5.13 Steady-state convergence

The absence of a time derivative from an equation written in steady-state form reduces the diagonal dominance, and hence convergence, of the resulting matrix equation, as discussed in Sec. 5.5 . Under-relaxation is therefore applied to eqn eqn, and eqn to promote convergence in the algorithm in Sec. 5.12 .

The eqn and eqn fields use equation under-relaxation with factors eqn and eqn, respectively. A value of 0.7 is commonly applied, decreasing to 0.5 for less convergent cases (and sometimes to 0.3 in compressible flow cases, beyond the scope of this book).

Under-relaxation of eqn is more subtle. The flux corrector requires that eqn is not under-relaxed to ensure eqn obeys mass conservation better. For the momentum corrector, field under-relaxation is subsequently applied to eqn with a factor eqn. To find an optimal eqn for the momentum corrector, we examine

Au = H(u) rp: \relax \special {t4ht=
This explicit momentum equation contains diagonal coefficients eqn and an off-diagonal contribution eqn from neighbour cell coefficients eqn and associated velocities eqn.

Convergence is compromised by the explicit nature of eqn. It can be more implicit by “adding and subtracting” eqn, where coefficients eqn are applied to eqn in the “owner” cells:

H(u) = X a u X a u X a u : |-------------N---------------{z----------------N----------N--} |--------------{z--N------------} H?(u) +A?u \relax \special {t4ht=
Combining Eq. (5.22 ) and Eq. (5.23) gives
 ? ? (A A )u = H (u) rp; \relax \special {t4ht=
which attempts to make Eq. (5.22 ) more implicit in eqn.4 The pressure equation derived from Eq. (5.24) is:
 ? r ---1---rp = r H--(u)- A A? A A? \relax \special {t4ht=
This corresponds to under-relaxation of Eq. (5.19 ) by eqn, where eqn. A momentum matrix with coefficients eqn is approximately diagonally equal since there is no time derivative. Since eqn represents the diagonal coefficients under-relaxed by eqn, eqn. Relating the expressions for eqn and eqn gives the optimal under-relaxation factor for eqn for convergence as
p = 1 u; \relax \special {t4ht=
leading to the popular choice of eqn and eqn.

Residual control

The algorithm in Sec. 5.12 uses a fixed number of solution steps eqn. In practice, eqn must be chosen to be large enough to reach an acceptable level of convergence. Once convergence is reached, the simulation should stop to avoid unnecessary computing cost.

A common stopping criterion applies a residual level for each equation, below which the equation is deemed to be converged. When all equations satisfy their respective residual controls, the simulation then stops.

Convergence can also be determined by monitoring any suitable metric, including objective measurements from the simulation, e.g. a force coefficient. When the metric no longer changes significantly over subsequent steps, the simulation is stopped.

4Jeff Van Doormaal and George Raithby, Enhancements of the SIMPLE method for predicting incompressible fluid flows, 1984.

Notes on CFD: General Principles - 5.13 Steady-state convergence