MODIFIED GREEN-GAUSS RECONSTRUCTION
3.1 Overview of Green–Gauss reconstruction
In this section, we present a brief overview of existing approaches for gradient com- putations which are based on the Gauss divergence theorem. For any vector field u, the Gauss divergence theorem states that the net outward flux through a closed surface is equal to the volume integral of the divergence of the vector field. Stated mathematically,
Z
Ω
∇ ·u dΩ =
Z
E(Ω)
u·n dS (3.1)
where, E(Ω) is the closed surface enclosing a volume Ω with n representing the unit outward normal to the surface (see Figure 3.1(a)). Substitutingu =φa, where a is any arbitrary constant vector and φ is a scalar field, we obtain after simplification,
Z
Ω
∇φ dΩ =
Z
E(Ω)
φn dS (3.2)
a)
c
f f
f
nf
nf
nf b)
f
∆r
c
nb
nf r
f
nf
f
nf
α f α
r r
Figure 3.1: (a) Cell geometry (b) nomenclature for non-orthogonal grid
One can obtain an expression for the centroidal gradient in a finite volume frame-
work by replacing the integrals in Eq. 3.2 by their discrete analogues. The volume integral defines the cell-averaged gradient and can be shown to be a second-order approximation to the centroidal value. We therefore have,
(∇φ)c= 1 Ω
Z
Ω
∇φdΩ +O(h2) (3.3) where h is representative of the cell dimension, and c represents the cell center.
For a polygonal (polyhedral) volume, the surface integral can be evaluated as the sum over all faces as,
Z
E(Ω)
φ n dS =X
f
Z
f
φ dS
nf (3.4)
where nf represents the outward unit face normal of a face f of the polygon (polyhedron). Employing a single-point Gauss quadrature to evaluate the integral over each face gives,
Z
f
φ dS =φf∆Sf +O(h2)∆Sf (3.5)
where, ∆Sf represents the face area. It must be realised that the single-point Gauss quadrature requires evaluating the scalar values only at the face center and is exact for constant and linear functions only, which explains the second-order error in Eq. 3.5. Using Eqs. 3.3 and 3.5 in Eq. 3.2 one can compute the gradient ofφ as,
(∇φ)c= 1 Ω
X
f
φf nf ∆Sf (3.6)
The expression for gradient computation defined by Eq. 3.6 must, therefore, be necessarily viewed as a numerical approximation to the first derivative which leads to consistent gradients on any mesh topology if the “true” values at the faces φf are known. This is seldom the case, with the solution known only at the cell centers and therefore needing some interpolation to obtain the face center values. Consequently, the reconstruction defined by Eq. 3.6has approximation errors that are dependent on the chosen interpolation strategy. The various existing approaches in practice, collec- tively referred to as GG reconstruction, merely differ in the methodology adopted to evaluate the face-centered value.
As stated earlier, all quantities are stored at the cell centers on collocated grids and therefore GG reconstruction necessitates the use of a suitable interpolation to obtain the face-centered values from the cell center quantities. One of the approaches, which is very prevalent, is to obtain the face values as the weighted average of the centroidal values in cells sharing the face using volume or inverse volume weighting or inverse distance weighting (linear interpolation) [13, 15]. A second approach em- ploys an analogous weighted interpolation to first estimate the quantities at nodes from those at the cells and then the face values are determined from the nodal values by simple averaging [14]. Both these class of approaches invariably require interpo- lation and the accuracy of the gradients are therefore dependent on the accuracy of the interpolation. It can be easily demonstrated that simple interpolation practices involving volume and/or inverse volume weighting would in general lead to gradient errors that do not diminish with grid refinement on genuinely unstructured grids thus rendering the gradient estimates inconsistent [15, 20]. Inverse distance-weighted (or linear) interpolation can lead to consistent gradients but only if the line joining the cell-centers passes through face center, which is not possible in general on arbitrary polygonal meshes. It must be pointed out that most interpolation approaches, except the Laplacian strategy of [16], the nodal least-squares approach in [14] and auxiliary control volume approach in [18] do not lead to consistent gradients on unstructured meshes. In essence, for Green–Gauss reconstruction to lead to consistent gradients, the interpolation of values from cell centers to face centers need to be at least second order accurate, and such approaches, as discussed are only a few and are not free from numerical issues and/or implementation either. Since the study in this Chapter is devoted to the development and investigation of a new alternative for gradient recon- struction derived from similar roots as GG reconstruction, we shall effect comparisons with the GG reconstruction to highlight the improvements from the proposed ap- proach. To this end, we shall assume that the face values are obtained from centroidal values using volume-weighted averaging and refer to this calculation methodology as SGG reconstruction. Mathematically, the interpolation can be expressed as,
φf = φC1ΩC1+φC2ΩC2
ΩC1+ ΩC2 (3.7)
where, C1 and C2 are the cells sharing the face f. It must be remarked that one can choose to use a linear interpolation for computing the face values, although a greater effort would be necessary to realise it on arbitrary polygonal meshes that could be both skewed and non-orthogonal. The use of linear interpolation for computing the face values in SGG reconstruction could be advantageous on few mesh topologies
but volume-weighted averaging is a more simpler choice in multi-dimensions and the conclusions drawn herein with this choice of interpolation for the SGG reconstruction are not different from those when other interpolation approaches are employed, except in a handful of cases.