Presets/cfd.xml

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:

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

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:

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

Optionally a diffusive term can be added to improve the stability (formulation known as ${\displaystyle\delta}$ -SPH).

Momentum equation

The momentum equation aproximation can be separated in 2 differential operators aproximation:

${\displaystyle\left\langle{\frac{d{\mathbf{u}}_{a}}{dt}}\right\rangle=-\left\langle{\frac{\nabla p_{a}}{\rho_{a}}}\right\rangle+\mu\left\langle\Delta{\mathbf{u}}_{a}\right\rangle+{\mathbf{g}},}$

such that:

${\displaystyle\left\langle{\frac{\nabla p_{a}}{\rho_{a}}}\right\rangle={\frac{1}{\gamma_{a}}}\left[\sum_{{b\in\Omega}}\left({\frac{p_{a}}{\rho_{a}^{2}}}+{\frac{p_{b}}{\rho_{b}^{2}}}\right)\cdot\nabla W\left({\mathbf{r}}_{b}-{\mathbf{r}}_{a}\right)m_{b}+\sum_{{b\in\partial\Omega}}\rho_{b}\left({\frac{p_{a}}{\rho_{a}^{2}}}+{\frac{p_{b}}{\rho_{b}^{2}}}\right)\cdot{\mathbf{n}}_{b}W\left({\mathbf{r}}_{b}-{\mathbf{r}}_{a}\right)S_{b}\right],}$

${\displaystyle\left\langle\Delta{\mathbf{u}}_{a}\right\rangle={\frac{1}{\gamma_{a}}}\sum_{{b\in\Omega}}K{\frac{\left({\mathbf{u}}_{b}-{\mathbf{u}}_{a}\right)\cdot\left({\mathbf{r}}_{b}-{\mathbf{r}}_{a}\right)}{\rho_{a}\rho_{b}\left|{\mathbf{r}}_{b}-{\mathbf{r}}_{a}\right|^{2}}}\nabla W\left({\mathbf{r}}_{b}-{\mathbf{r}}_{a}\right)m_{b};}$

where the constant $K=6,8,10$ for 1D, 2D and 3D respectively.

Time integration

A pseudo second-order time integration scheme is used in AQUAgpusph, composed by a predictor and a corrector stage.

The predictor, executed at the start of the time step, can be written as follows:

${\displaystyle\rho_{{n+1/2}}=\rho_{{n}}+\Delta t_{{n}}\,{\frac{d\rho_{{n}}}{dt}}}$

${\displaystyle{\mathbf{u}}_{{n+1/2}}={\mathbf{u}}_{{n}}+\Delta t_{{n}}\,{\frac{d{\mathbf{u}}_{{n}}}{dt}}}$

${\displaystyle{\mathbf{r}}_{{n+1/2}}={\mathbf{r}}_{{n}}+\Delta t_{{n}}\,{\mathbf{u}}_{{n}}+{\frac{\Delta t_{{n}}^{2}}{2}}\,{\frac{d{\mathbf{u}}_{{n}}}{dt}}}$

Regarding the corrector stage, computed after the derivatives evaluation described before, can be written as:

${\displaystyle\rho_{{n+1}}=\rho_{{n+1/2}}+{\frac{\Delta t_{{n+1}}^{2}}{2}}\left({\frac{d\rho_{{n+1}}}{dt}}-{\frac{d\rho_{{n}}}{dt}}\right)}$

${\displaystyle{\mathbf{u}}_{{n+1}}={\mathbf{u}}_{{n+1/2}}+{\frac{\Delta t_{{n+1}}}{2}}\left({\frac{d{\mathbf{u}}_{{n+1}}}{dt}}-{\frac{d{\mathbf{u}}_{{n}}}{dt}}\right)}$

Both, predictor and corrector are just applied to the fluid particles.

Usage

To use it just use the following include tag:

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

Where /usr/share/aquagpusph/ should be conveniently modified. For backward compatibility you can include /usr/share/aquagpusph/resources/OpenCLMain.xml instead, which is just redirecting to cfd.xml.

Remember that this preset should be the first included.

Variables defined

When this preset is loaded some variables are generated (on top of the default variables of AQUAgpusph):

