Open pipe termination in quiescent fluid
In the ideal open pipe termination limit discussed in the previous section, the radiation of sound from the open pipe termination was ignored.
This is the very low frequency limit. With increasing the frequency we get deviations. Firstly the inertia of the oscillating acoustic flow outside the pipe, around the open end, which implies that there is a finite pressure at the outlet of the pipe supplying the force needed for the acceleration of the fluid. As discovered by Bernoulli, this effect can be accounted for by assuming that the wave reflection occurs at a small distance 6 outside the pipe. This is called the end correction [Rayleigh (1954), Pierce (1990), Dowling and Ffowcs Williams (1983)]. The exact value of this end correction depends strongly on the geometry of the pipe termination. Whilst we get in the low frequency limit 6 = 0.61a [Levine and Schwinger (1948)] for a pipe with a radius a and infinitely thin walls (unflanged pipe) we get 6 = 0.82a for a flanged pipe (pipe end flush with a flat wall) [Morse and Ingard (1968), Peters et al. (1993)]. The order of magnitude of this end correction can be estimated by considering the pulsation of a sphere of radius a0. The end correction corresponds to the part of the solution (63) for the pressure field which does not carry energy (not in phase with iwa)
We find 6 = a0. Another way to look at this is to consider the incompressible part of the acoustic velocity (vr = i^a(a0/r)2 exp(iwt)) associated with the pulsation of the sphere and to integrate the radial component of the
momentum (p0(dv’r/dt = —(dp/dr)) to calculate the associated pressure on the surface of the sphere p(a0) = J Poш2d(a0/r)2dr = p0ui2aa0. It shows that for a 3-D radial flow the end correction is determined by the near field, which is incompressible.
This does not apply to a two-dimensional flow through a slit. Assuming an incompressible flow would result in an infinite large end correction, the near field is essentially compressible [Lesser and Lewis (1972)]. This illustrates the complexity of two-dimensional acoustic fields, as is discussed in Dowling and Ffowcs Williams (1983). In practice, this means that the use of a two-dimensional model for an unbounded flow can lead to unrealistically large radiation losses.
The fact that we hear music generated by a wind instrument is a clear demonstration that waves are radiated by the open pipe terminations (Figure 11). We now estimate the amplitude of these waves by coupling a plane wave propagation model in the pipe with a spherical wave emerging from the pipe termination. We assume that the frequency is so low that one can neglect compressibility in the region of the transition from plane waves to spherical waves.
In this case the mass conservation law implies:
Spv’x = 4nR2v’r (163)
where Sp is the pipe cross sectional area and R is the distance chosen such that it is small compared to the acoustic wave length, but large compared to the pipe radius (R > RSp). In terms of plane waves amplitude and spherical wave amplitude we have:
Sp(p+ – p-) = 4nR2 r f 1 + kR) exp(-*koR) – – i4n — ■ (164)
The conservation of acoustic energy over the same control volume yields:
sP(p+2 – pi2) -4п Щ – (165)
neglecting the phase of the waves and combined with (164) this implies:
Zr 1 — R (koa)2
Poco 1 + R 4
For a flanged pipe termination we have :
Zr 1 — R (koa)2
Poco 1 + R 2
Reflections on the flange (wall) double the radiation power (Figure 3).