Large Scale Structures and Instability Wave Model Theory

It is now generally recognized that the noise radiation from high speed jets in the peak noise direction is dominated by the noise generated by the large scale turbulent structures in the jet. Tam (1995a) provides an extensive review of the role played by the large scale structures in jet noise. Beginning with the experiments by McLaughlin and his research team at Oklahoma State University ( McLaughlin et al. (1975), McLaughlin et al. (1977)) numerous experimental observations of large-scale structures in both subsonic and supersonic jets at both high and low Reynolds number have been made.

The key observation of the large-scale turbulent structures in a high Reynolds number jet is that, though they occur in a non-deterministic fash­ion in space and time, any single occurrence is in the form of a train of structures of gradually increasing scale in the axial direction. The growth of the jet upstream of the end of the potential core, as in the case of the two-dimensional shear layer, is associated with the engulfment of ambient fluid and the ejection of high speed jet fluid induced by the large-scale struc­tures. The fact that the large-scale structures appear as a slowly-developing sequence suggests that they may be modeled in both a physical and math­ematical sense by a train of waves of gradually varying wavelength. If the growth and decay of the instability wave is included the large-scale struc­tures can be modeled as wavepackets. Tam (1971) was the first to demon­strate a direct link between the instability waves in the thin shear layers close to the jet exit and a radiating wave pattern. Chan and Westley (1973) also demonstrated the connection between the directional acoustic radiation from high speed jets and predictions based on a spatial stability analysis using a vortex sheet approximation to the jet flow. They showed good agree­ment between the computed and measured wavelengths and phase velocities of waves in the near fields of high speed helium jets.

In order to extend these ideas to jets with finite thickness, realistic jet profiles, several researchers used ideas originally proposed by Ko et al. (1970) who examined the development of finite amplitude instabilities in wakes. Chan (1974a), Chan (1974b), Chan (1975), Morris (1974), Mor­ris (1977), Liu (1974) and Tam (1975) predicted the evolution of fixed frequency instability waves in jet flows. Tam and Morris (1985) also used this general formulation to predict the development of tone-excited jets. The mean momentum and energy integral equations, with an eddy viscos­ity model to describe the dissipative action of the small-scale turbulence, were solved to describe the mean jet flow development. The local radial variation of the instability wave properties was obtained from a linear, in­viscid instability analysis, and the amplitude of the instability wave was determined by the integral kinetic energy equation for the wave.

Tam and Morris (1980) and Morris and Tam (1979) showed how the development of an instability wave could be coupled to the near and far sound fields of a two-dimensional shear layer and a jet respectively. The range of validity of these analyses was subsequently extended by Tam and Burton (1984). The local stability analysis of the jet flow that is used to describe the evolution of the instability waves, can be based either on a parallel mean flow approximation or some account can be taken of the effects of the relatively slow axial variation of the mean flow. The method of multiple scales (see Nayfeh (1973)) can be used for the latter purpose. In the method of multiple scales a series expansion is developed for the perturbations developing in the non-parallel mean flow. The expansion parameter є is a measure of the relative rate of change of the mean flow

in the axial to the cross stream directions. In a jet, є ~ dS/dx where S is a measure of the jet shear layer thickness. All the terms in the series expansion are required to have vanishing amplitude at large radial distances. However, Tam and Morris (1980) showed that the multiple scales expansion is not uniformly valid at large radial distances. A complete description of the unsteady field associated with the instability waves, including their acoustic radiation, can be obtained by the method of matched asymptotic expansions. The inner solution is given by the multiple scales solution. This solution involves an uneven stretching of the axial and radial coordinates to allow for the slow axial variation of the jet mean flow. However, there is no such preferred stretching required in the acoustic field. This implies that the coordinates should be stretched equally. The resulting series solution of the linearized equations of motion provides the outer solution. The matching of the two series solutions takes place in an intermediate region where both solutions can be used. For the jet case, the matching procedure is described by Tam and Burton (1984) and complete details are given by Dahl (1994). To lowest order, the pressure outside the jet is given by,

p (г, х,Ф, г) = f g (k) H^) [i (k) r] exp [i (kx + пф – wt)] dk, (45)

where,

A0 (єx) is the slowly axially varying amplitude of the leading order term in the inner solution, © (єx) /є provides the corresponding rapid phase varia­tion, k is an axial wavenumber and g (k) represents the Fourier transform of the axial variation of the inner instability wave solution. n is the azimuthal wavenumber, w is the instability wave radian frequency, and нП1 [ ] is the Hankel function of the first kind and order n.

