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On the Distribution and Scaling of Convective Wavespeeds in the Shear Layers of Heated, Supersonic Jets

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On the Distribution and Scaling of Convective Wavespeeds in the Shear Layers of Heated, Supersonic Jets 2D-5 9 June 30 - July 3, 2015 Melbourne, Australia ON THE DISTRIBUTION AND SCALING OF CONVECTIVE WAVESPEEDS IN THE SHEAR LAYERS OF HEATED, SUPERSONIC JETS Tobias Ecker Dept. of Aerospace and Ocean Engineering Virginia Tech 215 Randolph Hall, Blacksburg (VA), USA [email protected] K. Todd Lowe Wing F. Ng Dept. of Aerospace and Ocean Engineering Dept. of Mechanical Engineering Virginia Tech Virginia Tech 215 Randolph Hall, Blacksburg (VA), USA 440 Goodwin Hall, Blacksburg (VA), USA [email protected] [email protected] ABSTRACT reduction—a modest noise impact for such a miraculous The noise generated by supersonic plumes is of turbulence reduction. In contrast, the radiation efficiency growing concern given the enormous peak noise intensity — the amount of turbulent kinetic energy that will be radiated by tactical aircraft engines. A key component of converted to acoustic energy — scales nonlinearly with this noise is the enhanced radiation of mixing caused by eddy convective Mach number and offers considerably supersonic eddy speeds. As very little data exist for eddy more latitude for noise reduction. convection in high Reynolds number, supersonic plumes, The convection of turbulent fluctuations in shear our current ability to develop concepts that alter flows is directly related to the process of noise generation compressible eddy convective is limited. Herein we and propagation. Several theories based on acoustic present new experimental data and a phenomenological analogies derived by Lighthill (1952) directly relate the description of eddy wave-speeds in the developing shear pressure fluctuations and the noise intensity to fourth layer of supersonic heated jets. A new scaling of the order correlations and the eddy convective velocity. Bailly wavespeed in radial similarity coordinates is proposed et al. (1997) summarize the relevant aspects with regards which takes into account the influence of the ratio of static to the convective amplification factor for models by densities between the jet and ambient streams. The Ribner, Goldstein-Howes and Ffowcs Williams and frequency-dependent behavior of the convective velocity Maidanik. In all models the convective Mach number, reflects the process of high momentum, high velocity based on the eddy convective velocity can be identified as large-scale eddies pinching off from the potential core and a driver of the produced noise intensity and similarly the convecting into regions of locally reduced mean radiation efficiency (Papamoschou et al., 2014). velocities. In particular, we observe a structural change in The relative convective Mach number is defined as the wavespeed spectra at the end of the potential core. In ratio of the convective velocity to the sound speed, addition to high turbulence levels, the potential core (1) breakdown region can have enhanced eddy speeds, 푀푐 = (푈푗 − 푢푐)/푎푗 increasing noise radiation efficiency. where 푼풋 and 풂풋are the isentropic exit velocity and sound speed, respectively. For co-annular jet flows or shear layers, high speed stream often carries the subscript (1), INTRODUCTION whereas the lower speed stream is notated as (2). The Recent theoretical and computational work theoretical (symmetric) convective Mach number is (Papamoschou et al., 2014) has built upon earlier concepts defined (Papamoschou 1997): (Papamoschou, 1997) for peak noise reduction by altering the convective Mach number of eddies in the shear layer 푀푐 푠푦푚 ≡ (푈1 − 푈2)/(푎1 + 푎2) (2) of jet plumes. Papamoschou points out via acoustic For a single-stream jet at perfectly expanded condition analogy that noise in high speed plumes has two and same gas properties as the ambient gas, a relationship components — a scaling of the source provided by between the isentropic jet Mach number M and the turbulent fluctuation amplitudes and a radiation efficiency j contribution shown to be a function of the acoustic eddy symmetric convective Mach number 푀푐 푠푦푚 can be easily derived, convective Mach number, 푀a = 푈푐/푎, where the speed of sound 푎 is that of the surrounding ambient medium. While 푀푐 푠푦푚 = 푀푗/(1 + √휌푎/휌푗) (3) reducing the large-scale eddy turbulence in plumes is possible, considerable effort on the topic has, to date, revealing a simple dependence on the ratio of ambient yielded limited success. Further, the noise produced scales density to core density, 휌푎/휌푗. linearly with turbulence intensity. As such, a 50% Convective velocity may be considered at the integral reduction in turbulence intensity, equates to a 3 dB noise scale or as a function of frequency/wave-number (Wills, 1 1964). The frequency-dependent convective velocity, due Due to spatial aliasing, the estimated limiting maximum to the existence of differently sized eddies moving at Strouhal number for wave-speed measurement is different convective velocities within the turbulent field, 푆푟 < (푢 ⁄푈 )(퐷⁄2Δ푥) indicates a detachment between the turbulent