II - .... To a transcritical flow with hydraulic jump

1. By increasing the inflow velocity

In this case we tried to obtain an hydraulic jump. In this way we started from Q=8.858 m3/s, C=58, and we increased the inflow velocity. The consequence is that the Froude number will increase ( Fr = u / sqrt (g*h) ). In every case, we initialized the flow with a constant surface elevation.

Q=11 m3/s

The flow's still subcritical because the Froude number is lesser than one. So to have a transcitical flow, we must increase more the inflow velocity.

Q=15 m3/s

The flow is now transcritical at the point x=10 m because the Froude number is equal to one. At x=15 m, we have a hydraulic jumb, id. subcritical flow becomes subcritical (Fr<1).

Q=20 m3/s

In the first part of the flow (before the hump), this one is subcritical, but in the second part, the flow is supercritical. It's still supercritical at the outflow. But, we have imposed a boundary condition, and then this one isn't yet valid (because of the supercritical outflow). So our computation isn't right, because nothing have to be setted at the outflow.

2. By decreasing the outfow height

In this case we tried to obtain an hydraulic jump. In this way we started from the precedent conditions and we decreased the outflow surface elevation condition. The consequence is that the Froude number will increase when the height decrease. ( Fr = u / sqrt (g*h) ).

We computed two different cases, a supercritical case and a transcritical case :

a) Supercritical case ( Outflow surface elevation = 1 m ) :

Boundary conditions :

Initial condition : We use the analytical solution.

The bottom friction is conserved to a Chezy coefficient of 58.

TELEMAC shows that the flow after the hump is purely supercritical ( Fr > 1 ) :

In this case, the boundary condition set at the outflow is not valid. Indeed the method used to set boundary conditions is the method of caracteristics and in the supercritical case all the caracteristics go out of the domain. So, this case is not valid and as no physic reality. If we want to see a valid case with a supercritical flow we must set the outflow surface elevation in order to obtain a transcritical flow.

b) Transcritical case ( Outflow surface elevation = 1.5 m ) :

As above, we have here :

Boundary conditions :

Initial condition : We use the analytical solution.

The bottom friction is conserved to a Chezy coefficient of 58.

This graph shows that there are three regions in the channel : from the left to the right we have succesively, Fr<1, Fr>1and Fr<1. So it's a Transcritical regim. The boundary condition at the outflow is thus taken into account by the flow.


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