Las Nuevas Tendencias de Consumo

Diffusive fluxes require additional considerations for their discretization. The additional require- ments are due to the presence of the gradients in the diffusive fluxes defined in App. A. The literature is vast on this matter, so the following focuses only on unstructured multi-element meshes that are of interest for this thesis. In a cell-vertex context and using unstructured meshes as in AVBP, constructing an operator becomes even more cumbersome due to the dual- ity of the meshes already seen and because the stencil connectivity is harder to access. The type of dual cell is of type median dual as in [13] so the cell nodes are at the centroids of each cell but also at the center of the edges of the intersected faces as seen in Fig. 3.2a. The cell-vertex approximation is also known to be much better when compared to a cell-center approach as the number of operations scales with the number of edges while the cell centered approach scales with the number of elements. In 2D this does not imply a much larger cost but it does largely increase in 3D. Depending on the metrics, more specifically the type of elements, it has been shown by Barth [11] that triangular and tetrahedral type elements are conservative and additionally, the gradient is constant over the element. The objective behind such operators are also in Barth [11] and summarized in Puigt et al. [145]. The operator should indeed satisfy the following criteria to be efficient:

• Consistent, • Conservative, • Monotonic,

• Second-order accurate, • Insensitive to mesh quality, • and rely on a compact stencil.

3.3 Diffusion schemes in AVBP In AVBP there exist two operators, the FV 4 operator and the FE 2 operator. Using the previous criteria these two operators are compared in terms of performance. Consistency is guaranteed for the FV 4 in all cases but this is not the case for the FE 2 operator for certain connectivity as defined in Colin [35]. This however does not affect the quality of the results as shown in the same document. Both operators are built to be fully conservative and second-order accurate in all cases for regular meshes but neither of them is monotonic. This means that the appearance of chequerboard modes is expected. However, due to the Finite Element formulation and the more compact stencil, the FE 2 allows to diffuse the chequerboard modes contrarily to the FV 4 . This FE 2 operator was purposely developed to dissipate these chequerboard modes and is today the standard option for AVBP simulations. Both of these operators are detailed hereafter. It is important to note that for this section and to differentiate the nota- tion from previous sections, the superscript V is used to denote the viscous part of the problem.

3.3.1 FV 4 operator

The spatial discretization is evidently the same as for the convection schemes so the same hypotheses are taken into account when dealing with non-linear elements. The first operation to be performed is the obtention of the gradient at each node. To do this, the gradient is calculated in all primary cells represented in Fig. 3.4 using,

~ rUKe = 1 nd VKe X k Uk~sk, (3.49)

and redistributed to the nodes so that, ~ rU|j = 1 Vj X k VKe nv(Ke) ~ rUKe. (3.50)

Once the gradients are obtained at all neighbour nodes to j, the diffusive fluxes are recalcu- lated using both the conservative variables and the gradients previously calculated in the cells shaded in Fig. 3.4

Figure 3.4: Complete connectivity around node j for FV 4 . Gradients are required in all the primary cells and the divergence operator is applied only to the shaded cell.

RV_Ke = 1 ndVKe X k2Ke ~ F_kV _{· ~s}k, (3.51)

and redistributed once again using Eq. (3.50).

3.3.2 FE 2 operator

The passage to the 2 stencil is necessary to dissipate unwanted modes. The introduction of the Finite Element method instead of the Finite Volume approach that would be cheaper and easier to implement, is justified by the need to filter any diagonal modes that could appear [35]. Note that for triangular elements however the method degenerates into a Finite Volume scheme.

Instead of extending the stencil to the n + 2 neighbours as for the F V 4 , only the n + 1 neighbours are used as for the median dual cell. The calculation of the gradients is analogous to the one calculated for the fluxes for the family of convective TTG family schemes. These gradients are corrected for the non-linear elements for the set of primitive variables necessary to construct the viscous fluxes using the same operators. The divergence is then calculated around the dual cell of node j,

~ r · ~FV = 1 ndVj X j2⌦ ~ F_jV_|Ke~sj|Ke. (3.52)

This operator provides the necessary ingredients to avoid the apparition of diagonal modes, in no case excluding the appearance of other type of oscillations that are common in these simulations.

At this point, all of the operations either linked to the spatial or temporal operators have been detailed. Although most of the discussion has focused on the LW or TTG schemes that are mainly used for LES, other schemes are also implemented in AVBP. An important property that clearly differentiate all these numerical schemes relates to their stability limits.

Figure 3.5: Complete connectivity around node j for FE 2 . Gradients are required in all the primary cells and the divergence operator is applied only to the shaded cell.