Ordinary differential equations 19
4. Change the horizontal layers from 10 to 5.
5. Change the time step from 5 s to 50 s, and plot the discharge Q out
versus the water elevation z.
Compare the solution data obtained by running the modifications, derive
conclusions and discuss the significance of the various variables involved to
the estimation of the tank emptying time.
Example 2.2
A reservoir is receiving floodwaters from an upstream watershed.
As a result, water storage increases, and when the water surface
reaches a certain safety level, it discharges over a weir into the spillway. Given the following data, estimate the outflow hydrograph and
the time variation of the water stage in the reservoir:
Initial reservoir water stage = 50.0 m
Weir crest elevation = 95.0 m
Weir width = 100.0 m
Varying reservoir horizontal area = 100 z 2 ; (z is the vertical height
in metres)
Inflow hydrograph = 2000 1−

 

 
t
T d
; (T d = 40,000 s = 11.11 hr is the
flood duration)
The numerical scheme is given by Equations 2.5 and 2.6. A computational time step of 100 s has been selected, and the calculations terminate when there is no discharge over the weir.
The simulation results of the inflow and outflow hydrographs as well
as the water elevation are presented in Figure 2.5. The delay of the
outflow discharge due to the filling of the reservoir is evident. Also, as
it was expected, the peak of the water elevation was reached when the
inflow and outflow rates were equal.
Computer code 2.2
% Example 2.2 Flood Routing Through a Reservoir
% Td = Flood duration [s];
% Qo = Initial flood discharge [m^3/s];
% zo = Initial water elevation in reservoir [m];
% zs = Spillway crest elevation [m];
% B = Spillway width [m];
% Dt = Time step [s];
% nm = Number of integration steps;
% ns = Values of reservoir surfaces at different elevations;
% nf = Values of flood hydrograph at different times;
clc;clear all;close all;