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Parameter · 03 Model

Time dependency

model.time_dependency Step 03 · Model

Steady: the time derivative is dropped and the equations are solved iteratively (SIMPLE) until the mean of the pressure drop has converged. Transient: the flow is marched in physical time (PIMPLE, the transient incompressible solver) and the pressure drop is the time average over the last window of simulated time.

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Theory

A steady RANS solution of a muffler often does not settle completely: the jets out of tubes and holes keep the solution oscillating by 5-30 % from one iteration to the next. The program therefore reports the MEAN of the pressure drop over a window of iterations and stops when that mean has converged (Δp mean-stability tolerance), equation (1). This is cheap and is the default.

A transient run solves the time-dependent equations with an adaptive time step limited by the Courant number (Maximum Courant number). The pressure drop is the time average over the last averaging window, equation (2), and quality gate 2 checks that this average agrees with the one of the window before it, equation (3). Residuals are not judged in a transient run. It is incompressible only, and is much slower: the time step follows the smallest (boundary-layer) cells.

Equations

Δp0‾=1N∑i=n−N+1nΔp0,i\overline{\Delta p_0} = \frac{1}{N}\sum_{i=n-N+1}^{n} \Delta p_{0,i}
(1)
Δp0‾ t=1nW∑ti>tend−TWΔp0(ti)\overline{\Delta p_0}^{\,t} = \frac{1}{n_W}\sum_{t_i > t_{end} - T_W} \Delta p_0(t_i)
(2)
∣Δp0‾ t∣(tend−TW, tend]−Δp0‾ t∣(tend−2TW, tend−TW]∣∣Δp0‾ t∣(tend−TW, tend]∣<max⁡(ε, 1 %)\frac{\left\lvert \overline{\Delta p_0}^{\,t}\big|_{(t_{end}-T_W,\ t_{end}]} - \overline{\Delta p_0}^{\,t}\big|_{(t_{end}-2T_W,\ t_{end}-T_W]} \right\rvert}{\left\lvert\overline{\Delta p_0}^{\,t}\big|_{(t_{end}-T_W,\ t_{end}]}\right\rvert} < \max(\varepsilon,\ 1\,\%)
(3)
wheremeaningunit
Δp0,i\Delta p_{0,i}Total-pressure drop at iteration ii (mass-flow weighted)Pa
NNSamples in the averaging window (steady)-
nWn_WSamples in the last time window-
TWT_WAveraging window (transient)s
tendt_{end}Last simulated times
ε\varepsilonΔp mean-stability tolerance%

Default and rule

Default
steadyMeasured
Rule

Steady.

Source

Validation results from a residual-sensitivity study: on a coarse straight pipe the transient run gave 14.2 Pa against 15.4 Pa steady; on two mufflers the second-order steady window mean lay about 4 % above the time-accurate average. The transient defaults (Courant 2, 2 outer correctors, first-order turbulence convection) were the only combination that stayed stable in that study; Courant 5 with one outer corrector diverged on every case tested. PIMPLE and the solvers: (OpenFOAM user guide, 2026).

Valid range steady | transient (incompressible only)

Compressible

Transient is available for the incompressible analysis only; heat transfer and the mapped acoustic medium need a steady run.

Cost

Thousands of time steps are needed, each as expensive as a steady iteration; a real muffler run took most of a day. A transient run cannot be paused and continued.

References 1

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