Chapter 4. Non‐Uniform Flow in Open Channel

BFC21103 Hydraulics Chapter 4. Non‐Uniform Flow in Open Channel Tan Lai Wai, Wan Afnizan & Zarina Md Ali [email protected] February 2015 Learning Outcomes At the end of this chapter, students should be able to: i. Analyse the characteristics of hydraulic jump (rapidly‐varied flow) based on momentum equation ii. Analyse the characteristics of gradually‐varied flow BFC21103 Hydraulics Tan et al. ([email protected]) 4.1 Rapidly‐Varied Flow Occurs when the depth of flow change rapidly within short distance, e.g. hydraulic jump. Hydraulic jump occurs when supercritical flow changes suddenly to subcritical flow within a short distance. hydraulic jump y2 subcritical yc supercritical y1 1 2 BFC21103 Hydraulics Tan et al. ([email protected]) Datum 4.2 Hydraulic Jump Hydraulic jump only occurs if the upstream flow is supercritical, i.e. y1 < yc, and the downstream flow becomes subcritical flow, i.e. y2 > yc > y1 where, y1 = depth of flow just before the jump y2 = depth of flow just after the jump y1 and y2 are known as conjugate depths BFC21103 Hydraulics Tan et al. ([email protected]) Hydraulic jump in the laboratory flume BFC21103 Hydraulics Tan et al. ([email protected]) Hydraulic jump at the toe of spillway ‐ Itaipu dam, Brazil BFC21103 Hydraulics Tan et al. ([email protected]) Hydraulic jump downstream of sluice gate ‐ Harran canal, Turkey BFC21103 Hydraulics Tan et al. ([email protected]) Waves hitting sea wall in Depoe bay, Oregon U.S. BFC21103 Hydraulics Tan et al. ([email protected]) Surge waves due to fast flowing flood in Tangjiasan, China BFC21103 Hydraulics Tan et al. ([email protected]) Applications of Hydraulic Jump i. Energy dissipator i.e. reduce velocity and prevent erosion ii. Raise the water level for irrigation or water distribution purposes iii. Increase weight on apron by raising the depth of water to prevent uplift pressure iv. Mix chemical substance e.g. in water treatment process v. Aeration of flow, i.e. increase DO BFC21103 Hydraulics Tan et al. ([email protected]) Types of Jump Based on the Froude number before the jump Fr1 Fr1 = 1.0 − 1.7 → undular jump Fr1 = 1.7 − 2.5 → weak jump Fr1 = 4.5 − 9.0 → steady jump Fr1 = 2.5 − 4.5 Fr1 > 9.0 → oscillating jump → strong jump BFC21103 Hydraulics Tan et al. ([email protected]) Fr1 = 1.6 Energy dissipation = 45% to 70% BFC21103 Hydraulics Energy Tan et al. ([email protected]) dissipation up to 85% Momentum Equation Consider a hydraulic jump on a frictionless flat bed within a rectangular channel, F4 F1 y1 1 F1 y2 W F3 ∑ F = M2 − M1 2 F1 − F2 − F3 + F4 = ρQV2 − ρQV1 Since friction = 0 → F3 = 0 and flat bed F4 = Wsinθ = 0 BFC21103 Hydraulics Tan et al. ([email protected]) F1 − F2 = ρQV2 − ρQV1 y1 y2 ρgA1 − ρgA2 = ρQV2 − ρQV1 2 2 Dividing by ρgB, Since V1 = 1 2 1 2 qV2 qV1 y1 − y 2 = − g g 2 2 q q , and V2 = y2 y1 1 2 1 2 q2 q2 y1 − y 2 = − 2 2 gy2 gy1 q2 1 2 q2 1 2 + y1 = + y2 gy2 2 gy1 2 BFC21103 Hydraulics Tan et al. ([email protected]) Activity 4.1 Using the momentum force equation, draw the specific force curve if a hydraulic jump occurs within a rectangular channel with the discharge per unit width is 25 ft3/s. Given q = 25 ft3/s flows in a rectangular channel q2 1 2 + y Specific force is given as F = gy 2 Momentum Hydrostatic pressure flux BFC21103 Hydraulics Tan et al. ([email protected]) y (ft) F (ft2) 1.0 19.910 1.2 16.895 1.4 14.844 1.6 13.411 1.8 12.403 2.0 11.705 2.2 11.243 2.4 10.967 2.6 10.845 2.8 10.852 3.0 10.970 3.2 11.186 3.4 11.489 3.6 11.872 3.8 12.328 4.0 12.852 4.2 13.441 4.4 14.091 4.6 14.800 4.8 15.564 5.0 16.382 5.2 17.253 5.4 18.174 5.6 19.146 5.8 20.167 y (ft) 6 5 Subcritical flow Fr < 1 4 yc 3 2 1 Supercritical flow Fr > 1 0 0 5 10 Fmin F (ft2) 15 20 y1 and y2 with the same F are conjugate depths BFC21103 Hydraulics Tan et al. ([email protected]) Conjugate Depths Equation From the momentum equation of flow in a rectangular channel, q2 1 2 q2 1 2 + y1 = + y2 gy2 2 gy1 2 Rearranging, 2 2 2 q 2 q y22 − y12 = − gy1 gy2 2q 2 ⎛ y2 − y1 ⎞ (y2 − y1 )(y1 + y2 ) = ⎜⎜ ⎟⎟ g ⎝ y1 y 2 ⎠ 2q 2 (y1 + y2 )y1y2 = g 2 q It can be seen that Fr can be introduced since Fr2 = 3 gy BFC21103 Hydraulics Tan et al. ([email protected]) y2 2q 2 ( y1 + y 2 ) 2 = 3 y1 gy1 Division by y13 , y22 y2 2 + = 2 Fr 1 y12 y1 Note that solving y y2 + − 2Fr12 = 0 y y1 ax 2 + bx + c = 0 2 2 2 1 Solving for y2 gives y1 − b ± b2 − 4ac gives x = 2a 1 + 12 − 4(1)(− 2Fr12 ) y2 =− y1 2(1) ( y2 1 = − 1 + 1 + 8Fr12 y1 2 ) since y1 and y2 are positive values BFC21103 Hydraulics Tan et al. ([email protected]) Else if division is made by y23 , 1 + 1 + 8Fr22 y1 =− y2 2 y2 y1 > 1 or Note that for hydraulic jump to occur, <1 y1 y2 BFC21103 Hydraulics Tan et al. ([email protected]) Energy Loss There will be considerable loss of energy in hydraulic jump between sections 1 and 2 V12 2g EL V22 2g hydraulic jump y2 subcritical yc supercritical y1 1 2 BFC21103 Hydraulics Tan et al. ([email protected]) Datum Eo EGL Energy loss is calculated as E L = E1 − E 2 V12 ⎞ ⎛ V22 ⎞ ⎛ E L = ⎜ y1 + ⎟ − ⎜ y 2 + ⎟ 2g ⎠ ⎝ 2g ⎠ ⎝ For rectangular channel, it can be simplified as ⎛ q2 ⎞ ⎛ q2 ⎞ EL = ⎜⎜ y1 + ⎟ − ⎜⎜ y2 + ⎟ 2 ⎟ 2 ⎟ 2gy1 ⎠ ⎝ 2gy2 ⎠ ⎝ 1 q2 ⎛ 1 1 ⎞ ⎜⎜ 2 − 2 ⎟⎟ E L = ( y1 − y 2 ) + 2 g ⎝ y1 y 2 ⎠ 1 q 2 ⎛ y22 − y12 ⎞ E L = ( y1 − y 2 ) + ⎜⎜ 2 2 ⎟⎟ 2 g ⎝ y1 y 2 ⎠ BFC21103 Hydraulics Tan et al. ([email protected]) q2 1 Substituting = (y1 + y2 )y1 y2 g 2 ⎛ y22 − y12 ⎞ 1 ⎡1 ⎤ E L = ( y1 − y 2 ) + y1 y2 (y1 + y2 ) ⎜⎜ 2 2 ⎟⎟ ⎥⎦⎝ y y ⎠ ⎢ 2 ⎣2 1 2 4 y1 y2 (y1 − y2 ) + y1 y22 − y13 + y23 − y12 y2 EL = 4 y1 y 2 y23 + 3y12 y2 − 3y1 y22 − y13 EL = 4 y1 y 2 EL = (y2 − y1 )3 4 y1 y 2 which is expressed in meter BFC21103 Hydraulics Tan et al. ([email protected]) Power due to Energy Loss Power due to energy loss in unit Watt is given as PL = ρgQE L Height of Jump The height of jump is given as H j = y 2 − y1 BFC21103 Hydraulics Tan et al. ([email protected]) Length of Jump Based on Froude number upstream of the jump Fr1, Lj = 6.9(y2 − y1 ) for Fr1 ≤ 5.0 Lj = 6.1y2 for Fr1 > 5.0 BFC21103 Hydraulics Tan et al. ([email protected]) Activity 4.2 A spillway discharges flow at a rate of 7.75 m3/s/m. At the downstream horizontal apron, the depth of flow was found to be 0.5 m. What