Transient flow of gravity-driven viscous films over 3D patterned substrates: conditions leading to Wenzel, Cassie and intermediate states

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

“…Grid non-orthogonality activates in all schemes, (48), (49) and (50), a component that involves the approximate gradient ∇ a (it is activated in (49) also by unevenness or skewness). It is not hard to show that this component contributes O(1) to the truncation error if ∇ a is first-order accurate, and O(1/h) if it is zeroth-order accurate.…”

“…It is not hard to show that this component contributes O(1) to the truncation error if ∇ a is first-order accurate, and O(1/h) if it is zeroth-order accurate. On the undistorted Cartesian grids that were used for comparison the ∇ a terms of the schemes are not activated; in fact, schemes (48) and (49) reduce to the same simple formula there. The reason why we do not simply substitute ∇ a φ(c f ) for ∇φ(c f ) in Eq.…”

“…The reason why we do not simply substitute ∇ a φ(c f ) for ∇φ(c f ) in Eq. (47) instead of using schemes such as (48) and (49) is that this would allow spurious high-frequency (cell-to-cell) oscillations in the numerical solution φ. This occurs because the ∇ a calculation is mostly based on differences of φ values that are two cells apart, and so the gradient field ∇ a φ can be smooth even though φ oscillates from cell to cell.…”

“…This occurs because the ∇ a calculation is mostly based on differences of φ values that are two cells apart, and so the gradient field ∇ a φ can be smooth even though φ oscillates from cell to cell. Schemes such as (48) and (49) express the diffusive flux mostly as a function of the direct difference of φ across the face, φ(N f ) − φ(P ), and thus do not allow such oscillations to go undetected. The gradients ∇ a are used in an auxiliary fashion, in order to make the diffusive fluxes consistent.…”

“…The gradients ∇ a are used in an auxiliary fashion, in order to make the diffusive fluxes consistent. However, it has recently been shown [43][44][45] that schemes such as (48) and (49) can also be interpreted as equivalent to substituting an interpolated value of ∇ a φ for ∇φ(c f ) in (47) and adding a damping term which is a function of the direct difference φ(N f ) − φ(P ) and of the gradients ∇ a φ(P ) and ∇ a φ(N f ) and tends to zero with grid refinement.…”