39 Table 15. Test Case 3. Water Level Profile Comparison, Downstream. ....................... 39 Table 16. Small Slope Case Data. ............................................................................... 40 Table 17. Small Slope Case ISP location. .................................................................... 40 Table 18. Flat Slope Case Data. .................................................................................. 42 Table 19. Flat Slope Case ISP location Literature....................................................... 42 Table 20. “U” Shaped Data......................................................................................... 44 Table 21. U” Shaped Cases ISP location. .................................................................... 45 Table 22. Two slopes. Main legend. ............................................................................ 48 Tabla 23. Two slopes. Extra water profiles for Hydraulic Jump legend. ...................... 48 Table 24. Two slope Initial Case data ......................................................................... 49 Table 25. Two slope Initial Case ISP location. ............................................................ 49 Table 26. Two slope Case1 data. ................................................................................. 50 Table 27. Two slope Case1 ISP locations. ................................................................... 50 Table 28. Two slope Case 2 Data. ............................................................................... 51 Table 29. Two slope Case2 ISP locations. ................................................................... 51 Table 30. Two slope Case 3.1. Data. ........................................................................... 52 Table 31. Two slope Case 3.1. ISP locations. .............................................................. 52 Table 32. Two slope Case3.2. Data. ............................................................................ 53 Table 33. Two slope Case 3.2. ISP locations. .............................................................. 53 Table 34. Two slope Case 4 Data. ............................................................................... 54 Table 35. Two slope Case4 ISP locations. ................................................................... 54 Table 36. Two slope Case 5.1. Data. ........................................................................... 55 Table 37. Two slope Case5.1. ISP locations. ............................................................... 56 Table 38. Two slope Case 5.1. Hydraulic Jump Location. ........................................... 58 Table 39. Two slope Case 5.2 Data. ............................................................................ 59 Table 40. Two slope Case 5.2. ISP locations. .............................................................. 59 Table 41. Two slope Case 6 Data. ............................................................................... 60 Table 42. Two slope Case6 ISP locations. ................................................................... 61 Table 43. Two slope Case 6 Hydraulic Jump Location. ............................................... 61 Table 44. Two slope Case 7 Data. ............................................................................... 63 Table 45. Two slope Case 7 ISP locations. .................................................................. 63 Table 46. Two slope Case 8 Data. ............................................................................... 64 Table 47. Two slope Case 8 ISP locations. .................................................................. 64 ABSTRACT This document aims to find a good approximate solution of the water profile that occurs in open canals with a spatially varied flow concerning an increasing discharge. An introduction to the Spatially Varied Flow and its applications in natural and artificial canals is presented and a detailed deduction of the governing equation is done. The equation that describes this flow condition is an Ordinary Differential Equation. A numerical code is then written in order to solve the ODE that defines the water profile. The location of the internal boundary condition stands as a key step for a successful computation and a systematically procedure to determine it is described. All physically possible locations and their associated implications are also discussed. All possible situations regarding the Increasing Spatially Varied Flow occurring in a two slope rectangular canal are studied and the impossible situations are briefly explained. In the cases where a Hydraulic Jump occurs further computations are carried out in order to obtain a final water profile that fulfills both internal boundary conditions. Additionally, the water profile obtained in a street drainage canal is analyzed to understand the way it commonly fails. Key words: spatially varied flow, increasing discharge, control point, side spillway, street drainage canals. Bottom outlet canal reach A different decreasing SVF is the flow occurring over a bottom outlet. Figure 7. Bottom outlet DSVF. In this case the amount of outflow rate also needs to be determined. III. MATHEMATICAL DESCRIPTION OF THE SPATIALLY VARIED FLOW General laws that describe the flow along an open-canal are briefly defined in this section. Once the constitutive equations are introduced, the governing equation for the Spatially Varied Flow can be deduced. One-Dimensional Method of Flow analysis The presence of the boundaries in open canals causes the velocity vectors of the flow to have components in the three coordinate directions, not only on the longitudinal direction but also in the two normal directions to the flow. The same happens to the pressure gradient. In Figure 8 velocity distributions in different type of cross sections can be observed. In any kind of cross-section it is known that the velocity is zero at the solid boundaries and gradually increases as the distance to the boundary gets greater. Figure 8. Velocity Distribution. A Three-Dimensional Analysis considers these velocity profiles, resulting in a very complex analysis. However, it can be simplified and yet give meaningful results by considering a OneDimensional approach, the mean of the cross section velocities and pressure gradients for each cross section are taken as representative values. Mean velocity The mean velocity  in terms of the  direction can be defined as: =1 ·· .  