Dispersive and dissipation  properties of waves in channels with variable sections: what are we modelling? 

Mario Ricchiuto

In this talk we review some recent work on the modelling and simulation of non-hydrostatic flows in channels.
Two aspects are covered in the talk. 

The first is purely related to physical modelling.  Accounting for the flow kinematics in approximate models is very important 
whether it is to compute  the production and transport of sediments, or simply to account properly for secondary waves.
The latter introduce dispersive effects which are associated to the scale difference between the kinematics
of the internal flow layer, and the flow variation related to the larger scales of the hydrostatic free surface  propagation.
In the talk we will show some examples, largely overlooked in literature, of hydrostatic flows for which dispersive effects
are relevant, despite being  fully hydrostatic, and well simulated using the shallow water equations.
These effects, are purely geometrical, and have been observed many times both in natural and man made channels.
The kinematics associated to these waves are purely horizontal. 
A 1D dispersive model for these  waves is presented and discussed.

We then discuss the issue of designing numerical schemes for these equations. Inspired by the constraints of
hyperbolic problems many authors claim that energy dissipation must be ensured. In the last part of the talk
we will give some examples showing that both in absence and presence of physical friction, dissipation free methods
may  have radical advantages on dissipative ones when one is interested in the use of very coarse meshes, and long time  propagation.

References
R. Chassagne, A.G. Filippini, M. Ricchiuto and P. Bonneton, Dispersive and dispersive-like bores in channels with sloping banks,  Journal of Fluid Mechanics 870, pp. 595-616, 2019
B. Jouy, D. Violeau,  M. Ricchiuto, M.H. Le, One dimensional modelling of Favre waves in channels, Applied Mathematical Modelling  133, pp 170–194 2024
S. Gavrilyuk and M. Ricchiuto, A geometrical Green-Naghdi type system for dispersive-like waves in prismatic channels,  in revision on J.Fluid Mech ( https://arxiv.org/abs/2408.08625 )
H. Ranocha and M. Ricchiuto, Structure-preserving approximations of the Serre-Green-Naghdi equations in standard and hyperbolic form,  in revision on Num.Meth. for PDEs ( https://arxiv.org/abs/2408.02665 )