tailwater depth is needed to form a hydraulic jump? If a jump is formed, find its (i) type; (ii) length; (iii) head loss; and (iv) energy loss as a percentage of the initial energy. BFC21103 Hydraulics Tan et al. ([email protected]) Given q = 7.75 m3/s/m, y1 = 0.5 m q 7.75 = = 6.999 Fr1 = 3 3 gy1 9.81 × 0.5 ( Utilizing the conjugate depths equation, y2 1 = − 1 + 1 + 8Fr12 y1 2 ( ) 0.5 y2 = − 1 + 1 + 8 × 6.9992 2 y2 = 4.705 m (i) ) Based on the Fr1 = 6.999, the jump is a steady jump (4.5 < Fr1 < 9.0) BFC21103 Hydraulics Tan et al. ([email protected]) (ii) Since Fr1 = 6.999 > 5.0 Length of jump Lj = 6.1y2 Lj = 6.1 × 4.705 Lj = 28.70 m (iii) Head loss is given as EL = EL = (y2 − y1 )3 4 y1 y 2 (4.705 − 0.5)3 4 × 0.5 × 4.705 EL = 7.901 m BFC21103 Hydraulics Tan et al. ([email protected]) q2 (iv) Initial total energy is Eo = y1 + 2gy12 7.752 Eo = 0.5 + 2 × 9.81 × 0.52 Eo = 12.745 m EL 7.901 Percentage of energy loss = × 100% = 61.99% Eo 12.745 BFC21103 Hydraulics Tan et al. ([email protected]) Activity 4.3 A 25‐m wide spillway is discharging flow with velocity of 30 m/s at a depth of 1 m. Hydraulic jump occurs immediately downstream. Find the height of the jump and power loss due to the jump. Given B = 25 m, y1 = 1 m, V1 = 30 m/s 30 V1 = = 9.578 Fr1 = 9.81 × 1 gy1 Conjugate depths equation, ( y2 1 = − 1 + 1 + 8Fr12 y1 2 ( ) 1 y2 = − 1 + 1 + 8 × 9.5782 2 y2 = 13.055 m BFC21103 Hydraulics Tan et al. ([email protected]) ) (i) Height of jump H j = y2 − y1 H j = 13.055 − 1 H j = 12.055 m (ii) Energy loss EL = EL = (y2 − y1 )3 (12.055 − 1)3 4 y1 y 2 4 × 1 × 12.055 EL = 28.019 m Power due to energy loss PL = ρgQE L PL = 9810 × (25 × 1 × 30 ) × 28.019 PL = 206.15 MW BFC21103 Hydraulics Tan et al. ([email protected]) 4.3 Gradually‐Varied Flow A steady non‐uniform flow in a prismatic channel with gradual changes in its flow surface elevation. Examples: (i) Drawdown produced by sudden change in channel bed slope M2 control section S2 yo yc Mild slope Stee p sl ope Computations C omp u ta tion s BFC21103 Hydraulics Tan et al. ([email protected]) yo (ii) Backwater produced by increased in bed elevation M1 control section 1 control section 2 yo1 yo2 Mi l d s l o pe yc Computations Milder slope Computations BFC21103 Hydraulics Tan et al. ([email protected]) Lake Types of Slope yo So So < Sc yo > yc yo < yc yo = yc yo = ∞ yo = ∞ So > Sc or So = Sc So = 0 So < 0 → Type of slope Symbol Mild M → Steep S Critical C Horizontal H Adverse A → → → BFC21103 Hydraulics Tan et al. ([email protected]) Classification of GVF Profile Channel Mild slope Steep slope Critical slope Horizontal bed Adverse slope Region Condition Type 1 y > yo > yc M1 2 yo > y > yc M2 3 yo > yc > y M3 1 y > yc > yo S1 2 yc > y > yo S2 3 yc > yo > y S3 1 y > yo = yc C1 3 y < yo = yc C3 2 y > yc H2 3 y < yc H3 2 y > yc A2 3 y < yc A3 Classification of GVF Profile Slope Region 1 Region 2 M1 Mild M M2 yo yo yc yc y > yo > yc Region 3 yo yc yo > y > yc M3 yo > yc > y S1 yc yc Steep S S2 yo yo y > yc > yo Critical C yc yo = y yc > y > yo C1 c y > yo = yc − BFC21103 Hydraulics Tan et al. ([email protected]) yo S3 yc > yo > y yo = y C3 c yo = yc > y Slope Horizontal H Region 1 − Region 2 Region 3 H2 yc yc y > yc H2 yc > y A2 Adverse A − yc y > yc yc All curves in region 1 have positive slopes (backwater curves) All curves in region 2 have negative slopes (drawdown curves) BFC21103 Hydraulics Tan et al. ([email protected]) yc > y BFC21103 Hydraulics Tan et al. ([email protected]) Occurrence of Flow Profile (a) i. M1 profile Occurs due to obstruction to subcritical flow, e.g. weir, dam or other control structures. The profile extends to several kilometres upstream before approaching the normal depth. y > yo > yc M1 yo yc Mild slope BFC21103 Hydraulics Tan et al. ([email protected]) (a) ii. M2 profile Occurs when there is a sudden drop in the bottom of the channel, constriction of channel or channel outlet into reservoir. yo > y > yc M2 yo yc Mild slope BFC21103 Hydraulics Tan et al. ([email protected]) (a) iii. M3 profile Occurs when supercritical flow enters a mild slope channel, e.g. flow from a spillway or a sluice gate to a mild channel. yo > yc > y yo M3 yc Mild slope BFC21103 Hydraulics Tan et al. ([email protected]) (b) i. S1 profile Occurs when supercritical flow changes to pool of water (subcritical flow) due to obstruction such as weir or dam. y > yc > yo S1 yc yo Steep slope BFC21103 Hydraulics Tan et al. ([email protected]) (b) ii. S2 profile Occurs when flow from reservoir enter a steep slope or when there is a change from mild slope to steep slope. This profile is of shorter length. yc > y > yo yc yo S2 Steep slope BFC21103 Hydraulics Tan et al. ([email protected]) (b) iii. S3 profile Occurs when flow from reservoir enter a steep slope or when there is a change from mild slope to steep slope. This profile is of shorter length. yc > yo > y Ste S3 Steep slope yo yc ep e rs S3 lop e BFC21103 Hydraulics Tan et al. ([email protected]) Steep slope yo yc (c) C1 and C3 profiles Highly unstable and rarely occur, y > yo = yc and yo = yc > y (d) H2 and H3 profiles Occurs when the bed of mild slope becomes flatter. There is no region 1 since yo = ∞. y > yc H2 yc > y H3 yc Horizontal bed BFC21103 Hydraulics Tan et al. ([email protected]) Drop (e) A2 and A3 profiles Occurs when flow is on adverse slope, which is rare. These profiles occurs within a short length. yc > y A3 y > yc A2 yc Pool Drop Adverse slope BFC21103 Hydraulics Tan et al. ([email protected]) Activity 4.4 Determine the type of profile for the following flow. Sluice gate Sluice gate yc yo yo (a) yc (b) BFC21103 Hydraulics Tan et al. ([email protected]) yc > yo → S yo > yc → M Zone 3 → S3 Zone 3 → M3 Zone 1 → M1 Zone 1 → S1 S1 Sluice gate M1 S3 Sluice gate yc M3 yo (a) (b) BFC21103 Hydraulics Tan et al. ([email protected]) yo yc Activity 4.5 A rectangular channel with bottom width 4 m and bottom slope 0.0008 has discharge of 1.5 m3/s. Along the gradually‐varied flow in the channel, the depth at a section is found to be 0.3 m. Assuming Manning n = 0.016, determine the type of GVF profile. BFC21103 Hydraulics Tan et al. ([email protected]) Rectangular section, B = 4 m, So = 0.0008, n = 0.016, Q = 1.5 m3/s, y = 0.3 m. AR = 2 3 