24eaded1-b1b8-4333-9d6b-5fe8000e3271 579 5 7 decimal

GVF: STANDARD STEP EXAMPLE 2 - M2 Profile

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A trapezoidal channel with b = 20 ft, n = 0.025, z = 2, and So = 0.001 carries a discharge of

1000 ft3/s. If this channel terminates in a free overfall and there are no eddy losses, determine

the gradually varied flow profile by the trial-and-error step method.

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cusec ft 3^s / :

1. DATA

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Q 1000 ft 3^s / * :

S 0.001 :

B 20 ft * :

n 0.025 :

SS 2 :

x 011635574541 mat ft * :

percent (#1) ％ 1100 / :

Elev at start El.start 100 ft * :

Manning Coef. 
K = 1.00 for SI units; 
K = 1.49 for fps units KK 1.49 :

2. GENERAL EQUATIONS USED IN THE ANALYSIS

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Wetted Perimeter y P B 2 y * (1 SS 2^+ sqrt * + :

Hydraulic radius y R y A y P / :

Area y A y B y SS * + (* :

Velocity y V Q y A / :

Vel head y Vh y V (2^2 g.e * / :

Manning's Eqn y Vm KK n / y R (23 /^* (S sqrt * ft 13 /^s / (* :

Friction slope y Sf y V n * y R (23 /^ft 13 /^s / * KK * / (2^:

Total Energy y z H z y + y V (2^2 g.e * / + :

Top width y T B 2 SS * y * + :

<---- Eqn 1

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In this example, critical flow occurs at the control pont shown above. Hence, first we have to find y_c.

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Critical Flow Eqn for Trapezoidal section F1 Q 2^y.c T * g.e y.c A (3^* / 1 ≡ (:

Eqn to find Normal Depth yo) F2 Q y A y Vm * - (:

F1 Unknowns y.c 11 mat

F2 Unknowns y 11 mat

y.start F1 1 ft * FindRoot : ft 3.741

y.0 F2 1 ft * FindRoot : ft 6.243

y.1 y.start : ft 3.741

3. SAMPLE CALCULATIONS

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ΔXX x 1 el x 2 el x 1 el - x 3 el x 2 el - x 4 el x 3 el - 41 mat : ft 011623939041 mat

Z 0 x 2 el S * x 3 el S * x 4 el S * 41 mat : ft 0 0.116 0.355 0.745 41 mat

Section 1-Known y.1 ft 3.74102

y.1 Z 1 el H ft 5.2113

y.1 Sf 0.006751675

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Section 2-Guess depth: First start with "ystart". 
Then copy y_2_calc value here repeatedly until both are equal y.2 4.90419 ft * :

y.2 Z 2 el H 1.7518 m *

y.2 Sf 0.002492

Sf.avg_2 y.1 Sf y.2 Sf + 2 / : 0.00462185

h.f2 ΔXX 2 el Sf.avg_2 * : ft 0.53613

Calculated value H.2 y.1 Z 1 el H h.f2 + : ft 5.74739

Guess Value y.2 Z 2 el H ft 5.74739

Compare this with y2 y.2_calc H.2 Z 2 el - y.2 Vh - : ft 4.90419

y.2 y.2_calc - abs 0.0001 m * ≤ OK black light green 1 3 mat Retry red yellow 1 3 mat if

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Section 3-Guess depth: First start with "ystart". 
Then copy y_3_calc value here repeatedly until both are equal y.3 5.32557 ft * :

y.3 Z 3 el H 1.90921 m *

y.3 Sf 0.0018294

Sf.avg_3 y.2 Sf y.3 Sf + 2 / : 0.00216073

h.f3 ΔXX 3 el Sf.avg_3 * : ft 0.51641

Calculated value H.3 y.2 Z 2 el H h.f3 + : ft 6.2638

compare with y.3 Z 3 el H ft 6.2638

Compare this with y3 y.3_calc H.3 Z 3 el - y.3 Vh - : ft 5.32557

y.3 y.3_calc - abs 0.0001 m * ≤ OK black light green 1 3 mat Retry red yellow 1 3 mat if

Section 4-Guess depth: First start with "ystart". 
Then copy y_4_calc value here repeatedly until both are equal y.4 5.66462 ft * :

y.4 Z 4 el H ft 6.90305

y.4 Sf 0.0014487

Sf.avg_4 y.3 Sf y.4 Sf + 2 / : 0.00163909

h.f4 ΔXX 4 el Sf.avg_4 * : ft 0.63924

Calculated value H.4 y.3 Z 3 el H h.f4 + : ft 6.903

compare with y.4 Z 4 el H ft 6.903

Compare this with y4 y.4_calc H.4 Z 4 el - y.4 Vh - : ft 5.66462

y.4 y.4_calc - abs 0.0001 m * ≤ OK black light green 1 3 mat Retry red yellow 1 3 mat if

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Note:

