The Analysis of OpenFOAM® Scheme.

The following article is related to the investigation of numerical discretization schemes available in the OpenFOAM® software. As OpenFOAM® serves more than 30 discretization schemes for the advection term, only a few are mentioned here. The numerical investigation is related to the description gieven in »Equations and Condition«. The presented results are related to the OpenFOAM® version 5. It is worth to mention that the results might be different in newer versions since algorithms, numerics, or C++ coding might be improved. 

General Remark for the LimitedLinear Scheme.

First-to-Second-Order Accuracy

The Limited scheme (LimitedLinear) calculates a limiter that is applied to the field to avoid non-physical values, compare J.H. Ferziger and M. Perić. The numerical analysis below demonstrates that the applied limiter introduces the stability. This can be directly proved if the Linear scheme is used for the calculation. The calculated values of Φ are in the range of the defined interval [0:1]. Due to numerical errors, it is possible to get quantitative lower or higher values than the threshold. The order of magnitude of the error is in the range of 10-e13 (in this example). The comparison of this study shows that one should prefer the usage of limited schemes rather than the pure Linear scheme especially if the mesh is non-orthogonal. The LimitedLinear scheme blends between the Upwind and Linear scheme based on the applied limiter. As the limiter tends toward 1, the scheme is equal to the Upwind discretization. 

The Results of the LimitedLinear Scheme.

Structured Grid (0°)

20 x 20 numerical cells

LimitedLinear - Structured 0° 20x20

Structured Grid (0°)

40 x 40 numerical cells

LimitedLinear - Structured 0° 40x40

Structured Grid (0°)

80 x 80 numerical cells

LimitedLinear - Structured 0° 80x80

Data Analysis

Comparison

LimitedLinear - Structured 0° Data Analysis

Structured Grid (45°)

20 x 20 numerical cells

LimitedLinear - Structured 45° 20x20

Structured Grid (45°)

40 x 40 numerical cells

LimitedLinear - Structured 45° 40x40

Structured Grid (45°)

80 x 80 numerical cells

LimitedLinear - Structured 45° 80x80

Data Analysis

Comparison

LimitedLinear - Structured 45° Data Analysis

Unstructured Grid

20 x 20 numerical cells

LinearUpwind - Unstructured Grid 20x20

Unstructured Grid

40 x 40 numerical cells

LinearUpwind - Unstructured Grid 40x40

Unstructured Grid

80 x 80 numerical cells

LinearUpwind - Unstructured Grid 80x80

Data Analysis

Comparison

LinearUpwind - Unstructured Grid Data Analysis

Polygon Grid

20 x 20 numerical cells

LimitedLinear - Polygon Grid 20x20

Polygon Grid

40 x 40 numerical cells

LimitedLinear - Polygon Grid 40x40

Polygon Grid

80 x 80 numerical cells

LimitedLinear - Polygon Grid 80x80

Data Analysis

Comparison

LimitedLinear - Polygon Grid Data Analysis

 Grid Comparison

20 x 20 numerical cells

LimitedLinear - Grid Comparison 20x20

 Grid Comparison

40 x 40 numerical cells

LimitedLinear - Grid Comparison 40x40

 Grid Comparison

80 x 80 numerical cells

LimitedLinear - Grid Comparison 80x80