Qn S 1 2 o ⎛ Byo ⎞ 1.5 × 0.016 ⎟⎟ = (Byo )⎜⎜ 1 B + y 2 ⎝ o ⎠ 0.00082 2 3 ⎞ ⎛ (4 yo )⎜⎜ 4 yo ⎟⎟ = 0.8485 ⎝ 4 + 2y o ⎠ ⎛Q ⎞ y c = ⎜⎜ 2 ⎟⎟ ⎝B g⎠ 2 2 3 yo = 0.4261 m Since yo > y > y c → M2 BFC21103 Hydraulics Tan et al. ([email protected]) 1 3 ⎛ 1.5 ⎞ y c = ⎜⎜ 2 ⎟⎟ ⎝ 4 × 9.81 ⎠ 2 y c = 0.2429 m 1 3 Activity 4.6 A triangular channel has side slope 1(H):1(V), bed slope 0.001, and Manning roughness n = 0.015. If rate of flow is 0.2 m3/s (a) Determine the type of slope, and (b) Give the limit of depths of flow in regions 1, 2, and 3. BFC21103 Hydraulics Tan et al. ([email protected]) Triangular section, z = 1, So = 0.001, n = 0.015, Q = 0.2 m3/s AR = 2 3 Ac3 Q2 = Tc g Qn S (zy ) 1 2 o 2 ⎛ ⎞ 0.2 × 0.015 zy 2 o ⎜ ⎟ = (zyo )⎜ 1 2 ⎟ y + z 2 1 ⎝ o ⎠ 0.0012 0.22 = 2zy c 9.81 2 3 c 2 3 y c5 = 0.008155 ⎛ y ⎞ ⎟⎟ = 0.09487 (y )⎜⎜ ⎝ 2 2 yo ⎠ 2 o 2 o 2 3 y c = 0.3822 m y = 0.1897 8 3 o yo = 0.5361 m Since yo > y c → mild slope M BFC21103 Hydraulics Tan et al. ([email protected]) For M1 : y > 0.5361 m For M2 : 0.5361 m > y > 0.3822 m For M3 : y < 0.3822 m BFC21103 Hydraulics Tan et al. ([email protected]) Analysis of GVF Profile Two basic assumptions are involved in the analysis of GVF: 1. The pressure distribution at any section is hydrostatic. 2. The resistance to flow at any depth can be assumed using uniform‐flow equation, such as the Manning's equation, with the condition that the slope term to be used in the equation is the energy slope and not the bed slope. Thus, if in a GVF the depth of flow at any section is y, the energy slope Sf is: Sf = n2V 2 R 4 3 where R is the hydraulic radius of the section at depth y. BFC21103 Hydraulics Tan et al. ([email protected]) Differential Equation of GVF The total energy H of a gradually‐varied flow in a channel of small slope is: V2 H =z+y+ 2g V2 where the specific energy E = y + 2g V2 2g E line S f Wate y z Datum Energy So Schematic sketch of GVF BFC21103 Hydraulics Tan et al. ([email protected]) r su r face x Since the water surface varies in the longitudinal x‐direction, the depth of the flow and the total energy are functions of x. Differentiating total energy with respect to x, dH dz dy d ⎛ V 2 ⎞ = + + ⎜⎜ ⎟⎟ dx dx dx dx ⎝ 2g ⎠ Energy slope dH = −S f dx Bottom slope dZ = − So dx water‐surface slope relative to the channel bottom d ⎛ V 2 ⎞ d ⎛ Q 2 ⎞ dy ⎟ ⎜⎜ ⎟⎟ = ⎜⎜ Velocity term 2⎟ dx ⎝ 2g ⎠ dx ⎝ 2gA ⎠ dx Q 2 dA dy =− 3 gA dy dx BFC21103 Hydraulics Tan et al. ([email protected]) Since dA =T dy d ⎛ V 2 ⎞ Q2T dy ⎜⎜ ⎟⎟ = 3 dx ⎝ 2g ⎠ gA dx Differentiated energy equation can now be rewritten as dy ⎛ Q2T ⎞ dy − S f = − So + − ⎜⎜ 3 ⎟⎟ dx ⎝ gA ⎠ dx dy S o − S f = Rearranging, Q2T dx 1− 3 gA Dynamic equation of GVF BFC21103 Hydraulics Tan et al. ([email protected]) Other forms of dynamic equation of GVF (a) If K = conveyance at any depth y and Ko = conveyance corresponding to the normal depth yo, then Q for GVF K= Sf Q Ko = for uniform flow So S f K o2 = 2 So K If Z = section factor at depth y and Zc = section factor at the critical depth yc, 3 A Z2 = T Ac3 Q2 2 = and Z c = Tc g Q2T Z c2 Hence = 2 3 gA Z BFC21103 Hydraulics Tan et al. ([email protected]) Substituting into the GVF dynamic equation Sf ⎞ ⎛ ⎟ ⎜ 1− So ⎟ dy = So ⎜ ⎜ Q2T ⎟ dx ⎜1− 3 ⎟ ⎝ gA ⎠ ⎡ ⎛ K o ⎞2 ⎤ 1−⎜ ⎟ ⎥ ⎢ dy = So ⎢ ⎝ K ⎠2 ⎥ dx ⎢1 − ⎛ Z c ⎞ ⎥ ⎢⎣ ⎜⎝ Z ⎟⎠ ⎥⎦ This equation is useful in developing direct integration techniques. BFC21103 Hydraulics Tan et al. ([email protected]) (b) If Qo represents the normal discharge at a depth yo and Qc denotes the critical discharge at the same depth y, Qo = K So and Qc = Z g Using these definitions, the GVF dynamic equation in (a) can be rewritten as ⎛Q⎞ 1 − ⎜⎜ ⎟⎟ Qn ⎠ dy ⎝ = So 2 dx ⎛Q⎞ 1 − ⎜⎜ ⎟⎟ ⎝ Qc ⎠ 2 BFC21103 Hydraulics Tan et al. ([email protected]) (c) Another form of the GVF dynamic equation is dE = So − S f dx This equation is called the differential‐energy equation of GVF to distinguish it from the other GVF differential equations. This energy equation is very useful in developing numerical techniques for the GVF profile computation. BFC21103 Hydraulics Tan et al. ([email protected]) Analysis of GVF Profile Among the importance are: (a) determination of the effect of hydraulic structure to the flow; (b) inundation due to dam or weir construction; and (c) estimation of flood area. This course only considers the following methods: (a) Direct integration; (b) Numerical integration; and (c) Direct step. BFC21103 Hydraulics Tan et al. ([email protected]) Calculation of GVF Profile Gradually‐varied flow surface y1 Δx1 Changes in depth of flow y1+Δy1 y1+Δy2 Δx2 L yN+1 ΔxN dy can be calculated if: dx (a) y1 and yN+1 are known, or (b) L is known BFC21103 Hydraulics Tan et al. ([email protected]) A. Direct Integration Between two sections (x1, y1) and (x2, y2), M ⎫⎪ ⎧ ⎛ yc ⎞ J yo ⎪ [F (v2 , J ) − F (v1 , J )]⎬ x2 − x1 = ⎨(u2 − u1 ) − [F (u2 , N ) − F (u1 , N )] + ⎜⎜ ⎟⎟ So ⎪⎩ ⎪⎭ ⎝ yo ⎠ M where, y u= yo v =u N J N J= (N − M + 1) M, N = hydraulic exponents F(u, N) = varied‐flow function F(v, J) = same function as F(u, N) BFC21103 Hydraulics Tan et al. ([email protected]) y⎛ A dT ⎞ M = ⎜ 3T − ⎟ A⎝ T dy ⎠ 2y ⎛ dP ⎞ N = ⎜ 5T − 2R ⎟ 3A ⎝ dy ⎠ Section M N Rectangular 3 2 to 3.333 Trapezoidal 3 to 5 2 to 5.333 5 5.333 Triangular For trapezoidal channels, y⎞ y ⎛ 3⎜ 1 + 2z ⎟ 2z B⎠ − B M= ⎝ ⎛ 1 + z y ⎞ ⎛ 1 + 2z y ⎞ ⎜ ⎟ ⎟ ⎜ B⎠ ⎝ B⎠ ⎝ ⎛ y ⎞ 1 + z2 ⎛ 1 + 2z y ⎞ ⎟ 8 ⎜ ⎟ 10 ⎜⎝ B⎠ − ⎝B⎠ N= 3 ⎛1 + z y ⎞ 3 ⎡ ⎛ y ⎞ 1 + z2 ⎤ 1 2 + ⎜ ⎟ ⎜ ⎟ ⎥⎦ ⎢ B⎠ ⎝ ⎝B⎠ ⎣ BFC21103 Hydraulics Tan et al. ([email protected]) y 1 B y y and Do B 0. 5 z= z= 1.0 0.8 0.6 0.5 0.4 0.3 z = 0 (rectangular) 2.0 1 z z = 2.5 z=3 z=4 Circular z= z = 1.5 2 6.0 5.0 4.0 3.0 0.2 Do y 0.1 0.08 0.06 0.05 0.04 0.03 0.02 2.5 3.0 3.5 4.0 M 4.5 BFC21103 Hydraulics Tan et al. ([email protected]) 5.0 5.5 1.0 0.8 0.6 y y 0.5 0.4 and Do 0.3 B y 5 1. 5 z . 