A (k) = (k2 – p0M2w2)1/2 with 0 < arg (A)<n/2 (47)

where p0 is the ambient density nondimensionalized by the jet exit mean density. The pressure in the far field can be obtained by replacing the Hankel function in Eqn. (45) by its asymptotic form for large argument, introducing spherical polar coordinates centered on the jet exit with the polar axis aligned with the jet axis, and evaluating the resulting integral by the method of stationary phase (see Tam and Morris (1980)). The
stationary point is given by

ks = p^Mj ш cos в

where в is the polar angle. The mean square pressure in the far field is then given by

D (в) = Е2 NR_ = 4 g (ks)2 (49)

Thus, the far field directivity is determined by the magnitude of the compo­nent of the instability wave’s axial wavenumber spectrum that has a sonic phase velocity in the direction of the far field observer. This condition ap­plies in general and must be met by any source spectrum for noise radiation to occur.

Equation (49) is sufficiently simple that it provides an opportunity to examine the effect of the instability wave’s phase velocity and amplitude growth and decay on the far field directivity. Detailed calculations of the amplitude and phase evolution of a single frequency instability (see Tam and Burton (1984)) suggest that,

where A and a control the amplitude and its growth and decay rate respec­tively, xo is an arbitrary axial location for the wave’s maximum amplitude, and c is the nondimensional instability wave phase velocity (assumed to be constant). Then, from Eqn. (49) the far field directivity is given by,

where, Mc = cUj/ao. If the instability wave’s phase velocity is supersonic with respect to the ambient speed of sound ao, that is, Mc > 1, the far field directivity will have its peak amplitude at,

вреак = cos1 (1/Mc) . (52)

However, even if the instability wave travels at a subsonic phase velocity, relative to the ambient speed of sound, the instability wave or wavepacket can still radiate to the far field. In this case, the peak radiation direction will occur at 0 = 0. The amplitude is controlled by the value of a: the growth or decay factor of the wave amplitude. If a >> 1, D (в) varies slowly with 0. This is the case where either the growth or decay of the instability wave amplitude is very rapid. However, if a << 1, and the wave’s amplitude variation is slow, the far field sound pressure level would fall very rapidly with increasing в. Tam and Morris (1980) showed that if the growth and
decay of the instability wave is determined by linear theory, a is relatively small and the decrease in noise radiation levels with decrease in (shear layer) Mach number is very rapid. This suggests that the instability waves or large – scale structures are very inefficient noise radiators at convectively subsonic conditions. However, if the amplitude variation, most likely the decay of the large-scale structures, is controlled by a nonlinear process, then a rapid decay is possible and the radiation efficiency of the large scale structures would be significant. This issue is discussed again in Section 3.3.