Considering the Discharge as: =· This leads to a new definition of it: =· .  Therefore, two coefficients are introduced to adequately represent the non-uniformity of the velocity distribution through the transversal section. Boussinesq Coefficient  This coefficient is defined so that the momentum principle can be expressed in terms of V: The momentum flux is written as:=   besides, it’s known that: =    . , Thus it’s deducted that: =    .    Coriolis Coefficient  This coefficient is defined so that the Kinetic Energy can be expressed in terms of V: The Kinetic Energy is written as: ..=     besides, it’s known that: ..=       . , Thus it’s deducted that: =   ! ." #  ! These two coefficients are equal to 1 if the velocity distribution is indeed uniform and are greater that one in any other case. As the velocity distribution gets less uniform the coefficients get greater than 1. Continuity Equation It is the expression that represents the law of conservation of mass applied in an open canal flow. Considering a steady state flow of the SVF it’s noticed that the volumetric rate of flow is not constant along the canal since there is a lateral addition or withdrawal. Figure 9 illustrates the case of an increasing SVF. Figure 9. Increasing Spatially Varied Flow. Being  $ the initial discharge and % ∗  the rate of addition or substraction of discharge: ()= $ +% ∗ · * $ [1] Energy Equation The Bernouilli equation is used to represent the energy equation in the 1D Analysis for steady open canal flow: +=,+-·./01+·  23 In this paper 1≤0.15 and then the assumption./01~1 is always valid. Hence, +=,+-+·   8 [2] Which can be written as well as: +=,+-+· 9  8  [3] Momentum Equation The momentum principle in fluid mechanics is based on Newton’s second law of motion, stating that the external forces applied in a control volume are equal to the rate of change of the momentum. Figure 10 is used to deduce the expression for the momentum: Figure 10. Momentum Control Volume. Being :; the external forces applied to each considered section: <:=:;1−:;2=(  −  ) :;1=  ·ℎ ? ·@ and :;2=  ·ℎ ? ·@ Note that @=·3 and ℎ ? is the centroid and can be defined as: ℎ ? = A·  Then the momentum can be written as: =·ℎ ? + 9  8 [4] Friction Losses It’s considered that the friction losses are adequately represented by Manning’s formula: =  B C A/ ·E $/ [5] This expression links mean velocity, hydraulic radius and bottom slope making use of the Manning number, which is considered constant along the canal. Governing Equations for Inflow and Outflow In order to deduce the constitutive equations for the SVF, some assumptions are considered:  Incompressible fluid  Hydrostatic pressure, steady state solution.  1D Analysis  Manning friction looses  Prismatic canals, then F F*G =0. Inflow and Outflow cases will be studied separately; different equations are used to describe the flow in the canal. Flow with Increasing Discharge In this type of Spatially Varied Flow, an appreciable portion of the energy loss is due to the turbulent mixing of the added water and the water flowing in the canal. In most cases, these losses are relatively high and uncertain. For this reason the momentum equation is more convenient than the energy equation when modeling the flow. Figure 11 shows a control volume representing a ISVF: Figure 11. Increasing SVF Control Volume. From the Figure 11, on the direction parallel to the bottom slope it can be written that:   −  =H  −H  +I·0JK1−:L [6] Where:   ,  is the momentum at each cross section, H  , H  is the pressure force, I·0JK1 is the weight of control volume ∆ in the  direction , I·0JK1= @E $ ·∆. :L is the friction force, :L=@E O ·∆. Thus, ∆=−∆H+I·0JK1−:L [7] Dividing equation [7] by ∆ and taking limits as ∆→0: Q * =− R * +@E $ −@E O [8] with R * = STGUV * =@ G * Besides, from the 1D analysis it’s known that: =  =  /, Deriving this expression: Q * =·W 9 * · 9  −  ·  * ·    X [9] By definition: 9 * =% ∗  and  * =Y· FG F* + F F*G , being a prismatic canal:  * =Y· FG F* Thus equation [9 ]results in: Q * =·W% ∗ · 9  − 9    ·Y· FG F* X [10] Using equation [10], equation [8] can be rewritten as: G * = Z [ \Z ] \^ _[`∗ a \^ _[ a b [11] Equation [11] is the governing equation for the ISFV, it can be seen that is an ODE where G * is defined in terms of - and . Flow with Decreasing Discharge This type of flow can be understood as a flow diversion where the diverted water does not affect the energy head. The specific energy is not affected by the water leaving the main flow. Therefore the energy equation is suitable to solve this kind of flow. This can be sum up as an extra assumption:  The withdraw of water does not affect the energy content per unit of mass of the water in the canal. Differentiating the energy equation [3] with respect to : c * = d * + G * + e 8 W 9   · 9 * − ·9    ·  * X [12] Considering again that: 9 * =% ∗ the lateral discharge, and  * =Y· FG F* for prismatic canals. By definition: c * =−E O and d * =−E $ . Finally equation [12] can be simplified and rewritten as: G * = Z [ \Z ] \e _[`∗ a \e _[ a b [13] Equation [13] is the governing equation for the DSFV, it can be seen that is again an ODE where G * is defined in terms of - and . General Expression The same structure can be seen by taking a close look to both governing equations. A general expression is then written representing both cases: G * = Z [ \Z ] \fg _[`∗ a \g _ a b [14] Where: For a Decreasing SVF : µ =  and h=1 For an Increasing SVF: µ =  and h=2 Although both cases are represented in equation [14 ], this equation is not the ODE aimed to solve by numerical methods. Assumptions taken in each case differ a lot from the other. A clear example of these differences is found in % ∗ : the amount of inflow is known while the amount of outflow depends on the water level -, the sought