Theory

The Courant number of a cell is the number of cells the flow crosses in one time step, equation (1). With adjustTimeStep yes, OpenFOAM computes the largest Courant number of the mesh after each step and scales the next step towards $Co_{max}$ , growing by at most 20 % per step and never beyond Maximum time step, equation (2) (OpenFOAM user guide, 2026).

The largest Courant number occurs in the smallest cells, the boundary-layer cells next to fast walls, so these cells set the time step of the whole run: about $1.6\times10^{-5}$ s at Courant 1 on a 60 mm pipe.

PIMPLE with several outer correctors per step (PIMPLE outer correctors) tolerates Courant numbers above 1. A transient run always uses first-order upwind convection for $k$ and $\omega$ (k and ω convection scheme).

Equations

Co = \frac{\Delta t}{2V_P}\sum_f \lvert\phi_f\rvert

(1)

\Delta t^{n+1} = \min\!\left(\min\!\left(\lambda,\ 1 + 0.1\lambda,\ 1.2\right)\Delta t^{n},\ \Delta t_{max}\right), \qquad \lambda = \frac{Co_{max}}{Co^{n}_{max}}

(2)

where	meaning	unit
$Co$	Courant number of cell P	-
$\Delta t$	Time step	s
$V_P$	Cell volume	m³
$\phi_f$	Volume flux through face f	m³/s
$Co^{n}_{max}$	Largest Courant number of the mesh at step $n$	-
$\Delta t_{max}$	Maximum time step	s

Default and rule

Default

2Measured

Rule

2.

Source

Validation runs of the program (residual-sensitivity study):

Courant 5 with one outer corrector, an earlier default, diverged on every transient case tried, a plain verification pipe included: $\omega$ rose to about $10^{65}$ and the time step collapsed to about $10^{-13}$ s within a few hundred steps.
Courant 2-3.5 with 2 outer correctors and first-order $k$ , $\omega$ convection was stable on the pipe and on the mufflers.
On case V2 (a two-inlet single-chamber muffler with internal inlet and outlet tubes) at Courant 3.5, the run was stable through at least 0.15 s of simulated time, with steps of about $10^{-5}$ s.

Equation (2) is OpenFOAM's time-step adjustment as used by the transient incompressible solver.

Valid range > 0; validated from 2 to 3.5 with 2 outer correctors

Values

Above 0. With 1 outer corrector, keep it at or below 1.

Above 3.5

Not validated. Courant 5 diverged on every case of the study with the earlier settings.

References 1

OpenCFD Ltd. (2026). OpenFOAM user guide. openfoam.com. Web page

Theory

Equations

Default and rule

Valid range > 0; validated from 2 to 3.5 with 2 outer correctors

Related 5

References 1