Download Derivation - Advanced Hydrology - Lecture Slides and more Slides Aeronautical Engineering in PDF only on Docsity! Lecture 5: Derivation of S-curve and discrete convolution equations Module 3 Docsity.com • It is the hydrograph of direct surface discharge that would result from a continuous succession of unit storms producing 1cm(in.)in tr –hr • If the time base of the unit hydrograph is Tb hr, it reaches constant outflow (Qe) at T hr, since 1 cm of net rain on the catchment is being supplied and removed every tr hour and only T/tr unit graphs are necessary to produce an S-curve and develop constant outflow given by, Qe = (2.78·A) / tr where Qe = constant outflow (cumec) tr = duration of the unit graph (hr) A = area of the basin (km2 or acres) Module 3 Unit hydrograph S – Curve method Docsity.com Module 3 Unit hydrograph Time (hr) 2-hr UH 2-HR lagged UH’s Sum 0 0 0 1 75 75 2 250 0 250 3 300 75 375 4 275 250 0 525 5 200 300 75 575 6 100 275 250 0 625 7 75 200 300 75 650 8 50 100 275 250 0 675 9 25 75 200 300 75 675 10 0 50 100 275 250 0 675 11 25 75 200 300 75 675 Example Problem Contd… Docsity.com 0 100 200 300 400 500 600 700 800 0 2 4 6 8 10 12 14 Q (c fs ) Time (hr) S-curve 2 hr UH Lagged by 2 hr Draw your S-curve, as shown in figure below Make a spreadsheet with the 2-hr UH ordinates, then copy them in the next column lagged by D=2 hours. Keep adding columns until the row sums are fairly constant. The sums are the ordinates of your S-curve. Module 3 Unit hydrograph Example Problem Contd… Docsity.com Time (hr) S-curve ordinate S-curve lagged 3hr Difference 3-HR UH ordinate 0 0 0 0 1 75 75 50 2 250 250 166.7 3 375 0 375 250 4 525 75 450 300 5 575 250 352 216 6 625 375 250 166.7 7 650 525 125 83.3 8 675 575 100 66.7 9 675 625 50 33.3 10 675 650 25 16.7 11 675 675 0 0 Unit hydrograph Example Problem Contd… Docsity.com Unit hydrograph Contd…. P1 P2 P3 Input Pn U1 U2 U3 U4 U5 Un-m+1 n-m+1 Unit pulse response applied to P1 Unit pulse response applied to P2 n-m+1 Un-m+1 Output Qn Output ∑ − + = = m* n m n m 1 m 1 Q P U Combination of 3 rainfall UH Docsity.com The set of equations for discrete time convolution ∑ − + = = m* n m n m 1 m 1 Q P U n = 1, 2,…,N =1 1 1Q PU = +2 2 1 1 2Q P U PU = + +3 3 1 2 2 1 3Q P U P U PU −= + + +M M 1 M 1 2 1 MQ P U P U ..... PU + += + + + +M 1 M 2 2 M 1 M 1Q 0 P U ..... P U PU − − − − += + + + + + + +N 1 M N M M 1 N M 1Q 0 0 ..... 0 0 ..... P U P U − − += + + + + + + +N M 1 N M 1Q 0 0 ..... 0 0 ..... 0 P U Unit hydrograph Contd…. Docsity.com Solution • The ERH and DRH in table have M=3 and N=11 pulses respectively. • Hence, the number of pulses in the unit hydrograph is N-M+1=11-3+1=9. • Substituting the ordinates of the ERH and DRH into the equations in table yields a set of 11 simultaneous equations Module 3 Unit hydrograph − − = = =2 2 11 1 Q P U 1,928 1.93x404U 1,079 cfs/ in P 1.06 Similarly calculate for remaining ordinates and the final UH is tabulated below n 1 2 3 4 5 6 7 8 9 Un (cfs/in) 404 1,079 2,343 2,506 1,460 453 381 274 173 Example Problem Contd… Docsity.com
