There have been many attempts to develop mathematical models for the prediction of what is often referred to as "aerodynamic noise", in some cases acceptable correlation with experimental data has been achieved for particular situations but no universal theory has yet emerged. It is generally accepted that broad band sound will be generated in turbulent shear layers and Lighthill's theory (summarised in Goldstein (1976)) was developed to explain sound generated in the shear layers between jets and the surrounding air. Of greater interest in connection with aeroacoustics is the instability of the wakes of immersed bodies and the consequent formation of vortices as this is frequently the source of excitation of mechanical and/or acoustic resonances which often result in Sound Pressure Levels up to 165 dB being generated by flow at low subsonic velocities.
The resonances observed in the airflow model of the Hinkley Point circulators (see para 2, section 3, above) were described by Forster in a written contribution to the discussion of the paper by Rizk and Seymour (1964-5). The excitation was clearly shown to be associated with the wakes of the diffuser centrebody support spokes and a survey with a stethoscope identified a number of clearly defined acoustic modes. These observations led to a series of experimental investigations of flow induced resonances in various test rigs designed to facilitate investigations of the acoustic properties of the flow passages, the effects of mechanical vibration and the excitation mechanisms.
In the interests of simplicity, the initial tests were based on rectangular cascades of flat plates and included the basic case of a single plate spanning a rectangular passage. The first series of tests was designed to demonstrate that the resonances could be purely acoustic; two geometrically identical cascades, one with aluminium plates and the other with brass plates were tested. The two sets of plates had different natural frequencies of mechanical vibration but the frequencies of the resonances excited by both cascades were identical (Parker, 1966).
The acoustic mode of each resonance was explored with a probe microphone which disclosed that the wavelength along the cascade at each frequency corresponded to an effective acoustic velocity less than the "velocity of sound" and the amplitudes decreased progressively with distance upstream and downstream of the cascade. The development of a method of predicting the eigenfrequencies and modal distributions of the resonances (Parker, 1967) also provided an explanation of the fact that the acoustic energy was not dissipated upstream and downstream of the cascades. This amounted to finding solutions of the wave equation:
where C is the "velocity of sound", i.e. the velocity at which plane uniform sound waves propagate through the fluid. The required solutions are
for regions upstream and downstream of a straight cascade between end walls
and
for rotating waves in the regions upstream and downstream of annular cascades with high inside diameter/outside diameter ratios.
where
2 - 
2) 0.5/C
= 2 
C/L
and
is the frequency
The experiments showed that w was always less than W and therefore, for a resonance to build up n is real and the amplitude varies exponentially with increasing distance upstream and downstream of the cascade. The u component of the acoustic velocities may be obtained by differentiating with respect to x and integrating with respect to t from which it is seen that they are out of phase with the pressures. Integration of the product p(t)u(t) over a complete cycle shows that the net energy propagation in the x direction is zero. Similar procedures show that the net energy propagation in the y direction is zero for the cascade between end walls but non zero for travelling waves associated with annular cascades, in the latter case acoustic energy merely rotates continuously and there is again no overall loss.
Generally, analytical solutions for cascades have to be obtained by numerical methods and the above provides a way of dealing with the upstream and downstream computational boundaries. For practical annular cascades with inside/outside diameter ratios significantly less than 1.0, the solutions involve Bessel's functions and are more difficult to manage but the general conclusions regarding energy containment are the same. The experimental results associated with the above are reported in several papers (Parker, 1966, 1967a; Parker and Griffiths, 1968; Parker and Pryce, 1974). There have been a number of theoretical papers on straight cascades but, where the blade stagger is not zero, correlation with experiment is generally not good, mainly because incorrect assumptions are made as regards the configuration of the nodes upstream and downstream of the cascade. Experimentally the nodes are found in planes normal to the cascade while some theories assume (wrongly) that they are aligned with the blade chord. A further complication arises because, in cases of interest in turbomachines, the circumferential wavelength is not directly related to the circumferential pitch of the blades. This case can be dealt with by considering a single blade pitch with a travelling acoustic wave (the usual situation in a turbomachine resonance) and using a periodicity condition with an appropriate phase shift. This technique has been tested for a cascade of uncambered thin blades at zero stagger with the acoustic pressure (or velocity potential) represented as complex numbers but the results have not been published and, as far as is known, the method has not been developed for real cascades.
