I - From a subcritical flow...



1. Inviscid solution

In this part, we present the computation of the inviscid solution, with no bottom friction.

The boundary conditions are:

The initial conditions will be seted from two different ways:


  • Using a constant height:
  • To speciefied this, we enter this two lines in the cas200 case file:

    Then we launch the computation with 400 iterations and a step time of 0.1s.
    The results are plotted bellow:

    Surface elevation (Q=8.858 m3/s)

    Froude Number (Q=8.858 m3/s)

    Fluid velocity (Q=8.858 m3/s)

    Evolution of the suface elevation during the convergence

  • Using the analytical solution
  • To speciefied this, we enter this line in the cas200 case file:

    and we have to make a fortran file, which solve the equation:

    H0=h+Q2/(2gL2h2)

    This equation is solved by using the Newton-Raphson method. The program is repported below:

    c *****************************
    c * Methode de Newton-Raphson *
    c *****************************

    do i=1,NPOIN

    enddo

    The results are the same except for the convergence:

    Evolution of the suface elevation during the convergence

  • Interpretation
  • On the graphes "Evolution of the suface elevation during the convergence", we can notice that the convergence is faster for the computation initialize with the analytical solution. Indeed the variation of the surface elevation is about 0.015 m for the computation with the analytical solution, and about 0.09 for the constant initialization. We prefer start from the analytical solution because the solution of the problem is near this initialization.

    Comments on the numerical solutions:

    In the first part of the flow (before the hump) the surface elevation is constant. That's right (because there is no friction), but the constant doesn't correspond with the theory. The computation gave us a surface elevation of 2.01m, and the theory 2m.

    In the second part of the flow (after the hump) the surface elevation is constant and equal to 2m. This correspond with the analytical solution.

    So, between the 2 part of the flow, the computation gave us a dissipation (because the height before the hump is greater than the height after the hump). In theory, we don't have any dissipation because there is no friction, and the fluid is invicsid. So, the dissipation observed is due to the computation.

    On the Froud graph, we could see that: Fr<1. So the flow is always subcritical.



    2. Flow with friction

    In this part, we keep the same boundary conditions, but we introduce gradually bottom friction. By using the Maning Formula, we could find the Chezy number: C=Rh^(1/6)/n

    We computed all this cases which results are plotted below:

    Surface elevation (Q=8.858 m3/s; C=93)

    Velocity (Q=8.858 m3/s; C=93)

    Froude Number (Q=8.858 m3/s; C=93)

    Surface elevation (Q=8.858 m3/s; C=58)

    Velocity (Q=8.858 m3/s;C=58)

    Froude Number (Q=8.858 m3/s; C=58)

    Surface elevation (Q=8.858 m3/s; C=37)

    Velocity (Q=8.858 m3/s; C=37)

    Froude Number (Q=8.858 m3/s; C=37)

    Comments on the numerical solutions:

    The Froude Number increases with the friction (and so increases while the Chezy number increases). That makes the surface elevation decrease with the x-abscissa. That corresponds with the theory.

    Next: ... To a transcritical flow with hydraulic jump ->