A vertically-Lagrangian multilayer solver for free-surface flows

    This directory contains a hierarchy of modules which can be combined to solve an increasingly complex (and accurate) range of equations describing the motion of layers of incompressible fluids (illustrated below): from the hydrostatic Saint-Venant/shallow-water equations to the variable-density, incompressible Navier–Stokes equations with interfaces, Coriolis acceleration etc.

    The layered system and some of the associated fields

    The layered system and some of the associated fields

    The theoretical basis and main algorithms for this solver are described in Popinet, 2020.

    The modules are, in order of increasing complexity:

    • The hydrostatic solver approximates \displaystyle \begin{aligned} \partial_t h_k + \mathbf{{\nabla}} \cdot \left( h \mathbf{u} \right)_k & = 0,\\ \partial_t \left( h \mathbf{u} \right)_k + \mathbf{{\nabla}} \cdot \left( h \mathbf{u} \mathbf{u} \right)_k & = - gh_k \mathbf{{\nabla}} (\eta) \end{aligned}
    • The implicit free-surface extension adds time-implicit integration of the free-surface evolution.
    • The Coriolis extension adds the (horizontal) Coriolis acceleration.
    • The Boussinesq and isopycnal extensions add the buoyancy term due to small density variations i.e. the terms and equations in blue \displaystyle \begin{aligned} \partial_t \left( h \mathbf{u} \right)_k + \mathbf{{\nabla}} \cdot \left( h \mathbf{u} \mathbf{u} \right)_k & = - gh_k \mathbf{{\nabla}} (\eta) {\color{blue} - \mathbf{{\nabla}} (h q)_k + \left[ q \mathbf{{\nabla}} z \right]_k}\\ ...\\ {\color{blue}\partial_t \left( h T \right)_k + \mathbf{{\nabla}} \cdot \left( h \mathbf{u} T \right)_k} & {\color{blue} = 0,}\\ {\color{blue}q(z)} & {\color{blue} = \int_0^z g \Delta \rho(T) dz} \end{aligned}
    • The non-hydrostatic extension adds vertical momentum and the non-hydrostatic pressure i.e. \displaystyle \begin{aligned} \partial_t h_k + \mathbf{{\nabla}} \cdot \left( h \mathbf{u} \right)_k & = 0,\\ \partial_t \left( h \mathbf{u} \right)_k + \mathbf{{\nabla}} \cdot \left( h \mathbf{u} \mathbf{u} \right)_k & = - gh_k \mathbf{{\nabla}} (\eta) {\color{blue} - \mathbf{{\nabla}} (h \phi)_k + \left[ \phi \mathbf{{\nabla}} z \right]_k},\\ {\color{blue} \partial_t (hw)_k + \mathbf{{\nabla}} \cdot \left( hw \mathbf{u} \right)_k} & {\color{blue} = - [\phi]_k,}\\ {\color{blue} \mathbf{{\nabla}} \cdot \left( h \mathbf{u} \right)_k + \left[ w - \mathbf{u} \cdot \mathbf{{\nabla}} z \right]_k} & {\color{blue} = 0}, \end{aligned}

    Additional modules include:



    Stéphane Popinet. A vertically-Lagrangian, non-hydrostatic, multilayer model for multiscale free-surface flows. Journal of Computational Physics, 418:109609, May 2020. [ DOI | http | .pdf ]