The density near-field of a non-uniformly heated supersonic jet

Abstract

This work presents analyses of high-speed schlieren images that depict the spatio-temporal structure of near-field sound in uniformly and non-uniformly heated supersonic round jets. The non-uniformly heated jet has a concentrated region of locally lower total temperature flow around the centerline of an ideally expanded jet. Compared to the uniform jet, the non-uniform jet is shown to reduce jet noise by up to 2 ± 0.5 dB in the peak narrowband sound pressure level at polar angles upstream of the peak directivity. Space-time correlations are performed on frequency-filtered time series of fluctuating schlieren image intensities, an analog for the fluctuating near-field density gradients. The effect of path integration is evaluated using synthetic schlieren of the dominant azimuthal jet modes, which are simulated using the azimuthal basis function of the Fourier transform. Hydrodynamic structures are identified at low frequencies and are shown to be modified by the thermal non-uniformity at axial locations in the near- and far-nozzle regions. The mid-frequency range is dominated by convecting Mach waves that are decorrelated in the thermally non-uniform jet in the near- and far-nozzle regions. Correlations of the high frequency content capture the emission of an acoustic beam. Results indicate the perturbations induced by the thermal non-uniformity can persist far into the developing flow field and reduce the length scale of coherent structures in regions far from the nozzle exhaust. This suggests centerline base flow changes can be optimized to reduce the acoustic efficiency of unsteady flow structures present near strong noise-producing areas such as the potential core collapse region.

Graphical abstract

1 Introduction

A more complete understanding of the physics linking the energetic, turbulent sources in supersonic free jets to the significantly weaker radiated sound field is valuable to the aircraft industry and a necessary step toward optimizing noise reduction techniques. Jet noise reduction efforts are driven by several factors, which include renewed interest in supersonic transport over land (Huff et al. 2016) and reducing the hearing loss experienced by aircraft carrier personnel who work near tactical aircraft (Aubert and McKinley 2011).

In supersonic heated jets, the topic of the current study, turbulent structures can radiate noise via turbulent mixing noise and Mach wave radiation. Turbulent mixing noise is the sound produced by coherent structures that radiate sound as a function of their turbulent kinetic energy, convection speed, and correlation length and time scales (Papamoschou et al. 2014). When these structures convect with supersonic velocities, they produce intense Mach wave radiation in a manner analogous to supersonic flow over a wavy wall. Mach waves are super-directive and radiate at an angle based on the convection speed of the pressure disturbance (Tam and Chen 1994).

Turbulent structures can be described by two coexisting mechanisms, Kelvin–Helmholtz linear instabilities and Orr-type waves. Kelvin–Helmholtz (KH) linear instabilities dominate the developing shear layer upstream of the potential core terminus and remain coherent for long distances while convecting, growing, and ultimately decaying following breakdown of the potential core (Crighton and Gaster 1976). The disturbances from these waves are aligned with the direction of mean-flow shear and convect with constant phase velocities. Non-modal instabilities characterized by tilted structures and associated with the Orr mechanism, well-known for boundary layers, have been identified in regions within the nozzle and downstream of the potential core terminus (Schmidt et al. 2018; Pickering et al. 2020). These are often intertwined with the KH instability, as the tilted Orr motions provide optimal forcing for KH instabilities (Garnaud et al. 2013).

Fundamentally, jet noise reduction techniques function by shaping the base flow through changes to the nozzle boundary conditions or by introducing unsteady perturbations to alter the acoustically important characteristics of turbulent sources. Several noise reduction methods are highlighted below, each of which use distinct means to manipulate the nozzle boundary condition and impact different features (wavelength, amplitude, convection velocity) of the large-scale turbulence development.

Fluidic injection in the expansion contour of a supersonic nozzle simulates hard-walled corrugations and shapes the mean flow by inducing streamwise vorticity, providing noise benefits up to 5 dB in the over-all sound pressure level (OASPL) (Powers et al. 2013). Offset multi-stream nozzles create a locally thinner and thicker shear layer and reduce the convection speed of correlated structures on the thick side and in the vicinity potential core collapse (Stuber et al. 2019; Papamoschou and Phong 2017). This results in reductions up to 8 dB in the narrowband spectra at azimuthal locations aligned with the “thicker” shear layer (Henderson and Leib 2015), while an increase is observed on the “thinner” side (Henderson and Huff 2016). Inverted velocity profile (IVP) jets are multi-stream jets where the inner plume operates at a slower velocity than the bypass flow. These jets have demonstrated noise benefits up to 4 dB OASPL compared to jets with equivalent thrust, mass flow rate, and area (Tanna 1980). Noise reductions in IVP jets occur primarily at low frequencies and are attributed to slower convection velocities and increased axial turbulence levels (Tanna 1980).

Heated jets with thermal non-uniformities, the topic of the current study, are characterized by distorted total temperature exhaust profiles that create a locally “cooler” stream of slower velocity flow. Mayo et al. (2019) showed strong similarities in the axial turbulence development between heated jets with a radially offset cooler stream and offset multi-stream jets. Previous studies of centered thermal non-uniformities found amplitude reductions in the tail of the inferred near-field wavenumber spectra (Daniel et al. 2019b) and a decorrelation of Mach wave structures, which resulted in noise reductions of up to 2.5 dB OASPL (Daniel et al. 2019a).

While current noise reduction methods show promise, a more complete description of the relationship between the nozzle boundary condition and the characteristics of acoustically important turbulence is needed to optimize jet noise reduction. Efforts toward this goal include experimental measurements in the periphery of the shear layer, which have identified important characteristics of large-scale turbulence structures through their associated pressure fields.

The near-field, defined as the irrotational region directly outside of the shear layer, contains the beginnings of an acoustic field and hydrodynamic pressure fluctuations related to the convective signature of large-scale turbulent structures. The pressure fluctuations associated with these structures decay rapidly as $ky^{-6.67}$ (Arndt et al. 1997), where k is the wavenumber and y is the radial distance from the nozzle lip line. To fully capture the signature of the underlying turbulence, measurements must be taken as close as possible to the shear layer (Papamoschou and Phong 2017). For example, Tinney and Jordan (2008) measured the pressure near-field of co-axial subsonic jets using a linear microphone array and separated the hydrodynamic and acoustic components using a wavenumber-frequency analysis. The authors found nozzle serrations reduced the acoustic efficiency of coherent turbulence but did not induce any fundamental changes to their structure in downstream regions.

Schlieren imaging is an attractive method for near-field measurements, as it can measure density gradient fluctuations at locations extremely close to the shear layer with high spatial and temporal resolution. Furthermore, the density gradient can be directly related to the pressure field, meaning schlieren of the ‘density’ near-field will capture acoustic waves and hydrodynamic structures related to the footprint of large-scale turbulence.

