# Numerical computation of free surface flow in a channel with a hump at the bottom

Objectives :
The purpose of this work is to compute the free surface flow in an horizontal canal using the TELEMAC2D module of the TELEMAC software. This module solves the Saint-Venant equations for non-linear long waves. Different hydrodynamics regimes will be considered and a special attention will be paid to numerical boundary conditions.

Basic concepts :
Saint-Venant equations forms a system of non-linear hyperbolic equations and admit wave-like solutions travelling both upstream and downstream with celerity c in a frame moving with the fluid velocity. Depending on the relative magnitude of the fluid velocity and wave celerity  (Froude) the waves propagates upsteram or downstream in a fixed reference frame.
As we are going to compute only a solution invariant along the cross-flow direction, the problem is virtualy 1D (although the actual computation will be performed in 2D due to the intrinsic numerical formulation of TELEMAC 2D). In such a problem Saint-Venant equation admit two waves travelling in the absolute reference frame with a celerity of  U+c and U-c. Suppose now that U>0 two different situations may be encountered: U<c called subcritical case (F<1) and U>c called supercritical case (F>1).
Indeed, the froude number is a function of space so that a given flow may be subcritical at some place and supercritical elsewhere. Such a flow is sometimes called transcritical.

Geometry and boundary conditions :
We consired a channel with a crectangular cross-section which is 20 m long and 10 m wide. At mid-lenght a hump is created at the channel bottom. This hump has a height of 0.2 m and has a parabolic shape.
The lateral walls of the channel are supposed to shear-stress free whereas at the bottom we will consider both the case with or without shear-stress. The are two open boundaries at the inflow and the outflow where different boundary conditions have to be imposed depending on the flow regime.

From a subcritical flow to a transcritical flow with a hydraulic jump :

Fortunately, there is an analytical solution fro the problem under study, when ther is no shear stress at the bottom.
The computation will be started at t = to from the analytical solution, for a subritical flow.

What are the boundary conditions to be imposed the open inflow and outflow boundaries ?

Explain why we prefer starting from the analytical solution (The answer may be given after trying another set of initial conditions).

Plot the fluid velocity, surface elevation and Froude number. Comment the numerical solution

From now plan to introduce gradually bottom friction at the bottom (no friction at the walls to preserve the 1D character of the solution).

Starting from the inviscid solution, compute new solutions with increasing friction, inflow and outflow condition being unchanged. Plot the fluid velocity, surface elevation and Froude number. Comments.

As we wnat to produce a transcritical flow the fluid velocity at the inflow boundary should be increased. Keeping the bottom friction at a given value, increase gradually the inflow speed (or discharge rate either). Plot the fluid velocity, surface elevation and Froude number. Comments.

Apart from the fluid velocity,  is there another parameter that could be used to create a supercritical region in the flow ?

Are the boundary conditions at the open boundaries still valid (if the response is no, what
to do so?)?

Removing the hydraulic jump :

Keeping the fluid velocity at the inflow boundary at its previous highest value, we would like to decrease the bottom friction gradualy. Compute the solution for different bottom friction values. Comments.

Are the boundary conditions at the open boundaries still valid (if the response is no, what
to do so?)?

Conclude.

Ressources :

The Telemac configuration files are located under  the directory :
/logiciels-mfn/INTRANET/optmfn/fsfho/fichiers_fournis/canal

Telemac software and Telemac manuals (paper version)

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Dominique Astruc (astruc@imft.fr)