0 = 1 B z= z z= 1 ) ular tang (rec 2.0 z=0 6.0 5.0 4.0 3.0 Circula r z=2 z = 2.5 z=3 z=4 0.2 Do y 0.1 0.08 0.06 0.05 0.04 0.03 0.02 2.0 2.5 3.0 3.5 N 4.0 4.5 BFC21103 Hydraulics Tan et al. ([email protected]) 5.0 5.5 Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.201 0.221 0.241 0.262 0.282 0.200 0.221 0.241 0.261 0.282 0.200 0.220 0.241 0.261 0.281 0.200 0.220 0.240 0.261 0.281 0.200 0.220 0.240 0.260 0.281 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.303 0.324 0.344 0.366 0.387 0.302 0.323 0.343 0.364 0.385 0.302 0.322 0.343 0.363 0.384 0.301 0.322 0.342 0.363 0.383 0.301 0.321 0.342 0.362 0.383 0.301 0.321 0.341 0.362 0.382 0.300 0.321 0.341 0.361 0.382 u 0.00 0.02 0.04 0.06 0.08 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.10 0.12 0.14 0.16 0.18 0.100 0.120 0.141 0.161 0.181 0.100 0.120 0.140 0.161 0.181 0.100 0.120 0.140 0.160 0.181 0.100 0.120 0.140 0.160 0.180 0.20 0.22 0.24 0.26 0.28 0.202 0.223 0.243 0.264 0.286 0.201 0.222 0.242 0.263 0.284 0.201 0.221 0.242 0.262 0.283 0.30 0.32 0.34 0.36 0.38 0.307 0.329 0.350 0.373 0.395 0.305 0.326 0.348 0.370 0.392 0.304 0.325 0.346 0.367 0.389 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 0.405 0.426 0.448 0.470 0.492 0.404 0.425 0.446 0.468 0.489 0.403 0.424 0.445 0.466 0.488 0.403 0.423 0.444 0.465 0.486 0.402 0.423 0.443 0.464 0.485 0.517 0.540 0.563 0.587 0.612 0.514 0.536 0.559 0.583 0.607 0.511 0.534 0.556 0.579 0.603 0.509 0.531 0.554 0.576 0.599 0.508 0.529 0.551 0.574 0.596 0.506 0.528 0.550 0.572 0.594 0.644 0.657 0.671 0.684 0.698 0.637 0.650 0.663 0.676 0.690 0.631 0.644 0.657 0.669 0.683 0.627 0.639 0.651 0.664 0.677 0.623 0.635 0.647 0.659 0.672 0.620 0.631 0.643 0.655 0.667 0.617 0.628 0.640 0.652 0.664 0.712 0.727 0.742 0.757 0.772 0.703 0.717 0.731 0.746 0.761 0.696 0.709 0.723 0.737 0.751 0.689 0.703 0.716 0.729 0.743 0.684 0.697 0.710 0.723 0.737 0.680 0.692 0.705 0.718 0.731 0.676 0.688 0.701 0.713 0.726 u 0.40 0.42 0.44 0.46 0.48 0.418 0.441 0.465 0.489 0.514 0.414 0.437 0.460 0.483 0.507 0.411 0.433 0.456 0.478 0.502 0.408 0.430 0.452 0.475 0.497 0.407 0.428 0.450 0.472 0.494 0.50 0.52 0.54 0.56 0.58 0.539 0.565 0.592 0.619 0.647 0.531 0.556 0.582 0.608 0.635 0.525 0.550 0.574 0.600 0.626 0.521 0.544 0.568 0.593 0.618 0.60 0.61 0.62 0.63 0.64 0.676 0.691 0.707 0.722 0.738 0.663 0.677 0.692 0.707 0.722 0.653 0.666 0.680 0.694 0.709 0.65 0.66 0.67 0.68 0.69 0.754 0.771 0.787 0.805 0.822 0.737 0.753 0.769 0.785 0.802 0.724 0.739 0.754 0.769 0.785 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 0.766 0.781 0.796 0.811 0.827 0.757 0.772 0.786 0.802 0.817 0.750 0.764 0.779 0.793 0.808 0.744 0.758 0.772 0.786 0.800 0.739 0.752 0.766 0.780 0.794 0.857 0.874 0.892 0.911 0.930 0.844 0.861 0.878 0.896 0.914 0.833 0.849 0.866 0.883 0.901 0.823 0.839 0.855 0.872 0.889 0.815 0.830 0.846 0.862 0.879 0.808 0.823 0.838 0.854 0.870 0.970 0.992 1.015 1.039 1.064 0.950 0.971 0.993 1.016 1.040 0.934 0.954 0.974 0.996 1.019 0.919 0.938 0.958 0.979 1.001 0.907 0.925 0.945 0.965 0.985 0.896 0.914 0.932 0.952 0.972 0.887 0.904 0.922 0.940 0.960 1.091 1.119 1.149 1.181 1.216 1.065 1.092 1.120 1.151 1.183 1.043 1.068 1.095 1.124 1.155 1.024 1.048 1.074 1.101 1.131 1.007 1.031 1.055 1.081 1.110 0.993 1.015 1.039 1.064 1.091 0.980 1.002 1.025 1.049 1.075 u 0.70 0.71 0.72 0.73 0.74 0.841 0.859 0.878 0.898 0.918 0.819 0.837 0.855 0.874 0.893 0.802 0.819 0.836 0.853 0.871 0.787 0.804 0.820 0.837 0.854 0.776 0.791 0.807 0.823 0.840 0.75 0.76 0.77 0.78 0.79 0.939 0.961 0.984 1.007 1.031 0.912 0.933 0.954 0.976 0.998 0.890 0.909 0.929 0.950 0.971 0.872 0.890 0.909 0.929 0.949 0.80 0.81 0.82 0.83 0.84 1.056 1.083 1.110 1.139 1.170 1.022 1.047 1.072 1.099 1.128 0.994 1.017 1.041 1.067 1.093 0.85 0.86 0.87 0.88 0.89 1.202 1.236 1.273 1.312 1.355 1.158 1.190 1.224 1.260 1.300 1.122 1.151 1.183 1.217 1.254 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 1.189 1.225 1.266 1.311 1.363 1.163 1.197 1.236 1.279 1.328 1.140 1.173 1.210 1.251 1.297 1.120 1.152 1.187 1.226 1.270 1.103 1.133 1.166 1.204 1.246 1.467 1.545 1.644 1.707 1.783 1.423 1.497 1.590 1.649 1.720 1.385 1.454 1.543 1.598 1.666 1.352 1.417 1.501 1.553 1.617 1.322 1.385 1.464 1.514 1.575 1.296 1.355 1.431 1.479 1.536 1.959 2.106 2.355 2.931 ∞ 1.880 2.017 2.250 2.788 ∞ 1.812 1.940 2.159 2.663 ∞ 1.752 1.873 2.079 2.554 ∞ 1.699 1.814 2.008 2.457 ∞ 1.652 1.761 1.945 2.370 ∞ 1.610 1.714 1.889 2.293 ∞ 2.399 1.818 1.572 1.428 1.327 2.184 1.649 1.419 1.286 1.191 2.008 1.506 1.291 1.166 1.078 1.856 1.384 1.182 1.065 0.982 1.725 1.279 1.089 0.978 0.900 1.610 1.188 1.007 0.902 0.828 1.508 1.107 0.936 0.836 0.766 u 0.90 0.91 0.92 0.93 0.94 1.401 1.452 1.508 1.572 1.645 1.343 1.390 1.442 1.500 1.568 1.294 1.338 1.386 1.441 1.503 1.253 1.294 1.340 1.391 1.449 1.218 1.257 1.300 1.348 1.403 0.950 0.960 0.970 0.975 0.980 1.730 1.834 1.968 2.052 2.155 1.647 1.743 1.865 1.943 2.040 1.577 1.666 1.780 1.851 1.936 1.518 1.601 1.707 1.773 1.855 0.985 0.990 0.995 0.999 1.000 2.294 2.477 2.792 3.523 ∞ 2.165 2.333 2.621 3.292 ∞ 2.056 2.212 2.478 3.097 ∞ 1.001 1.005 1.010 1.015 1.020 3.317 2.587 2.273 2.090 1.961 2.931 2.272 1.984 1.817 1.698 2.640 2.021 1.756 1.602 1.493 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 0.955 0.868 0.802 0.748 0.703 0.866 0.785 0.723 0.672 0.630 0.790 0.714 0.656 0.608 0.569 0.725 0.653 0.598 0.553 0.516 0.668 0.600 0.548 0.506 0.471 0.749 0.713 0.681 0.652 0.626 0.665 0.631 0.601 0.575 0.551 0.595 0.563 0.536 0.511 0.488 0.535 0.506 0.480 0.457 0.436 0.485 0.457 0.433 0.411 0.392 0.441 0.415 0.392 0.372 0.354 0.693 0.669 0.647 0.627 0.608 0.602 0.581 0.561 0.542 0.525 0.529 0.509 0.490 0.473 0.458 0.468 0.450 0.432 0.417 0.402 0.417 0.400 0.384 0.369 0.355 0.374 0.358 0.343 0.329 0.316 0.337 0.322 0.308 0.295 0.283 0.591 0.574 0.559 0.531 0.505 0.509 0.494 0.480 0.454 0.431 0.443 0.429 0.416 0.392 0.371 0.388 0.375 0.363 0.341 0.322 0.343 0.331 0.320 0.299 0.281 0.305 0.294 0.283 0.264 0.248 0.272 0.262 0.252 0.235 0.219 u 1.03 1.04 1.05 1.06 1.07 1.779 1.651 1.552 1.472 1.405 1.532 1.415 1.325 1.252 1.191 1.340 1.232 1.149 1.082 1.026 1.186 1.086 1.010 0.947 0.895 1.060 0.967 0.896 0.838 0.790 1.08 1.09 1.10 1.11 1.12 1.346 1.296 1.250 1.210 1.173 1.138 1.091 1.050 1.013 0.980 0.977 0.935 0.897 0.864 0.833 0.851 0.812 0.777 0.746 0.718 1.13 1.14 1.15 1.16 1.17 1.139 1.108 1.079 1.052 1.027 0.949 0.921 0.895 0.871 0.848 0.805 0.780 0.756 0.734 0.713 1.18 1.19 1.20 1.22 1.24 1.003 0.981 0.960 0.922 0.887 0.827 0.807 0.788 0.754 0.723 0.694 0.676 0.659 0.628 0.600 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 0.351 0.334 0.318 0.304 0.290 0.304 0.288 0.274 0.260 0.248 0.265 