However, few studies have used schlieren to extract quantitative information from high-speed jet flows. Murray and Lyons (2016) measured shock propagation angles from shadowgraph images of a supersonic jet to determine the convection speed and axial location of turbulent structures. Tinney and Schram (2019) applied proper orthogonal decomposition (POD) to a single horizontal row of pixels from high speed schlieren images of a Mach 3 jet. The authors extracted the most energetic spatial POD eigenvectors to predict the far-field sound using the one-dimensional wave equation. Berry et al. (2017) applied POD to time-resolved schlieren measurements of a supersonic multi-stream rectangular jet equipped with an aft deck and described distinct flow structures related to vortices downstream of the aft deck. Here, an aft deck is a flat plate that simulates an embedded engine exhausting over an aircraft surface. Akamine et al. (2021) performed spectral proper orthogonal decomposition (SPOD) of high-speed schlieren images of an unheated Mach 1.8 jet. The authors found that the subdominant multilobe SPOD structures were related to intermittent wave packets.

The above studies represent the few which demonstrate that while schlieren is a relatively simple measurement, it can be a powerful analysis tool when used to extract meaningful, quantitative information from the near-field. In the present study, schlieren measurements are used to describe the temporal evolution of structures in the density near-field of a supersonic heated jet. Specifically, space-time correlations of frequency-filtered schlieren images depict the temporal evolution of hydrodynamic structures and acoustic waves in the near-field. A comparison of a thermally uniform and non-uniform jets indicate a centered thermal non-uniformity decreases the size of correlated near-field structures at locations near the potential core collapse. A space-frequency coherence analysis supports this conclusion, showing a reduction in the axial coherence of low frequency near-field content.

The remainder of this paper is organized as follows. Section 2 provides a description of the Virginia Tech heated supersonic jet rig, the experimental apparatus used to generate the centered thermal non-uniformity, and a characterization of the far-field acoustic benefit. Section 3 presents key results describing the spatio-temporal behavior of the density near-field using space-time correlation and space-frequency coherence analyses. A summary of the results and their impact follows in Sect. 4.

2 Methodology

2.1 Experiment apparatus

Experimental measurements of supersonic heated jets with uniform and thermally non-uniform nozzle boundary conditions were performed using the Virginia Tech heated supersonic jet rig. In both cases, a converging-diverging nozzle with exit diameter D = 0.0381 m generated a perfectly expanded, heated Mach 1.5 flow with a nozzle pressure ratio of NPR = 3.67 and Reynolds number of $\mathrm{Re}_{D}$ = 8.5 $\times 10^{5}$. The nozzle expansion contour was designed using the method of characteristics to produce a nearly shock-free flow and is a scaled version of the one used in experiments performed at the Pennsylvania State University (Kuo et al. 2014).

A total temperature non-uniformity was generated by introducing a secondary stream of unheated air into the centerline of the heated jet plenum via a secondary converging nozzle that terminated 2.23D upstream of the primary nozzle exhaust plane (Fig. 1). Once introduced, the unheated air stream accelerated through the supersonic nozzle and generated a concentrated volume of locally lower total temperature fluid along the jet centerline. The jet with temperature non-uniformity is defined as the NUC (“non-uniform centered”) jet. The uniform temperature profile was achieved by shutting off the unheated air flow through the internal, secondary nozzle. Flow variables of the uniform heated jet are defined by the subscript j while the unheated and heated streams of the NUC jet are indicated by s (secondary) and p (primary), respectively. See Daniel et al. (2019b) for an in-depth description of the centered non-uniformity hardware and the Virginia Tech facility.

Fig. 1

Diagram of the a thermal non-uniformity geometry. TTR contours for the b uniform and c NUC jets

Flow conditions of the heated and unheated streams were continuously monitored and recorded. Total temperature and total pressure measurements were made using a National Instruments 9213 thermocouple module in a NI cDAQ-9184 chassis and a Scanivalve Corp. ZOC17IP/8Px-APC pressure transducer, respectively. The mass flowrate of the unheated flow $\dot{m_{s}}$ was measured using a Lambda-Square orifice meter. The total temperature profiles of the NUC and uniform jets were characterized near the nozzle exhaust plane (x/D = 0.04) using total temperature probe measurements taken over a period of 5 seconds at 676 points spread over a 1.1D$\times$1.1D grid. The total temperature ratios (TTR), NPR, and ratios of the mass flow ${\dot{m}}$ and thrust F, are calculated from these measurements using isentropic relations and are reported in Table 1.

Table 1 Experimental conditions

The average total temperature profile at the nozzle exhaust plane of the uniform and NUC jet is presented in Fig. 1b, c. The temperature distribution of the uniform and heated portion of the NUC jet is nearly constant, with a 2.4% root-mean-square variation. In the NUC jet, a concentrated spot of fluid with an average temperature of TTR = 1.2 is generated by the unheated air plume introduced upstream along the heated jet plenum centerline.

2.2 Far-field narrowband spectra

The acoustic far-field of the uniform and NUC jets was characterized using a ground array comprised of five PCB 378C10 microphones located on a 100D radius polar arc at observer angles $\theta _{0}$ from 125$^{\circ }$ to 145$^{\circ }$ in 5$^{\circ }$ increments. The PCB microphones have a flat frequency response up to 30 kHz, or $\mathrm{St}$ = 1.9. Here, $\mathrm{St}$ is the Strouhal number, a non-dimensional frequency defined as $\mathrm{St}=fD/U_j$, where $U_j$ is the isentropic jet exit velocity and D is the jet diameter. Signals were amplified and low-pass filtered at 100kHz using a PCB model 482C series signal conditioner. All channels were simultaneously digitally sampled using a 16-bit NI PIXe-6358 module at 200 kHz for 7s over a ± 10 V bipolar range, yielding $1.4 \times 10^{6}$ data points per microphone. Measurements were performed outside of the facility with the ground microphone array set on a concrete pad surrounding the test cell. This ground microphone array technique follows SAE standard AIR 1672B. See Quinn et al. (2019) for details on the ground array and comparisons of measurements made at the Virginia Tech facility to others in the literature.

Fig. 2

Far-field narrowband spectra (a) and OASPL (b

Single-sided spectral densities were calculated using Welch’s method with a 50% overlap and a Hanning window, resulting in 1499 records and a frequency resolution of $\mathrm{St}$ = 0.009. Corrections for atmospheric absorption due to humidity effects and a conversion to Strouhal number scaling were applied to the uncorrected sound pressure levels in the same manner as in Daniel et al. (2019b). The spectra in Fig. 2a are offset by 20 dB for clarity, where the thickness of each line represents the 95% confidence interval of the measurement.

The narrowband spectra in Fig. 2a demonstrate that the thermal non-uniformity reduces the far-field sound by 2 $\,\pm \,$0.5 dB compared to the uniform jet. Reductions occur over a polar range of 130$^{\circ }$–135$^{\circ }$ and Strouhal numbers ranging from 0.1 to 0.7. Smaller, but still statistically significant reductions occur at angles just downstream (140$^{\circ }$) and upstream (125$^{\circ }$) of this area. The OASPL in Fig. 2b shows a similar trend, with a maximum reduction of 2.5 dB OASPL at 135$^{\circ }$.