variable. Therefore it is believed that analyzing each type of Spatially varied Flow separately would lead to simpler computations, deal with both flows as a general case would not be a practical approach. In this paper the subject of study is the Spatially Varied Flow with Increasing discharge, so the equation dealt with is equation [11]. IV. NUMERICAL APPROACH OF THE SPATIALLY VARIED FLOW Once the flow has been characterized by a governing equation, equation [11], a numerical code is written, which computes the water profile for any given cross section’s canal geometry as well as for any given bottom slope, as long as the assumption ./01~1 is acceptable. The water profile can be simply computed by integrating the governing equation along the canal, considering the appropriated internal boundary conditions. This integration is approximated using a numerical method, which is a discrete method that computes an approximate solution using a certain ∆ in a great amount of discrete points. The key step to carry out the integration successfully is to determine the Integration Starting Point, the numerical integration should be started here. If the canal has two reaches an Integration Starting Point is independently found for each of them. The Integration Starting Point is usually located in the so called Control Point. The Control Point is the only position where the critical depth - ? can occur, which is a water level easily to be computed and it does not depend on the reach slope. Its location depends on the combination of all the parameters affecting the flow such as the slope, the cross-section geometry, the roughness coefficient. If the Control Point is found along the canal reach the Integration Starting Point is located there and the water level in this position is the critical depth. If the Control Point is located elsewhere, the Integration Starting Point is found in the upstream or downstream extreme of the reach and the value of the water level has to be determined. Control Point The possible locations of the Control Point are here discussed, their associated Integration Starting Point location and value are also presented. The Control Point is located where the Transitional Profile line and the Critical Depth line cross. Each condition is represented by one equation, both equations are introduced in the next section. There are basically three possible situations: 1) Control Point Located Downstream the Canal. In this case the Transitional profile does not reach the Critical depth line in the canal length and the position of the Control Point is found downstream the canal. In this case the Integration Starting Point is located in the downstream extreme, the value of the water level might be the Critical depth in the case of a single slope canal. If another canal reach is found downstream, the value of the water level is given by the initial water level of the second reach which is a subcritical flow. In any case the flow is subcritical along the reach studied. Figure 12. Integration direction if Control Point located downstream. 2) Control Point Located In the Canal In this case it exists a point from the canal reach  such that both lines cross. The Control Point is located along the canal and so the Integration Starting Point is located at this position. The integration is carried out upstream and downstream the Control Point. The flow is subcritical at the beginning of the canal or reach, goes through a critical depth, and proceeds being supercritical. Note that since the amount of water increases along the canal, it’s not intuitive to determine if the supercritical depth is higher or lower than the subcritical flow occurred some meters upstream. Figure 13. Integration direction if Control Point Located in the canal reach. 3) Control Point Located Upstream the Canal In this case the Critical depth line reaches the Transitional opofile upstream the reach domain, this can only occur if the Initial Discharge  i is not zero. The Control Point is located upstream the canal, in other words. The Starting Integration Point is the initial position of the canal or reach and its condition is determined by the previous reach in the case of two slope canal. The flow is supercritical along the canal. Figure 14. Integration direction if Control Point Located Upstream Critical Depth Line Given a canal cross section’s geometry and a certain discharge, the critical depth is defined as the water depth such that the water is flowing with the minimum specific energy. Other canal properties such as the bed slope and the roughness do not influence the critical flow condition. This condition can be expressed as: j G =0, being =-+ 9     then j G =1− e9  8  ·Y The Froude Number associated to the flow is defined as: :k  = e9  8  ·Y Thus, :k=  l3 Y Hence, the critical flow condition can be described as the water level - such that the Froude Number is one. :k ? = 9 m8  n no = 1 Rearranging the terms a new expression that characterizes the critical flow is obtained:   − 8  b =0 [15] Note that equation [15] depends on , which is variable along the canal in Spatially Varied flow conditions. In opposition to the Gradually Varied Flow it can be seen that for the SVF the Critical depth line is not a straight line parallel to the bed slope. No term regarding the bottom slope or the roughness coefficient does appear in equation [15], the critical depth line is independent on the bottom slope canal. The interaction between the transitional profile and the critical depth line are shown in the Figure 16: Figure 16. Test Case 1.B. Critical Depth and Transitional profile. This leads to a difference of 2.7| on the, which can be also expressed as: .œ • ·100= 6.9%. By analogy on the - axis it leads to a difference of0.24|, which can be also expressed as: $.