Numerical Simulation of Positive Surge Moving Upstream
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3.2 Boundary Conditions
The channel is subdivided into N (in the present case, it is 50) reaches of equal length,
and there are N + 1 computational nodes. Thus, there are 2(N + 1) unknowns for
each new time step. For the interior nodes, 2(N−1) equations are obtained using
finite difference techniques and two at boundary nodes, and remaining two equations are provided by applying principles of method of characteristics [19]. In case
of subcritical flow, a boundary condition must be defined for both the upstream and
downstream boundaries whereas for supercritical flow, none is required at the downstream boundary and two boundary conditions required for the upstream boundary.
For subcritical flow, the minus characteristic equation will be applied at the upstream
boundary, whereas the plus characteristic boundary equation at the downstream
boundary [4]. In supercritical flow, both the plus and minus characteristics boundary
equations will be applied at the upstream boundary. In the present simulation for
downstream boundary, discharge is considered zero.
3.3 Initial Conditions
At the start of the simulation (t = 0), the values of the dependent variables (V and y
or Q and A) for all nodes are referred to as initial conditions.
4 Experimentation
Experiments were conducted at the Hydraulic and Water Resources Engineering
Laboratory, Department of Civil Engineering, Indian institute of Technology,
Kharagpur in the multipurpose tilting flume apparatus (Fig. 1) having width, 0.075 m,
depth, 0.3 m, and length, 4.9 m. An Agilent data acquisition system was connected to
pressure sensors which were placed at different locations at the bottom of the flume.
The pressure sensors were used to measure pressure variation during flow of water
with respect to time, which in turn gave the changes in depths. Each pressure sensor
had a capacity of 0.5 m. The Manning’s roughness coefficient was estimated to be
0.008. The sidewalls of the apparatus were made of transparent material. In the experiments, the pressure sensors were calibrated initially, and a suitable equation relating
the sensor output in millivolts (mV) versus water depth (m) was obtained which was
used in converting the electrical signals to equivalent water depths. Figure 2 shows
a photograph of a moving surge, and Fig. 3 indicates the location of the pressure
sensors at the bottom of the hydraulic flume.
The experiments were conducted for the following two conditions:
287
3.2 Boundary Conditions
The channel is subdivided into N (in the present case, it is 50) reaches of equal length,
and there are N + 1 computational nodes. Thus, there are 2(N + 1) unknowns for
each new time step. For the interior nodes, 2(N−1) equations are obtained using
finite difference techniques and two at boundary nodes, and remaining two equations are provided by applying principles of method of characteristics [19]. In case
of subcritical flow, a boundary condition must be defined for both the upstream and
downstream boundaries whereas for supercritical flow, none is required at the downstream boundary and two boundary conditions required for the upstream boundary.
For subcritical flow, the minus characteristic equation will be applied at the upstream
boundary, whereas the plus characteristic boundary equation at the downstream
boundary [4]. In supercritical flow, both the plus and minus characteristics boundary
equations will be applied at the upstream boundary. In the present simulation for
downstream boundary, discharge is considered zero.
3.3 Initial Conditions
At the start of the simulation (t = 0), the values of the dependent variables (V and y
or Q and A) for all nodes are referred to as initial conditions.
4 Experimentation
Experiments were conducted at the Hydraulic and Water Resources Engineering
Laboratory, Department of Civil Engineering, Indian institute of Technology,
Kharagpur in the multipurpose tilting flume apparatus (Fig. 1) having width, 0.075 m,
depth, 0.3 m, and length, 4.9 m. An Agilent data acquisition system was connected to
pressure sensors which were placed at different locations at the bottom of the flume.
The pressure sensors were used to measure pressure variation during flow of water
with respect to time, which in turn gave the changes in depths. Each pressure sensor
had a capacity of 0.5 m. The Manning’s roughness coefficient was estimated to be
0.008. The sidewalls of the apparatus were made of transparent material. In the experiments, the pressure sensors were calibrated initially, and a suitable equation relating
the sensor output in millivolts (mV) versus water depth (m) was obtained which was
used in converting the electrical signals to equivalent water depths. Figure 2 shows
a photograph of a moving surge, and Fig. 3 indicates the location of the pressure
sensors at the bottom of the hydraulic flume.
The experiments were conducted for the following two conditions: