Theory

$k$ and $\omega$ must stay positive. A second-order scheme that overshoots near the steep gradients of a jet edge can produce negative values, which the solver clips ("bounding"), and this slows convergence. Every steady run therefore starts with first-order bounded Gauss upwind for $k$ and $\omega$ for Upwind iterations before the switch iterations, then restarts with the scheme chosen here (OpenFOAM user guide, 2026):

limitedLinear 1: a TVD blend of central and upwind interpolation with the limiter of equation (1); it returns to upwind where the solution is not smooth, so it stays bounded.
linearUpwind: the upwind value extrapolated with a cell-limited gradient (limitedGrad), equation (2). Less limited, sharper jets; on a poor mesh $k$ and $\omega$ can go negative and be clipped.
upwind: first order throughout, equation (3).

Steady runs use all of them in the bounded form (the continuity error of the iteration subtracted, see Momentum (U) convection scheme). The gradients of $k$ and $\omega$ are cell-limited too, because the SST cross-diffusion term uses $\nabla k\cdot\nabla\omega$ directly.

A transient run always uses first-order Gauss upwind for $k$ and $\omega$ , whatever this setting says.

Equations

\phi_f = \psi(r)\,\phi_f^{lin} + \left(1 - \psi(r)\right)\phi_C, \qquad \psi(r) = \max\!\left(0,\ \min(2r,\ 1)\right)

(1)

\phi_f = \phi_C + (\mathbf{x}_f - \mathbf{x}_C)\cdot(\nabla\phi)_C

(2)

\phi_f = \phi_C

(3)

where	meaning	unit
$\phi$	$k$ or $\omega$	m²/s², 1/s
$\phi_f$	Value interpolated to face f	m²/s², 1/s
$\phi_C$	Value of the upwind cell	m²/s², 1/s
$\phi_f^{lin}$	Central (linear) interpolation	m²/s², 1/s
$\psi$	Limiter, with $r$ the smoothness ratio of Momentum (U) convection scheme	-
$\mathbf{x}_f, \mathbf{x}_C$	Face and upwind cell centres	m

Default and rule

Default: auto: limitedLinear 1Computed by the programMeasured
Rule: Auto = limitedLinear 1, for both analysis types. A typed linearUpwind gets a warning for a compressible run.
Source: limitedLinear 1 is the program's default for the muffler jets, chosen from validation runs as robust and bounded. In the program's residual-sensitivity study the transient solver with limitedLinear 1 for $k$ and $\omega$ diverged on every case tested, a plain pipe included ( $\omega$ reaching about $10^{65}$ ); with first-order upwind it was stable. This is the basis of the fixed first-order scheme of transient runs. Schemes: (OpenFOAM user guide, 2026).

Valid range auto | limitedLinear | linearUpwind | upwind; steady runs only

Choice

Auto (limitedLinear 1), limitedLinear, linearUpwind, upwind. Not used by transient runs.

linearUpwind for compressible

Near high-Mach jets $k$ and $\omega$ can go negative and be clipped.

References 1

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

Theory

Equations

Default and rule

Valid range auto | limitedLinear | linearUpwind | upwind; steady runs only

Related 4

References 1