— . ·100=9.23%. These differences are not tolerated. It’s considered that the tabular solution for the Control Point, which is in this case the Integration Starting Point, is a rougher approximation than the numerical one since the discretization is done considering bigger space steps. However, a further analysis comparing the method presented in this paper with other technical literature is needed. The water profile and the value of the Froude number associated can be seen in the graphs obtained from the numerical code: Figure 17. Test Case 1.A. Water Profile and Bottom Slope. Figure 18. Test Case 1.A. Froude Number. Note that to guarantee stability when integrating, it’s needed to start from - a little bit higher or lower than - ? =2.36|: Upstream section: - ¡R =- ? ·(1+0.06)=2.5| Dowstream section: - ¡R =- ? ·(1−0.03)=2.28| This operation though causes small disturbances that can be found Figure 17 and Figure 18. Figure 18 gives relevant information about the type of flow that occurs in each position  of the canal. First the flow is subcritical, Froude <1, until it gets to the Control Point where the flow gets critical, Froude =1. Finally it continues as supercritical flow, Froude>1, until the end of the canal. Test Case 1.B.: The case is repeated for % ∗ =3 £  ¤£ ⁄ , in this case only the results concerning the Integration Starting Point are presented since it’s considered that a further analysis would get to the same conclusions as the previous case. By analogy Subramanya (1982) obtains the Integration Starting Point at: Integration Starting Point x 56  | y 4  | Table 4. Test Case 1.B. ISP location Literature. Similar result is obtained by using the numerical code: Integration Starting Po int x 51 . 7  | y 3 . 61  | Table 5. Test Case 1.B. ISP location. Which correspond again to the Control Point location. Figure 19. Test Case 1.B., Critical Depth and Transitional profile. Again, a significant difference is presented when comparing the value. This leads to a difference of 4.3| on the, which can be also expressed as: —. ¦ ·100= 7.7%. By analogy on the - axis it leads to a difference of0.39|, which can be also expressed as: $.• — ·100=12.25%. Again these differences are considered too great to be accepted due to the method used by Subramanya (1982). Test Case 2. Jain (2001) finds the Control Point location by giving random values of  and checking if both sides of the equation match by a trial an error procedure. The governing equation is also solved with the same procedure. Jain (2001) uses the following data: Cana l Data Geometry Trapezoidal Canal Bottom Width ƒ = 3  | Tangent Inv . XS | = 0 . 5 Bed Slope E i = 0.15 Up. Initial Discharge  i =  0  |  / 0 Lateral Discharge % =  4   |  / 0 / | C ana l Length ˜ = 122  | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 6. Test Case 2 Data. Jain (2001) finds the Integration Starting Point at: Integration Starting Point x 55  | y 5 . 88  | Table 7. Test Case 2 ISP location Literature. The Integration Starting Point using the numerical code is located at: Integration Starting Point x 53 . 06  | y 5 . 81  | Table 8. Test Case 2 ISP location. Figure 20. Test Case 2. Critical Depth and Transitional profile. This leads to a difference of 0.04| on the coordinate, which can also be expressed as: .— ¦¦ ·100=2.54%. By analogy on the - axis, it leads to a difference of0.02|, which can also be expressed as: $.$œ ¦.§§ ·100=1.19%. The flow is Subcritical until it reaches the Control Point at =49.94| then it turns Supercritical. Small differences can be observed comparing the two methods, these differences can be accepted, it can be considered that both methods lead to the same result. The water profile and the value of the Froude number associated to each water level care shown: Figure 21. Test Case 2. Water Profile and Bottom Slope. Figure 22. Test Case 2. Froude Number. Note that to guarantee stability when integrating, it’s needed to start from - a little bit higher or lower than - ? =5.81|, in other words, it’s needed to get away from the critical depth. Upstream section: - ¡R =- ? ·(1+0.01)=5.87| Downstream section: - ¡R =- ? ·(1−0.01)=5.75| The flow is subcritical until it reaches the Control Point at =53.06| and then it turns supercritical. A Water Profile Comparison Once the Control Point in found, Jain (2001) proceeds with the water profile computation for certain positions . The results are compared: Values located upstream the Integration Starting Point: x [m] Jain (2001 ) [m] Numerical c ode [m] Difference [m] x = 0 y = 2.6 y = 2.92 0.32 x = 25 y = 4.7 y = 4.76 0.06 Table 9. Test Case 2. Water Level Profile Comparison, Upstream. Values located downstream the Integration Starting Point: x [m] Jain (2001 ) [m] Numerical code [m] Difference [m] x = 90 y = 6.74 y = 6.75 0.01 x = L =122 y = 7.38 y = 7.38 0.00 Table 10. Test Case 2. Water Level Profile Comparison, Downstream. Comparing the values obtained by the two methods, it’s observed that the greatest difference is: 0.32|which can also be expressed in a percentage as: $. . ·100=12.3%. Test Case 3. The same case is presented in Chow (1959) and French (1985). In this case a similar tabular procedure used by Subramanya (1982) is used to compute the Control Point as well as the water profile. The data used in this case is: C ana l Data Geometry Trapezoidal Canal Bottom Width ƒ = 3  | Tangent Inv. XS | = 0 . 5 Bed Slope E i = 0.1505 Up. Initial Discharge  i =  0  |  / 0 Lateral Discharge  i =  3 . 72   |  / 0 / | Cana l Length ˜ = 122  | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 11. Test Case 3 Data. The authors find the Integration Starting Point at: Integration Starting Point x 49 . 98  | y 5 . 39  | Table 12. Test Case 3 ISP location literature. Similar result is obtained by using the Numerical Code: Integration Starting Point x 49 . 94  | y 5 . 41  | Table 13. Test Case 3 ISP location. The computations are shown in Figure 23: Figure 23. Test Case 3. Critical depth and Transitional Profile. This leads to a difference of 0.04| on the, which can be also expressed as: $.$— —•.—§ ·100=0.08%. By analogy on the - axis it leads to a difference of0.02|, which can be also expressed as: $.$ ¦.