Grazing correction

For a perforated tube: lowers the discharge coefficient of the holes according to the flow that grazes along the tube, taken from the solved flow, and updates $K$ during the run. With the option off, $K$ stays at $K_0 = 1/(\sigma C_{D0})^2$ .

Last updated 2026-11-03

Theory

Flow running along a perforated wall deflects the jets and narrows their effective area, so the discharge coefficient drops as the ratio of hole velocity $V_i$ to grazing velocity $V_\infty$ falls. Rogers and Hersh measured this for flow from the grazing stream into a cavity (inflow, equation 2) and from a cavity into the stream (outflow, equation 3) (Rogers and Hersh, 1975). The program floors the ratio at their lowest measured value, 0.004, and limits the result by the no-grazing value, equation (1).

From the solved flow, every face of the wall gives a normal velocity $U_n$ (positive = out of the tube, which is Rogers and Hersh's inflow) and a grazing speed (Grazing velocity). Each face gets its own $K_f$ . The faces are combined into one $K$ for the wall with the weight of the power each face dissipates, $\lvert U_n\rvert^3 A$ , equation (4), so that the uniform $K$ removes the same power at the same face velocities. The new value is under-relaxed in log space (K relaxation).

When $K$ is evaluated:

1. Once at the end of the upwind start-up phase; the value is written into the restart that the switch to the final schemes makes anyway. 2. During the final phase, every K check interval iterations from the written samples, at most Maximum K evaluations evaluations in all. 3. A check that moves the relaxed $K$ by more than 2 % stops the solver, writes the new $K$ into the saved pressure field and restarts it; the first check within 2 % ends the checks.

The pressure-drop averaging window starts after the last real restart. Every evaluation is recorded in logs/perforate_k.json.

Equations

C_{d,f} = \min\left(C_{De},\ C_{D0}\right), \qquad r = \max\left(\frac{V_i}{V_\infty},\ 0.004\right), \qquad V_i = \frac{\lvert U_n\rvert}{\sigma}

(1)

C_{De} = 0.80\,r^{0.47} \quad (U_n > 0\text{: out of the tube, into the cavity})

(2)

C_{De} = 0.97\,r^{0.534} \quad (U_n < 0\text{: from the cavity into the tube})

(3)

K = \frac{\sum_f K_f\,\lvert U_{n,f}\rvert^3 A_f}{\sum_f \lvert U_{n,f}\rvert^3 A_f}, \qquad K_f = \frac{1}{(\sigma\,C_{d,f})^2}

(4)

where	meaning	unit
$V_i$	Mean velocity in the holes	m/s
$V_\infty$	Grazing (core) velocity	m/s
$U_n$	Normal velocity on the wall face, + out of the tube	m/s
$\sigma$	Open-area ratio	-
$C_{D0}$	No-grazing discharge coefficient	-
$A_f$	Area of face $f$	m²
$K$	Loss coefficient used for the whole wall	-

Default and rule

Default

onMeasured

Rule

On.

Source

Correlations: Rogers and Hersh eqs. (33) and (35), measured on a 12.7 mm square-edged orifice, $L/D = 1$ , $V_\infty$ 24-85 m/s, $V_i/V_\infty$ 0.004-1 (Rogers and Hersh, 1975).

The program checks $K$ rather than restarting at every update because every solver restart disturbs SIMPLE whether or not $K$ changes. On the two-baffle test muffler the first iteration after a restart showed a pressure residual of about 0.4, a continuity error of about 0.9 and $\Delta p_0$ 17 % high for about 40 iterations. Validation on the same muffler (4 perforated tubes): $K$ moved 0.09 % at the first check, so no restart was made, and $\Delta p_0$ was 2229 Pa, the same as with a restart at every update.

Valid range on | off

Outside the tested range

Not modelled: the effect of $L/D$ on inflow and of the boundary-layer thickness that Rogers and Hersh report. A grazing speed above 85 m/s, or more than 5 % of the face below the ratio 0.004, gives a note in the K record.

Steady only

A transient run keeps $K_0$ ; the correction is not applied there.

Perpendicular baffles

Where the whole flow crosses the holes ( $V_i \ge V_\infty$ ) the grazing coefficient never goes below $C_{D0}$ and $K$ stays near $K_0$ .

References 1

Rogers, T. and Hersh, A. S. (1975). The effect of grazing flow on the steady state resistance of square-edged orifices. 2nd Aeroacoustics Conference, AIAA Paper 75-493. DOI 10.2514/6.1975-493

Theory

Equations

Default and rule

Valid range on | off

Related 7

References 1