Analytical integration of 0th, 2nd, and 4th order polynomial filtering functions on unstructured grid for dispersed phase fraction computation in an Euler–Lagrange approach

This paper presents an analytical approach to evaluate the volume integrals emerging during dispersed phase fraction computation in Lagrangian–Eulerian methods. It studies a zeroth, second, and fourth order polynomial filtering function in test cases featuring structured and unstructured grids. The analytical integration is enabled in three steps. First, the divergence theorem is applied to transform the volume integral into surface integrals over the volumes’ boundaries. Secondly, the surfaces are projected alongside the first divergence direction. Lastly, the divergence theorem is applied for the second time to transform the surface integrals into line integrals. We propose a generic strategy and simplifications to derive an analytical description of the complex geometrical entities such as non-planar surfaces. This strategy enables a closed solution to the line integrals for polynomial filtering functions. Furthermore, this paper shows that the proposed approach is suitable to handle unstructured grids. A sine wave and Gaussian filtering function is tested and the fourth order polynomial is found to be a good surrogate for the sine wave filtering function as no expensive trigonometric evaluations are necessary.