• ·100=0.37%. Really small differences can be observed comparing the two methods; both methods lead to the same result. The water profile and the value of the Froude number associated can be seen in the following figures: Figure 24. Test Case 3, Water Profile and Bottom Slope. Figure 25.Test Case 3, Froude Number. Note that, again, to guarantee stability when integrating it’s needed to start from - a little bit higher or lower than- ? =5.41|: Upstream section: - ¡R =- ? ·(1+0.02)=5.52| Downstream section: - ¡R =- ? ·(1−0.01)=5.36| This operation is not noticed in Figure 24 neither in Figure 25 because the deviation in very small, 2%. Again the flow is subcritical unti it reaches the Control Point at =49.94| to finally flow in a supercritical regime. Water Profile Comparison Once the Integration Starting Point in found, Chown (1959) proceeds with the water level computation. The results are compared. Values located upstream the Integration Starting Point: x [m] Chow (1959 ) [m] Numerical code [m] Difference [m] x = 0 y = 2. 62 y = 2.70 0.08 x = 3 y = 3.01 y = 3.05 0.04 x = 7.6 y = 3.43 y = 3.47 0.04 x = 15.3 y = 3.95 y = 4.00 0.05 Table 14. Test Case 3. Water Level Profile Comparison, Upstream. Values located downstream the Integration Starting Point: x [m] Chow (1959 ) [m] Numerical c ode [m] Difference [m] x = 60.9 y = 5.68 y = 5.72 0.04 x = 76.2 y = 6.07 y = 6.12 0.05 x = 91.5 y = 6.41 y = 6.44 0.03 x = 106.8 y = 6.72 y = 6.75 0.03 x = L = 122 y = 6.96 y = 7,02 0.06 Table 15. Test Case 3. Water Level Profile Comparison, Downstream. Comparing the values obtained by the two methods, it’s observed that the greatest difference is: 0.06|which can also be expressed in a percentage as: $.$ œ.$ ·100=0.85%. It is a really small difference, the numerical code computes the same approximation as Chow (1959). Discussion Once the results implementing the Test Cases on the numerical code are compared with the results found by the technical literature, it can be concluded that the numerical code works correctly and it finds a good approximation of the ODE that governs the Increasing SVF. However, some sensitivity is detected when running the code in Matlab regarding the initial approximations used for fzero functions. All the cases shown until now represent the same type of open canal: a trapezoidal canal with a great slope and a length of 100 m or more. Henceforth the numerical code is used to compute new cases not found in literature that represent different kind of situations with civil engineering applications. VI. WATER PROFILE: ONE SLOPE CASES Small bed slope canal Here it’s considered the case where the bottom slope ,So, is such that the Control Point is located downstream the given canal, i.e. ¡R >˜ . Hence, the Starting Integration Point is located at  ¡R =˜ and the integration can be carried out upstream covering the whole canal length. For this case Test Case 3 is modified by diminishing E $ so that the condition mentioned above is fulfilled. All the other Inputs are considered the same as Test Case 3. The data in this case is: Canal Data Geometry Trapezoidal Canal Bottom Width ƒ = 3  | Tangent Inv. XS | = 0 . 5 Bed Slope E i = 0.09 Up. Initial Discharge  i =  0  |  / 0 Lateral Discharge % i =  3 . 72   |  / 0 / | C anal Length ˜ = 122  | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 16. Small Slope Case Data. Integration Starting Point location: Integration Starting Point x 122  | y 8 . 56  | Table 17. Small Slope Case ISP location. Figure 26. Small Slope Case Critical depth and Transitional profile. It can be seen that the Control Point would be located downstream ˜=122|, but since the canal is shorter, the integration is started at =122|. . It is considered that the outlet is free, the water level reaches the Critical depth at the end of the canal. The water profile is computed integrating upstream the Integration Starting Point. The resulting water profile as well as the value of the Froude number associated to each water level can be seen in the graphs obtained from the numerical code, Figure27 and Figure 28: Figure 27. Small Bed Slope. Water Profile and Bottom Slope. Figure 28. Small Bed Slope Case 4. Froude Number. The flow is Subcritical along the entire canal and the Froude number increases continuously in the flow direction reaching the value of 1 at ˜=122|. Flat slope canal Finally to conclude with the Trapezoidal Canals an extreme case where the bed slope is zero is presented here. For this case Test Case 3 is modified using E $ =0. All the other inputs are considered the same as Test Case 3. The grey cells correspond to impossible situations which are briefly explained while the possible cases are covered in detail. For the following cases some notation needs to be mentioned for a good understanding when looking at the results presented in this document: Reach 1. Subcritical Flow. Reach 1. Supercritical Flow. Reach 2. Subcritical Flow. Reach 2. Supercritical Flow. Bottom Slope Canal. Table 22. Two slopes. Main legend. Control Point figures as well as Froude number figures are not presented in this section, it is considered that at this stage of the proceedings the flow characteristics can be understood without this support graph. However, both figures can be found in Appendix 4 in needed. A complementary legend is here shown which is used in some of the cases presented in this section, when a Hydraulic Jump needs to be computed: Reach 2. Supercritical Flow from R1. Reach 1 and 2 . Subcritical Flow from R2. Reach 1 and 2. Conjugated Supercritical Flow from R2. Tabla 23. Two slopes. Extra water profiles for Hydraulic Jump legend. Initial Case Table 24 shows the data used to compute the Initial Case. C anal Data Geometry Rectangular Canal Width ƒ = 4 | Bed Slope E i = 0.05 Up. Initial Discharge  i =  0  |  / 0 Lateral Discharge % ∗ =  0 . 