Presets/cfd

Introduction

cfd.xml is the main preset to be loaded in order to perform Fluid Dynamics simulations.

Governing equations

The following governing equations are approximately solved with this preset (called conservation of mass equation, momentum equation, and equation of state):

${\displaystyle{\frac{d\rho}{dt}}=-\rho{\mathrm{div}}\left({\mathbf{u}}\right),}$

${\displaystyle{\frac{d{\mathbf{u}}}{dt}}={\mathbf{g}}+{\frac{{\mathrm{div}}\left({\mathbb{T}}\right)}{\rho}},}$

$p=p\left(\rho\right);$

where ${\displaystyle\rho}$ and ${\displaystyle{\mathbf{u}}}$ are the density and the velocity respectively, ${\displaystyle{\mathbf{g}}}$ is an external volumetric force field, $p$ is the pressure, and ${\displaystyle{\mathbb{T}}}$ is the stress tensor for a Newtonian fluid:

${\displaystyle{\mathbb{T}}=\left(-p+\lambda{\mathrm{tr}}\left({\mathbb{D}}\right)\right){\mathbb{1}}+2\mu{\mathbb{D}},}$

with ${\displaystyle\lambda}$ and ${\displaystyle\mu}$ the viscous coefficients and ${\displaystyle{\mathbb{D}}}$ the rate of strain tensor:

${\displaystyle{\mathbb{D}}={\frac{\nabla{\mathbf{u}}+\nabla{\mathbf{u}}^{T}}{2}}.}$

As it has been stated in the 3^rd governing equation (formerly Equation Of State, or EOS), we need to stablish a relation between the density and the pressure:

$p={\frac{c_{s}^{2}\rho_{0}}{\gamma}}\left(\left({\frac{\rho}{\rho_{0}}}\right)^{\gamma}+1\right),$

where $c_{s}$ is the sound speed (at least 10 times the maximum speed expected during the simulation), ${\displaystyle\rho_{0}}$ the density of reference of the fluid, and ${\displaystyle\gamma}$ an incompressibility factor (1 is recommended, but it can be increased to get better incompressibility properties).

Numerical modelling

Conservation of mass equation

As usual in SPH the fluid domain is discretized in a set of particles, such that the aproximate solution of the conservation of mass equation can be written as:

${\displaystyle{\frac{d\rho_{a}}{dt}}={\frac{1}{\gamma_{b}}}\left[\sum_{{b\in\Omega}}\left({\mathbf{u}}_{b}-{\mathbf{u}}_{a}\right)\cdot\nabla W\left({\mathbf{r}}_{b}-{\mathbf{r}}_{a}\right)m_{b}+\sum_{{b\in\partial\Omega}}\rho_{b}\left({\mathbf{u}}_{b}-{\mathbf{u}}_{a}\right)\cdot{\mathbf{n}}_{b}W\left({\mathbf{r}}_{b}-{\mathbf{r}}_{a}\right)S_{b}\right],}$

where $a$ and $b$ are particle indexes, ${\displaystyle{\mathbf{r}}_{a}}$ and ${\displaystyle{\mathbf{r}}_{b}}$ their positions, $m_{b}$ the mass of the particle $b$ , ${\displaystyle{\mathbf{n}}_{b}}$ and $S_{b}$ the outward normal and the surface of the boundary element $b$ , and ${\displaystyle\gamma_{b}}$ a renormalization factor:

${\displaystyle\gamma_{b}=\sum_{{b\in\Omega}}W\left({\mathbf{r}}_{b}-{\mathbf{r}}_{a}\right){\frac{m_{b}}{\rho_{b}}},}$

Using cfd.xml

To use it just use the include tag:

<Include file="/usr/share/aquagpusph/resources/Presets/domain.xml" />

Where /usr/share/aquagpusph/ should be conveniently modified. For backguard compatibility you can include /usr/share/aquagpusph/resources/OpenCLMain.xml instead (it is just including cfd.xml).

This preset should be included before any other one.