The acoustic benefit of the NUC jet is not related to a reduced thrust, as comparisons are made at matched static thrust and mass flow rate conditions. Furthermore, the noise reduction is greater than what would be achieved by a fully mixed equivalent jet. As described in Daniel et al. (2019b), an equivalent uniform velocity profile is found by area weighting $U_\mathrm{p}$ and $U_\mathrm{s}$. In jets with $U>600 \,\mathrm{m/s}$, the radiated sound will scale as $U^3$ (Ffowcs Williams 1963) and as $U^8$ for low-speed jets (Lighthill 1954). If the thermally uniform jet is used as a reference, the expected noise reduction is 0.3–0.8 dB depending on the velocity scaling used.

The noise reduction achieved by the centered thermal non-uniformity is related to the secondary shear layer created by the velocity difference between the heated and unheated portions of the jet. The difference in temperature between the cooler secondary plume $T_\mathrm{s}$ and surrounding annulus of heated fluid from the primary jet $T_\mathrm{p}$ drives a velocity difference between the two streams whose ratio scales as the ratio of the local sound speed, or square-root of the temperature ratio, i.e., $U_{\mathrm{s}}/U_{\mathrm{p}}\propto \sqrt{T_\mathrm{s}/T_\mathrm{p}}$.

The velocity differential induces centerline turbulence that creates small, local flow field changes and influences the global jet flow development, suggesting turbulence source mechanisms may be altered over a range of axial locations. This hypothesis motivates the rest of the study, which uses schlieren imaging to depict the temporal evolution of acoustic and hydrodynamic features in the near-field. Schlieren of the uniform and NUC jets are then compared to asses structural changes in the hydrodynamic and acoustic fields that begin to explain the acoustic benefit of the thermal non-uniformity.

2.3 Near-field schlieren

Fig. 3

Schlieren images of the uniform (a, b) and NUC jets (c, d). The cross-correlations are calculated between a probe point at and every point inside the red rectangle. The coherence is calculated between and points along

The spatio-temporal evolution of acoustic and hydrodynamic structures in the near-field is examined using schlieren images. A schlieren image is created by the refraction of collimated light rays as they pass through a flow volume due to variations in the refractive index n of the fluid. When the collimated light rays are focused using collection optics, and a knife edge is placed at the focal point, the resulting image represents a path-integrated measure of changes in n. In gases, the index of refraction is related to the density by $n=1+\kappa \rho$, where $\kappa$ is the Gladstone-Dale constant (Settles 2001). In this way, schlieren images provide a path-integrated measure of the density gradient. This effect is shown in Fig. 3, where the lighter intensity pixels represent the change in the density gradient across the wavefront of a Mach wave.

Images sensitive to the fluctuating radial density gradient, $\partial \rho '/\partial y$, are produced using a Z-type schlieren system composed of a point light source, two parabolic mirrors, and a horizontal knife edge. A point light source was created using DC powered ABET 150 Watt Xenon Arc Lamp and four razor blades configured to form a pinhole. A Phantom v2512 high-speed camera recorded schlieren images at 110kHz and an exposure time of 1$\mu s$, resulting in a set of 40,000 images with a resolution of 512$\times$320 pixels and a Nyquist frequency of $\mathrm{St}$ = 3.5. Images were spatially calibrated using a calibration target located a known distance away from a datum at the nozzle exhaust. The spatial resolution of all images presented is 0.018D per pixel. For each axial station, measurements of the NUC and uniform jets were taken back-to-back without shutting down the heated supersonic jet rig or turning off any schlieren equipment. This was done to reduce the influence of calibration factors, such as changes in the illumination uniformity, when comparisons of normalized statistics are made.

The density near-field is investigated using a field of view that includes both the shear layer and the ambient fluid directly outside as shown in Fig. 3. The schlieren images depict the large range of turbulent length scales in the jet shear layer and intense Mach wave radiation produced by sources with supersonic convection velocities. The lower temperature core flow in the non-uniform jet is observed along the centerline until $x/D=5$ and begins to mix out at locations further downstream.

Measurements are made in the near- and far-nozzle regions to compare the behavior of turbulent structures in significantly different regions of the developing jet. The far-nozzle region is centered on the end of the potential core, an area known to produce intense noise and to have increased convection velocities compared to upstream stations (Ecker et al. 2017). In the near-nozzle region, base flow changes induced by the thermal non-uniformity are clearly present (Daniel et al. 2019b). However, at near-nozzle locations, it becomes difficult to resolve structures in both space and time due to their small wavelengths. At the current framerate (110 kHz), a feature traveling at $0.8U_\mathrm{j}$ (Schmidt et al. 2018) is displaced by $\sim$ 0.1D between consecutive frames. Therefore, features captured by the schlieren inside of the shear layer that are smaller than 0.1D will not be time resolved while structures outside of the shear layer that convect at the sound speed will be time-resolved for lengths greater than 0.08D.

The space-time structure of acoustic and hydrodynamic features in the near-field is examined by calculating the cross-correlation $R_{12}$ between pixels in the schlieren images. First, the fluctuating schlieren intensity is found by $A'(x,y,t)=A(x,y,t)-<A(x,y,t)>$, where A(x, y, t) is the instantaneous signal recorded by the camera and $<A(x,y,t)>$ represents the temporally averaged image intensity. Both the instantaneous signal and the mean flow image contain contributions from small thermal variations in the ambient fluid around the jet. These thermal variations are low frequency in nature and removed using a high-pass, fast Fourier transform (FFT) filter with a cutoff frequency of St = 0.03. The FFT filter functions by taking the Fourier transform of the time series for each pixel in the schlieren image, setting the spectral amplitudes outside the desired frequency range to zero, and then taking the inverse FFT to obtain a frequency filtered time history.

The cross-correlation $R_{12}$ is calculated between a probe point located at x, y and every point in the schlieren field of view. The probe point and the area used in the cross-correlation analysis are defined by the red dot and the region inside the red rectangle in Fig. 3. The cross-correlation is calculated as

$$\begin{aligned} R_{12}(x,y;\xi ,\zeta ,\tau )=E[A'(x,y,t)A'(x+\xi ,y+\zeta , t+\tau )]. \end{aligned}$$

(1)

Here, $E[\cdot ]$ is the ensemble average over the entire time series and $\xi , \zeta , \tau$ are separations in x, y, t, respectively. For the sake of comparisons, the cross-correlation function is normalized as

$$\begin{aligned} \begin{aligned}&\rho _{12}(x,y;\xi ,\zeta ,\tau ) \\&\quad =\frac{R_{12}(x,y;\xi ,\zeta ,\tau )}{\sqrt{R_{11}(x,y;0,0,0)R_{22}(x,y;\xi ,\zeta ,0)}}, \end{aligned} \end{aligned}$$

(2)

where $\rho _{12}$ is the normalized correlation coefficient function and $R_{11}$, $R_{22}$ are the auto-correlation functions at the probe point and every point in the field of view.