0.250 0.237 0.225 0.214 0.233 0.219 0.207 0.196 0.185 0.205 0.193 0.181 0.171 0.162 0.329 0.316 0.304 0.293 0.282 0.278 0.266 0.256 0.246 0.236 0.237 0.226 0.217 0.208 0.199 0.204 0.194 0.185 0.177 0.169 0.176 0.167 0.159 0.152 0.145 0.153 0.145 0.138 0.131 0.125 0.330 0.320 0.310 0.288 0.269 0.273 0.263 0.255 0.235 0.218 0.227 0.219 0.211 0.194 0.179 0.191 0.184 0.177 0.161 0.148 0.162 0.156 0.149 0.135 0.123 0.139 0.133 0.127 0.114 0.103 0.119 0.113 0.108 0.097 0.087 0.251 0.236 0.222 0.209 0.198 0.203 0.189 0.177 0.166 0.156 0.165 0.153 0.143 0.133 0.125 0.136 0.125 0.116 0.108 0.100 0.113 0.103 0.095 0.088 0.082 0.094 0.086 0.079 0.072 0.067 0.079 0.072 0.065 0.060 0.055 u 1.26 1.28 1.30 1.32 1.34 0.856 0.827 0.800 0.776 0.753 0.694 0.669 0.645 0.623 0.603 0.574 0.551 0.530 0.510 0.492 0.482 0.461 0.442 0.424 0.408 0.410 0.391 0.373 0.357 0.342 1.36 1.38 1.40 1.42 1.44 0.731 0.711 0.692 0.675 0.658 0.584 0.566 0.549 0.534 0.519 0.475 0.459 0.444 0.431 0.418 0.393 0.378 0.365 0.353 0.341 1.46 1.48 1.50 1.55 1.60 0.642 0.627 0.613 0.580 0.551 0.505 0.492 0.479 0.451 0.425 0.405 0.394 0.383 0.358 0.335 1.65 1.70 1.75 1.80 1.85 0.525 0.501 0.480 0.460 0.442 0.403 0.382 0.364 0.347 0.332 0.316 0.298 0.282 0.267 0.254 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 0.117 0.110 0.104 0.092 0.083 0.094 0.088 0.082 0.073 0.065 0.076 0.070 0.066 0.058 0.051 0.062 0.057 0.053 0.046 0.040 0.050 0.046 0.043 0.037 0.032 0.098 0.089 0.082 0.076 0.070 0.075 0.068 0.062 0.057 0.052 0.058 0.052 0.047 0.043 0.039 0.045 0.040 0.036 0.033 0.029 0.035 0.031 0.028 0.025 0.022 0.028 0.024 0.022 0.019 0.017 0.089 0.083 0.078 0.059 0.046 0.065 0.060 0.056 0.041 0.031 0.048 0.044 0.041 0.029 0.022 0.036 0.033 0.030 0.021 0.015 0.027 0.024 0.022 0.015 0.010 0.020 0.018 0.017 0.011 0.007 0.015 0.014 0.012 0.008 0.005 0.037 0.031 0.022 0.017 0.013 0.025 0.020 0.014 0.010 0.008 0.017 0.013 0.009 0.006 0.005 0.011 0.009 0.006 0.004 0.003 0.008 0.006 0.004 0.002 0.002 0.005 0.004 0.002 0.002 0.001 0.004 0.003 0.002 0.001 0.001 u 1.90 1.95 2.00 2.10 2.20 0.425 0.409 0.395 0.369 0.346 0.317 0.304 0.292 0.273 0.251 0.242 0.231 0.221 0.202 0.186 0.188 0.178 0.169 0.154 0.141 0.147 0.139 0.132 0.119 0.107 2.3 2.4 2.5 2.6 2.7 0.326 0.308 0.292 0.277 0.264 0.235 0.220 0.207 0.195 0.184 0.173 0.160 0.150 0.140 0.131 0.129 0.119 0.110 0.102 0.095 2.8 2.9 3.0 3.5 4.0 0.252 0.241 0.230 0.190 0.161 0.175 0.166 0.158 0.126 0.104 0.124 0.117 0.110 0.085 0.069 4.5 5.0 6.0 7.0 8.0 0.139 0.122 0.098 0.081 0.069 0.088 0.076 0.058 0.047 0.040 0.057 0.048 0.036 0.028 0.022 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 2.2 2.4 2.6 2.8 3.0 3.2 u 9.0 10.0 20.0 0.060 0.053 0.023 0.033 0.028 0.011 0.019 0.016 0.005 0.011 0.009 0.003 0.006 0.005 0.001 0.004 0.003 0.001 BFC21103 Hydraulics Tan et al. ([email protected]) 3.4 3.6 3.8 4.0 0.002 0.002 0.000 0.001 0.001 0.000 0.001 0.001 0.000 0.000 0.000 0.000 Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 u 0.00 0.02 0.04 0.06 0.08 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.10 0.12 0.14 0.16 0.18 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.20 0.22 0.24 0.26 0.28 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.30 0.32 0.34 0.36 0.38 0.300 0.321 0.341 0.361 0.381 0.300 0.320 0.340 0.361 0.381 0.300 0.320 0.340 0.360 0.381 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8 0.400 0.420 0.441 0.461 0.481 0.400 0.420 0.440 0.460 0.480 0.400 0.420 0.440 0.460 0.480 0.400 0.420 0.440 0.460 0.480 0.400 0.420 0.440 0.460 0.480 0.501 0.522 0.542 0.563 0.584 0.501 0.521 0.542 0.562 0.583 0.501 0.521 0.541 0.562 0.582 0.500 0.521 0.541 0.561 0.582 0.500 0.520 0.541 0.561 0.581 0.500 0.520 0.541 0.561 0.581 0.606 0.617 0.628 0.638 0.649 0.605 0.615 0.626 0.636 0.647 0.604 0.614 0.625 0.635 0.646 0.603 0.613 0.624 0.634 0.645 0.602 0.612 0.623 0.633 0.644 0.602 0.611 0.622 0.632 0.643 0.601 0.611 0.622 0.632 0.642 0.660 0.672 0.683 0.694 0.706 0.658 0.669 0.680 0.691 0.703 0.656 0.667 0.678 0.689 0.700 0.655 0.666 0.676 0.687 0.698 0.654 0.665 0.675 0.686 0.696 0.653 0.664 0.674 0.685 0.695 0.653 0.663 0.673 0.684 0.694 u 0.40 0.42 0.44 0.46 0.48 0.402 0.422 0.443 0.463 0.484 0.401 0.421 0.442 0.462 0.483 0.401 0.421 0.441 0.462 0.482 0.400 0.421 0.441 0.461 0.481 0.400 0.420 0.441 0.461 0.481 0.50 0.52 0.54 0.56 0.58 0.505 0.527 0.548 0.570 0.592 0.504 0.525 0.546 0.567 0.589 0.503 0.523 0.544 0.565 0.587 0.502 0.522 0.543 0.564 0.585 0.60 0.61 0.62 0.63 0.64 0.614 0.626 0.637 0.649 0.661 0.611 0.622 0.633 0.644 0.656 0.608 0.619 0.630 0.641 0.652 0.65 0.66 0.67 0.68 0.69 0.673 0.685 0.697 0.709 0.722 0.667 0.679 0.691 0.703 0.715 0.663 0.675 0.686 0.698 0.710 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8 0.711 0.723 0.734 0.746 0.757 0.709 0.720 0.732 0.743 0.754 0.708 0.719 0.730 0.741 0.752 0.706 0.717 0.728 0.739 0.750 0.705 0.716 0.727 0.737 0.748 0.773 0.786 0.798 0.811 0.824 0.769 0.781 0.794 0.806 0.819 0.766 0.778 0.790 0.802 0.815 0.763 0.775 0.787 0.799 0.811 0.761 0.773 0.784 0.796 0.808 0.760 0.771 0.782 0.794 0.805 0.845 0.860 0.875 0.890 0.906 0.838 0.852 0.867 0.881 0.897 0.832 0.846 0.860 0.874 0.889 0.828 0.841 0.854 0.868 0.883 0.824 0.837 0.850 0.863 0.877 0.820 0.833 0.846 0.859 0.873 0.818 0.830 0.842 0.855 0.869 0.923 0.940 0.959 0.978 0.999 0.913 0.930 0.947 0.966 0.986 0.904 0.921 0.937 0.955 0.974 0.897 0.913 0.929 0.946 0.964 0.892 0.907 0.922 0.939 0.956 0.887 0.901 0.916 0.932 0.949 0.882 0.896 0.911 0.927 0.943 u 0.70 0.71 0.72 0.73 0.74 0.735 0.748 0.761 0.774 0.788 0.727 0.740 0.752 0.765 0.779 0.722 0.734 0.746 0.759 0.771 0.717 0.729 0.741 0.753 0.766 0.714 0.725 0.737 0.749 0.761 0.75 0.76 0.77 0.78 0.79 0.802 0.817 0.831 0.847 0.862 0.792 0.806 0.820 0.834 0.849 0.784 0.798 0.811 0.825 0.839 0.778 0.791 0.804 0.817 0.831 0.80 0.81 0.82 0.83 0.84 0.878 0.895 0.912 0.931 0.949 0.865 0.881 0.897 0.914 0.932 0.854 0.869 0.885 0.901 0.918 0.85 0.86 0.87 0.88 0.89 0.969 0.990 1.012 1.035 1.060 0.950 0.970 0.990 1.012 1.035 0.935 0.954 0.973 0.994 1.015 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8 0.994 1.016 1.040 1.066 1.095 0.984 1.004 1.027 1.052 1.080 0.975 0.995 1.016 1.040 1.067 0.967 0.986 1.007 1.030 1.055 0.960 0.979 0.999 1.021 