Discharge coefficient C_D (acoustics)

The ratio of the effective to the geometric open area of the holes. Flow through a sharp-edged hole contracts to a vena contracta, so the hole behaves as if it were smaller. The coefficient is used by the Elnady impedance model, by the thin-plate option "use C_D" and by the bias-flow term.

Last updated 2026-11-03

Theory

Because the jet issuing from a sharp-edged hole is narrower than the hole, the effective open-area ratio of the plate is $\sigma C_D$ . The coefficient enters the acoustic models as follows:

Elnady model (Elnady et al., 2010): the complete hole impedance (viscous tube term and end corrections) is divided by $C_D$ .
Bias flow: the added resistance is $1.15\,M_b/C_D$ , equation (1).
Thin-plate physics baseline: the hole impedance is divided by $C_D$ when Thin plate: use C_D is enabled.

On Auto, the program interpolates $C_D$ bilinearly in a table of hole diameter (0.6-5 mm) and wall thickness (0.5-2.5 mm), equation (2). Outside the table the nearest edge value is used and a note is written to the report.

In a CFD + acoustics case with a perforated tube, the coefficient is entered once, on the flow section of the card (Discharge coefficient C_D0). The same value then also sets the steady-flow loss coefficient $K = 1/(\sigma C_D)^2$ .

Equations

\zeta_h \propto \frac{1}{C_D}\left[\frac{ik\,t}{F(K'd)} + \frac{ik\,\delta\,f_{int}}{F(Kd)}\right], \qquad \theta_{bias} = \frac{1.15\,M_b}{C_D}

(1)

C_D = (1 - f_t)(1 - f_d)\,T_{i,j} + (1 - f_t)f_d\,T_{i,j+1} + f_t(1 - f_d)\,T_{i+1,j} + f_t f_d\,T_{i+1,j+1}

(2)

where	meaning	unit
$C_D$	Discharge coefficient	-
$M_b$	Bias (through-hole) Mach number	-
$T_{i,j}$	Table value at thickness row i, diameter column j	-
$f_t, f_d$	Fractional position of t and d between the bracketing table rows and columns	-

Default and rule

Default: Auto: bilinear interpolation in the hole-diameter × wall-thickness tableComputed by the program
Rule: Auto (table interpolation). A value entered by the user is applied unchanged in every model that uses $C_D$ .
Source: The table contains the DC-flow discharge coefficients measured by Elnady as a function of hole diameter and wall thickness: Table 1 of Paper III (Elnady and Boden, 2004) of his doctoral thesis (Elnady, 2004). The measurements cover hole diameters of 0.6-2.4 mm and wall thicknesses of 0.8-2.4 mm; the remainder of the table is Elnady's extrapolation, which shows that above a hole diameter of 2.5 mm $C_D$ depends on the wall thickness only. The table values are not reproduced here; refer to the thesis. The way $C_D$ enters the hole impedance follows (Elnady et al., 2010), equations (6)-(7).

Valid range 0.05 ≤ C_D ≤ 1

Values

0.05 to 1. The table values lie between approximately 0.7 and 0.9.

Extrapolated values

For hole diameters outside 0.6-2.4 mm or wall thicknesses outside 0.8-2.4 mm, the table value is an extrapolation. Outside the table limits the edge value is used, and the report says so.

Theory

Equations

Default and rule

Valid range 0.05 ≤ C_D ≤ 1

Related 7

References 3