Flow Through the Reed Orifice

• The flow through the reed orifice is typically determined using the Bernoulli equation for steady, laminar flow of an incompressible fluid (Fletcher and Rossing, 1991; Saneyoshi et al., 1987; Thompson, 1979; Worman, 1971; Backus, 1963; Wilson and Beavers, 1974).

• Relating pressure ($p$) and velocity ($v$) at any two points of height $y$ within a continuous tube of flow, the Bernoulli equation is given by
    $$p_1 + \frac{1}{2} \rho \, v^{2}_{1} + \rho \, g \, y_1 = p_2 + \frac{1}{2} \rho \, v^{2}_{2} + \rho \, g \, y_2,$$ (20)
  where $\rho$ is fluid density and $g$ is the acceleration of gravity. This expression is based on continuity of volume flow and conservation of energy.

  Figure 9: An approximate reed orifice geometry.

  

• For the reed geometry as approximated in Fig. 9, the upstream cavity is viewed as a large tank of constant or slowly varying pressure $p_u$ and essentially zero volume flow $u_u$.

• Application of the Bernoulli equation, assuming no appreciable change in height, leads to an expression for the flow through the reed orifice of the form
    $$u = w \, x \left(\frac{2 \, \vert p_{\Delta} \vert}{\rho} \right)^{1/2} \mbox{sgn($p_{\Delta}$)},$$ (21)
  where $w$ and $x$ are the width and height of the reed channel, respectively, and the sign function is
  $$\mbox{sgn($p_{\Delta}$)} = \left\{ \begin{array}{cl} +1 & \mbox{if $p_{\Delta} \ge 0$} \\ -1 & \mbox{otherwise}. \end{array} \right.$$

• An example flow curve determined using Eqs. (21) is shown in Fig. 10.

  Figure 10: Flow characteristic through the reed orifice.

  

• Although the flow expression of Eq. (21) is derived for steady flow, its use is generally assumed to be valid under oscillating conditions as well (referred to as “quasi-static”). The validity of this assumption has been questioned by results of da Silva et al. (2007) and others.

• The acoustic resistance to flow presented by the orifice is inversely proportional to the slope of the flow curves in Fig. 10 (Fletcher and Rossing, 1991). For low values of $p_{\Delta}$, the resistance is given by a small positive value. At very high pressure differences, the reed is blown closed against the mouthpiece facing and all flow ceases.

• The threshold blowing pressure, which occurs at the flow peak, can be determined from Eqs. (21) and (19) and is roughly equal to 1/3 the closure pressure. Between the threshold and closure pressures, the flow resistance is negative and the reed mechanism functions as an acoustic generator.