2.1 Laboratory arrangement

The experiments were conducted in the Anechoic Jet Laboratory (AJL) at The University of Mississippi’s National Centre for Physical Acoustics. The facility was purpose built for the study of high-temperature, supersonic jet noise. Descriptions of the design and capabilities of the facility can be found in Ponton, Seiner, Ukeiley & Jansen (Reference Ponton, Seiner, Ukeiley and Jansen2001) and Murray & Jansen (Reference Murray and Jansen2012). Recent upgrades to the facility include burner hardware modifications and instrumentation upgrades to enhance monitoring of the jet operating conditions (Murray & Jansen Reference Murray and Jansen2014). Details relevant to the current work are summarized below.

The AJL jet rig uses propane combustion to reach jet temperature equivalent to full-scale jets. The gaseous propane is injected into the jet rig and burned in a swirl-can style combustor housed well upstream of the nozzle. The burner section is followed by a ceramic flow straightener and settling chamber leading into a $5.3:1$ contraction. The end of the contraction is threaded to accept the nozzle assembly.

The walls, floor and ceiling of the AJL are covered with fibreglass wedges to create a fully anechoic environment suitable for jet noise measurements. Acoustic data were not acquired in the current experiment. Instead, the floor wedges were removed to allow flow diagnostic equipment to be placed in the room. A precision three-axis traverse system was place in the AJL and laser aligned with the jet exhaust. The traverse was used to perform a stagnation pressure probe survey of the jet to inform the positioning of the two shadowgraph imaging regions. The shadowgraph system was supported on floor-mounted tripods. The traverse system remained in the room during the acquisition of shadowgraph images, but the pressure probe was removed from the flow.

2.2 Description of the model-scale jet nozzles

The nozzle assembly used for this experiment consisted of three components: a centrebody, a short extension and a convergent/divergent nozzle. The centrebody represents the non-rotating core in a turbo-jet engine and is intended to reproduce a realistic internal flow leading up to the nozzle contraction. The extension represents the augmenter section in a typical afterburning engine and places the nozzle throat at a realistic scaled distance downstream from the centrebody. The nozzle itself is modular with twelve internal segments captured by a locking ring.

Two sets of segments were uses in the nozzle assembly for this study. The baseline inserts have flat panels in the expansion section of the nozzle to mimic the variable geometry nozzles typically found on supersonic aircraft. The convergent/divergent (C–D) nozzle has a 50 mm exit diameter and a design Mach number of 1.74 based on the ratio between the throat and exit areas. The second nozzle was formed by replacing the flat panels with contoured inserts (COINs). The COINs nozzle has a smaller exit area but the same throat area as the baseline nozzle, resulting in a design Mach number of 1.55. Photographs of the two nozzles are shown in figure 2.

Figure 2.

Photographs of the two model-scale C–D nozzles used in the experiment: (a) baseline nozzle, (b) nozzle with COINs jet noise reduction concept.

Table 1.

Jet operating conditions.

2.3 Test conditions and jet characteristics

Both nozzles were operated with the same stagnation conditions for the experiment. Table 1 lists the measured and calculated quantities describing the jet parameters using $j$ , $\infty$ , 0 as subscripts for the jet exit, ambient and stagnation conditions, respectively. The measured quantities are averaged over the entire experiment. The jet exit conditions are calculated using standard isentropic relations. The ratio of specific heats, ${\it\gamma}$ , is determined with an iterative method taking the area ratio of the nozzle into account (Murray & Jansen Reference Murray and Jansen2014). Dynamic viscosity is calculated using Sutherland’s law.

The jet stagnation and ambient temperatures were measured with K-type thermocouples using an ioTech TempScan/1100 model to an accuracy of $\pm 0.5$  K. The stagnation and ambient pressures were measured with an Esterline model 9016 pressure scanner with an accuracy of $\pm 0.5$  kPa.

Figure 3.

Composite schlieren images showing features of the jets studied: (a) baseline jet, (b) COINs jet.