The coherence length scale of near-field structures are examined using the space-frequency coherence $\gamma ^2_{12}$, calculated between the probe point located at x, y and points along a line parallel to the shear layer. The locations of the probe point and the moving point are shown by the red dot and the line in Fig. 3. The coherence is calculated as

$$\begin{aligned} \gamma ^2_{12}(x,y;\xi ,\zeta ,f)=\frac{|{G_{12}(x,y;\xi ,\zeta ,f)}|^2}{G_{11}(x,y;f), G_{22}(x,y;\xi ,\zeta ,f)}. \end{aligned}$$

(3)

The auto-spectral density functions $G_{11}$, $G_{22}$, and the cross-spectral density function $G_{12}$, are estimated across frequencies f as

$$\begin{aligned} & G_{{11}} (x,y;f) = \frac{1}{{\pi f_{s} NN_{{rec}} \overline{{w^{2} }} }} \\ & \quad \times \sum\limits_{{p = 1}}^{{N_{{rec}} }} {\mathcal{F}} \{ A^{{\prime p}} (x,y,t)\} {\mathcal{F}}^{*} \{ A^{{\prime p}} (x,y,t)\} \\ & G_{{22}} (x,y;\xi ,\zeta ,f) = \frac{1}{{\pi f_{s} NN_{{rec}} \overline{{w^{2} }} }} \\ & \quad \times \sum\limits_{{p = 1}}^{{N_{{rec}} }} {\mathcal{F}} \{ A^{{\prime p}} (x + \xi ,y + \zeta ,t)\} {\mathcal{F}}^{*} \{ A^{{\prime p}} (x + \xi ,y + \zeta ,t)\} \\ & G_{{12}} (x,y;\xi ,\zeta ,f) = \frac{1}{{\pi f_{s} NN_{{rec}} \overline{{w^{2} }} }} \\ & \quad \times \sum\limits_{{p = 1}}^{{N_{{rec}} }} {\mathcal{F}} \{ A^{{\prime p}} (x,y,t)\} {\mathcal{F}}^{*} \{ A^{{\prime p}} (x + \xi ,y + \zeta ,t)\} . \\ \end{aligned}$$

(4)

Here, ${\mathcal{F}}\{\cdot \}$ is the discrete Fourier transform, $^*$ represents the complex conjugate, and $\overline{w^2}$ is the RMS of the window function (Glegg and Devenport 2017). Spectral averaging was performed using Hanning windowed records with 50% overlap and a record length of N = 100, yielding $N_{rec}$ = 799 records and a frequency resolution of St = 0.07.

The averaging uncertainty in the correlation coefficient and coherence is calculated (Bendat and Piersol 1988) as

$$\begin{aligned} \delta [\rho _{12}]= & {} \frac{2}{\sqrt{NN_{rec}}}(1-\rho _{12}^2) \end{aligned}$$

(5)

$$\begin{aligned} \delta [\gamma ^2_{12}]= & {} \frac{ 2\sqrt{2}\gamma _{12}}{\sqrt{N_{rec}}}(1-\gamma ^2_{12}). \end{aligned}$$

(6)

The averaging uncertainty in the correlation coefficient is on the order of $10^{-2}$, which is much smaller than the change in the contour levels of the correlations in Figs. 6, 8, 9 and 10. The maximum averaging uncertainty in the coherence is $\delta [\gamma ^2_{ab}]$ = 0.04.

The dominant uncertainty in the schlieren measurement is related to how well the statistical structures calculated from a path integrated measure of the density gradient represent the acoustically important structures in the pressure near-field. This uncertainty is addressed by first demonstrating the relationship between the fluctuating pressure and the density gradient field, which are related through the isentropic relation

$$\begin{aligned} \frac{\rho '}{\rho } = \left( \frac{p'}{p}\right) ^{1/\gamma }, \end{aligned}$$

(7)

where $\gamma$ is the ratio of specific heats and $\rho '$, $p'$ are the density and pressure perturbations around $\rho$, p, the mean density and pressure (Anderson 2003). The radial gradient of the fluctuating density is found using the speed of sound, defined as $c^2=\partial p' /\partial \rho '=\gamma p/\rho$, and taking the partial derivative of (7),

$$\begin{aligned} \frac{\partial \rho \prime }{\partial y}=\frac{1}{c^2}\frac{\partial p^{\prime }}{\partial y}\left( \frac{p'}{p}\right) ^{\frac{1-\gamma }{\gamma }}. \end{aligned}$$

(8)

The fluctuating pressure is solved for as a function of density as follows:

$$\begin{aligned} p'=p \left( c^2\frac{\partial \rho '}{\partial y} /\frac{\partial p^{\prime }}{\partial y} \right) ^{\frac{\gamma }{\gamma -1}}. \end{aligned}$$

(9)

Equation (9) shows the fluctuating pressure in the jet near-field is proportional to the density gradient as $p' \propto (\frac{\partial \rho '}{\partial y} /\frac{\partial p^{\prime }}{\partial y})^{\frac{\gamma }{\gamma -1}}$. This indicates observations regarding the structure of the density near-field will also apply to the near-pressure field.

The impact of the path integration must also be accounted for. Schlieren images provide path-integrated values, meaning space-time correlations of these images do not represent the evolution of structures within a single streamwise/radial plane, but instead describe statistical structures integrated over a line perpendicular to the imaging plane.

The effect of path integration can be assessed by simulating the dominant azimuthal modes of the jet and comparing synthetic schlieren images generated from each mode to profiles of the radial gradient taken from a centered streamwise plane. As shown in Tinney and Jordan (2008), Fourier basis functions exhibit characteristics very similar to POD modes of the near-pressure field. Therefore, the first four azimuthal modes of the pressure near-field are simulated using the basis function of the polar Fourier transform. Here, we follow the derivation outlined in Wang et al. (2008) and define the basis function in a separation-of-variables form as

$$\begin{aligned} \varPsi _{nm}(r,\phi )=R_{nm}(r)\varPhi _m(\phi ), \end{aligned}$$

(10)

where $\varPhi _m(\phi )$ and $R_{nm}(r)$ are the angular and radial parts of the basis function over radius r and azimuthal angle $\phi$ for azimuthal modes m and eigenvalues n. The expression for the angular part of the basis function is simply

$$\begin{aligned} \varPhi _m(\phi )=\frac{1}{\sqrt{2 \pi }} e^{i m \phi }. \end{aligned}$$

(11)

The normalized radial function, defined over the finite interval [0, a], is

$$\begin{aligned} R_{nm}=\frac{1}{\sqrt{N_n^{(m)}}}J_m(k_{nm}r), \end{aligned}$$

(12)

where $N_n^{(m)}$ is a normalization factor and $J_m(k_{nm}r)$ are the m-th order Bessel functions that are mutually orthogonal for the set of wavenumbers $k_{nm}$. The values of $k_{nm}$ over which $J_m(k_{nm}r)$ are mutually orthogonal can be defined by imposing boundary conditions using Sturm-Liouville (S-L) theory. The resulting equation describing the boundary condition is given as follows:

$$\begin{aligned} J_m(x_{nm}) \cos \beta - x_{nm}J^{\prime }_m(x_{nm})\sin \beta =0 \end{aligned}$$

(13)

where $x_{nm}=k_{nm}a$ are the non-negative solutions to (13) and $\beta \in [0,\pi )$. The normalization factor $N_n^{(m)}$ can be found as:

$$\begin{aligned} N_n^{(m)}=\frac{a^2}{2}\left[ J_m^{\prime 2}(x_{nm})+\left( 1-\frac{m^2}{x^2_{nm}}J^2_m(x_{nm})\right) \right] . \end{aligned}$$

(14)

In S-L theory, the boundary conditions are specified by defining $\beta$. Here, we use a zero-value boundary condition with $\sin \beta =0$, reducing (13) to $J_m(x_{nm})=0$ and (14) to

$$\begin{aligned} N_n^{(m)}=\frac{a^2}{2}J^2_{m+1}(x_{nm}). \end{aligned}$$

(15)

With the radial and azimuthal parts of the basis function defined the $n=1$, $m=0,1,2,3$ azimuthal Fourier modes can be calculated as shown in Fig. 4a–d.