So far, only a single frequency instability wave has been considered. At low to moderate Reynolds numbers, such as in the experiments by McLaughlin et al. (1975), McLaughlin et al. (1977), a single frequency or azimuthal mode number can be excited easily. At high Reynolds num­bers the jet turbulence has a broadband spectrum. Tam and Chen (1979) developed a stochastic model for the large-scale turbulent structures in a two-dimensional shear layer in terms of a random superposition of the shear layer instability waves. The spectrum and two-point statistics were shown to be dominated by the most unstable mode. As described in Section 4.3 below, Tam (1987) used this stochastic description of the turbulence to develop a prediction scheme for broadband shock-associated noise. Morris et al. (1990) also used a broadband excitation of instability waves in a shear layer to simulate the axial evolution of the shear layer growth rate and the turbulence spectrum. This model was able to predict the absolute value of the shear layer growth rate for a wide range of velocity ratios and Mach numbers, without the need to specify any empirical model constants. These ideas were extended to jets by Viswanathan and Morris (1992). All these models demonstrate that the turbulence spectrum at the largest scales is controlled by the large-scale structures. In addition, the models are weakly nonlinear: that is, there is no significant interaction between different fre­quency and azimuthal mode number components. Thus, the broadband noise spectrum can be considered as a random superposition of the contri­butions of individual frequency and azimuthal mode number components. In closing this discussion it is important to note that the noise radiation by the large-scale structures or instability waves is broadband, though its peak will be at relatively low frequencies, so it is not readily distinguished from the noise radiation by the fine-scale turbulence. However, the two mecha­nisms do generate different spectral shapes in the far field. This is discussed further in the next section.

3.3 Similarity Spectra and the Two Mechanisms of Turbulent Mixing Noise

Two mechanisms have been recognized in the generation of turbulent mixing noise. The first is associated with the large-scale structures in the jet. The second is related to the more random spatial and temporal behav­ior of the smaller turbulent scales. The prediction of jet noise radiation by the large scales can be determined within the framework of an instability wave model as described in Section 3.2. In very high speed jets there is a direct weakly nonlinear coupling between the flow disturbances created by the large-scale structures and the acoustic field they generate. At lower jet velocities, a strong nonlinear mechanism is likely to be needed for the spatial and temporal behavior of the large-scale structures to generate the wavenumbers necessary for noise radiation. However, at this time, no first principles theory exists for noise radiation by large-scale structures in con­vectively subsonic jets. On the other hand, the smaller-scale turbulence, when viewed in terms of its wavenumber and frequency content, is able to generate radiating components. But the energy-containing components of the turbulence are not efficient noise radiators in convectively subsonic jets. The theoretical description of this noise generation process is the one found in almost all models based on the acoustic analogy, described in Section 3.1, as well as the model proposed by Tam and Auriault (1999), to be described in the next section.

Since the radiation by the large-scale structures involves a direct con­nection between the jet turbulence and the acoustic field, this process does not experience mean flow/acoustic interaction effects. On the other hand, noise from fine-scale turbulence, being a more local process within the tur­bulent jet plume, is subject to refraction as the generated sound propagates through the sheared mean flow. Both mechanisms generate sound over a broad range of frequencies, though the large-scale structure noise is strongest at low frequencies and in directions close to the downstream jet axis.

In the mixing layer of a high Reynolds number turbulent jet, there is no intrinsic length scale. Furthermore, molecular viscosity is not important, except as an energy sink at the smallest scales. So high Reynolds number jets exhibit a dynamic, inviscid behavior. Hence, there is also no intrinsic time scale in this region of the jet. Experimental measurements have shown that the mean flow as well as the turbulence statistics exhibit self-similarity. Tam et al. (1996) contended that the noise from the fine scale turbulence is also generated in the same region of the jet, where the flow properties are similar. Based on these observations, they proposed that the noise spectra of both the large and small scale noise components should also exhibit self­similarity. They reasoned further that the absence of a time (or frequency)
scale implied that the frequency f must be scaled by fL, the peak frequency of the large turbulent structures noise spectrum or fF the peak frequency of the fine-scale turbulence noise spectrum.

Tam et al. (1996) expressed the jet noise spectrum S as a sum of the two independent noise components, in the following similarity form,

(53)

F(f/ fL) and G(f/fF) are the similarity spectra associated with the large – scale and fine-scale turbulence, respectively and r is the radial distance to the observer from the jet exit. These spectrum functions have been normalized such that F(1) = G(1) = 1. A and B, the amplitudes of the two spectra, and the peak frequencies fL and fF are functions of the jet operating conditions and direction of radiation.

