.. Hydroflow documentation file, created by Pierre-Yves Gousenbourger on Wed Aug 2, 2023. .. _physics-about: Governing equations =================== Hydroflow relies on a depth-averaged representation of various flow situations: pure hydrodynamical flows, flows involving sediment transport and morphological evolution, depth-averaged turbulence, porosity approach, ... The core governing equations are the shallow-water equations that can be complemented with additional source terms, equations or specific modules to represent the different flows. The resulting available models are listed in the table below. All models derive from the shallow-water equations (Saint-Venant model - SV). Additional modules comprise source terms due to turbulence that can be computed using different turbulence models (see dedicated section), or a bank failure operator. Not all modules are compatible with all models. In the current release, only the basic SV model is available, the other models will be issued soon. ================ ======================== ======== ======== ===================== Category Name of the model Nickname Num eqs Compatible modules ================ ======================== ======== ======== ===================== Hydrodynamical Saint-Venant SV 3 Turbulence Hydrodynamical Single porosity SP 3 / Morphodynamical Saint-Venant - Exner SVE 4 Bank-failure operator, Multiple grain sizes Morphodynamical Two-layer 2L 6 Bank-failure operator Morphodynamical Two-phase/two-layer 2P2L 8 Bank-failure operator Morphogeological Saint-Venant - Richards SVR / ================ ======================== ======== ======== ===================== Hydrodynamical models --------------------- Saint-Venant model (SV) ^^^^^^^^^^^^^^^^^^^^^^^ This 2DH model (two-dimensions horizontal) aims to simulate the hydrodynamics of (non-)transient flows. The fluid column is made of one layer of pure water (one phase) of height :math:`h` located above the fixed bed, denoted as the bedrock :math:`z_{br}`. This bedrock level cannot change and there is not any exchange of mass (and hence of momentum) at the water and bed surfaces. It constitutes the base model. .. image:: ../pics/hydrodynamic_model.png :width: 400 :alt: Hydrodynamical model The core governing equations are the shallow-water equations that can be written in two-dimensional conservative vector form, as follows: .. math:: \frac{\partial \bf{U}}{\partial t} + \frac{\partial \bf{F(U)}}{\partial x} + \frac{\partial \bf{G(U)}}{\partial y} = \bf{S} where .. math:: \bf{U}=\begin{pmatrix} h \\ uh \\ vh \end{pmatrix} = \begin{pmatrix} h \\ q_x \\ q_y \end{pmatrix} \; \bf{S}=\begin{pmatrix} 0 \\ gh(S_{0x}-S_{fx}) \\ gh(S_{0y}-S_{fy}) \end{pmatrix} .. math:: \bf{F}=\begin{pmatrix} q_x \\ \frac{q_x^2}{h}+g\frac{h^2}{2} \\ \frac{q_xq_y}{h} \end{pmatrix} \; \bf{G}=\begin{pmatrix} q_y \\ \frac{q_xq_y}{h} \\ \frac{q_y^2}{h}+g\frac{h^2}{2} \end{pmatrix} \; with :math:`\bf{U}` the vector of conserved variables, :math:`\bf{F}` and :math:`\bf{G}` the vectors of fluxes in the :math:`x` and :math:`y` directions, and :math:`\bf{S}` the vector of source terms. In these vectors, :math:`h` is the water depth, :math:`q_x` and :math:`q_y` the unit-dicharges in the :math:`x` and :math:`y` directions respectively, :math:`u` and :math:`v` the velocity componenents, :math:`S_{0x}` and :math:`S_{0y}` the bed slopes in the :math:`x` and :math:`y` directions, and :math:`S_{fx}` and :math:`S_{fy}` the friction slopes in these two directions, calculated using Manning's formula: .. math:: S_{fx}=\frac{n^2 u \sqrt{u^2+v^2}}{h^{4/3}} \;\; S_{fy}=\frac{n^2 v \sqrt{u^2+v^2}}{h^{4/3}} Porosity model (P) ^^^^^^^^^^^^^^^^^^ Coming soon... Turbulence model (T) ^^^^^^^^^^^^^^^^^^^^ Coming soon... Morphodynamical models ---------------------- Saint-Venant - Exner model (SVE) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This model is dedicated to transient flows with bedload sediment transport. Below the water column of depth :math:`h` lies a moveable bed layer of finite depth :math:`h_s`. This bed layer is a mixture of water