Next, synthetic schlieren is generated from the Fourier basis functions following the method of Hay et al. (2019), which relates the deflection of a light ray on an image plane, i.e. a schlieren image, to the acoustic pressure field. The curvature of light rays is described by:

$$\begin{aligned} \frac{\partial ^2 y}{\partial z^2}=\frac{1}{n}\frac{\partial n}{\partial y}, \end{aligned}$$

(16)

where n is the refractive index of the gas. The refractive index and its gradient in terms of the pressure field are found using the isentropic gas relation in (7) and $n=1+\kappa \rho$,

$$\begin{aligned} \begin{aligned} n&=1+\kappa \rho \left( 1+\frac{p^{\prime }}{p}\right) ^{1/\gamma } \\&= 1+\kappa \rho \left( 1+\frac{p^{\prime }}{\gamma p} \right) +O\left( \kappa \rho \frac{p^{\prime 2}}{p^2}\right) \\&\quad \frac{\partial n}{\partial y} =\frac{\kappa }{c^2}\frac{\partial p^{\prime }}{\partial y} + O\left( \kappa \rho \frac{p^{\prime 2}}{p^2}\right) \end{aligned} \end{aligned}$$

(17)

The light ray deflection on the $x-y$ image plane is found by substituting (17) into (16), neglecting quadratic terms related to the small values $p'/p$ and $\kappa \rho$, and integrating over the path in the z-direction as

$$\begin{aligned} \epsilon (y)=\frac{\kappa }{c^2} \int _{-a}^a \frac{\partial p^{\prime }}{\partial y} dz. \end{aligned}$$

(18)

Equation 18 relates the fluctuating pressure field to the light ray deflection $\epsilon (y)$ that generates a schlieren image. This relationship is used to generate synthetic schlieren of each azimuthal mode by replacing the pressure field in (18) with the basis function $\varPsi _{nm}$ yielding,

$$\begin{aligned} \epsilon _{n,m}(y)=\frac{\kappa }{c^2} \int _{-a}^a \frac{\partial \varPsi _{nm}}{\partial y} dz. \end{aligned}$$

(19)

Fig. 4

Basis functions for a m=0, b m=1, c m=2, and d m=3 azimuthal Fourier modes ($-1<\frac{\partial \varPsi _{nm}}{\partial y} / \Vert \frac{\partial \varPsi _{nm}}{\partial y}\Vert <1$). Plots (e–h) to the right of each basis function compare line profiles at $z=0$ for each Fourier mode () to synthetic schlieren () integrated across z. Both line profiles are normalized by their respective maximum, ($\epsilon _{n,m} / \Vert \epsilon _{n,m}\Vert$, $\frac{\partial \varPsi _{nm}}{\partial y} / \Vert \frac{\partial \varPsi _{nm}}{\partial y}\Vert$). The right column i–l shows the difference between the normalized synthetic schlieren and line profiles ($\varDelta =\epsilon _{n,m} / \Vert \epsilon _{n,m}\Vert -\frac{\partial \varPsi _{nm}}{\partial y} / \Vert \frac{\partial \varPsi _{nm}}{\partial y}\Vert$) for multiple azimuthal rotations of the basis function. The range of rotations are defined by () in (b–d)

The effect of the path integration is evaluated by comparing the synthetic schlieren () to the radial gradient of the basis function taken along the line at $z=0$ () in Fig. 4e–h. For comparison, the synthetic schlieren and the line profiles are normalized by their respective maximum ($\epsilon _{n,m} / \Vert \epsilon _{n,m}\Vert$, $\frac{\partial \varPsi _{nm}}{\partial y} / \Vert \frac{\partial \varPsi _{nm}}{\partial y}\Vert$). The difference between the normalized synthetic schlieren and radial gradient profiles ($\varDelta =\epsilon _{n,m} / \Vert \epsilon _{n,m}\Vert -\frac{\partial \varPsi _{nm}}{\partial y} / \Vert \frac{\partial \varPsi _{nm}}{\partial y}\Vert$) are shown in Fig. 4i–h. For higher order modes, the effect of rotating the basis function is examined by plotting $\varDelta$ for multiple azimuthal rotations. The limits of the rotation are defined by the black dashed line in Fig. 4b–d.

In the $m=0,1$ modes, the path integration primarily effects the outer edges of the profile, where the largest changes in the shape are a shift of the outer peaks toward $y/a=0$. For higher modes, the effects of the path integration become more apparent as the three-dimensionality of the mode shape increases. In the $m=2,3$ cases, the shape of the synthetic schlieren deviates from the line profile and the location of the outer peaks and troughs again shift toward $y/a=0$.

When the basis function is rotated azimuthally, the shape of the line profile at $z=0$ and the synthetic schlieren maintain the same shape as shown in Fig. 4j–l. The exception are line profiles along axes of symmetry, where the radial gradient will be zero. The synthetic schlieren in these cases will also be zero due to the symmetry of the basis function.

The similarity between the synthetic schlieren and the radial gradient profiles for $m=0,1$ basis functions suggest the path integration has a small effect on schlieren measurements of the jet near-field. The $m=0,1$ azimuthal modes are the dominant contributors to the jet near-field, where the energy contained in the axisymmetric mode increases with jet Mach number (Cavalieri et al. 2011; Tam and Chen 1994; Jordan and Colonius 2013). Although the effect of the path integration increases for $m>2$, these modes make up a smaller amount of the total energy in the pressure field and will only have a small impact on the schlieren image.

The dominate effect of the path integration is for structures to appear closer to the jet centerline. The Fourier basis functions of the dominate azimuthal modes in Fig. 4 show the profiles from path integrated images are shifted toward the centerline compared to the profile in a single streamwise plane. This effect will impact correlations of schlieren images by shifting correlated structures toward the centerline.

3 Spatio-temporal behavior of the density near-field

3.1 Space-frequency coherence

The coherence between points in the schlieren image show the frequency dependent length scales of near-field structures and describe the modifications induced in them by the thermal non-uniformity. The space-frequency coherence of the fluctuating radial density gradient is calculated between probes points at $y/D=0.8, 1.4$, $x/D=3, 7$ and points taken along a line parallel to the shear layer as shown in Fig. 3. The location of the shear layer edge is approximated using a spreading angle of $6^{\circ }$, which was determined from the PIV measurements of Mayo et al. (2019).

