Equation and Conditions

Referred to literature R. Schwarze, a comparison and analysis of numerical schemes will be done. A convective flow field in a defined domain (see picture 1.1) will be applied, and the scalar value of Φ is analyzed. The equation which is solved is an ordinary transport equation with time derivation, convective, diffusion and source terms. Since we focus on the convective term and the influence of schemes, mesh coarseness and orientation we can simplify the following equation:

\[\frac{\partial \rho \phi}{\partial t} + \nabla \bullet (\rho \textbf{U} \phi) - \nabla \bullet (D \nabla \phi) = S_\phi\]

 with the following definitions:

    - The system will be solved till time derivation is zero (steady-state condition)
    - Diffusion coefficient D = 0
    - Incompressible fluid and therefore no temperature-dependent density (constant)
    - No source terms

Hence, we can re-write the equation:

\[\nabla \bullet (\rho \textbf{U} \phi) = 0\]

 Thus, this equation and the assumptions/boundary condition lead us to a physical system which is shown in picture 1.1. Without diffusion, the scalar Φ must not mix with up while going through the domain. We will find a hard separation (dashed line) of the values. Now we will investigate into the mixing due to numerical schemes, numerical discretization and mesh dependency. Therefore, three different mesh refinements, volume types (hexahedral, tetrahedral) and for hexahedral meshes, the orientation of the mesh edges are used. The solver used is an included in the standard OpenFOAM® libraries (scalarTransportFoam). Three meshes are defined, and the discretization is applied to the outer edges:

    20 x 20 cells
    40 x 40 cells
    80 x 80 cells

Important notice: All simulations are done with the same control settings. Hence, it could occur that some schemes will blow up due to the time derivation and instability, although this scheme will produce good results if we choose a smaller time step. This behavior is because some dimensionless numbers like the Courant or the Fourier number have to be in a certain range. If we go above a limit, we have in general two options a) decreasing the time step; b) numerical stabilization using different methods. This is the reason why we can observe different results by using the same time step. Jósef Nagy also discussed that phenomenon from Linz. The test case is built as given below. The different meshes can be checked out below too.

Case Setup

Structured Mesh

(transport under 0°)

Structured Mesh 20 x 20
Structured Mesh 40 x 40
Structured Mesh 80 x 80

Structured Mesh

(transport under 0°)

Structured Mesh 20 x 20
Structured Mesh 40 x 40
Structured Mesh 80 x 80

Unstructured Mesh

Unstructured Mesh 20 x 20
Untructured Mesh 60 x 60
Unstructured Mesh 80 x 80

Polygon Mesh

Polygon Mesh 20 x 20
Polygon Mesh 40 x 40