# Incompressible Fluid Application

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## General Description

### 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.

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

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

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

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

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

## 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.

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

$+(\rho\partial_{t}\mathbf{u}-\mu\Delta\mathbf{u} + \rho\mathbf{a}\cdot\nabla\mathbf{u}+\nabla p,\mu\Delta\mathbf{v} + \rho\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q)_{\tau1,t} +(\rho\nabla\cdot\mathbf{u},\nabla\cdot\mathbf{v})_{\tau2}$

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