The LimitedLimitedLinear Scheme.

The following article is related to the Limited scheme, implemented in OpenFOAM®. The article will describe the implementation to understand the usage and how the code works in that particular area. The LimitedScheme Schemes are related to the limited surface Interpolation schemes and apply a limiter to the calculation. These schemes are also known as NVD/TVD schemes (Normalized Variable Diminishing and Total Variation Diminishing). Thus, in OpenFOAM®, all LimitedScheme schemes are related to the limited surface interpolation schemes (such as the Upwind scheme). However, compared to the Upwind or LinearUpwind scheme, a limiter is applied. As already said, these schemes represent the NVD/TVD scheme section.

General Remarks.

The LimitedLimitedLinear Scheme (Second Order Scheme).

The limitedLimitedLinear scheme is based on the limitedSchemes surface interpolation schemes and calculates a limiter equal to the limited scheme. However, the user can specify a lower and upper bound for the field of interest to avoid non-physical values, compare J.H. Ferziger and M. Perić, and make the solution more stable. The numerical analysis below demonstrates - similar to the limitedLinear scheme - that the applied limiter introduces stability. The calculated values of Φ are always in the range of the defined interval [0:1]. Even though higher values of Φ can be reached, the bound help to improve stability and accuracy. Instead of using the limiteLimitedLinear scheme, one can use the limitedLimitedLinear01 scheme, which is built for fields in which the range of the quantity is defined in the range from zero to one.

The Results.

Of The Numerical Analysis.

Structured Grid (0°)

20 x 20 numerical cells

Structured Grid (0°)

40 x 40 numerical cells

Structured Grid (0°)

80 x 80 numerical cells

Data Analysis

Comparison

Structured Grid (45°)

20 x 20 numerical cells

Structured Grid (45°)

40 x 40 numerical cells

Structured Grid (45°)

80 x 80 numerical cells

Data Analysis

Comparison

Unstructured Grid

20 x 20 numerical cells

Unstructured Grid

40 x 40 numerical cells

Unstructured Grid

80 x 80 numerical cells

Data Analysis

Comparison

Polygon Grid

20 x 20 numerical cells

Polygon Grid

40 x 40 numerical cells

Polygon Grid

80 x 80 numerical cells

Data Analysis

Comparison

Grid Comparison

20 x 20 numerical cells

Grid Comparison

40 x 40 numerical cells

Grid Comparison

80 x 80 numerical cells