NAME	TYPE	LENGTH	DESCRIPTION
g	vec	1	Gravity acceleration, ${\displaystyle{\mathbf{g}}}$ . Null by default
cs	float	1	Sound speed $c_{s}$
rho_min	float	1	A minimum value for the density. 0 by default
rho_max	float	1	A maximum value for the density. $2.25\times 10^{{10}}$ by default
r_min	vec	1	Minimum position of the particles. It is an automatically computed value, and therefore setting it is useless
r_max	vec	1	Maximum position of the particles. It is an automatically computed value, and therefore setting it is useless
gamma	float*	n_sets	Incompressibility factor of the EOS, ${\displaystyle\gamma}$
refd	float*	n_sets	Density of reference, ${\displaystyle\rho_{0}}$
visc_dyn	float*	n_sets	Dynamic viscosity, ${\displaystyle\mu}$
delta	float*	n_sets	${\displaystyle\delta}$ -SPH factor
imove	unsigned int*	N	Moving flag: 1 for Fluid particles, 0 for Sensors, -1 for Fixed particles, -2 for Elastic bounce particles, -3 for Boundary integral elements
normal	vec*	N	Normal of the particles, $n$ , only relevant for non fluid particles
u	vec*	N	Velocity of the particles, ${\displaystyle{\mathbf{u}}}$
dudt	vec*	N	Acceleration of the particles, ${\displaystyle{\frac{d{\mathbf{u}}}{dt}}}$
rho	float*	N	Density of the particles, ${\displaystyle\rho}$
drhodt	float*	N	Density variation rate of the particles, ${\displaystyle{\frac{d\rho}{dt}}}$
m	float*	N	Mass of the particles (area for the boundary integral elements), $m$
p	float*	N	Pressure of the particles, $p$
id_in	unsigned int*	N	Backup of id for the predictor-corrector
iset_in	unsigned int*	N	Backup of iset for the predictor-corrector
imove_in	int*	N	Backup of imove for the predictor-corrector
r_in	vec*	N	Backup of r for the predictor-corrector
normal_in	vec*	N	Backup of normal for the predictor-corrector
u_in	vec*	N	Backup of u for the predictor-corrector
dudt_in	vec*	N	Backup of dudt for the predictor-corrector
rho_in	float*	N	Backup of rho for the predictor-corrector
drhodt_in	float*	N	Backup of drhodt for the predictor-corrector
m_in	float*	N	Backup of m for the predictor-corrector
p_in	float*	N	Backup of p for the predictor-corrector
grad_p	vec*	N	Resulting ${\displaystyle{\frac{\nabla p}{\rho}}}$
lap_u	vec*	N	Resulting ${\displaystyle{\frac{\Delta{\mathbf{u}}}{\rho}}}$
div_u	float*	N	Resulting ${\displaystyle\rho\nabla\cdot{\mathbf{u}}}$
lap_p	float*	N	Resulting ${\displaystyle\Delta p}$

Definitions

The following definitions are made (see Definitions to learn more about this):

NAME	VALUE	DESCRIPTION
DIMS	dims, i.e. 2 for the 2D version, 3 for the 3D version	Number of dimensions
H	h	Kernel length
CONW	1/h^dims	Constant multiplier for the kernel $W_{{ab}}$ evaluation
CONW	1/h^(dims+2)	Constant multiplier for the kernel gradient ${\displaystyle\nabla W_{{ab}}}$ evaluation.
SUPPORT	2.f	Kernel support, i.e. the number of times h needed to vanish it.

Tools

The following tools are set to be executed each time step (see Tools to learn more about this):

NAME	TYPE	DESCRIPTION
Predictor	kernel	Integration scheme predictor stage.
Link-List	link-list	Link list computation. It is giving information to sort the particles by the cell index.
Backup id	copy	Save id into id_in before sorting it.
Backup iset	copy	Save iset into iset_in before sorting it.
Backup imove	copy	Save imove into imove_in before sorting it.
Backup normal	copy	Save normal into normal_in before sorting it.
Backup m	copy	Save m into m_in before sorting it.
Sort	kernel	All the data fields are sorted using the data from the link list.
Backup dudt	copy	Save dudt into dudt_in for the Corrector stage, before computing the new interactions.
Backup drhodt	copy	Save drhodt into drhodt_in for the Corrector stage, before computing the new interactions.
Reinit grad_p	set	Set ${\displaystyle{\frac{\nabla p_{a}}{\rho_{a}}}={\mathbf{0}}}$ .
Reinit lap_p	set	Set ${\displaystyle\Delta p_{a}=0}$ .
Reinit div_u	set	Set ${\displaystyle\rho_{a}\nabla\cdot{\mathbf{u}}_{a}=0}$ .
Reinit lap_u	set	Set ${\displaystyle{\frac{\Delta{\mathbf{u}}_{a}}{\rho_{a}}}={\mathbf{0}}}$ .
Reinit shepard	set	Set ${\displaystyle\gamma_{a}=0}$ .
Interactions	kernel	Performs the interactions between fluid particles. It is also performing some interpolation tasks in the sensors and boundary elements.
NonFluidSet	kernel	Set the values of pressure and density in the boundary elements.
BoundaryIntegrals	kernel	Compute the effect of the boundary elements on the fluid particles.
Rates	kernel	Set the variation rates ${\displaystyle{\frac{d{\mathbf{u}}_{a}}{dt}}}$ and ${\displaystyle{\frac{d\rho_{a}}{dt}}}$ from the computed ${\displaystyle{\frac{\nabla p_{a}}{\rho_{a}}}={\mathbf{0}}}$ , ${\displaystyle\Delta p_{a}=0}$ , ${\displaystyle\rho_{a}\nabla\cdot{\mathbf{u}}_{a}=0}$ , ${\displaystyle{\frac{\Delta{\mathbf{u}}_{a}}{\rho_{a}}}={\mathbf{0}}}$ and ${\displaystyle\gamma_{a}=0}$ .
ElasticBounce	kernel	Avoid the walls trespassing applying an elastic bounce.
Corrector	kernel	Integration scheme corrector stage.
FixedTimeStep	set_scalar	Set the time step as ${\displaystyle\Delta t=Co{\frac{h}{c_{s}}}}$ , with $Co$ as the courant number.
t = t + dt	set_scalar	Increment by ${\displaystyle\Delta t}$ the time variable t.
iter += 1	set_scalar	Increment by ${\displaystyle\Delta t}$ the iteration variable iter.

Reports

This preset is not adding new reports (see Reports).