Velocity relaxation

Under-relaxation of the momentum equation of the steady solver. On auto with SIMPLEC it is 0.7 during the first-order start-up phase and 0.9 after it; with plain SIMPLE and for the compressible solver it is 0.7 throughout.

Last updated 2026-11-03

Theory

The momentum equation is relaxed implicitly: the diagonal coefficient of every cell is divided by $\alpha_U$ and the matching part of the old value is moved to the right-hand side, equation (1). At convergence ( $\mathbf{U}_P = \mathbf{U}_P^{old}$ ) the added terms cancel, so relaxation changes the path, not the solution. A smaller $\alpha_U$ makes the matrix more diagonally dominant and each iteration a smaller step (Patankar and Spalding, 1972).

The steady run has two phases (Upwind iterations before the switch): a start-up phase with first-order turbulence convection, and the final phase.

Under SIMPLEC the auto value rises from 0.7 to 0.9 at the restart between the phases: the field is established by then and the larger step converges faster. The program rewrites the relaxation in fvSolution at that restart.
A typed value applies to both phases.
After a divergence fallback (plain SIMPLE, p 0.3), U stays at 0.7: 0.9 together with $\alpha_p = 0.3$ is the unstable pairing the program warns about.

The full table of auto values for every solver is in Pressure relaxation.

Equations

\frac{a_P}{\alpha_U}\,\mathbf{U}_P = \sum_{nb} a_{nb}\mathbf{U}_{nb} + \mathbf{b} + \frac{1-\alpha_U}{\alpha_U}\,a_P\,\mathbf{U}_P^{old}

(1)

where	meaning	unit
$a_P$	Diagonal coefficient of cell P in the momentum matrix	kg/s (per unit density)
$a_{nb}$	Coefficients of the neighbouring cells	kg/s (per unit density)
$\mathbf{U}_P$	Velocity of cell P	m/s
$\mathbf{U}_P^{old}$	Velocity of the previous iteration	m/s
$\mathbf{b}$	Source terms, including the pressure gradient	N (per unit density)
$\alpha_U$	Velocity relaxation factor	-

Default and rule

Default: auto: 0.7 in the start-up phase, 0.9 after it (SIMPLEC); 0.7 (plain SIMPLE, compressible); 0.9 (transonic SIMPLE)Computed by the programMeasured
Rule: Auto. SIMPLEC (incompressible default): 0.7 / 0.9; plain SIMPLE and the compressible SIMPLE solver: 0.7; transonic SIMPLE: 0.9 in both phases; pseudo-transient: not relaxed.
Source: Validation on the two-inlet test muffler (see Pressure relaxation): SIMPLEC with U 0.9 gave the lowest pressure residual (1.1e-2 after 600 iterations) and the steadiest pressure drop; with U 0.7 the residual was 1.6e-2. The lower start-up value 0.7 is a program choice that gives the crude start-up field the more robust setting.

Valid range 0 < α ≤ 1

Values

$0 < \alpha_U \le 1$ . With plain SIMPLE the program warns when a typed $\alpha_p + \alpha_U$ exceeds 1.1.

References 1

Patankar, S. V. and Spalding, D. B. (1972). A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. International Journal of Heat and Mass Transfer, 15(10), 1787-1806. DOI 10.1016/0017-9310(72)90054-3

Theory

Equations

Default and rule

Valid range 0 < α ≤ 1

Related 4

References 1