The second-order realizable moment advection scheme developed in Laurent and Nguyen, (2017) is extended to the case of unstructured grids with cells of arbitrary shape. The necessary modifications to the scheme and the conditions under which the scheme ensures the realizability of the advected moment set are presented. The implementation of the scheme in the OpenFOAM ® CFD toolbox is verified comparing the results obtained in one-dimensional test cases involving moment sets well inside the moment space, and at the boundary of the moment space. Results obtained with the proposed scheme are compared to the corresponding analytical solution. The scheme is then tested considering two-dimensional cases of pure moment advection with an imposed irrotational velocity field. First, a quadrilateral grid is considered to determine the order of the scheme and compare it to the results reported in Laurent and Nguyen (2017) with the same grid resolution. Then, the accuracy of the scheme on two-dimensional triangular grids is determined.