frequency ℎ푖ℎ 푐 푗 (6) and wave-number spectra and divergence from Taylor’s where D is the nozzle diameter. Alternatively to the frozen turbulence hypothesis. spectral approach, a narrow band-pass filter approach can Fisher and Davies (1964) studied the validity of be used to generate narrowband time signals for time Taylor’s hypothesis in shear flows via two-point cross- delay cross-correlation processing (e.g., Fisher and Davies correlation using hot wire anemometry and found a clear 1964). frequency dependence of the convective velocity from In the present work, a very high repetition rate, multi- their experimental results. Davis et al. (1964) studied the point velocimetry instrument based upon the time- subsonic round jet and found flares of strong fluctuations resolved Doppler global velocimetry (TR-DGV) technique of (−푢′) adjacent to the core (region I in figure 1) and (Ecker et al. 2014a; 2014b) is used to obtain convective ′ (+푢 ) in the outer region (region II in figure 1). These wave-speed measurements in heated, supersonic jets. The observations are consistent with subsequent studies that processing methods developed are discussed to follow. provide the established distribution of mean and integral convective velocities within a jet shear layer (figure 1). Large eddies strongly influence this observed structure. DATA PROCESSING In order to determine the sensitivity of wave-speed estimates to different data processing parameters, studies using Monte-Carlo simulations were performed. The model signals are constructed from a one- dimensional model turbulent power spectrum (Pope 2000) with each signal given a random phase spectrum. The mean velocity and the turbulence intensity of the dataset used were set at 650 m/s and 10%, respectively. The phase of this dataset is then evolved based on the separation between four flow sensors using an empirical fit given by Morris and Zaman (2010), thus creating a non-linear phase angle in the cross-spectrum between stations. The separation distances were the same as the actual physical sensor spacing used in this study. Two processing techniques were considered: (1) reconstruction based on the phase angle, (2) digital band- pass filtering in combination with time delay cross- correlation processing. Figure 2 shows the reconstructed convective velocity profile using both the phase method Figure 1. (a) Mean and convective velocity distribution and the band-pass filter method for two signal to noise ratios. over the jet radius. (b) Indication of the two different The processing parameters investigated are: (1) ratio convective velocity regions trough the jet domain. of averaged sets length S to dataset length N and (2) ratio of window length W (at 50% overlap) to subset length S at a fixed dataset length (푁 = 100푘 samples). Further (3) Previous studies (de Kat et al., 2013) have shown the signal to noise ratio (SNRlog), (4) the type of window limitations in determining wave-number dependent (Square, Hanning, Hamming) and the (5) magnitude of the convective velocities due to the effect of low pass filtering convective wave speed uc. Table 1 presents all the cases of PIV on the power spectra. Restrictions due to the considered in this sensitivity analysis. characteristics of the phase spectrum are analyzed and The results indicate that a small subset length and a general limiting conditions for experimental parameters small window size lead to lowest RMS errors. Only a not only limited to PIV are summarized. Kerhervé et al. minimal sensitivity to the mean convective velocity, (2004b) demonstrated the frequency dependency of the which imposes a time shift between the signals at the two turbulence scales in sub- and supersonic jet flows from stations, has been noted. Errors tend to be slightly higher their non-evenly sampled laser Doppler velocimeter for higher convective velocities as the phase angles are (LDV) data. The study indicated that limitations due to shallower and therefore more sensitive to noise. noise and data-rate appear to influence the spectra and the Convective velocities at very low frequencies were found derived phase angles. to be inherently uncertain and frequencies below 푓푙표푤 = The frequency-dependent wave-speed may be found 0.1(푈푗⁄2Δ푥) were excluded from the statistics. from the phase difference between the signals at stations 푖 The narrowband filter method was implemented by and 푗, applying a narrow band-pass filter to the data set before 푢 (푓) = 2휋푓훥푥/휙 (푓) (4) performing time delay cross-correlation processing. The 푐 푖푗 digital band-pass filter used is based on a fast Fourier where the phase is found from the cross-spectrum, 퐺 , 푖푗 transform (FFT) method. The cases considered in the 휙 (푓) = 푡푎푛−1(퐼푚[퐺 ]/푅푒푎푙[퐺 ]) (5) analysis of the performance of the filter are summarized in 푖푗 푖푗 푖푗 table 2. 2 The dependence of the RMS error on SNRlog and EXPERIMENTAL APPARATUS AND FACILITY window type is shown in figure 3. The phase method The laser-based time-resolved DGV (TR-DGV) requires very high SNRlog to be effective, whereas the filter concept used in this study has frequency response to 125 method is very robust even down to SNRlog = 5 dB.
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