1.045 1.149 1.191 1.245 1.279 1.319 1.129 1.169 1.220 1.252 1.290 1.112 1.150 1.198 1.228 1.264 1.097 1.134 1.179 1.208 1.242 1.085 1.119 1.163 1.190 1.222 1.073 1.107 1.148 1.174 1.205 1.409 1.487 1.617 1.917 ∞ 1.371 1.443 1.565 1.845 ∞ 1.338 1.406 1.520 1.780 ∞ 1.310 1.373 1.481 1.725 ∞ 1.285 1.345 1.446 1.678 ∞ 1.263 1.320 1.416 1.635 ∞ 1.244 1.298 1.389 1.596 ∞ 1.033 0.736 0.610 0.537 0.486 0.951 0.669 0.551 0.483 0.436 0.870 0.611 0.501 0.438 0.394 0.803 0.562 0.459 0.399 0.358 0.746 0.519 0.422 0.366 0.327 0.697 0.481 0.390 0.337 0.300 0.651 0.448 0.361 0.311 0.277 u 0.90 0.91 0.92 0.93 0.94 1.087 1.116 1.148 1.184 1.225 1.060 1.088 1.117 1.151 1.188 1.039 1.064 1.092 1.123 1.158 1.021 1.045 1.072 1.101 1.134 1.007 1.029 1.054 1.081 1.113 0.950 0.960 0.970 0.975 0.980 1.272 1.329 1.402 1.447 1.502 1.232 1.285 1.351 1.393 1.443 1.199 1.248 1.310 1.348 1.395 1.172 1.217 1.275 1.311 1.354 0.985 0.990 0.995 0.999 1.000 1.573 1.671 1.838 2.223 ∞ 1.508 1.598 1.751 2.102 ∞ 1.454 1.537 1.678 2.002 ∞ 1.001 1.005 1.010 1.015 1.020 1.417 1.036 0.873 0.778 0.711 1.264 0.915 0.766 0.680 0.620 1.138 0.817 0.681 0.602 0.546 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8 0.332 0.290 0.258 0.232 0.211 0.300 0.261 0.231 0.207 0.188 0.273 0.236 0.208 0.186 0.168 0.250 0.215 0.189 0.168 0.151 0.229 0.196 0.172 0.152 0.136 0.220 0.204 0.189 0.176 0.165 0.194 0.179 0.165 0.154 0.143 0.172 0.158 0.145 0.135 0.125 0.153 0.140 0.129 0.119 0.110 0.137 0.125 0.114 0.105 0.097 0.123 0.112 0.102 0.094 0.086 0.181 0.170 0.161 0.153 0.145 0.155 0.146 0.137 0.130 0.123 0.134 0.126 0.118 0.111 0.105 0.117 0.109 0.102 0.096 0.090 0.102 0.095 0.089 0.083 0.078 0.090 0.084 0.078 0.072 0.068 0.080 0.074 0.068 0.064 0.059 0.138 0.131 0.125 0.114 0.104 0.116 0.110 0.105 0.095 0.086 0.099 0.093 0.089 0.080 0.072 0.085 0.080 0.076 0.067 0.061 0.073 0.069 0.065 0.057 0.051 0.063 0.059 0.056 0.049 0.044 0.055 0.052 0.048 0.042 0.037 u 1.03 1.04 1.05 1.06 1.07 0.618 0.554 0.504 0.464 0.431 0.535 0.477 0.432 0.396 0.366 0.469 0.415 0.374 0.342 0.315 0.415 0.365 0.328 0.298 0.273 0.370 0.324 0.290 0.262 0.239 1.08 1.09 1.10 1.11 1.12 0.403 0.379 0.357 0.338 0.321 0.341 0.319 0.299 0.282 0.267 0.292 0.272 0.254 0.239 0.225 0.252 0.234 0.218 0.204 0.192 1.13 1.14 1.15 1.16 1.17 0.305 0.291 0.278 0.266 0.254 0.253 0.240 0.229 0.218 0.208 0.212 0.201 0.191 0.181 0.173 1.18 1.19 1.20 1.22 1.24 0.244 0.235 0.226 0.209 0.195 0.199 0.191 0.183 0.168 0.156 0.165 0.157 0.150 0.138 0.127 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8 0.065 0.059 0.054 0.050 0.045 0.055 0.049 0.045 0.041 0.037 0.046 0.041 0.037 0.034 0.031 0.039 0.035 0.031 0.028 0.025 0.033 0.030 0.026 0.024 0.021 0.052 0.048 0.044 0.041 0.038 0.042 0.038 0.035 0.033 0.030 0.034 0.031 0.029 0.026 0.024 0.028 0.025 0.023 0.021 0.019 0.023 0.021 0.019 0.017 0.016 0.019 0.017 0.015 0.014 0.013 0.046 0.043 0.040 0.035 0.030 0.036 0.033 0.031 0.026 0.023 0.028 0.026 0.024 0.020 0.017 0.022 0.021 0.019 0.016 0.013 0.018 0.016 0.015 0.012 0.010 0.014 0.013 0.012 0.010 0.008 0.011 0.010 0.010 0.008 0.006 0.026 0.023 0.020 0.017 0.015 0.019 0.017 0.014 0.013 0.011 0.014 0.012 0.010 0.009 0.008 0.011 0.009 0.008 0.007 0.006 0.008 0.007 0.006 0.005 0.004 0.006 0.005 0.004 0.004 0.003 0.005 0.004 0.003 0.003 0.002 u 1.26 1.28 1.30 1.32 1.34 0.182 0.170 0.160 0.150 0.142 0.145 0.135 0.126 0.118 0.110 0.117 0.108 0.100 0.093 0.087 0.095 0.088 0.081 0.075 0.069 0.079 0.072 0.066 0.061 0.056 1.36 1.38 1.40 1.42 1.44 0.134 0.127 0.120 0.114 0.108 0.103 0.097 0.092 0.087 0.082 0.081 0.076 0.071 0.067 0.063 0.064 0.060 0.056 0.052 0.049 1.46 1.48 1.50 1.55 1.60 0.103 0.098 0.093 0.083 0.074 0.077 0.073 0.069 0.061 0.054 0.059 0.056 0.053 0.046 0.040 1.65 1.70 1.75 1.80 1.85 0.067 0.060 0.054 0.049 0.045 0.048 0.043 0.038 0.035 0.031 0.035 0.031 0.027 0.024 0.022 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8 0.007 0.006 0.005 0.004 0.003 0.005 0.004 0.004 0.003 0.002 0.004 0.003 0.003 0.002 0.001 0.003 0.002 0.002 0.001 0.001 0.002 0.002 0.001 0.001 0.001 0.004 0.003 0.003 0.002 0.002 0.003 0.002 0.002 0.001 0.001 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.000 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.002 0.002 0.002 0.001 0.000 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.001 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 u 1.90 1.95 2.00 2.10 2.20 0.041 0.038 0.035 0.030 0.025 0.028 0.026 0.023 0.020 0.016 0.020 0.018 0.016 0.013 0.011 0.014 0.012 0.011 0.009 0.007 0.010 0.009 0.008 0.006 0.005 2.3 2.4 2.5 2.6 2.7 0.022 0.019 0.017 0.015 0.013 0.014 0.012 0.010 0.009 0.008 0.009 0.008 0.006 0.005 0.005 0.006 0.005 0.004 0.003 0.003 2.8 2.9 3.0 3.5 4.0 0.012 0.010 0.009 0.006 0.004 0.007 0.006 0.005 0.003 0.002 0.004 0.004 0.003 0.002 0.001 4.5 5.0 6.0 7.0 8.0 0.003 0.002 0.001 0.001 0.000 0.001 0.001 0.000 0.000 0.000 0.001 0.000 0.000 0.000 0.000 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 4.2 4.6 5.0 5.4 5.8 6.2 u 9.0 10.0 20.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 BFC21103 Hydraulics Tan et al. ([email protected]) 6.6 7.0 7.4 7.8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 8.2 8.6 9.0 9.4 9.8 u 0.00 0.02 0.04 0.06 0.08 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.000 0.020 0.040 0.060 0.080 0.10 0.12 0.14 0.16 0.18 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.100 0.120 0.140 0.160 0.180 0.20 0.22 0.24 0.26 0.28 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.200 0.220 0.240 0.260 0.280 0.30 0.32 0.34 0.36 0.38 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 0.300 0.320 0.340 0.360 0.380 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 8.2 8.6 9.0 9.4 9.8 u 0.40 0.42 0.44 0.46 0.48 0.400 0.420 0.440 0.460 0.480 0.400 0.420 0.440 0.460 0.480 0.400 0.420 0.440 0.460 0.480 0.400 0.420 0.440 0.460 0.480 0.400 0.420 0.440 0.460 0.480 0.50 0.52 0.54 0.56 0.58 0.500 0.520 0.540 0.561 0.581 0.500 0.520 0.540 0.560 0.581 0.500 0.520 0.540 0.560 0.580 0.500 0.520 0.540 0.560 0.580 0.500 0.520 0.540 0.560 0.580 0.60 0.61 0.62 0.63 0.64 0.601 0.611 0.621 0.632 0.642 0.601 0.611 0.621 0.631 0.641 0.601 0.611 0.621 0.631 0.641 0.600 0.611 0.621 0.631 0.641 0.600 0.610 0.621 0.631 0.641 0.65 0.66 0.67 