Fig. 5

Space-frequency coherence the fluctuating radial density gradient of the uniform (a, c) and NUC (b, d) jets. Coherence of the uniform and NUC jets at $St=0.2$ , and $St = 0.4$ (e, f)

The space-frequency coherence in Fig. 5a–d exhibits extensive axial coherence at low frequencies, where sharp edges on either side of the coherence contour represent the end of the schlieren field of view. As the frequency increases, the coherence length scale rapidly decreases.

The coherence contours in Fig. 5 qualitatively resemble the space-frequency coherence plots of Tinney and Jordan (2008), who calculated the coherence between near-field microphones in a linear array. The authors contributed the shape of the near-field coherence contour to the frequency dependent boundary between the hydrodynamic and acoustic fields. As described by Arndt et al. (1997), rapidly decaying hydrodynamic fluctuations dominate low frequencies at very small distances from the shear layer. These hydrodynamic fluctuations represent the convective ” ‘footprint’ of the coherent turbulence and have large coherence length scales. As the frequency increases, the hydrodynamic field gives way to an acoustic field, which is comparatively less organized and will have shorted axial coherence length scales.

However, an important distinction between the space-frequency coherence plots in Tinney and Jordan (2008) and Fig. 5 is the presence of Mach waves in the near-field of the current study. Mach waves are super-directive acoustic waves produced by the supersonic convection of turbulent sources. They are highly organized and have longer axial coherence length scales compared the more disorganized acoustic field produced by coherent turbulence, i.e., turbulent mixing noise. The large coherence lengths scales at low frequencies in Fig. 5 represent contributions from hydrodynamic fluctuations and highly coherent Mach waves. As the frequency increases, the reduction in coherence length represents a decrease in the coherence of Mach waves and transition to a field dominated by turbulent mixing noise.

The space-frequency coherence of the NUC and uniform jets is compared in Fig. 5e, f at $\mathrm{St}\,=\,0.2, 0.4$. For both frequencies, the axial coherence decreases by 1D and 1.7D for the $x/D=3$ and $x/D=7$ locations, respectively. This change is consistent with the observed reduction in the far-field narrowband spectrum, which occurs over the same frequency range. However, it is unclear if the reduced axial coherence is related to changes in Mach waves or hydrodynamic structures in the near-field. In the next section, space-time correlations of schlieren images separate these near-field structures and examine how they are modified by the thermal non-uniformity.

3.2 Space-time correlations

Fig. 6

Space-time correlations of the density near-field of the uniform jet in the (a) near-nozzle ($x/D=3$) and (b) far-nozzle ($x/D=7$) region. Included are the location of the shear layer () and the probe point (x)

The space-time evolution of near-field structures is described using the cross-correlation of schlieren images. The cross-correlation is calculated between the same probe points used in the space-frequency coherence and every point in the region defined by the red rectangle in Fig. 3.

The correlations in Fig. 6 take the form of a highly organized traveling wave packet and represent Mach waves like those observed in the conditional POD analysis of Schmidt and Schmid (2019). At negative time lags, $\tau ^*<0$, the shape of the Mach wave structure stays relatively constant, implying it is produced by a long-lived wave packet that originates upstream of the probe point. At positive time lags, $\tau ^*\ge 0$, the correlation strength increases and the shape of the structures are slightly modified, suggesting the correlations are more complex than they first appear.

The changes at positive time lags are related to the presence of sources other than the traveling Mach waves, such as hydrodynamic structures and turbulent mixing noise. These features are present in the correlations and interact constructively and destructively with each other to produce the features in Fig. 6. In the next section, these structures are separated by frequency filtering schlieren images.

3.3 Frequency-filtered space-time correlations

Fig. 7

Auto-spectra calculated at the probe point (), and at locations radially outward (, ) for (a) $x/D=3$ and (b) $x/D=7$. Vertical lines represent cutoff frequencies used to frequency filter schlieren images

The different near-field features that contribute to the space-time correlation in Fig. 6 are separated by frequency filtering schlieren images. First, the frequency range dominated by hypodynamic structures is identified using the auto-correlation of the density near-field. The auto-correlation is calculated at probe points located axially at $x/D= 3, 7$ and at increasing distances from the shear layer.

The auto-spectra in Fig. 7 display a double hump feature that has been previously observed in near-field pressure measurements (Arndt et al. 1997; Kuo et al. 2013) and represents the hydrodynamic and acoustic components of the near-field. Arndt et al. (1997) demonstrated low frequencies in the near-field are dominated by hydrodynamic fluctuations that roll off in intensity and give way to an acoustic field as the frequency increases. The authors showed the hydrodynamic and acoustic fields are clearly divided at non-dimensional wavenumbers in the range $1< ky < 2$ (Arndt et al. 1997), where y is the distance between the probe point and the nozzle lip line. In Fig. 7b, the trough at $St=0.15$ represents this frequency dependent boundary and is equivalent to a non-dimensional wavenumber of $ky=1.6$, which is in the range predicted by Arndt et al. (1997).

The auto-spectra of the schlieren images capture additional features consistent with near-field microphone-based measurements in the literature. In Fig. 7b, the boundary between the hydrodynamic and acoustic fields shifts to lower frequencies for auto-spectra at locations further from the shear layer to maintain a constant ky. Furthermore, the peak of the acoustic hump shifts to lower frequencies as the probe is moved downstream from $x/D=3$ to $x/D=7$. This trend is also observed in Kuo et al. (2013) and is related to the increased size of turbulent structures at downstream locations.

The division between the acoustic and hydrodynamic fields at $x/D=3$ (Fig. 7a) is not as clearly defined. While there is a distinct low frequency hump, the spectra flatten out over a large frequency range before the acoustic peak. The shape of the auto-spectra can be contributed to the axial location of the probe. In the near-nozzle region, shear layer structures are very small, and the intensity of the hydrodynamic fluctuations is decreased relative to the acoustic. The reduction in the relative hydrodynamic energy reduces the magnitude of the hydrodynamic peak and blurs the division between regimes.

The auto-spectra of the schlieren images capture features in the near-field that agree with pressure measurements in the literature. Therefore, the location of the trough is used to define a frequency band dominated by hydrodynamic fluctuations, denoted as region I in Fig. 7. Hydrodynamic structures are isolated by low-pass filtering every pixel in the schlieren image using a FFT-based filter with cut off frequency of $\mathrm{St}\,=\,0.15$. Although the location of the frequency-dependent division for the $x/D=3$ case is uncertain, schlieren images are filtered using the same cutoff frequency used in the $x/D=7$ case, $\mathrm{St}\,=\,0.15$. All subsequent frequency filtering is performed in the same manner, where FFT filters are applied to every pixel in the schlieren image.

