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Rotation-symmetric pressure distribution
In this example, you will learn how to

solve a rotation-symmetric problem one-dimensionally
perform a convergence test against an analytical solution
apply the Rotational Extrusion filters in ParaView for a two-dimensional visualization of the one-dimensional results

Result. With the Rotational Extrusion and the Warp By Scalar filters in ParaView,
the pressure distribution of this example looks as shown in the following picture:

    
          Fig.1  - Rotation-symmetric pressure distribution on a disc (warped to 3D). 
    

Table of contents. This description is structured as follows:


Rotation-symmetric pressure distribution
Problem setup
Mathematical model
Discretization


Implementation & Post processing
Part 1: Rotation-symmetric one-phase flow simulation setup
Part 2: Main program flow
Part 3: Post-processing with ParaView


Problem setup
We consider a single-phase problem that leads to a rotation-symmetric pressure distribution.
The following figure illustrates the setup:

    
          Fig.2  - Setup for the rotation-symmetric problem. The pressure boundary conditions are shown by the colored lines and the simulation domain is depicted in grey.
    

This could, for example, represent a cross section of an injection/extraction well in a homogeneous
and isotropic porous medium, where the well with radius 
r_1 is cut out and the
injection/extraction pressure p_1 is prescribed as a Dirichlet boundary condition. At the outer
radius r_2
, we set the pressure p_2
. In the polar coordinates r
 and \varphi, the
solution to this problem is independent of the angular coordinate \varphi and can be reduced to
a one-dimensional problem in the radial coordinate r. Therefore, in this example, we want to
solve the problem on a one-dimensional computational domain as illustrated by the orange line in
the above figure.

Mathematical model
In this example we are using the single-phase model of DuMu^x, which considers Darcy's law to relate
the Darcy velocity 
\textbf u
 to gradients of the pressure p. In the case of rotational
symmetry, the mass balance equation for the fluid phase can be transformed using polar coordinates:

-\frac{1}{r} \frac{\partial}{\partial r} \left( r  \frac{\varrho k}{\mu} \frac{\partial p}{\partial r} \right) = 0,


where we identify the Darcy velocity in radial direction 
u_r = -\frac{k}{\mu} \frac{\partial p}{\partial r},
and where k
 is the permeability of the porous medium, \mu is the dynamic viscosity of the
fluid, and \varrho
 is the fluid density.

Discretization
We employ a finite-volume scheme to spatially discretize the mass balance equation shown above.
Let us consider a discretization of the one-dimensional domain into control volumes
K_i = \left[ r_i, r_{i+1} \right]. The discrete equation describing mass conservation inside a control volume
K_i
 is obtained by integration and reads:

    - 2 \pi r_{i+1} \left( \varrho u_r \right)_{r_{i+1}}
    + 2 \pi r_i \left( \varrho u_r \right)_{r_i}
    = 0.


For this type of equation, the implementation of the finite-volume schemes in DuMu^x is based on
the general form:

\sum_{\sigma \in \mathcal{S}_K} | \sigma | \left( \varrho \textbf u \cdot \textbf n \right)_\sigma = 0,


where 
\sigma are the faces of the control volume and where the notation
( \cdot )_\sigma
 was used to denote quantities evaluated for a face \sigma.
The area of a face is denoted with | \sigma |. Thus, comparing the two equations
we identify | \sigma | = 2 \pi r_\sigma for the case of rotational symmetry
on a disc. Here, r_\sigma refers to the radius at which the face is situated
in the one-dimensional discretization.
In DuMu^x, this is rotational extrusion is approximated by using modified control volume
volumes and control volume face areas for the mid-point integration rule.

Implementation & Post processing

Part 1: Rotation-symmetric one-phase flow simulation setup


➡️ Click to continue with part 1 of the documentation


Part 2: Main program flow


➡️ Click to continue with part 2 of the documentation


Part 3: Post-processing with ParaView


➡️ Click to continue with part 3 of the documentation