FLUENT 6 - Slow convergence along centerline for polar grids (O-grids)

It is known that FLUENT converges very slowly along the centerline of 3D polar grids. These types of grids are generated in some preprocessors such as GridGen and ICEM.

This problem represents some limitations of the FLUENT segregated solver for O-type grids with a high degree of anisotropy. With the standard settings, FLUENT is not able to obtain convergence near the axis. The explanation is the following:

FLUENT's segregated solver in STEADY STATE mode requires underrelaxation for the momentum equations to facilitate the work of pressure-velocity coupling schemes like SIMPLE, SIMPLEC, or PISO. This causes very high diagonal dominance in the matrix on coarse levels of the algebraic multigrid (AMG) solver for the momentum equations, and the near-axis cells are almost completely detached from the other cells inside the AMG.

The workaround is to modify the conditions in the solver such that the segregated pressure-velocity coupling algorithm will allow application of momentum underrelaxation equal to 1.0.

The simplest method of overcoming this problem is to use the transient solver with the SIMPLEC or PISO pressure-velocity coupling algorithms and an underrelaxation factor of 1.0 for momentum.

A more computationally efficient method for steady flow simulations is to include the pseudo-transient terms in the transport equations for momentum and turbulence quantities in the form of a user-defined source term. This allows the pseudo-transient term to act as some sort of underrelaxation to prevent divergence, instead of the underrelaxation factors in the individual transport equations.

To set up such a problem,

1. Establish the flow field everywhere except near the centerline
2. compile the UDF below
3. hook the UDF to the transport equations for U, V, W, k, and epsilon
4. change the P-V coupling to SIMPLEC
5. Set all under-relaxation factors to 1
6. Iterate

The value of delta_t (the pseudo-timestep) may need to be varied to suit the specific conditions of the problem. Also, to speed up convergence, the AMG can be forced to visit more coarse levels by applying the V or F type of AMG cycles for the velocity components (optional).

#include "udf.h"

static real delta_t = 1e-5;

real rho = C_R(c,t);
real source;

source = 0.0;
dS[eqn] = -rho/delta_t;

return source;

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