# Incompressible Fluid Application

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| [[Image:cylinder_vel.jpg|400px]] | | [[Image:cylinder_vel.jpg|400px]] | ||

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+ | | An offshore platform subjected to waves. Problem solved using edgebased levelset with 3000000 elements | ||

+ | |||

|} | |} | ||

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The aim of this application is to solve the well known set of '''Navier-Stokes''' equations. The problem suffers from severe '''locking''' and/or '''instability''' using linear FEM. | The aim of this application is to solve the well known set of '''Navier-Stokes''' equations. The problem suffers from severe '''locking''' and/or '''instability''' using linear FEM. | ||

− | + | <math> | |

− | + | ||

− | + | \partial_{t}\mathbf{u}-\nu\Delta\mathbf{u} + \mathbf{u}\cdot\nabla\mathbf{u}+\nabla p = \mathbf{f} \quad \text{in} \quad \Omega, ]0,T[ | |

+ | </math> | ||

− | + | <math> | |

+ | \quad \quad \quad \quad \quad \nabla\cdot\mathbf{u} = 0 \quad \text{in} \quad \Omega, ]0,T[ | ||

+ | </math> | ||

+ | <math> | ||

+ | \mathbf{u} = \mathbf{u_{0}} \quad \text{in} \quad \Omega, t=0 | ||

+ | </math> | ||

+ | |||

+ | <math> | ||

+ | \mathbf{u} = \mathbf{0} \qquad \text{in} \Gamma, t\in ]0,T[ | ||

+ | </math> | ||

+ | |||

+ | |||

+ | Different approaches could be chosen to solve this problem. '''Fractional step''', '''Subgrid scale stabilization''', '''GLS''' are among the others. | ||

+ | |||

+ | |||

+ | Some '''references''' to these methods are: | ||

+ | |||

+ | 1)''Stabilized finite element approximation of transient incompressible flows using orthogonal subscales | ||

+ | Ramon Codina | ||

+ | Computer Methods in Applied Mechanics and Engineering | ||

+ | Vol. 191 (2002), 4295-4321'' | ||

=== Numerical approach === | === Numerical approach === | ||

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Every physical problem is solved defining many different ingredients. | Every physical problem is solved defining many different ingredients. | ||

Try to be quite schematic. | Try to be quite schematic. | ||

+ | ==== Elements ==== | ||

+ | {| class="wikitable" width="100%" style="text-align:left; background:#d0d9dd; border:0px solid #e1eaee; font-size:100%; -moz-border-radius-topleft:0px; -moz-border-radius-bottomleft:0px; padding:0px 0px 0px 0px;" valign="top" | ||

+ | !Element | ||

+ | !Methodology | ||

+ | !Time Scheme | ||

+ | !Geometry | ||

+ | |-style="background:#F1FAFF;" | ||

+ | | [[FLUID]] | ||

+ | | [[Fractional step]] | ||

+ | | Forward/Backward Euler | ||

+ | | 2D,3D Geometries | ||

+ | |-style="background:#F1FAFF;" | ||

+ | | [[ASGS]] | ||

+ | | [[Variational multiscale]] | ||

+ | | Generalized <math>\alpha</math> | ||

+ | | 2D,3D Geometries | ||

+ | |-style="background:#F1FAFF;" | ||

+ | | [[Fluid2DGLS_expl]] | ||

+ | | [[Least square]] | ||

+ | | Runge-Kutta | ||

+ | | 2D,3D Geometries | ||

+ | |} | ||

== Theory == | == Theory == | ||

+ | |||

+ | |||

+ | == ASGS(Algebraic Sub Grid Scale) == | ||

+ | The basic idea of this method is to approximate the effect of the | ||

+ | continuous solution which can not be resolved by the finite element mesh on | ||

+ | the discrete finite element solution. | ||

+ | |||

+ | <math> | ||

+ | (\rho \partial_{t}\mathbf{u},\mathbf{v})+ \mu(\nabla\mathbf{u},\nabla\mathbf{v})+(\rho\mathbf{a}\cdot\nabla\mathbf{u},\mathbf{v})-(p,\nabla\cdot\mathbf{v})+(q,\nabla\cdot\mathbf{u}) | ||

+ | </math> | ||

+ | |||

+ | <math> | ||

+ | +(\rho\partial_{t}\mathbf{u}-\mu\Delta\mathbf{u} + \rho\mathbf{a}\cdot\nabla\mathbf{u}+\nabla p,\mu\Delta\mathbf{v} + \rho\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q)_{\tau1,t} | ||

+ | +(\rho\nabla\cdot\mathbf{u},\nabla\cdot\mathbf{v})_{\tau2} | ||

+ | </math> | ||

+ | |||

+ | <math> | ||

+ | =\langle\mathbf{f},\mathbf{v}\rangle+(\mathbf{f},\mu\Delta\mathbf{v} + \rho\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q))_{\tau1,t} | ||

+ | </math> | ||

== Using the Application == | == Using the Application == | ||

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== Programming Documentation == | == Programming Documentation == | ||

+ | |||

+ | [[PureConvectionEdgeBased]] | ||

[[Category: Applications]] | [[Category: Applications]] |

## Latest revision as of 12:18, 7 June 2010

## Contents |

## General Description

An offshore platform subjected to waves. Problem solved using edgebased levelset with 3000000 elements |

### Theory

The aim of this application is to solve the well known set of **Navier-Stokes** equations. The problem suffers from severe **locking** and/or **instability** using linear FEM.

Different approaches could be chosen to solve this problem. **Fractional step**, **Subgrid scale stabilization**, **GLS** are among the others.

Some **references** to these methods are:

1)*Stabilized finite element approximation of transient incompressible flows using orthogonal subscales*
Ramon Codina
Computer Methods in Applied Mechanics and Engineering
Vol. 191 (2002), 4295-4321

### Numerical approach

All numerical details here.

This is a part quite open, depending on the application we are considering.

Every physical problem is solved defining many different ingredients. Try to be quite schematic.

#### Elements

Element | Methodology | Time Scheme | Geometry |
---|---|---|---|

FLUID | Fractional step | Forward/Backward Euler | 2D,3D Geometries |

ASGS | Variational multiscale | Generalized α | 2D,3D Geometries |

Fluid2DGLS_expl | Least square | Runge-Kutta | 2D,3D Geometries |

## Theory

## ASGS(Algebraic Sub Grid Scale)

The basic idea of this method is to approximate the effect of the continuous solution which can not be resolved by the finite element mesh on the discrete finite element solution.