0.68 0.69 0.652 0.662 0.673 0.683 0.694 0.652 0.662 0.672 0.683 0.693 0.651 0.662 0.672 0.682 0.692 0.651 0.661 0.672 0.682 0.692 0.651 0.661 0.671 0.681 0.692 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 8.2 8.6 9.0 9.4 9.8 u 0.70 0.71 0.72 0.73 0.74 0.704 0.715 0.726 0.736 0.747 0.704 0.714 0.725 0.735 0.746 0.703 0.713 0.724 0.734 0.745 0.702 0.713 0.723 0.734 0.744 0.702 0.712 0.723 0.733 0.744 0.75 0.76 0.77 0.78 0.79 0.758 0.769 0.780 0.792 0.804 0.757 0.768 0.779 0.790 0.802 0.756 0.767 0.778 0.789 0.800 0.755 0.766 0.777 0.788 0.799 0.754 0.765 0.776 0.787 0.798 0.80 0.81 0.82 0.83 0.84 0.815 0.827 0.839 0.852 0.865 0.813 0.825 0.837 0.849 0.862 0.811 0.823 0.835 0.847 0.860 0.810 0.822 0.833 0.845 0.858 0.809 0.820 0.831 0.844 0.856 0.85 0.86 0.87 0.88 0.89 0.878 0.892 0.907 0.921 0.937 0.875 0.889 0.903 0.918 0.933 0.873 0.886 0.900 0.914 0.929 0.870 0.883 0.897 0.911 0.925 0.868 0.881 0.894 0.908 0.922 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 8.2 8.6 9.0 9.4 9.8 u 0.90 0.91 0.92 0.93 0.94 0.954 0.972 0.991 1.012 1.036 0.949 0.967 0.986 1.006 1.029 0.944 0.961 0.980 0.999 1.022 0.940 0.957 0.975 0.994 1.016 0.937 0.953 0.970 0.989 1.010 0.950 0.960 0.970 0.975 0.980 1.062 1.097 1.136 1.157 1.187 1.055 1.085 1.124 1.147 1.175 1.047 1.074 1.112 1.134 1.160 1.040 1.063 1.100 1.122 1.150 1.033 1.053 1.087 1.108 1.132 0.985 0.990 0.995 0.999 1.000 1.224 1.275 1.363 1.560 ∞ 1.210 1.260 1.342 1.530 ∞ 1.196 1.243 1.320 1.500 ∞ 1.183 1.228 1.302 1.476 ∞ 1.165 1.208 1.280 1.447 ∞ 1.001 1.005 1.010 1.015 1.020 0.614 0.420 0.337 0.289 0.257 0.577 0.391 0.313 0.269 0.237 0.546 0.368 0.294 0.255 0.221 0.519 0.350 0.278 0.237 0.209 0.494 0.331 0.262 0.223 0.196 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 8.2 8.6 9.0 9.4 9.8 u 1.03 1.04 1.05 1.06 1.07 0.212 0.173 0.158 0.140 0.123 0.195 0.165 0.143 0.127 0.112 0.181 0.152 0.132 0.116 0.102 0.170 0.143 0.124 0.106 0.094 0.159 0.134 0.115 0.098 0.086 1.08 1.09 1.10 1.11 1.12 0.111 0.101 0.092 0.084 0.077 0.101 0.091 0.083 0.075 0.069 0.092 0.082 0.074 0.067 0.062 0.084 0.075 0.067 0.060 0.055 0.077 0.069 0.062 0.055 0.050 1.13 1.14 1.15 1.16 1.17 0.071 0.065 0.061 0.056 0.052 0.063 0.058 0.054 0.050 0.046 0.056 0.052 0.048 0.045 0.041 0.050 0.046 0.043 0.040 0.036 0.045 0.041 0.038 0.035 0.032 1.18 1.19 1.20 1.22 1.24 0.048 0.045 0.043 0.037 0.032 0.042 0.039 0.037 0.032 0.028 0.037 0.034 0.032 0.028 0.024 0.033 0.030 0.028 0.024 0.021 0.029 0.027 0.025 0.021 0.018 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 8.2 8.6 9.0 9.4 9.8 u 1.26 1.28 1.30 1.32 1.34 0.028 0.025 0.022 0.020 0.018 0.024 0.021 0.019 0.017 0.015 0.021 0.018 0.016 0.014 0.012 0.018 0.016 0.014 0.012 0.010 0.016 0.014 0.012 0.010 0.009 1.36 1.38 1.40 1.42 1.44 0.016 0.014 0.013 0.011 0.010 0.013 0.012 0.011 0.009 0.008 0.011 0.010 0.009 0.008 0.007 0.009 0.008 0.007 0.006 0.006 0.008 0.007 0.006 0.005 0.005 1.46 1.48 1.50 1.55 1.60 0.009 0.009 0.008 0.006 0.005 0.008 0.007 0.006 0.005 0.004 0.006 0.005 0.005 0.004 0.003 0.005 0.004 0.004 0.003 0.002 0.004 0.004 0.003 0.003 0.002 1.65 1.70 1.75 1.80 1.85 0.004 0.003 0.002 0.002 0.002 0.003 0.002 0.002 0.001 0.001 0.002 0.002 0.002 0.001 0.001 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 BFC21103 Hydraulics Tan et al. ([email protected]) Varied‐flow function for positive slopes F(u, N) (Chow, 1959) N 8.2 8.6 9.0 9.4 9.8 u 1.90 1.95 2.00 2.10 2.20 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.001 0.000 0.000 0.001 0.001 0.000 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2.3 2.4 2.5 2.6 2.7 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2.8 3.0 4.0 5.0 6.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 7.0 8.0 9.0 10.0 20.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 BFC21103 Hydraulics Tan et al. ([email protected]) Steps in direct integration method: 1. Calculate yo and yc 2. Determine N and M 3. Calculate J 4. Calculate u and v 5. Find F(u, N) and F(v, J) 6. Calculate length of the reach BFC21103 Hydraulics Tan et al. ([email protected]) Activity 4.7 A very wide river with Manning roughness n = 0.035 has uniform depth of 3.0 m and longitudinal slope of 0.0005. Based on direct integration method, estimate the length of nonuniform flow produced by a weir that caused the water surface to increase as much as 1.5 m upstream of weir. yo = 3 m 4.5 m So = 0.0005; n = 0.035 BFC21103 Hydraulics Tan et al. ([email protected]) Step 1. Calculate yo and yc yo = 3.0 m 1 1 q = yoR 3 So2 n 2 1 3 12 q = y o So n (For very wide channel, R ≈ y) 5 1 1 q= × 3 3 × 0.00052 0.035 5 q = 3.987 m2 /s ⎛ q 2 ⎞ ⎛ 3.9872 ⎞ y c = ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ g ⎠ ⎝ 9.81 ⎠ 1 3 1 3 y c = 1.175 m y = 3 m to 4.5 m > yo = 3 m > y c = 1.175 m → M1 profile BFC21103 Hydraulics Tan et al. ([email protected]) Step 2. Determine M and N 10 M = 3 and N = 3 Step 3. Calculate J N 3.333 = = 2.500 J= (N − M + 1) (3.333 − 3 + 1) Step 4. Calculate u1, u2, v1, and v2 u1 = y1 3.003 = = 1.001 yo 3 v1 = u = 1.001 N J 1 3.333 2.5 = 1.001 u2 = y2 4.5 = = 1.5 yo 3 v2 = u = 1.5 N J 2 BFC21103 Hydraulics Tan et al. ([email protected]) 3.333 2.5 = 1.717 Step 5. Find F(u1, N), F(u2, N), F(v1, J), and F(v2, J) F (u1 , N ) = F (1.001, 3.333) = 1.907 F (u2 , N ) = F (1.5, 3.333) = 0.1884 F (v1 , J ) = F (1.001, 2.5) = 2.786 F (v2 , J ) = F (1.717, 2.5) = 0.3342 Step 6. Calculate length of channel reach M ⎫⎪ ⎧ ⎛ yc ⎞ J yo ⎪ [F (v2 , J ) − F (v1 , J )]⎬ x2 − x1 = ⎨(u2 − u1 ) − [F (u2 , N ) − F (u1 , N )] + ⎜⎜ ⎟⎟ So ⎪⎩ ⎪⎭ ⎝ yo ⎠ M 3 ⎫ 3 ⎧ 1 . 