Resonances of a similar nature have also been observed in "egg box" type flow straighteners and protective grids (Spruyt, 1970) and also in passages containing heat exchanger tube banks (Parker, 1978; Burton, 1980; Blevins, 1986; Weaver and Fitzpatrick, 1987; Fitzpatrick, 1985). The essential feature of all the observed acoustic resonances has been that the flow passages contained immersed bodies which reduced the effective velocity of sound propagation in directions normal to the flow (and sometimes in the flow direction as well), eigenvalue solutions of the wave equation for such situations have been found by using exponential decay along the passage as the computational boundary condition of the regions upstream and downstream of the immersed bodies.
Acoustic resonances have been observed in axial flow compressors. They consist of waves rotating round the annulus containing the blades and in the spaces immediately upstream and downstream and are, in many ways similar to resonances in single cascades. The first reported case occurred in the rotor - stator combination of a low speed single stage research compressor (Parker, 1967b, 1968) and similar resonances have been found in a four stage low speed research compressor (Camp, ??). Although still not fully understood, acoustic resonances which are believed to have caused blade failures have been observed in high speed, multistage compressors associated with aero-engine research and development but, for commercial reasons, the details have not been published.
To define the modes in a compressor it is necessary to specify the number of "lobes" (complete cycles) of the rotating wave and the axial variation of amplitude which appears to be related to the location of the source of excitation but invariably includes exponential decay upstream and downstream of the bladed region. It is obvious that, when attempting to predict the resonance frequencies, the effect of the blades (both stators and rotors) have to be taken into account but it would not be practical to treat each blade separately so methods of specifying representative homogeneous properties of the fluid in the annulus are required. An attempt to do this by allowing for a reduction in the effective velocity of sound due to scattering by the blades and the effects of the temperature rise through the stages (Parker, 1984) produced results which did not compare well with the limited experimental data available from full scale multistage machines. The discrepancies suggest that acoustic absorption is an important factor which should be included in the governing equation and that it will also be necessary to identify the blade rows which generate the excitation. Identification of potential sources of excitation requires a comprehensive knowledge of the incidence at which each blade row is operating, the resulting vortex shedding characteristics of the blades and the downstream interactions as vortices convect through the succeeding blade rows.
The only source of energy which can drive a flow induced resonance is the fluid flow itself and a corresponding pressure drop along the flow passage has been measured (Parker and Pryce, 1974). The excitation mechanism is a self induced oscillation involving a feed-back loop which can be described in general terms but the variables are difficult to quantify. Figure 1 indicates the variables and connections between them which will have to be considered in constructing a mathematical model of the loop which must also include the effects of acoustic and mechanical damping on amplitude and phase as well as vortex convection times.
Examination of the wakes of bodies immersed in a steady fluid flow show fluctuations at an average frequency which is related to (generally increasing monatonically with) the flow velocity but in the short term there are random variations in the frequency and correlation lengths are of the same order of magnitude as the wake thickness. Experimental observations (for example, Parker (1969)) have shown conclusively that, when a resonance is excited, the wake fluctuations and vortex shedding become highly correlated over large distances and are locked to the frequency of the resonance which is at or close to the eigenfrequency of the mode. It is also generally observed that the effect on the flow manifests itself in the form of shed vortices which can be observed by flow visualisation with stroboscopic illumination. The same techniques have shown that a resonance also affects any separation bubbles which may occur in the flow, variations in the reattachment position being correlated in the same way (Parker and Welsh, 1983).