The experimental results of Oertel (Reference Oertel1980) indicate that the convective Mach numbers of the three instability waves follow simple algebraic formulas: $M_{c}^{\prime }=(U_{j}+a_{j})/$ $(a_{j}+a_{\infty })$ , $M_{c}=U_{j}/(a_{j}+a_{\infty })$ , $M_{c}^{\prime \prime }=(U_{j}-a_{j})/(a_{j}+a_{\infty })$ and $M_{c}^{\prime }<M_{c}<M_{c}^{\prime \prime }$ . These are listed in table 1 for reference. The only mode with supersonic convective Mach number is $M_{c}^{\prime }$ , which corresponds to the Kelvin–Helmholtz instability mode according to the inviscid instability theory of Tam & Hu (Reference Tam and Hu1989). This suggests that the Mach wave pattern should only have one Mach wave angle for these jet conditions. Interestingly, the associated Mach wave propagation angle of ${\it\phi}_{c}^{\prime }=\cos ^{-1}(1/M_{c}^{\prime })=49.5^{\circ }$ gives a convective velocity ratio relative to the jet exit velocity of only $U_{c}^{\prime }/U_{j}=0.65$ . However, most direct experimental measurements of the convection velocity give ratios in the range of 0.7–0.8 (Norum & Seiner Reference Norum and Seiner1982; Troutt & McLaughlin Reference Troutt and McLaughlin1982; Tinney et al. Reference Tinney, Ukeiley and Glauser2008). Taking $U_{c}=0.8U_{j}$ would yield ${\it\phi}\approx 58^{\circ }$ . The same difference between observed ${\it\phi}$ and predicted ${\it\phi}_{c}^{\prime }$ was indicated by Seiner et al. (Reference Seiner, Ponton, Jansen and Lagen1992).

The baseline nozzle’s area ratio is consistent with a design Mach number of 1.74. Therefore, the operating conditions yield an over-expanded jet with a predominant standing shock pattern in the jet plume as shown in the composite schlieren photograph in figure 3(a). The conic-section nozzle generates a characteristic double-shock pattern. One diamond-shaped shock pattern originates near the nozzle throat, and a second shock pattern originates near the nozzle exit.

The COINs nozzle effectively contours the expansion section of the nozzle according to the method of characteristics resulting in a nearly shock-free jet. The COINs shapes were designed for the nozzle pressure ratio (NPR) and jet temperature used for the experiment. The COINs nozzle has the same throat area as the baseline nozzle, so the exit area of the COINs nozzle is smaller in agreement with a design Mach number of 1.55. The COINs also produce streamwise vorticity resulting from the encroachment of the lobe shapes into the core flow (see figure 2 b). The reduction in shock strength and the vorticity generation are evident in the shadowgraph images shown in figure 3(b). The impact on jet noise is a drastic reduction in broadband shock associated noise and also a decrease in the peak noise (Murray & Jansen Reference Murray and Jansen2012). The mechanism for peak noise reduction is the primary interest of the present work.

Figure 4 shows the stagnation pressure decay along the centreline of the jet for the baseline and COINs nozzles. The baseline nozzle, being over expanded at this NPR, is a mixed flow (both subsonic and supersonic regions are present) with strong shocks. The stagnation pressure begins to decay smoothly for the baseline nozzle around $x/D=7$ . This agrees well with empirical estimates of the potential core length for overexpanded jets (Varnier Reference Varnier2001). The COINs nozzle is close to perfectly expanded at this NPR with a more uniform stagnation pressure in the potential core region. The streamwise vortices generated by the contoured inserts enhance mixing and result in a shortened potential core length. The stagnation pressure for the COINs nozzle begins to decay smoothly around $x/D=5$ .

Figure 4.

Comparison of the stagnation pressure along the jet centreline for the baseline and COINs nozzles. Vertical lines indicate apparent potential core lengths based on visual inspection.

2.4 Retro-reflective shadowgraph imaging system

A retro-reflective shadowgraph system was assembled to image not only the jet but also the emission and propagation of shock fronts into the ambient region out to $y/D_{j}=7$ (seven jet diameters laterally from the jet axis). Two regions were imaged: one near the nozzle and another downstream of the potential core. The direct shadowgraphy imaging technique described in detail by Hargather & Settles (Reference Hargather and Settles2009) was chosen as the measurement method because of its simplicity, large field of view and sensitivity to sharp refractive disturbances such as shock waves. A rod mirror arrangement was selected for the light source and camera to be optically coincident to avoid any image doubling effects. An illustration of the set-up is shown in figure 5.

