{"id":46396,"date":"2021-09-23T13:55:37","date_gmt":"2021-09-23T13:55:37","guid":{"rendered":"https:\/\/www.simscale.com\/?page_id=46396"},"modified":"2025-06-02T15:58:45","modified_gmt":"2025-06-02T15:58:45","slug":"non-orthogonal-correctors","status":"publish","type":"page","link":"https:\/\/www.simscale.com\/docs\/simulation-setup\/numerics\/non-orthogonal-correctors\/","title":{"rendered":"CFD Numerics: Non-Orthogonal Correctors"},"content":{"rendered":"\n

The non-orthogonality of a mesh is defined as the angle made by the vector joining two adjacent cell centers across their common face and the face normal.<\/p>\n\n\n\n\n\n\n

\n
\"orthogonality<\/a>
Figure 1: A vector perpendicular to the face shared by both cells creates an angle \u03b8 in relation to the vector connecting the cell centroids of cell 1 and neighbor cell 2. This is the Non-Orthogonality angle.\\(^1\\)<\/figcaption><\/figure>\n<\/div>\n\n\n

As in real-life problems due to complex models, the numerical mesh is rarely orthogonal. So the non-orthogonality correction must be made for stability and accuracy. On the Scheme level, this is done under Numerics, where the Surface Normal Gradients and laplacian terms are corrected.<\/p>\n\n\n\n

For Meshes with non-orthogonality less than < 65-70, like the skewed cell in Figure 1, where the angle, \u03b8, is sufficiently steep, the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) solution algorithm is adequate for calculating the fluxes correctly. However, for bad quality meshes, an additional correction to account for mesh non-orthogonality is available in both SIMPLE and PISO algorithms in the standard OpenFOAM solver applications.<\/p>\n\n\n\n

When the mesh is bad (non-orthogonality > 75-80), the pressure calculation can be corrected by running several additional iterations of the pressure equation (inner loop). This is schematically represented below:<\/p>\n\n\n

\n
\"non
Figure 2: These corrections are the \u201cNon-Orthogonal corrector\u201d loops defined.<\/figcaption><\/figure>\n<\/div>\n\n\n

So for a good mesh, with no need for non-orthogonal correctors, the default SIMPLE algorithm is performed, that includes the following steps:<\/p>\n\n\n\n

    \n
  1. Momentum Equation;<\/li>\n\n\n\n
  2. Pressure Equation;<\/li>\n\n\n\n
  3. Check continuity;<\/li>\n\n\n\n
  4. Energy Equation;<\/li>\n\n\n\n
  5. Turbulence Equations etc.<\/li>\n<\/ol>\n\n\n\n

    In case of a bad mesh, with two non-orthogonal correctors, two extra steps are added when solving the pressure equation during the SIMPLE algorithm, and the sequence is formed as follows:<\/p>\n\n\n\n

      \n
    1. Momentum Equation<\/li>\n\n\n\n
    2. Pressure Equation;\n