They accomplished this by treating only the Cartesian-like constant coefficient terms in the governing equations implicitly and updating the remaining terms iteratively. The present study proposes an extension to these studies by allowing for an arbitrary 3-D coordinate mapping. The use of collocation, rather than Galerkin or Tau, derivatives allows the implicit part of the algebraic system to include coefficients representing non-constant coordinate metrics. The collocation discretization method is described below. A multi-level iterative approach is then described for solving the resulting nonlinear algebraic system.
Automatic summation is assumed for repeating indices.
The covariant and contravariant base vectors are related to the covariant and contravariant forms of the metric tensor, respectively, and to the Jacobian of transformation for the generalized mapping of Eq. Contravariant components of the velocity vector are used in both the momentum and continuity equation, and contravariant components of the momentum equation are used to represent it as three scalar equations. Appropriate wall boundary conditions on velocity, along with the specification of the mean pressure, complete the continuous problem. For a staggered grid arrangement, the velocity is represented at different points than the pressure, and sets of collocation points are selected for both variables in each coordinate direction. The momentum equation is solved at the velocity points, and the continuity equation is solved at the pressure points.