Frequencies $\mathrm{St}>0.15$ represent an acoustic field containing Mach waves emitted by the supersonic convection of turbulent structures and turbulent mixing noise produced by coherent turbulent sources. In Tinney and Jordan (2008), a rapid decrease in the coherence length was shown to represent the transition between the hydrodynamic field and a less organized acoustic field. However, in the current study, Mach waves are present in the near-field and will have long coherence lengths scales as shown in unfiltered cross-correlation in Fig. 6. The rapid decrease in the coherence length in Fig. 5 represents a decrease in the influence of Mach waves and a transition to a more disorganized acoustic field.

Near-field fluctuations dominated by Mach waves are found by bandpass filtering schlieren images using cutoff frequencies defined by the end of the hydrodynamic band (region I) and the contraction in the coherence length scale in Fig. 5. This mid-frequency band, noted as region II in Fig. 7, starts at $\mathrm{St}\,=\,0.15$ and ends at $\mathrm{St}\,=\,1.35$ and $\mathrm{St}\,=\,0.75$ for the $x/D=3$ and $x/D=7$ cases, respectively.

Cross-correlations are also taken of high-frequency near-field content by high-pass filtering schlieren images using cutoff frequencies defined by the upper frequency limit of region II. The high-pass filtered schlieren images, defined as region III, contain a more disorganized acoustic field where the influence of Mach waves is reduced and structures representing turbulent mixing noise may be viewed.

Fig. 8

Space-time correlations of the low frequency density near-field content (region I) for the uniform jet in the (a) near-nozzle ($x/D=3$) and (b) far-nozzle ($x/D=7$) region. Included are the location of the shear layer () and the probe point (x)

Cross-correlations of the low frequency content (region I) in Fig. 8 capture the space-time evolution of hydrodynamic structures. The correlations form semi-circular waves that travel along the shear layer edge, represented by the dash-dot line. The convection velocity of each structure is estimated by tracking the movement of the peak correlation, yielding $0.4U_j$ and $0.48U_j$ for the $x/D=3$ and $x/D=7$ cases, respectively. The subsonic convections indicate the structures are indeed hydrodynamic and not acoustic waves.

The correlation abruptly changes shape across the shear layer edge, forming a compact wave packet inside the jet that may represent a coherent turbulent structure. Hydrodynamic fluctuations are produced by turbulent shear layer structures, so correlations of hydrodynamic waves should also capture the turbulent source. However, the interpretation of correlations inside the jet shear layer is complicated by several factors. First, path integrated measurements in the shear layer introduce uncertainties related to the large density gradients. Second, a turbulent source and the acoustic wave it produces will have the same wave vector (Glegg and Devenport 2017) and therefore look the same when viewed through a cross-correlation. Saltzman et al. (2021) demonstrated this using 50 kHz velocimetry measurements in the Virginia Tech heated supersonic jet rig at the same nominal conditions ($M_\mathrm{s}\,=\,1.5, \mathrm{TTR}\,=\,2$) as the present study. The authors compared space-time correlations of low-pass filtered velocimetry measurements in the shear layer to low-pass filtered schlieren measurements of the near-field and found the time and length scales of both correlations were similar.

Fig. 9

Space-time correlations of the mid frequency density near-field content (region II) for the uniform jet in the (a) near-nozzle ($x/D=3$) and (b) far-nozzle ($x/D=7$) region. Included are the location of the shear layer (), the probe point (x), and the locus of the correlation maxima ()

In Fig. 9, cross-correlations of bandpass filtered schlieren images (region II) describe the space-time evolution of Mach waves. The correlations have a wave packet-like structure and exhibit minimal temporal evolution at negative time lags. At time lags $\tau ^*>0$, the correlation amplitude increases, and the size of the wave packet lobes grow while maintaining a similar shape. The changes at positive time lags represent the emission of a Mach wave from the probe point that propagates with an ejection angle shown by the locus of the peak correlation, denoted by the dotted lines in Fig. 9. The emitted Mach wave propagates with ejection angles of $\theta _{0}=145 ^{\circ }$ and $\theta _{0}=135 ^{\circ }$ for the $x/D=3$ and $x/D=7$ correlations, respectively. These angles are similar to the peak noise directivity indicated by the OASPL polar plot shown in Fig. 2 ($\theta _{0}=140 ^{\circ }$), and indicate that these large wavelength structures represent the intense Mach waves that dominate the far-field at shallow angles to the jet axis.

Furthermore, the correlations representing Mach wave structures extend into shear layer. This is a result of collimated light rays traveling through Mach waves located at different azimuthal angles around the jet that are located vertically ‘closer’ to the jet centerline. The causes the Mach wave structures in the correlations to appear at locations inside the shear layer. This effect is also show by the synthetic schlieren in Fig. 3, where the path integration shifts the peak of the pressure gradient toward the jet centerline.

Below the lip line ($y/D=0.5$), the correlations change shape, forming vertical ‘blobs’ that deviate from the well-defined wave-packet structure. These lip line structures may represent turbulent structures that produce Mach waves. However, as with the correlations in Fig. 8, schlieren measurements at these locations are difficult to interpret due to the large density gradients and how turbulent sources and acoustic waves will look the same when viewed through a correlation.

Fig. 10

Space-time correlations of the high-frequency density near-field content (region III) for the uniform jet in the (a) near-nozzle ($x/D=3$) and (b) far-nozzle ($x/D=7$) region. Included are the location of the shear layer (), the probe point (x), and the locus of the correlation maxima ()

The correlations of the high frequency fluctuations (region III) are shown in Fig. 10. At positive time lags $\tau ^*>0$, the correlations take the shape of an acoustic wave emitting from the shear layer. The wave is slightly curved, suggesting turbulent mixing noise contributes more to the overall correlation shape than super-directive Mach waves.

At negative time lags, the correlated structure is almost entirely inside the shear layer, with the locus of the peak correlation traveling inside the jet and parallel to the shear layer edge. As the time delay approaches $\tau ^*=0$, the peak correlation ejects from the shear layer as an acoustic wave is emitted. Although there are uncertainties regarding the identity of correlations in the jet, Fig. 10 appears to depict the refraction of an acoustic wave through the shear layer.

The directivity of the emitted wave can be predicted using Snell’s law. First, it is assumed the wave propagates toward the shear layer at a small angle of $\theta _i=5 ^{\circ }$. The wave speed inside the shear layer is calculated by tracking the peak correlation value and found to be $0.74U_j$ and $1.15U_j$ for the x/D = 3 and x/D = 7 case, respectively. Knowing the convection speed outside the shear layer is the sound speed, the transmission angle is calculated as $\theta _t=140 ^{\circ }$ and $\theta _t=119 ^{\circ }$ for the $x/D=3$ and $x/D=7$ cases, respectively. These estimated refraction angles are similar to the propagation angle of the acoustic beams in the correlations in Fig. 10, which travel at $\theta _{0}=135^{\circ }, 115^{\circ }$ for correlations at $x/D=3$ and $x/D=7$, respectively.

Correlations of the frequency filtered schlieren images have identified acoustically important near-field structures in different regions of the developing jet. In the next section, these features are compared for the uniform and NUC jets to evaluate changes in the turbulence source mechanisms over a range of axial locations.

