Multi-fidelity Acoustic Modelling
Jet noise modelling has been an important topic of research since Lighthill (1952). For fast-turn-around-time sound predictions, low-order methods are used where acoustic modelling is coupled with Reynolds Averaged Navier-Stokes (RANS) equations (Mani et al. 1978; Goldstein 2002, 2003; Goldstein and Leib 2008; Khavaran, Bridges 2012, Leib and Goldstein 2011). One of the most advanced acoustic analogy models is due to Goldstein (2002,2003) who showed that the Navier-Stokes equations can be rewritten as a set of `base’ flow equations plus a formally linear set of equations for a`residual’ component and that the latter is governed by a form of the linearized Navier-Stokes equations with source terms whose strengths are represented by a generalized Reynolds stress. Goldstein and Leib (2008) used the Goldstein analogy approach to develop a jet noise prediction scheme with information about the turbulence obtained from an industry-standard RANS modelling tool.
As recent experiments suggest (Harper-Bourne, 2003; Bridges, 2014), analogously to the similarity of the jet mean flow data which can be collapsed to ‘universal’ fit functions, such as in (Witze, 1974), the fourth-order velocity correlation functions, which are the effective noise sources in accordance with the acoustic analogy, appear to collapse to some universal shapes too for a range of jet conditions and frequencies. Most recently, Semiletov and Karabasov (draft report, 2014) have used Large Eddy Simulation (LES) as a post-processing tool to perform a detailed analysis of the sound sources of two high-subsonic single-stream static cold and hot jets. These jets correspond to the conditions of a recent QinetiQ experiment (which data are still not publically available). In the former work, a novel implementation of the generalized acoustic analogy, which uses the available LES grid in space and time at full resolution, has been developed. Unlike in the previous works, such as in (Karabasov et al 2013; Karabasov and Sandberg, 2014), the new implementation is completely free from the modelling assumptions such as the functional behavior of the covariance of fluctuating Reynolds stress or correlation scale dependence on frequency. This new implementation of the acoustic analogy based on LES is more robust than integral methods such as based on the Ffowcs Williams - Hawkings approach and its acoustic solution converges faster than those acoustic analogy methods which are based on fourth-order statistics of the fluctuating flow. The acoustics predictions of this acoustic analogy model are within 2-3dB from the experiment. As a further analysis of the jet noise sources has shown, the source data appear to collapse to similar non-dimensional curves as the previous experimental results suggest. Moreover, the sound source of the hot jet (covariance of the enthalpy fluctuation stress) obeys a similar scaling as the cold jet source (covariance of the fluctuating Reynolds stress).
When combining the G&L2008 spreading jet propagation model with the low-order RANS-based source models, in the spirit of Leib &Goldstein 2011 source model, the following has been observed: (i) the far-field sound predictions are quite sensitive to the calibration parameters; these parameters are not always available from the experimental flow data, especially for the hot jets, and need to be calibrated with respect to the far field sound spectra, and (ii) the problem of the source calibration based on the far-field sound spectra does not seem to have a unique solution: very different sets of the source calibration parameters may possibly lead to the far-field noise spectra which are the same within 2-3dB. Clearly, this uncertainty in source modelling should be reduced if the resulting low-order model is to be used for jet noise predictions in design optimisation studies as well as to provide useful insights about the effective jet noise sources.
The uncertainty of the low-order jet noise modelling comes from a certain number of assumptions made about the turbulent flow statistics. For example, such assumptions may involve modelling of the fourth-order velocity correlation functions based on the statistical quantities obtainable from RANS such as turbulent kinetic energy, energy dissipation rate and the jet meanflow velocity and its gradients. Since the experimental flow field measurements available are typically limited, inevitably, the low-order models have to rely on the far-field data for defining the rest of the calibration coefficients. Typically, the model calibration is performed by the model developer, who then has to work a ‘human optimiser’ in the sense of selecting the model parameters which correspond to the best fit to the experimental data available. As the low-order model becomes more complex to incorporate more physics, such as variable correlation amplitudes and space scales depending on the source directivity as well as cusp points of the correlation curves and negative correlation loops observed experimentally, more coefficients are required. In turn, this makes the whole process of model development very complicated and can also lead to sub-optimal solutions. This is where the automatic optimisation algorithms (e.g. Toropov, 1989; Toropov and Yoshida, 2005; Polynkin and Toropov, 2012) can help the developers by not only freeing up their time but also providing the model coefficients which are optimal over a given set of jet noise data. To guide the process of low-order models development, the high fidelity models, such as those based on LES (e.g. Faranosov et al, 2013), can be used to more accurately constrain the rage of variation of the calibration parameters.
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