1. Manual Iteration steps are shown above.

3. If number of sections are more, using the following Main Program is very easy.

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Main Program

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The basic logic is balancing Total Energy (TE) between 2 adjacent sections by repeated iterations

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TE is a function of z & y

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TE y z H ≡ z y + y V (2^2 g.e * / + ≡

xx sl StdStep j 1 xx rows range j 1 ≡ Δxx j el 0 : zz j el 0 : yy j el y.start : TE1 j el yy j el zz j el H : TE2 j el TE1 j el : sf j el yy j el Sf : sf_avg j el 0 : hf j el Δxx j el sf_avg j el * : ar j el yy j el A : pr j el yy j el P : hr j el yy j el R : vv j el yy j el V : vh j el yy j el Vh : dif j el 0 : NN j el 0 : 151 line Δxx j el xx j el xx j 1 - el - : zz j el zz 1 el xx j el sl * + : NN j el 0 : yg yy j 1 - el : TE1 j el yg zz j el H eval : sf j el yg Sf eval : sf_avg j el 0.5 sf j 1 - el sf j el + (* eval : hf j el Δxx j el sf_avg j el * eval : TE2 j el TE2 j 1 - el hf j el + eval : ΔTE 0.0001 ft * : TE1 j el TE2 j el - abs ΔTE ≥ yg TE2 j el zz j el - yg Vh - eval : sf j el yg Sf eval : sf_avg j el 0.5 sf j 1 - el sf j el + (* eval : hf j el Δxx j el sf_avg j el * eval : TE2 j el TE2 j 1 - el hf j el + eval : TE1 j el yg zz j el H eval : dif j el TE1 j el TE2 j el - : NN j el NN j el 1 + : 81 line while yy j el yg : 121 line if ar j el yy j el A : pr j el yy j el P : hr j el yy j el R : vv j el yy j el V : vh j el yy j el Vh : Elv j el yy j el zz j el + El.start + : 71 line for A xx ft / - yy ft / ar ft 2^/ pr ft / hr ft / vv ft s / / vh ft / Elv ft / TE1 ft / augment eval : A A Δxx ft / - hf ft / TE2 ft / dif ft / NN augment eval : 31 line :

See Eqn 1

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Calculations at control point

1. y_start known

2. TE1 = TE2 Total Energy using y_start

3. Similarly, calculate other initial canal parameters

4. dif = 0 Diffrence in TE at start

5. NN = 0 Number of iterations at start

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<--- yg = yy_j-1

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Guess Depth = Depth in previous station

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<--- TE1 = Total Energy (TE) corresponding to 'yg' using Eqn 1

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<--- sf = Friction slope corresponding to 'yg'

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<--- sf_avg = Avg. Friction slope between two sections

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<--- hf = Head loss due to friction between two stations

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<--- TE2 = TE after deducting "hf" from TE in previous station

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<--- ΔTE = Tolerance for energy balancing in iterations

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<--- Start of 'while' loop

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y TE z - V 2^2 g.e * / - ≡

<--- 'yg' re-calculated from TE2 within 'while' loop, from Eqn 1

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By re-calculating 'yg', Iteration continues while abs(TE2-TE1) ≥ ΔTE

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<--- yy = final 'yg'

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Calculation of other final canal parameters

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<--- Resulting matrix

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title x y A P R V Vh Elv H Δx hf H_Calc Dif NN augment :

Ans x S StdStep :

Ans2 title Ans stack :

NN = Number of iterations

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NOTE: The "while loop" solves by repeated iterations, and hence, trial calculations are not shown.

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RESULTS TABLE Ans2 x y A P R V Vh Elv H Δx hf H_Calc Dif NN 0 3.741 102.811 36.73 2.799 9.727 1.47 103.741 5.211 00 5.211 00116 - 4.904 146.182 41.932 3.486 6.841 0.727 105.02 5.747 116 - 0.536 5.747 08355 - 5.326 163.235 43.817 3.725 6.126 0.583 105.681 6.264 239 - 0.516 6.264 05745 - 5.665 177.468 45.333 3.915 5.635 0.493 106.41 6.903 390 - 0.639 6.903 03514 mat

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Results from manual iterations for comparison (See SAMPLE CALCULATIONS)

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y.1 ft 3.741

y.2 ft 4.904

y.3 ft 5.326

y.4 ft 5.665

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To find flow profile:

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Depth at furthest station 
from control point y.min Ans 2 col min m * : 3.741 m *

Normal depth y.0 ft 6.243

Ratio of Normal depth to depth 
at furthest station (≤90%) Ratio y.0 y.min / : ％ 50.9

y.0 y.min / 90 ％ * ≤ Calulations OK black light green 1 3 mat Error: y > 90% x uniform depth; reduce last value of 'x' in input red yellow 1 3 mat if

Note:

On the U/S side as the depth increases, and the water surface approaches

the Normal Depth Line (NDL) asymptotically.

Hence, this location is usually taken as the point where % .

In this example, we get a M2 profile.

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