A Fully Conservative 2D Model over Evolving Geometries

Ricardo CanelasMaster degree student

IST 14.12.09

Teton Dam 1976

Objectives:

To develop a 2D fully conservative model for the propagation of discontinuous flows

over evolving geometries

Tackle the domain definition and mesh generation issues, define a database structure susceptible to

be well articulated with the discretization procedure.

Gmsh(http://www.geuz.org/gmsh/)

Winged Edge Data Structure

Use Gmsh as routine to:

generate the initial mesh

generate subsequent refined meshes using a background mesh technique that is built according to a non-real time evaluation of the spatial variation of hydrodynamic variables (height and velocities) and of the morphological parameters (slopes)

Gmsh can use a simple scripting language as I/O

Possible to integrate in another code the tasks of generating meshing domains, outputting results, and generating new meshes

Merging of altimetric information (DTM) with the “flat” mesh

Efficient terrain surface discretization using a “parametric” space: a simple projection in Cartesian coordinates

One condition: the DTM is a regular grid, for fast interpolation

Example on an idealized surface

Domain definition Gmsh Delaunay triangulation

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DTM merging

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Generation of anisotropic meshes with background meshing techniques on Gmsh

Calibrate initial mesh characteristic lenghts(generalized size of na element around a point) on each node according to defined criteria:

spacial variation of hidrodinamic variables (height and velocities) and morphological parameters (slopes)

Example: linear variation of charactistic lenghts acording to third coordinate

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Mesh topology data structure

Winged Edge data structure

Basic element is edgeHighly redundantConstant time queriesRelatively small memory

requirements

Define discretization procedure for an uncoupled solution

Use of Flux-vector Splitting Finite Volume Method:

- Evaluate the PDEs, in their integral form over any discrete cell and balance the fluxes through the cell edges, in an attempt to estimate the real continuous solution.- Flux-vector Splitting considers a linear separation of the flux

Complete Conservation Equations

Total mass

Momentum in x direction

Momentum in y direction

Sediment mass in transport layer

Closure equations needed for hb, ub, τb, Cb* and Λ – granular dynamics and numerical simulation

Equation discretization

Full system in a compact form

Wereis the independant conservative variables vector

is the primitive variables vector

and are the flux vectors in x and y direction

is the source terms vector

Trough proper integration,

and evaluation of the integral form,

the final expression for the computation of flux trough element edges becomes

[Ferreira, 2009]

Geomorphology – code integration in the uncoupled case

Development of 2D code that allows the computation of bed and lateral erosion and the integration of debris volume derived from geotechnical failure in the flow, compatible with the FVM nature of the hydrodynamical code is of major importance in this work.

Bed erosion – Formulation

Equation for the mass conservation of sediments in the bed

Closure equations (derived from granular dinamics)

Equilibrium concentration

Adaptation lenght

Bed erosion – discretization problems

ΔZb is evaluated in the conservation equation for the bed of the system, at the barycenter of each element

Compatibility problems in the edges due to diferential erosion in adjacent elements – must devise a conservative way to force compatibility

ΔZb1ΔZb2

Time step

Free surface level remais constant, velocities are computed again in each cell to acomodate volume change

Geotechnical failiure

Geotechnical failiure represents a big contribuition of solid material to the flow in the case of dam break, and should be evaluated carefully.

The initial aproach will be comparing each element maximum gradient with the critical value and performing a rotation of the element on a normal to the line of maximum gradient, fixed on the lowest node of the element.

Geotechnical Failiure model

i>icrit

Θ=i-icrit

ΔZ1

ΔZ2

Geotechnical Failiure model

Compatibilized elements Volume to integrate on the flow on the next time

Model Limitations

-Instantaneous failiure and colapse;-Accuracy dependant on element size, computed not regarding this fact;

Advantages

-Easy to implement;-Low computing load

Model Validation

-Actual case study with results produced by a 1D model is available for direct comparison