175 2.5 ⎛ ⎞ L= ⎟ × [0.3342 − 2.786]⎬ ⎨(1.5 − 1.001) − [0.1884 − 1.907] + ⎜ 0.0005 ⎩ 3 ⎝ 3 ⎠ ⎭ L = 12569.05 m BFC21103 Hydraulics Tan et al. ([email protected]) yo = 3 m y = 3.003 m 4.5 m So = 0.0005; n = 0.035 L = 12569.05 m BFC21103 Hydraulics Tan et al. ([email protected]) B. Numerical Integration Section All sections Rectangular Very wide channel (Chezy) Equations used ⎡ ⎛ K So ⎢1 − ⎜⎜ o dy ⎢⎣ ⎝ K ave = Q 2T dx 1− 3 gA ⎞ ⎟⎟ ⎠ ⎛ ⎛ y ⎜1−⎜ o ⎜ ⎜⎝ y ave dy = So ⎜ dx ⎜ 1 − ⎛⎜ y c ⎜ ⎜y ⎝ ⎝ ave 3 ⎞ ⎞⎟ ⎟⎟ ⎠ ⎟ 3 ⎟ ⎞ ⎟ ⎟⎟ ⎟ ⎠ ⎠ 2 ⎤ ⎥ ⎥⎦ ⎡ ⎛ K ⎞2 ⎤ So ⎢1 − ⎜⎜ o ⎟⎟ ⎥ dy ⎢⎣ ⎝ Kave ⎠ ⎥⎦ = 3 dx ⎛ y ⎞ 1 − ⎜⎜ c ⎟⎟ ⎝ yave ⎠ BFC21103 Hydraulics Tan et al. ([email protected]) ⎡ Q 2T ⎤ ⎢ 1− 3 ⎥ dy ⎢ gA ⎥ dx = So ⎢ ⎛ K o ⎞2 ⎥ ⎢ 1 − ⎜⎜ ⎟⎟ ⎥ ⎢⎣ ⎝ K ave ⎠ ⎥⎦ ⎡ ⎛ y ⎞3 ⎤ ⎢ 1 − ⎜⎜ c ⎟⎟ ⎥ dy ⎢ ⎝ yave ⎠ ⎥ dx = So ⎢ ⎛ K o ⎞ 2 ⎥ ⎢1 − ⎜⎜ ⎟⎟ ⎥ ⎢⎣ ⎝ K ave ⎠ ⎥⎦ ⎡ ⎛ y ⎞3 ⎤ ⎢1 − ⎜⎜ c ⎟⎟ ⎥ dy ⎢ ⎝ yave ⎠ ⎥ dx = So ⎢ ⎛ y o ⎞ 3 ⎥ ⎢1 − ⎜⎜ ⎟⎟ ⎥ ⎣⎢ ⎝ yave ⎠ ⎥⎦ Section Very wide channel (Manning) Equations used 10 ⎞ ⎛ ⎜ ⎛ yo ⎞ 3 ⎟ ⎟⎟ ⎟ 1 − ⎜⎜ ⎜ y dy = So ⎜ ⎝ ave ⎠ 3 ⎟ dx ⎜ ⎛ yc ⎞ ⎟ ⎟⎟ ⎟ ⎜ 1 − ⎜⎜ yave ⎠ ⎟ ⎜ ⎝ ⎠ ⎝ BFC21103 Hydraulics Tan et al. ([email protected]) 3 ⎤ ⎡ ⎢ 1 − ⎛⎜ y c ⎞⎟ ⎥ dy ⎢ ⎜⎝ yave ⎟⎠ ⎥ dx = ⎢ 10 ⎥ So ⎢ ⎛ yo ⎞ 3 ⎥ ⎟⎟ ⎥ ⎢1 − ⎜⎜ ⎣ ⎝ yave ⎠ ⎦ Activity 4.8 A very wide river with Manning roughness n = 0.035 has uniform depth of 3.0 m and longitudinal slope of 0.0005. Based on numerical integration, estimate the length of nonuniform flow produced by a weir that caused the water surface to increase as much as 1.5 m upstream of weir. Use N = 4 steps. yo = 3 m 4.5 m So = 0.0005; n = 0.035 BFC21103 Hydraulics Tan et al. ([email protected]) Calculate yo and yc yo = 3.0 m 1 1 q = yoR 3 So2 n 2 1 3 12 q = y o So n (For very wide channel, R ≈ y) 5 1 1 q= × 3 3 × 0.00052 0.035 5 q = 3.987 m2 /s ⎛ q 2 ⎞ ⎛ 3.9872 ⎞ y c = ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ g ⎠ ⎝ 9.81 ⎠ 1 3 1 3 y c = 1.175 m y = 3 m to 4.5 m > yo = 3 m > y c = 1.175 m → M1 profile BFC21103 Hydraulics Tan et al. ([email protected]) 3 ⎤ ⎡ ⎢ 1 − ⎛⎜ y c ⎞⎟ ⎥ dy ⎢ ⎜⎝ yave ⎟⎠ ⎥ So = 0.0005, yo = 3.0 m, yc = 1.175 m and dx = ⎢ 10 ⎥ So ⎢ ⎛ yo ⎞ 3 ⎥ ⎟⎟ ⎥ ⎢1 − ⎜⎜ ⎣ ⎝ yave ⎠ ⎦ Δy (4.5 − 3) dy = = = 0.375 m N 4 y (m) yave (m) 4.5 ‐ 4.125 4.3125 ⎛ yc ⎞ ⎟⎟ 1 − ⎜⎜ ⎝ yave ⎠ 0.9798 4.125 ‐ 3.750 3.9375 0.9734 0.5960 1224.9 3.750 ‐ 3.375 3.5625 0.9641 0.4361 1658.2 3.375 ‐ 3.0 3.1875 0.9499 0.1830 3893.7 3 ⎛ yo ⎞ ⎟⎟ 1 − ⎜⎜ ⎝ yave ⎠ 0.7017 BFC21103 Hydraulics Tan et al. ([email protected]) 10 3 L = Σdx dx (m) 1047.2 7824.0 5 4 y (m) 3 2 3.75 m 3.375 m 3m 4.125 m 4.5 m 1 0 0 1000 2000 3893.7 3000 4000 5000 1658.2 x (m) 6000 7000 1224.9 1047.2 L = 7824.0 m BFC21103 Hydraulics Tan et al. ([email protected]) 8000 9000 C. Direct Step Method dE = So − S f dx Rearranging Between two sections dx = dE So − S f E 2 − E1 dx = 1 ⎡ So − (S f 1 + S f 2 )⎤ ⎥⎦ ⎣⎢2 ⎛ V12 ⎞ V22 ⎞ ⎛ ⎜ y 2 + ⎟ − ⎜ y1 + ⎟ 2g ⎠ ⎝ 2g ⎠ dx = ⎝ ⎡ ⎛ 2 2 2 2 ⎞⎤ 1 ⎜ n V1 n V2 ⎟⎥ ⎢ So − ⎜ 4 + 4 ⎟ ⎢2 ⎜ ⎥ ⎟ 3 3 R2 ⎠⎥⎦ ⎢⎣ ⎝ R1 BFC21103 Hydraulics Tan et al. ([email protected]) Activity 4.9 A 200 m wide channel conveys flow at uniform depth of 3 m on a 0.0005 slope and Manning n = 0.035. Based on direct step method, determine the type and length of GVF flow produced by a weir which has caused the upstream flow to be elevated as much as 1.5 m. Use N = 4 steps. yo = 3 m 4.5 m So = 0.0005; n = 0.035 BFC21103 Hydraulics Tan et al. ([email protected]) Calculate yo and yc yo = 3.0 m 1 1 q = yoR 3 So2 n 2 1 3 12 q = y o So n (For very wide channel, R ≈ y) 5 1 1 q= × 3 3 × 0.00052 0.035 5 q = 3.987 m2 /s ⎛ q 2 ⎞ ⎛ 3.9872 ⎞ y c = ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ g ⎠ ⎝ 9.81 ⎠ 1 3 1 3 y c = 1.175 m y = 3 m to 4.5 m > yo = 3 m > y c = 1.175 m → M1 profile BFC21103 Hydraulics Tan et al. ([email protected]) So = 0.0005, yo = 3.0 m, yc = 1.175 m and dx = dy = Δy (4.5 − 3) = = 0.375 m N 4 1 2 3 y (m) R (m) V (m/s) 4 dE So − S f 5 6 7 V2 (m) 2g E (m) dE (m) Sf ×10−4 8 Sf ×10−4 9 So − S f ×10−4 10 dx (m) 4.5 4.5 0.8860 0.04001 4.540 1.294 4.125 4.125 0.9665 0.04761 4.173 0.367 1.730 1.512 3.488 1052.2 3.75 3.75 1.0632 0.05761 3.808 0.365 2.377 2.054 2.946 1239.0 3.375 3.375 1.1813 0.07112 3.446 0.362 3.377 2.877 2.123 1705.1 3.0 3.0 1.3290 0.09002 3.090 0.356 5.001 4.189 0.811 4389.6 BFC21103 Hydraulics Tan et al. ([email protected]) L = Σdx 8385.9 5 4 y (m) 3 2 3.75 m 3.375 m 3m 4.125 m 4.5 m 1 0 0 1000 2000 4389.6 3000 4000 5000 1705.1 x (m) 6000 7000 1239.0 1052.2 L = 8385.9 m BFC21103 Hydraulics Tan et al. ([email protected]) 8000 9000 Activity 4.10 A 10 m wide channel conveys flow at uniform depth of 3 m on a 0.0005 slope and Manning n = 0.035. Based on direct step method, determine the type and length of GVF flow produced by a weir which has caused the upstream flow to be elevated as much as 1.5 m. Use N = 4 steps. yo = 3 m 4.5 m So = 0.0005; n = 0.035 BFC21103 Hydraulics Tan et al. ([email protected]) Calculate yo and yc yo = 3.0 m 1 1 q = yoR 3 So2 n 2 1 1 30 ⎞ 3 ⎛ q= × 3 × ⎜ ⎟ × 0.00052 0.035 ⎝ 16 ⎠ 2 q = 2.914 m2 /s ⎛ q 2 ⎞ ⎛ 2.9142 ⎞ y c = ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ g ⎠ ⎝ 9.81 ⎠ 1 3 1 3 y c = 0.9530 m y = 3 m to 4.5 m > yo = 3 m > y c = 0.953 m → M1 profile BFC21103 Hydraulics Tan et al. ([email protected]) So = 0.0005, B = 10 m, yo = 3.0 m, yc = 1.175 m and dx = q = 2.914 m2 /s dE So − S f Δy (4.5 − 3) dy = = = 0.375 m N 4 1 2 3 y (m) R (m) V (m/s) 4.5 4 5 6 7 8 Sf ×10−4 9 So − S f ×10−4 V2 (m) 2g E (m) dE (m) Sf ×10−4 2.368 0.6476 0.02138 4.521 4.125 2.260 0.7064 0.02543 4.150 0.371 3.75 2.143 0.7771 0.03078 3.781 3.375 2.015 0.8634 0.03799 3.0 1.875 0.9713 0.04808 10 dx (m) 2.061 1.845 3.155 1175.9 0.369 2.677 2.369 2.631 1402.5 3.413 0.368 3.588 3.133 1.867 1971.1 3.048 0.365 4.999 4.294 0.706 5170.0 1.628 BFC21103 Hydraulics Tan et al. ([email protected]) L = Σdx 9719.5 5 4 y (m) 3 2 3.75 m 3.375 m 3m 4.125 m 4.5 m 1 0 0 1000 2000 5170.0 3000 4000 5000 1971.1 x (m) 6000 7000 1402.5 1175.9 L = 9719.5 m BFC21103 Hydraulics Tan et al. ([email protected]) 8000 9000 Assignment #4 Q1. A river 6 m wide conveys flow at a normal depth 2.0 m along a slope of 0.0002 and n = 0.035, upstream of a water fall. The flow reduces to a depth of 1.9 m just before the fall. Taking N = 3 steps, determine: (i) type of flow profile; (ii) length of gradually‐varied flow profile produced based on the method of numerical integration; and (iii) sketch of the flow profile. BFC21103 Hydraulics Tan Lai Wai ([email protected]) Q2. A river 6 m wide conveys flow at a normal depth 2.0 m along a slope of 0.0002 and n = 0.035, upstream of a water fall. The flow reduces to a depth of 1.9 m just before the fall. Using direct step method with N = 5 steps, determine: (i) type of flow profile; (ii) length of gradually‐varied flow profile produced; and (iii) sketch of the flow profile. ‐ End of Question ‐ BFC21103 Hydraulics Tan Lai Wai ([email protected]) THANK YOU BFC21103 Hydraulics Tan Lai Wai ([email protected])