# Incompressible Fluid Application

## Contents |

## General Description

ADVERTISMENT STYLE no numerical details!!! |

### 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 | Geometry |
---|---|

FractionalStep | 2D,3D Geometries |

SubgridScale | 2D,3D Geometries |

Fluid2DGLS_expl | 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.

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(\rho \mbox{}\partial_{t}\mathbf{u},\mathbf{v})&+ \mu\mbox{}(\nabla\mathbf{u},\nabla\mathbf{v})+(\rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{u},\mathbf{v})-(p,\nabla\cdot\mathbf{v})+(q,\nabla\cdot\mathbf{u})\nonumber \\ &+(\rho\mbox{}\partial_{t}\mathbf{u}-\mu\mbox{}\Delta\mathbf{u} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{u}+\nabla p,\mu\mbox{}\Delta\mathbf{v} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q)_{\tau1,t}\nonumber \\ &+(\rho\mbox{}\nabla\cdot\mathbf{u},\mbox{}\nabla\cdot\mathbf{v})_{\tau2}=\langle\mathbf{f},\mathbf{v}\rangle+(\mathbf{f},\mu\mbox{}\Delta\mathbf{v} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q))_{\tau1,t}