and sediment and rests on a non-moveable bed layer (bedrock) located at a level :math:`z_{br}`. This model is the simplest model that can be used to simulate flows involving sediment transport and bed level evolution. It is suitable for flows involving only bedload (no suspension) with a transport layer of limited height (hence no intense sheet-flow or debris-flow). .. image:: ../pics/morphodynamic_model.png :width: 400 :alt: Morphodynamical model The governing equations are the shallow-water equations of the SV model, complemented with a sediment mass conservation equation, namely the Exner equation: .. math:: \frac{\partial z_b}{\partial t} + \frac{1}{\epsilon_0} \frac{\partial q_{sx}}{\partial x} + \frac{1}{\epsilon_0} \frac{\partial q_{sy}}{\partial y} = 0 where :math:`z_b = z_{br}+h_s` is the erodible bed elevation, and :math:`q_{sx}` and :math:`q_{sy}` the :math:`x` and :math:`y` components of the sediment discharge :math:`q_s`. This sediment discharge is calculated using a closure formulation for the transport capacity, e.g. that of Meyer-Peter & Müller: .. math:: q_s = \sqrt{g(s-1)d^3} (\tau_b - 0.047)^{1.5} with :math:`s=\rho_s/\rho_w` the sediment density and :math:`\tau_b = \sqrt{ghS_f}` the bed shear stress calculated using the Manning formula. Two-layer model (2L) ^^^^^^^^^^^^^^^^^^^^ Coming soon... Two-phase/two-layer model (2P2L) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The goal of this model is to simulate transient flows highly-laden with solid particles, e.g. dam-flushes. In these situations, simple bedload models such as the SVE model are not appropriate for their hypotheses are far from being met (fluid column of pure water, no momentum exchanges between water and sediment, etc.). To cope with the distinct behaviours of suspension and bedload and the momentum exchanges between water and sediment, this model extends the SVE model to a two-phase and two-layer approach. Above the non-erodible bed lies a moveable bed layer of height :math:`h_s`, still a mixture of water and sediment, just like for the SVE model. However, instead of coping with the bedload transport through a sediment discharge, the bed elevation is modified by a mass exchange rate at the interface between the moveable bed layer and the fluid column. The solid concentration of this layer is constant so that an exchange of sediment always implies an exchange of water as well. Above the moveable bed layer, the fluid column is divided into two layers :math:`h_{sb}` and :math:`h_{ss}`, which are seen as a heterogeneous mixture of water and sediment. The solid concentrations in these two layers are variable so that the mass exchange rate at their interface only deals with sediment. Furthermore, two major hypotheses are made. First, the sediment in the upper layer :math:`h_{ss}`, less concentrated, is supposed to be transported at the same velocity as that of water, so there is no drag between the two phases in the upper layer :math:`h_{ss}` while drag is considered in the lower layer :math:`h_{sb}` where the concentration is higher. Secondly, water velocity is averaged over the whole fluid column, rather than over each layer. Hence, the model rests on two velocity vectors only: one for the sediment in the lower layer and one for the water phase. .. image:: ../pics/physics_two_phase_two_layers.png :width: 400 :alt: Two-phase/two-layer morphodynamical model The 2P2L model is based on eight governing equations and many closure formulations, while the SVE model rested on four governing equations and one closure formulation. The 2P2L model is hence way more computationally expensive and it is probably not worth using it for flows with a limited bedload transport. Similarly, the governing equations are depth-averaged mass and momentum conservation equations associated with water and sediment, but they cope with additional physical processes such as drag and sediment inertia. Morphogeological model ---------------------- Saint-Venant - Richards model (SVR) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Coming soon...