5   |  / 0 / | C ana l Length ˜ = 100 | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 24. Two slope Initial Case data The Initial Case consists simply on a subcritical flow along the entire canal, the Control Point is located downstream the end of the canal. Hence the Integration Starting Point is located at: Integration Starting Point x 100 m y 2.52 m Table 25. Two slope Initial Case ISP location. Integrating the canal governing equation upstream the Integration Starting Point the following water profile is obtained: Figure 35. Two slope Initial Case Water Profile From this case the following cases are derived by modifying the bottom slope. The canal is divided in two equal reaches 50 m long. Case 1 Firstly the slope of the Reach 2 is decreased. The following results are thus expected: - The Control Point 1 position is expected to be found downstream Reach 1 since Reach 1 can be seen as a shorter version of the Initial Case. - Meanwhile, the position of Control Point 2 is expected to be found downstream since Reach 2 can be seen as a shorter and flatter version of the Initial Case. The data used for this case is: Canal Data Geometry Rectangular Canal Width ƒ = 4  | Bed Slopes E i 1 = 0.05 E i 2 = 0.01 Up. Initial Discharge  i =  0  |  / 0 Lateral Discharge % ∗ =  0 . 5   |  / 0 / | C ana l Length ˜ = 100 | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 26. Two slope Case1 data. The Integration Starting Points are found at: Integration Starting Points x1 50 m y1 - x2 100 m Y2 2.52 m Table 27. Two slope Case1 ISP locations. Remark that no information is given regarding the water level - at =50|, the position of Integration Starting Point 1. This is so since the water level at this position is not - ? as it would be it if Reach 1 was an independent canal. The water level is given by the integration of the water profile in Reach 2. In other words, Reach 1 water level depends on Reach 2, the flow along the canal depends on a boundary condition found at the end of the canal. The integration of the entire canal is done in the upstream direction. Hence the water profile is: Figure 36. Two slope Case1 Water Profile. The flow is subcritical along the whole canal. Case 2 This case is the opposite case of Case 1, the slope of Reach 1 is decreased while the slope of Reach 2 is kept equal to the Initial Case’s slope. By analogy to Case 1, the same assumptions can be made. The canal is expected to be subcritical in all its length since the only modification done from the Initial Case is decreasing the slope of the reach of the canal. The data used for this case is: Cana l Data Geometry Rectangular Canal Width ƒ = 4 | Bed Slopes E i 1 = 0.01 E i 2 = 0.05 Up. Initial Discharge  i =  0  |  / 0 Lateral Discharge % ∗ =  0 . 5   |  / 0 / | C ana l Length ˜ = 100 | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 28. Two slope Case 2 Data. The integration Starting Points are again found at: Integration Starting Points x1 50 m y1 - x2 100 m Y2 2.52 m Table 29. Two slope Case2 ISP locations. Again the flow is subcritical along the canal and the water level at the Transition Slope Point depends on the downstream condition. Although the Integration Starting Point is the same, the Water Profile takes a different shape: Figure 37. Two slope Case 2 Water Profile. Case 3.1. The previous case, Case 2, is slightly modified by increasing the bottom slope of Reach 2 such as the Control Point 2 is now located at Reach 2. In this case the data used is: C ana l Data Geometry Rectangular Cana l Width ƒ = 4 | Bed Slopes E i 1 = 0.01 E i 2 = 0.06 Up. Initial Discharge  i =  0  |  / 0 Lateral Discharge % ∗ =  0 . 5   |  / 0 / | C ana l Length ˜ = 100 | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 30. Two slope Case 3.1. Data. The integration Starting Points are located at: Integration Starting Points x1 50 m y1 - x2 75.9 m Y2 2. 09 m Table 31. Two slope Case 3.1. ISP locations. The integration is also started at Reach 2, upstream and downstream the Integration Starting Point 2, which is in this case Control Point 2. After that the integration can be continued in Reach 1 using as the Integration Starting Point (boundary condition) the water level found from Reach 2. The water profile is then: Figure 38. Two slope Case 3.1. Water Profile. The flow is subcritical until it reaches the Control Point 2 at =76.3| to go on as a supercritical flow. Case 3.2. The same situation could be given being Reach 1 slope higher than Reach 2 Slope. However, the range of slope values that give such situation is very narrow. One of these cases is: C ana l Data Geometry Rectangular Canal Width ƒ = 4 | Bed Slopes E i 1 = 0.065 E i 2 = 0.057 Up. Initial Discharge  i =  0  |  / 0 Lateral Discharge % ∗ =  0 . 5   |  / 0 / | C ana l Length ˜ = 100 | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 32. Two slope Case3.2. Data. The Positions of the Integration Starting Points are: Integration Startig Points x1 50 m y1 - x2 91 m Y2 2.36 m Table 33. Two slope Case 3.2. ISP locations. In this case, if the entire canal would have Reach’s 1 Slope the Control Point would be located between the Transition Slope Point and the end of the canal. In other words Control Point 1 is located along Reach 2 and so the Integration Starting Point 1 is the Transition Slope Point. Reach 2 slope is very similar to Reach 1 slope and so Control Point 2 is located a little bit downstream the supposed Control Point 1, but still in Reach 2. The computations of this case would be very similar to the ones for Case 3.1. Case 4 Here the previous case, Case 3.1., is slightly modified by increasing the bottom slope of Reach 2 such as the Control Point 2 in found upstream the Transition Slope Point. In Case 4 the data used is: Canal Data Geometry Rectangular Cana l Width ƒ = 4 | Bed Slopes E i 1 = 0.01 E i 2 = 0.07 Up. Initial Discharge  i =  0  |  / 0 Lateral Discharge % ∗ =  0 . 