It is usual to describe of the excitation of the resonances as "vortex shedding excitation" though some workers suggest that the discrete vortices may be a consequence of, rather than the principal factor in, the excitation mechanism. Whatever the truth of this argument, flow visualisation and measurements using anemometers or pressure sensors indicate that vortices are shed in all cases. Welsh and others at Commonwealth Scientific and Industrial Reasearch Organization (CSIRO) in Australia (Welsh et al., 1984; Stoneman et al., 1985; Stokes and Parker, 1988) have developed computational methods of predicting the excitation of resonances by integrating the acoustic energy generated by the convection of vorticity through the acoustic fields around the trailing edges of flat plates and aerofoils by using Howe's equation (Howe, 1975).
When the shed vortices convect downstream they can generate further acoustic energy at the shedding frequency if they interact with the acoustic and velocity fields around other bodies in the flow. This generates acoustic energy at the vortex shedding frequency and the phase, relative to the excitation due to shedding, varies according to the distance downstream and the convection velocity. The consequence is that the velocity range over which a resonance is excited varies with the spacing between the shedding and interacting bodies and, in many cases, excitation occurs over two velocity ranges separated by a range with no resonance (Legerton et al., 1991a, 1991b). Downstream interaction also creates an apparent resonance effect in non enclosed flows in which the frequency varies with the spacing. Powerful, discrete tones were observed when various bodies ranging from flat plates and single round wires to gauzes and honeycomb structures were placed downstream of a plate in an open jet. As the spacing was increased it was found that the frequency varied between upper and lower limits in a series of ranges in which the frequency decreased progressively and then increased stepwise giving a saw tooth relationship between spacing and frequency (unpublished tests at the CSIRO, 1981).
A particularly important aspect of downstream interaction occurs in axial flow compressors where vortices shed from one blade row interact with the blades in succeeding rows (Parker and Stoneman, 1985, 1987) It has been found that the relationship between compressor speed and the frequencies of resonances which occur in both single stage research compressors and full scale multistage compressors can appear as multiple "families" of resonances which change if the axial spacing between blade rows is varied.
One aspect of aeroacoustics which requires considerable further research is the problem of predicting the frequency (and therefore mode) which will be excited at any particular flow velocity in configurations such as cascades of compressor blades. Experiments with round bars and flat plates with semicircular edges suggest that, in non resonant conditions, the frequency can be estimated by assuming a Strouhal number of about 0.2. Many writers appear to believe that this is a universal value provided the linear dimension is adjusted for boundary layer or wake thickness but, for most other profiles this is far from reliable, for example, for plates with square edges the Strouhal number varies with chord/thickness ratio, (Parker and Welsh, 1983; Stoneman et al., 1985; Stokes and Parker, 1988).
Where the "natural" shedding frequency (sometimes referred to as the Strouhal frequency) is known, it provides a rough guide to the lowest velocity at which a particular resonance will be excited but, once the resonance has been established, it normally persists over a considerable velocity range with a small increase in frequency as shown in Figure 2.
The higher velocities correspond to lower Strouhal numbers and there is no reliable way of estimating the upper velocity limit (i.e. the lowest Strouhal number) as it appears to depend on several factors such as acoustic absorption and convective effects which have not generally been quantified. The situation is further complicated by a form of hysteresis effect so the actual resonance excited at any particular time often depends on the history of the preceding velocity variation.
The following movie segments show the wake behind a flat plate with semi-circular edges in a passage with a rectangular cross section. Smoke is introduced upstream and is illuminated by a stroboscopic light driven at a frequency slightly below the tunnel resonance frequency. The flow velocity is increased progressively from a value below that at which the resonance is excited. At first the vortices are uncorrelated (Movie 1 (.mpeg)), when the resonance starts the vortex shedding becomes highly correlated and locks on to the resonance (Movie 2 (.mpeg)). The velocity is then reduced slowly and the shedding reverts to natural, uncorrelated shedding when the resonance ceases (Movie 3 (.mpeg)).
