Broadband Shock-Associated Noise Prediction
A good representation of the large-scale turbulence structures is the next required step in the development of predictions for broadband shock – associated noise. Tam (1987) proposed a formal mathematical theory starting from the equations of motion. Tam and Chen (1979), in their study of plane mixing layers, had developed a stochastic model to describe the large-scale turbulence structures. In this approach, the large structures are represented by a superposition of the instability wave modes of the flow, with the amplitudes of the instability waves represented by stochastic random functions possessing similarity properties. In the initial mixing region of the axisymmetric jet, self-similarity applies and hence the same argument can be invoked. Here, only a general description of the stochastic theory is provided. Complete details are given by Tam (1987) and Tam (1995a).
Tam (1987) decomposed the flow variables into four parts, consisting of the time-averaged mean, perturbations associated with the turbulence structures, perturbations associated with the shock cell structure and the time-dependent disturbances that are generated as a consequence of the interactions between the large structures and the quasi-periodic shock structures. These interaction terms, which are responsible for shock noise generation, can be determined from the solution of a boundary value problem. Tam developed expressions for the noise power spectrum, both for the near and far fields. The complete formal solution requires extensive computations, which renders it impractical. So, Tam introduced a similarity source model for the interaction terms and developed a semi-empirical theory with four empirical constants. The values of the empirical constants were chosen by comparison with the measurements of Norum and Seiner (1982a).
Tam (1987) showed good agreement with experiment for both the spectral levels and directivity of broadband shock-associated noise for both under – and over-expanded supersonic jets. A sample comparison at several radiation angles was shown in Figure 10. There is good overall agreement, with the predicted peak frequencies at all angles following the measured trend. The calculated spectra also reproduce the reduction in the halfwidth of the dominant peak, as the observer angle moves towards the jet inlet. Tam also showed good comparisons of the near-field OASPL with experiments. For practical airplane applications, such as the prediction of the impinging shock noise on the fuselage, this capability is very valuable.
Tam (1990) extended his theory for slightly imperfectly expanded jets to moderately imperfectly expanded Mach numbers. Based on the measurements of Norum and Seiner (1982a), as shown in Fig. 11, Tam noted that the dependence of broadband shock noise on jet Mach number for underexpanded jets is quite different from that for overexpanded jets. The expression for the intensity of shock-associated noise, given by the slightly imperfectly expanded theory, Is rc (Mj – Mj) , is strictly valid only for small deviations of the fully expanded Mach number from the design Mach number. To increase the range of applicability of the theory for a broader Mach number range, for Mach numbers slightly less than that of local maximum point C to slightly greater than point B in Fig. 11, Tam (1990) made modifications to the stochastic theory. Primarily, this approach involved the proper prescription of the shock cell strength, since the turbulence spectrum and the shock cell wavelength are not affected significantly by the degree of imperfect expansion. He noted that the effect on spectral shape was consequently unimportant and that the degree of imperfect expansion only affected the spectral level. Suitable expressions for the broadband shock – associated noise, for both cold and hot jets, were shown to provide very good agreement with measured spectra for jets operated at strongly off – design conditions.
Tam (1991) included the effects of flight on broadband shock-associated noise, through additional considerations of changes in the noise source as well as the effect of flight on the convection speed of the large scale structures. He argued that due to the thick boundary layer on the nozzle external surface, the shock cell strength would not be modified to a great extent and hence could be approximated by that for the static case: especially for low flight Mach numbers. Further, he assumed that the same similarity source spectrum adopted for the static case, with account taken of the increased shock-cell spacing and increased convection speed for the large-scale structures due to the co-flowing stream, would be valid. He developed modified expressions for the noise power spectrum and peak frequency for the flight case. This formulation provided the correct trends of a reduction in peak frequency, a narrowing of the spectral peak, and the appearance of higher order peaks with increasing flight speed. Tam (1992), through a transformation of the co-ordinate system, developed expressions for the calculation of broadband shock noise as measured by a ground observer in a typical flyover noise test. This expression contains a term in the form of the familiar Doppler shift, but without a high power of convective amplification factor.
Tam and Reddy (1996) adapted the stochastic noise theory for the prediction of broadband shock noise from rectangular nozzles. The flow and shock cell structure of supersonic rectangular jets is different from those of round jets. As noted out by Tam (1988), for rectangular nozzles with straight sidewalls, two different shock cell systems are set up in the flow. One is the familiar system formed outside the nozzle and the other one originates inside the nozzle, close to the nozzle throat. Because of the second shock cell system, broadband shock noise is generated even at the so-called design Mach number of the nozzle. So, an additional term was added to the expression for the shock cell strength, to account for the second shock cell system. Tam (1988) had already developed a vortex sheet model for the description of the shock cell spacing from elliptic and rectangular jets. This expression was modified empirically to account for the finite thickness of the mixing layer. Furthermore, the convection velocity of the large-scale structures was changed to be 0.55Uj, instead of the typical value of 0.7Uj used for circular jets. With these modifications, good agreement with the measurements of Ponton et al. (1986) was shown for rectangular nozzles with different aspect ratios.
The primary difficulty with the stochastic broadband shock noise theory is that it does not provide a connection to the flow. So, calculations for different geometries requires a reformulation of the model parameters. To overcome this difficulty, Morris and Miller (2010) developed a broadband shock-associated noise (BBSAN) model that uses input from RANS CFD calculations to provide the properties of the shock cell structure and the characteristic scales of the turbulence. The model is formulated as an acoustic analogy based on the linearized Euler equations. The far field pressure is determined from a convolution of the equivalent source terms and the vector Green’s function for the linearized Euler equations. In the first version of the model the effects of mean flow refraction were neglected, since BBSAN radiation occurs primarily towards smaller inlet angles or towards the sideline. The evaluation of the spectral density depends on a model for the shock cell structure and the second order two-point velocity correlation of the turbulence. Note that it is the fourth order cross correlation of the velocity that is needed for the source modeling of fine-scale turbulence noise in the traditional acoustic analogy framework. In addition, the shock cell structure is represented by its axial wavenumber spectrum. In this way the model has much in common with Tam (1987).
Calculations were presented by Morris and Miller (2010) for circular and rectangular jets using the same empirical parameters for all cases. Figure 17 shows a prediction for a circular convergent nozzle with Mj = 1.5 and Tt/Ta = 1.0. The observer is at an inlet angle of 60° at R/D = 100. The total BBSAN prediction is shown with the black line and compares well with measurements – especially in the peak BBSAN frequency range. Curves are also shown for the contribution to the total spectrum from the turbulence interaction with different peaks in the shock cell wavenumber spectrum. These interactions give multiple smaller peaks at higher frequencies, which are also seen in the measurements. The same parameters are used to predict the BBSAN for a rectangular jet with aspect ratio 1.75. This is shown in Figure 18. The agreement is very good. Note how the mixing noise overwhelms the BBSAN at small angles to the jet downstream axis in this case. This model has been extended to dual stream jets. In this case, the presence of the outer high speed fan stream does give rise to propagation effects. Miller and Morris (2011) consider dual stream jets operating off-design and include adjoint solutions to the vector Green’s function to account for refraction effects. It should be noted that Tam et al. (2009) has extended his model for the BBSAN peak frequencies for a dual stream jet.