5   |  / 0 / | Canal Length ˜ = 100 | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 34. Two slope Case 4 Data. Integration Starting Points 1 and 2 are located at: Integration Starting Points x1 50 m y1 1.59 m x2 50 m Y2 1.59 m Table 35. Two slope Case4 ISP locations. The flow in the whole Reach 2 in then supercritical while the flow in Reach1 is subcritical. That is to say, the Integration Starting Point 1 and 2 are the same point, the Transition Slope Point. This situation is possible since Reach 1 is flat enough so that Control Point 1 would be located downstream the Transition Slope Point while Reach 2 is steep enough so Control Point 2 would be located upstream the Transition Slope Point. In the Transition Slope Point the water level is known, it is the Critical Depth - ? , in this particular case - ? =1.59|. Therefore the water profile is computed upstream (Reach 1) and downstream (Reach 2) this Integration Starting Point: Figure 39. Two slope Case4 Water Profile. Case 5.1. In this case the Initial Case is modified so that the location of Control Point 1 is located along Reach 1, so that it corresponds to the Integration Starting Point 1. This can be done by increasing the bottom slope. The Integration Starting Point 2 is kept at the end of the canal; in this case is chosen a flatter slope than the Initial Case’s slope so that the results are clearer. The following results are expected: - Reach 1 is integrated upstream and downstream Integration Starting Point 1. - Reach 2 is integrated upstream the Integration Starting Point 2. Therefore both water profiles do not match at the Transition Slope Point and a further analysis needs to be carried out. For this reason an extra figure is shown to make the reasoning clearer. The water profile is shown, and once the conclusion is obtained, the bottom slope is shown for a general idea of the whole problem. The data used for this case: Canal Data Geometry Rectangular Cana l Width ƒ = 4 | Bed Slopes E i 1 = 0.09 E i 2 = 0.04 Up. Initial Discharge % ∗ =  0  |  / 0 Lateral Discharge  i =  0 . 5   |  / 0 / | Canal Length ˜ = 100 | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 36. Two slope Case 5.1. Data. The Integration Starting Points are found at: Integration Starting Points x1 20 . 11 m y1 0.86 m x2 100 m Y2 2.52 m Table 37. Two slope Case5.1. ISP locations. The water profile considering each reach is then: Figure 40. Two slope Case5.1. Independent Water Profiles. It can be seen that water profiles from both reaches do not match at all. Moreover, the flow gets supercritical in Reach 1 while the flow in Reach 2 is subcritical. It can be deduced then that a Hydraulic Jump is occurring in order to match the conditions from both reaches, the Hydraulic Jump is the hydraulic solution to overcome this discontinuity. The position of the Hydraulic Jump can be found in Reach 1 or in Reach 2. Hydraulic Jump in Reach 2 Firstly it is assumed that the location of the Hydraulic Jump is in Reach 2: - The conjugate depth line of the water profile in Reach 2 is computed, which corresponds to a supercritical flow. - A new water profile is computed in Reach 2 using as the Integration Starting Point the water level at the Transition Profile given by Reach 1, since the Integration Starting point corresponds to a supercritical flow, the water Profile computed is also in supercritical flow conditions. Hence, if the Hydraulic Jump does exist in Reach 2, it is located where both lines cross. The following figure shows the just mentioned computations: Figure 41. Two slope Case5.1. Hydraulic Jump in R2. In Figure 40 it can be seen that the water profile computed from Reach 1, the green line, suffers some instabilities, so this behavior lacks of physical sense. This senseless tendency can be explained as follows: the water level increases until reaching the Critical Depth (at =53.1|) from that point the integration loses stability. The derivate is FG F* →∞ , this can be easily seen in equation [16], when = ? the denominator is zero and so the derivate tends to ∞. Taking a careful look to the Runge-Kutta Numerical Method it can be seen that once FG F* →∞, then  • →∞, and that’s the reason why the integration loses its physical sense. Starting from the supercritical condition given by Reach 1 the flow has the tendency to increase until it reaches the critical condition, it wouldn’t have any physical sense that the flow would turn more supercritical in order to cross the conjugate line so that the Hydraulic Jump could occur. Thus the Hydraulic jump is not located in Reach 2. Hydraulic Jump in Reach 1 The Hydraulic Jump is then sought is Reach 1 using the same procedure: - The water profile from Reach 2 is lengthened in Reach 1. This is done integrating upstream using the condition given by the water profile in Reach 2, which is a subcritical flow. - The conjugate of this new water profile is computed, that corresponds to a supercritical flow. Hence, if the Hydraulic Jump does exist in Reach 1, it is located where the conjugate water profile crosses the supercritical water profile from Reach 1. The following table shows the just mentioned computations: Case 8 A similar case of Case 7 in shown here, the same characteristics are sought while permuting the bottom slopes. In this Case: Canal Data Geometry Rectangular Cana l Width ƒ = 4 | Bed Slopes E i 1 = 0.07 E i 2 = 0.09 Up. Initial Discharge % ∗ =  0  |  / 0 Lateral Discharge  i =  0 . 5   |  / 0 / | Canal Length ˜ = 100 | Boussinesq coefficient  = 1 Manning Coefficient K = 0 . 015 Table 46. Two slope Case 8 Data. The Integration Starting Points are located at: Integration Starting Points x1 44.94 m y1 1.47 m x2 50 m Y2 - m Table 47. Two slope Case 8 ISP locations. Note that the same reasoning is used to justify the lack of information regarding the water level in Integration Starting Point 2, its value depends on the upstream flow, in Reach 1. The water profile is shown in Figure 51: Figure 51. Two slope Case 8 Water Profile with bottom slope. As well as Case 7, in this case the flow is subcritical the first meters of Reach 1 until it reaches the Control Point 1 and it turns supercritical. Impossible Cases Finally the grey cells that represent impossible cases that cannot occur in reality are briefly presented in the next section. I-1 Being So1>So1, this case would represent a Subcritical flow in Reach 1 and a Supercritical flow in Reach 2. This would be an extreme case of Case 3.2. , with Control Point 1 located at the Transition Slope Point and Control Point 2 also located at the Transition Slope Point, so So1=So2, which is not considered here since it’s supposed So1>So2. I-2 Being So1<So1, this case would represent a Subcritical-Supercritical flow in Reach 1 and a Subcritical flow in Reach 2. Assuming that the flow is Subcritical-Supercritical is Reach 1, a steeper slope, So2, would lead to a more Supercritical flow, there is no chance that given these conditions a Subcritical flow could occur in Reach 2. I-3 Being So1<So1, this case would represent a