Vortex sound theory and whistling
5.4 Powell/Howe analogy
The quantitative relationship between vortex shedding and sound production was first established by Powell (1964). His approach was limited
to free field conditions and low Mach numbers. Howe [Howe (1975), Howe (1998), Howe (2002)] proposed a generalization of this approach to arbitrary Mach numbers, which is valid for confined flows. In its most general form it implies a numerical solution of a complex convective wave equation [Doak (1995), Musafir (1997)] and it is mostly used at low Mach numbers. In most cases it is used for the analysis of the sound production based on an energy corollary, which we are looking at.
In section (5.2) we have seen that the choice of variable is important in an analogy, because it determines the approximations that are intuitively reasonable. In section (6.4) we have seen that in the presence of a frictionless mean flow, the total enthalpy fluctuation is a natural variable. Following equation (31):
dt + VB = — (w + v) + f (190)
we see that the Corriolis acceleration (w x v) acts as a source of sound if we define the acoustic velocity U as the time-dependent part of the potential flow in a Helmholtz decomposition of the flow velocity:
v = V(p0 + ) + Vx ф. (191)
This yields the definition proposed by Howe (1984) for the acoustic velocity:
U = V<f’ . (192)
For low Mach number flows Howe (1984) proposes the use of the following approximation for the time average acoustic power < P > produced by a flow:
< P >= —po < I (w x v) • U dV > . (193)
This corresponds to the use of the energy corollary (64) assuming f = —p0(w x v). This intuitive statement gives an excellent insight into the sound production associated with vortex shedding in low Mach number flows, which is due to the fact that vorticity is a conserved quantity in 2D frictionless flows. We therefore have an intuition for the dynamics of vortices in such flows [Prandtl (1934), Milne-Thomson (1952), Paterson (1983), Saffman (1992)].
A drawback of the vortex sound theory is that it stresses the dipole character of the sound source: V – (w x v). Unlike the analogy of Lighthill, it does not impose a quadrupole character to the sound field. Thus, in order
to apply this analogy to flows, such as a free jet, one has to use analogies as proposed by Mohring (1978) or Schram and Hirschberg (2003), which do take this aspect into account. In our discussions we limit ourselves to applications with a dominant dipole source term. In such cases the formulation of Howe (1984), as given in equation (193) can be used.