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0494bdd2
Commit
0494bdd2
authored
Jun 18, 2020
by
Tom Fischer
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[docu] Add docu for one test example of PVCDBC.
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web/content/docs/benchmarks/liquid-flow/PVCDBC_1_ts_2.png
web/content/docs/benchmarks/liquid-flow/PVCDBC_1_ts_2.png
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web/content/docs/benchmarks/liquid-flow/primary-variable-constrain-dirichlet-boundary-condition.pandoc
...ry-variable-constrain-dirichlet-boundary-condition.pandoc
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web/content/docs/benchmarks/liquid-flow/PVCDBC_1_ts_2.png
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0494bdd2
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web/content/docs/benchmarks/liquid-flow/primary-variable-constrain-dirichlet-boundary-condition.pandoc
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+++
date = "2020-06-18T11:33:45+01:00"
title = "Primary variable constraint Dirichlet-type boundary condition"
weight = 171
project = "Parabolic/LiquidFlow/SimpleSynthetics/PrimaryVariableConstraintDirichletBC/cuboid_1x1x1_hex_1000_Dirichlet_Dirichlet_1.prj"
author = "Thomas Fischer"
[menu]
[menu.benchmarks]
parent = "liquid-flow"
+++
{{< data-link >}}
## Problem description
We start with the following parabolic PDE:
$$
\left( c \rho_R + \phi \frac{\partial \rho_R}{\partial p}\right) \frac{\partial
p}{\partial t} - \nabla \cdot
\left[ \rho_R \frac{\kappa}{\mu} \left( \nabla p + \rho_R g \right) \right]
- Q_p = 0.
$$
where
- $c$ ... a constant that characterizes the storage as a consequence that the
solid phase is changing
- $\rho_R$ ... the density
- $p$ ... pressure
- $t$ ... time
- $\kappa$ ... the intrinsic permeability tensor of the porous medium
- $\mu$ ... is the pressure dependent dynamic viscosity
- $Q_p$ ... source/sink terms
In order to obtain a unique solution it is necessary to specify conditions on
the boundary $\Gamma$ of the domain $\Omega$.
The benchmark at hand should demonstrate the primary variable constraint
Dirichlet-type boundary condition. Here, the size of the sub-domain, the
Dirichlet-type boundary condition is defined on, is variable and changes
according to a condition depending on the value of the primary variable.
$$
\Gamma^\ast_D = \{ x \in \mathbb{R}^d, x \in \Gamma_D, \text{Condition}(p(x)) \}
$$
## Examples:
On the left (x=0) and right side (x=1) of the domain $\Omega = [0,1]^3$ the
usual Dirichlet-type boundary conditions are set, i.e.,
$$
p = 1, \quad x=0 \qquad \text{and} \qquad p = 1\quad x=1
$$
The initial condition $p_0$ is set to zero. Additionally, a primary variable
constraint Dirichlet-type boundary condition (PVCDBC) is specified:
$$
p = -0.1, \quad \text{for}\quad z = 1 \quad \text{and}\quad p(x,y,1) > 0.
$$
At the beginning of the simulation the PVCDBC is inactive. Because of the
'normal' Dirichlet-type boundary conditions the pressure is greater than zero
after the first time step and the PVCDBC is activated in the second time step.
The effect is depicted in the figure:
{{< img src="../PVCDBC_1_ts_2.png" >}}
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