Subcritical-Supercritical flow in Reach 1 followed by a Subcritical-Supercritical flow in Reach 2. Again, assuming the flow is Subcritical-Supercritical is Reach 1, a steeper slope, So2, would lead to a more Supercritical flow where its Control Point would be located upstream the Transition Slope Point. VIII. CONCLUSIONS AND FUTURE DEVELOPMENT This document finds an approximate solution for the water profile for one slope and two slope canals for a spatially varied flow with an increasing discharge. The procedure has been kept as general as possible. The numerical code is ready to be implemented on any kind of geometry cross-section, the only step that should be carried out would be writing the function that expresses the area, the canal width and the wet perimeter in terms of water level. The numerical code has been successfully validated using some cases found in technical literature. The same structure could be used to analyze the spatially varied flow with decreasing discharge. In this case some assumptions regarding the amount of outflow rate should be studied and finally the numerical code could be modified to find the water profile. The numerical code used to solve the Ordinary Differential Equation works correctly despite of the great sensitivity shown regarding first approximations. The numerical code could be improved to avoid such inconveniences. However, it cannot be forgotten that the numerical code is considered a mathematical tool to solve a physical problem, thus the code itself is not considered as the main objective of this document. This document can be seen as a kick-off study of the spatially varied flow with increasing discharge. The general approach of two slope cases can be considered the first step to analyze multiple slope cases with this type of flow. The interaction between both reaches is described and its possible implications are carefully explained. A further study might find the approximate solution of the water profile for a K−slope canal. The challenge would be found in writing the code that considers independently K−1 interactions to later compute a final water profile. All physical concepts could be extracted from this document as the problem was dealt as general as possible. REFERENCES - Subramanya, K.(1982). Flow in Open Channels. Tata McGraw-Hill Publishing Company Limited, New Delhi - French,R. (1985). Open-Channel Hydraulics. McGraw-Hill Publishing, New York. - Jain, S. 2001. Open-Channel Flow. United States of America: John Wiley & Sons, Inc, New York. - Chow, V. (1959). Open-Channel Hydraulics. McGraw-Hill Book Company Limited, New York. - Jeppson, R. (2011). Open Channel Flow. CRC Press, USA. - Montes, S. (1998). Hydraulics of Open Channel Flow. ASCE Press, USA. - Smith,K (1867). Control Point in lateral spillway cannel. ASCE Press, USA. - May R. et al.(2003). Hydraulic designs of side weris. Thomas Telford, London. APPENDICES APPENDIX 1. Control Point computation. In this appendix the intersection of Transitional Profile and Critical Depth Line is described, this computations using Matlab require a careful look. Matlab doesn’t deal with explicit expressions, there are two ways to find the intersection between this two functions: by plotting them discretely and finding an approximation of the intersection point (,) or by finding the two adjacent points between which the intersection occurs and computing the intersection point as a point between the two adjacent points The second method is used in the numerical code written. Note that one of the equations dealt with, equation [xTP] expresses  = (). For this reason the discretization is considered using ∆ steps. Defining a certain ∆, two vectors   and   are computed, which correspond to the value of  equations [xTP] and [CD] for different values of . A  vector is created with all the values of  where the computation is carried out. The Difference vector () =   ()−  () is computed as well as the product vector: ()= ()·( +1), where  denotes a generic component of  vector. Note that the relative position of the two functions is not know in advance, in order to keep generality, the sign of the product of the differences is the determinant value in opposition of the value of the difference itself. The first  where this difference changes its sign, i.e. the first negative coordinate of  vector, corresponds to   . The intersection has already occurred. Figure below illustrates the situation described. Then two new variables are defined:   =  and   =  The   is defined as   = ! "#$ ! %&' ( ) ,   (  ) and   (  ) are then computed. Then   =   (  )−  (  ) is computed. - If the absolute value of   < + ,being + a certain tolerance previously defined, the computations are stop and it is considered the intersection to be found at   . In this paper it is used + = 0.05/ - If   ·  < 0, then   =  , since the intersection is found between   and   . - If   ·  > 0, then   =  , since the intersection is found between   and   . In the case where   ≥ + a iteration process is started using the new definition of   or   until the condition is fulfilled. APPENDIX 2. “U” Shaped Area Computations. The Area and the Width are computed for an “U” shaped canal. The cross-section of this canal can be analyzed as a rectangle plus a semi circumference. Area Computation For  < 2 : The main challenge to compute the area of semicircular bottom relies on expressing the area in terms of , the water depth. First the area is expressed in terms of 3. From figure below and considering that the area of the flow section is the area of the section minus the area of the triangular portion: 4 = )2)·23− )22·673 ·28963 The previous equation can be rewritten in terms of  because: 8963 = :;! ;< which lead to 3 = =>8896:;! ;< Finally: 4 =  )2):2·=>8896:;! ;<−sin(2·=>8896:;! ;<)< For  ≥ 2: The area can be simply expressed as: 4 = B;C )+22 ·(−2) Width Computation For  < 2 : The width (D) can be computed in terms of 3 : D = 22 ·673 Then: D = 22 ·sin(=>8896:;! ;<) For  ≥ 2: The width is: D = 22 APPENDIX 3. Main instructions of the numerical code. APPENDIX 4. Control Point Locations and Froude Number for Two Slope Cases. Case 1 Case 2