To understand vortex shedding from cascades of aerofoil blades it would seem logical to start by examining isolated aerofoils in non resonant conditions before progressing to resonant conditions. However we find that this, apparently simple case, is not really understood and there is no reliable way of predicting vortex shedding frequency for a given aerofoil at a given incidence. The complexity of this topic is illustrated by a "ladder effect" first reported by Patterson et al. (1973) and illustrated in Figure 3.
Arbey and Bataille (1983) proposed an explanation based on "vortical disturbances" developing in the boundary layers and convecting towards the trailing edge to develop into the shed vortices. The steps between the lines result from changes in the number of developing vortices present between the point of origin and the separation point.
The application of flow stability theory has been shown to offer a promising line of approach to the prediction of aerofoil vortex shedding frequency and comparison of predicted shedding frequencies with measurements in a water tunnel gave good results for laminar flow (Gorman et al., 1992). An attempt to extend the comparison to higher Reynolds Numbers using a specially built, low turbulence wind tunnel was less successful. The tunnel had solid walls and the original intention was that wake profile and vortex shedding frequency measurements would be made at velocities at which no resonances were excited to validate the stability analysis over a range of Reynolds Number and then to progress to examination of resonant conditions. In the event it was found that, at low Reynolds Numbers, vortices were shed at frequencies which increased continuously with velocity but, at all velocities from that at which the shedding frequency was the same as the first acoustic resonance up to the maximum attainable in the tunnel, resonances were excited at all velocities as shown in Figure 4.
Each of the segments starting slightly below the tunnel resonance and ending slightly above are similar to the single resonances found with flat plates where the frequency variation is known to be associated with a progressive phase change between the acoustic field and the vortex shedding (Parker, 1969; Parker et al., 1993). The discontinuities are believed to be due to step changes in the number of vortices in the boundary layer similar to that suggested as an explanation of Patterson's ladder effect (Arbey and Bataille, 1983) but there was no evidence of any ladder effect in the low velocity region where conditions might have been expected to correspond to Patterson's data. There are also short ranges where frequencies well removed from the tunnel resonance are excited, these have not been explained.
In many situations the geometry of the vortex shedding body can be altered without prejudicing the performance of the equipment of which it forms a part and this has often provided a solution to the problem. One example in which the leading edge was modified involved resonances generated in a protective grid placed in front of large ventilation fans in a long road tunnel (Fitzpatrick, 1985). The problem was solved by welding round bars along the leading edges to produce flow separation. It is, however, more common to modify the trailing edges, one profile which has been very successful in this is shown in Figure 5.
The shape is similar to one developed in the course of an experimental investigation of trailing edge shapes to suppress mechanical vibration in water turbine runners (Donaldson, 1956) but has been used to suppress acoustic resonances in cascade corner vanes, spokes in annular diffusers and other configurations.
In other situations a cure similar to the strakes used to prevent wind driven oscillations of steel chimneys has been used. This has been found to be effective in suppressing noise in many situations such as the bars in car roof racks. A rather unusual example was an eight bladed windmill a Heckington in Lincolnshire (UK) which was renovated some years ago. There was no problem while the mill was operating but, in high winds the mill is stopped by turning the slats which form the sails to present open grids which allow the wind to blow straight through. Unfortunately this results in each sail forming a straight cascade of plates which resonated due to vortex shedding and became a serious environmental noise problem which could not be cured by banning operation of the mill. It was in fact cured by wrapping lengths of stout cord in coarse spirals round each slat and sticking them to the surfaces with paint. This problem, which is known as "trumpeting", is reputed to have occurred with several other windmills.
1. Introduction
4. Flow Noise
Next Section: 5. Mechanical Vibration
6. Absorption
8. Discussion