Tam et al. (1996) also recast this equation in decibel form as,

10bg)=10bg(£F(і>lG(і))-20bg(D) (54)

where pref is the standard reference pressure (2 x 10-5 N/m2) of the decibel scale. If this hypothesis is true, then this equation would be valid for any jet operating condition and radiation angle. At large aft angles where the large-scale structure noise is dominant, and in the forward quadrant where the fine scale structure is dominant, the above equation reduces to a simpler form with the measured spectra characterized by either the large-scale or fine-scale component, F or G.

Tam et al. (1996) investigated the jet noise database acquired with round nozzles, operated at supersonic Mach numbers, at NASA Langley’s Jet Noise Laboratory (JNL). This database consisted of narrow-band data with a 122 Hz bandwidth that covered a Mach number range of 1.37 to 2.24 and a total temperature ratio range of 1.0 to 4.9. From a selected subset of this database, they developed two empirical similarity spectrum functions and determined that the empirical spectra fitted the measured spectra over the entire range of Mach numbers and temperature ratios. Figure 16 shows the shape of these two similarity spectra, 10 log F and 10 log G, plotted on a decibel scale as a function of log (f / fpeak). The shapes of the two similarity spectra are very different. The spectrum associated with the large-scale structures (10 log F) has a narrow peak and drops off linearly, while the spectral shape associated with the fine-scale turbulence (10 log G) has a broader peak and a more gradual roll off away from the peak. The correct

Tam (1998a) also examined the supersonic noise data of Yamamoto et al. (1984) from a variety of round nozzles: convergent, C-D, conver­gent with a plug, C-D with a plug, and a 20-chute C-D suppressor. Good agreement between the similarity spectra and measurements for both static and wind-on conditions, was found. This study indicates that the turbu­lent mixing noise of supersonic jets from not-too-complex nozzle geometries consist of two components. In addition, the shapes of the noise spectra of supersonic jets are very similar regardless of the nozzle geometry.

More recently, Tam and Zaman (2000) carried out some simple experi­ments and measured noise from elliptic (AR=3.0) and rectangular (AR=3.0, 8.0) nozzles, a circular nozzle with 2 and 4 tabs, and a six-lobed mixer noz­zle. All the test points were restricted to unheated jets at subsonic Mach numbers. The results showed that, in general, turbulent mixing noise from
non-circular subsonic jets can also be described by the two similarity com­ponents. However, it should be borne in mind that the quality of this data set is questionable. One interesting result of this study is that the noise field of the lobed mixer was very different from those from the other simpler ge­ometries. The lobed mixer, designed to enhance perimeter mixing with the ambient fluid, divides the main jet into many thin sheets at the nozzle exit. As such, flow from this geometry does not apparently support large-scale structures comparable in scale to the equivalent circular diameter. The test results indicated that the noise spectrum at large aft angles, where the large-scale structure noise typically dominates, fitted the spectrum associ­ated with fine-scale turbulence. Tam and Zaman (2000) concluded that the nozzle geometry, by suppressing the development of large-scale structures, effectively eliminated the noise associated with this component. In the ab­sence of this component, the fine-scale noise component becomes dominant at all radiation angles.

In summary, there is strong evidence that two self-similar spectra can be used to characterize the radiated noise spectra for a wide range of jet operating conditions. This applies to supersonic jets, from both circular and other simple non-circular nozzle geometries as well as cold subsonic jets. The spectra at lower angles to the jet inlet from heated subsonic jets can also be characterized by the fine-scale similarity spectrum. At angles close to the jet downstream axis, the spectral shape changes at higher temperatures, as noted by Viswanathan (2004). An important point to remember is that all these experimental data are restricted to single stream nozzles. However, in closing this section, it should be noted that other explanations for the different spectral shapes between the sideline and large aft angles are available. For example, it has been argued that a combination of convective amplification, Doppler frequency shift, and mean flow/acoustic interaction effects, could result in the observed spectral changes with angle. This hypothesis is the basis for the jet noise model developed by Morfey and Szewczyk (1978) as well as the MGB and MGBK methods discussed in Section 3.1. It remains to be determined whether one or other of these descriptions of turbulent mixing noise generation, or a combination of the two, are valid.