Mean-flow convection

Adds the convection of sound by the mean flow of the CFD solution to the acoustic equation: waves travel faster downstream and slower upstream, and the port powers include the flow. Off by default: on the model tested it cost about four times the sweep time for a change of 0.04 dB.

Last updated 2026-11-11

Theory

With this option the acoustic unknown is still the pressure. The mapped mean velocity $\mathbf{U}(\mathbf{x})$ of the flow solution (Acoustic medium) enters the weak form through two additional matrices on the gas elements, equation (1): a symmetric one, $K_{UU}$ , and a skew one, $G$ , multiplied by $\mathrm{i}\omega$ . Packing and perforate terms are unchanged. The local Mach number is clipped at 0.5 (a fast local jet; the model is linear acoustics on a given mean flow). The system matrix is no longer symmetric, which makes it more expensive to factorise.

At the ports the mean flow has a normal Mach number $M$ (positive when leaving).

The axial wavenumbers and the modal power change, equations (2) and (3); for the plane mode the power factor is $(1+M)^2$ downstream and $(1-M)^2$ upstream (Morfey, 1971).
The acoustic mesh is refined to $N_\lambda/(1 - M_{max})$ elements per wavelength (Elements per wavelength), because a wave travelling against the flow is shorter.
The lossless energy balance and the passivity check become advisory: sound exchanges energy with a non-uniform flow.

The model is exact for a uniform flow; in a jet, the refraction of sound by the shear layer is not represented.

Equations

\left[K - K_{UU} - \omega^2 M + \mathrm{i}\omega G\right]\mathbf{p} = \mathbf{f}, \qquad K_{UU} = \int \frac{(\mathbf{U}\cdot\nabla p)(\mathbf{U}\cdot\nabla q)}{\rho c^2}\,dV, \qquad G = \int \frac{(\mathbf{U}\cdot\nabla p)\,q - p\,(\mathbf{U}\cdot\nabla q)}{\rho c^2}\,dV

(1)

\kappa = \sqrt{k^2 - (1 - M^2)k_t^2}, \qquad k_z = \frac{-Mk \pm \kappa}{1 - M^2}

(2)

w = \frac{S\,\mathrm{Re}(\kappa)}{2\rho\omega}\left(\frac{k}{k - Mk_z}\right)^2

(3)

where	meaning	unit
$K, M$	Stiffness and mass matrices of the Helmholtz equation	-
$\mathbf{U}$	Mapped mean velocity (local Mach at most 0.5)	m/s
$p, q$	Pressure and test function	Pa, -
$\rho, c$	Local density and speed of sound	kg/m³, m/s
$k$	Wavenumber $\omega/c$	1/m
$k_t$	Transverse wavenumber of the port mode	1/m
$M$	Normal Mach number of the mean flow at the port, + leaving	-
$k_z$	Axial wavenumber of the wave leaving (+) or entering (-)	1/m
$w$	Power per squared amplitude of the mode	W/Pa²
$S$	Port area	m²

Default and rule

Default

offMeasured

Rule

Off. The option would be enabled by default only if it cost less than about twice the sweep time and improved the results; on case V4 it cost about four times the sweep time and changed the transmission loss by at most 0.04 dB.

Source

The equations are the convected Helmholtz operator with the pressure as unknown, as derived for this program; the port power factor is that of (Morfey, 1971).

Measured cost and effect:

Case V4 (a 0.3 m expansion chamber with an extended inlet tube; 43.8k unknowns, no packing or perforates, so the system changes from real to complex): sweep 75.6 s instead of 18.5 s (4.1 times); the transmission loss changes by at most 0.04 dB (rms 0.02 dB) at its largest Mach number, 0.028.
Case V1 (a 2.0 m multi-chamber muffler with a perforated pipe and three absorptive packings; its system is already complex because of the packing, 119.5k unknowns): one factorisation 2.9 s instead of 2.0-2.4 s, and 1.20 GB instead of 0.79 GB.

Validation: a uniform-flow duct at Mach 0.2 transmits with $\lvert B\rvert = 1$ and the phase $e^{-\mathrm{i}kL/(1 \pm M)}$ within 0.02 rad, with a transmission loss below 0.02 dB and an inlet power factor of $(1+M)^2$ .

Valid range on | off; needs fluid flow and the acoustic medium from the CFD fields

Verified

Run on a real muffler with the mapped CFD medium (2026-11-11): at a duct Mach number of about 0.03 it changed the transmission loss by less than 0.05 dB and cost about 1.6 times the run time. Worth the cost only where the duct Mach number is above about 0.1.

Requirements

Needs fluid flow and the acoustic medium "From CFD fields" (which needs the steady compressible solver); refused otherwise.

Cost

1.3-1.5 times the time with packing or perforates, about 4 times without, and more memory.

No insertion loss

Switches the insertion loss and the radiated SPL off. No outlet termination in the program is reliable with flow: part of the sound reaching the pipe end is absorbed by vortex shedding at the edge instead of being radiated, and published results cover only low frequencies ( $ka \le 0.5$ ) and cold jets. The full reasoning is given in Compute the insertion loss. The transmission loss (anechoic ports) is computed as usual.

References 1

Morfey, C. L. (1971). Acoustic energy in non-uniform flows. Journal of Sound and Vibration, 14(2), 159-170. DOI 10.1016/0022-460X(71)90381-6

Theory

Equations

Default and rule

Valid range on | off; needs fluid flow and the acoustic medium from the CFD fields

Related 4

References 1