3.4 Comparison of NUC and uniform correlations

Fig. 11

Comparison of constant contours (− 0.1, 0.2) of the space-time correlations of the low frequency components (region I) in the density near-field of the Uniform and NUC jet in the (a) near-nozzle ($x/D=3$) and (b) far-nozzle ($x/D=7$) region. Included are the location of the shear layer (), and the probe point (x)

Fig. 12

Comparison of constant contours (− 0.1, 0.2) of the space-time correlations of the mid frequency components (region II) in the density near-field of the Uniform and NUC jet in the (a) near-nozzle ($x/D=3$) and (b) far-nozzle ($x/D=7$) region. Included are the location of the shear layer (), the probe point (x), and the locus of the correlation maxima ()

Correlation coefficients of the frequency-filtered fields of the NUC and uniform jets are compared in Figs. 11, 12 and 13 using constant contour levels of − 0.1 and 0.2. Correlations in Fig. 11 show that hydrodynamic structures in the NUC jet have shortened correlation length scales compared to the uniform jet. The effect occurs at both axial stations and impacts the hydrodynamic wave features and the structures inside the jet shear layer. Mach wave correlations of the NUC jet in Fig. 12 also exhibit reduced correlation lengths compared to the uniform case for regions downstream and less so for upstream locations. Similarly, correlations of the high frequency content in Fig. 13 show the spatial extent of structures in the shear layer and the acoustic beam in the near-field are reduced in the NUC case. However, the decorrelation is only appreciable in the far-nozzle region, with no observed difference in the upstream location.

Fig. 13

Comparison of constant contours (− 0.1, 0.2) of the space-time correlations of the high frequency components (region III) in the density near-field of the Uniform and NUC jet in the (a) near-nozzle ($x/D=3$) and (b) far-nozzle ($x/D=7$) region. Included are the location of the shear layer (), the probe point (x), and the locus of the correlation maxima ()

The reduced spatial extent of correlations imply that the perturbations induced by the thermal non-uniformity reduce the length scale of the hydrodynamic structures and large-scale instabilities that radiate Mach waves at locations far from the nozzle exhaust. This result is significant, as it indicates that the perturbations induced by the thermal non-uniformity persist within the potential core and decrease the correlation length scales of noise-producing instabilities at locations far downstream.

The radiation efficiency of turbulent structures can be decreased by reducing four fundamental factors: the turbulent kinetic energy (TKE), the convection velocity, and the correlation time and length scales (Papamoschou et al. 2014). The 2.5 dB noise reduction observed in the NUC case is not driven by reduction in TKE, as a 50% reduction is needed to reduce radiated noise by 3 dB (Papamoschou et al. 2014) and was not observed in the PIV measurements of Daniel et al. (2019b). A change in convection velocity also does not contribute to the acoustic benefit as the locus of the peak correlations across all cases are similar, indicating the directivity, and by extension the convection velocity, has not changed significantly. Therefore, the acoustic efficiency of the turbulence is reduced by a decrease in the correlation length scale of turbulent features in both the near- and far-nozzle regions.

The observed reduction in the correlation length scale is a major finding given, as Papamoschou (2018) states, “While it is possible to disorganize the turbulence in the vicinity of the nozzle, the extent to which structures at large distances from the nozzle can be affected is not clear, given the natural tendency of the shear layer to self-organize into large vortical motions (Fiedler 1988).” The essence of this statement is consistent with the results of Tinney and Jordan (2008), who compared the near pressure fields of multi-stream subsonic jets with nozzle serrations to a non-serrated baseline case. The authors found serrations reduced the axial coherence of acoustic and hydrodynamic components in the near-nozzle region and did not affect structures at locations downstream of $x/D=3$.

The current results indicate that perturbations along the jet centerline are effective at reducing the axial coherence of turbulent structures in the far-nozzle region. Centerline perturbations are likely shielded from the outer shear layer by the irrotational fluid of the jet potential core, allowing them to convect over long distances and interact directly with the shear layer at locations far downstream.

In terms of thrust loss, centerline perturbations may also be a more cost-effective method to reduce noise as compared to reduction techniques which alter the shear layer directly at the nozzle exhaust. The centerline boundary condition could be further optimized to target the intense noise sources near the potential core collapse region. The area near the potential core collapse region of the jet is a strong producer of noise due to the large correlation length scale of the turbulence and the presence of the strong intermittent noise events near the core collapse region (Fisher et al. 1977; Morrison and McLaughlin 1979; Juvé et al. 1980). Therefore, for a given amount of thrust loss, concentrating thermal reductions around the jet centerline may yield the most cost-effective acoustic benefits.

Lastly, the noise benefit of a centered thermal non-uniformity may scale with the temperature-driven velocity deficit. Evidence comes from Brès et al. (2019), who performed LES simulations of over-expanded jets with non-uniform temperature profiles. In the computational work, the non-uniform jets studied had large temperature differences, with the streams of the thermally non-uniform jet operating at TTR’s of 7 and 3. The authors found the non-uniform jet had a maximum reduction of 4 dB in the overall sound pressure as compared to the uniform case. This, as well as evidence from the current work, suggests larger noise reductions could be achieved by increasing the temperature difference, which could be attained in full-scale engines operating at afterburning conditions.

4 Summary and conclusion

The structure of the density near-field of heated supersonic jets was investigated using schlieren imaging. The effect of path integration was evaluated by simulating the jet pressure field using the azimuthal basis function of the Fourier transform. Synthetic schlieren generated using the basis functions showed, for the dominant azimuthal modes ($m\,=\,0,1$), the shape the path integrated profile was similar to the pressure gradient taken along a line in a centered streamwise plane. The main effect of the schlieren was to shift the shape of the profile inward toward the jet centerline. This effect was observed in the space-time correlations, where correlations representing Mach waves were observed inside the jet shear layer.

The coherence calculated between points in the near-field showed extensive coaxial coherence at low frequencies, indicating a region dominated by hydrodynamic fluctuations and Mach waves. As the frequency increased, a rapid decrease in coherence length represented the transition to a more disorganized acoustic field. A comparison of the space-frequency coherence from the NUC and uniform jets showed the thermal non-uniformity reduced the axial coherence of low frequency structures.

Structures present in the near-field were further investigated using space-time correlations of frequency filtered schlieren images. Correlations of low frequency fluctuations depicted hydrodynamic waves traveling along the shear layer at subsonic speeds. Band-pass filtered images were dominated by super-directive Mach wave structures and correlations of high frequency displayed the emission of an acoustic beam from the shear layer. The introduction of a thermal non-uniformity was found to reduce the size of correlations representing hydrodynamic waves and Mach waves in the near-nozzle regions ($x/D\,=\,3$ and at axial locations near the potential core collapse (x/D = 7). The results indicate that base flow changes induced by the introduction of a centered thermal non-uniformity have a direct impact on the correlation length scale of coherent turbulence and large-scale instabilities at locations far from the nozzle exit. These changes reduce the strength of radiated Mach waves and result in noise reductions up to 2 ± 0.5 dB in the peak narrowband spectral sound pressure levels. These findings are significant as they suggest centerline boundary condition changes may be an efficient way to reduce jet noise.