As light passes through the measurement region, the light rays bend due to inhomogeneities in the refractive index, $n$ . Letting $z$ be the optical axis with $x$ and $y$ in the plane normal to $z$ , the angular deflection of a light ray is given by the path integral of the refractive index gradient. In the $x$ direction this becomes ${\it\varepsilon}_{x}=1/n\int \partial n/\partial x\,\text{d}z$ . The refractive index is linearly related to the density, ${\it\rho}$ , according to $n-1={\it\kappa}{\it\rho}$ , where ${\it\kappa}$ is the Gladstone–Dale constant. Certainly, the angular deflection is the result of all refractive gradients along the path of integration. However, if ${\it\varepsilon}$ were constant everywhere the result would be a constant deflection of all the light rays yielding no change in the illumination of the screen and hence no shadows. Rather, it is the gradients in ${\it\varepsilon}$ that result in the forming of light or dark regions in the image. Therefore, the direct shadowgraph method observes gradients in the deflection angle, and by extension the Laplacian, of the refractive index. As a result, direct shadowgraphy is often selected to image shock fronts where $\partial ^{2}n/\partial x^{2}$ is very large. This is in contrast to the schlieren technique where a knife edge is employed resulting in the measure of the total deflection angle making it more sensitive to weaker disturbances (Settles Reference Settles2001).

The direct shadowgraphy’s sensitivity to strong deflection gradients is used here to investigate shock-like waves in the emitted noise from the jet that appear as sharp, distinct streaks in the shadowgraph images. As the shock fronts spread outward from the jet axis they come into sharp focus on the spherical surface where they are tangent to the diverging light rays. Three such rays are illustrated in figure 5. Two shock front types are illustrated in the figure. For an axisymmetric jet mode, the resulting shocks would appear as concentric circles on a plane normal to the jet axis. For a helical jet mode, the same axial cut would yield a spiral shape. From the cameras point of view, the surface of sharp focus is projected onto the screen, giving a side view of the jet. Waves from an axisymmetric mode would be symmetric on the top and bottom of the jet, while helical models would yield alternating wave patterns. In the present work, only the region above the jet is imaged, so no distinction is made between axisymmetric and helical modes of the source.

Figure 5.

Illustration of the retro-reflective shadowgraph optical set-up.

The curvature of the surface of sharp focus can introduce distortion depending on the magnification and object distances as illustrated in figure 6 (Dewey Reference Dewey2001). The object plane is parallel to the image plane inside the camera located at $C$ . For a wave front emerging from $G$ some distance $r$ from the camera, the point $T$ follows the surface of sharp focus illustrated in figure 5. When the wave front reaches a radius of $a$ , the refractive image of the wave front will appear to be at $P$ and will be projected onto the screen, giving an image height of $H$ at a distance $R$ from the camera. The correct wave front radius, $a$ , is linearly related to the apparent size, $h$ , geometrically.

(2.1) $$\begin{eqnarray}a=h\cos {\it\theta}=\frac{H\,r}{R}\,\frac{R}{\sqrt{H^{2}+R^{2}}}.\end{eqnarray}$$

The first fraction in the product is the linear magnification. The second fraction is a stretching factor that depends on $H$ .

Figure 6.

Geometry of the diverging ray optical system for the discussion of magnification and stretching.

In the set-up used here, the camera is approximately 180 cm from the jet axis and the retro-reflective screen is positioned opposite the camera approximately 135 cm from the jet axis. The circle of illumination on the screen is 66 cm in diameter, so the maximum image height $H$ is 33 cm. With $R=315~\text{cm}$ , the stretching factor is 0.995. This indicates that the distortion for this laboratory set-up is negligible, and no corrections for this effect were required in the analysis.

The set-up of the system is shown in figure 7 and follows that of Hargather & Settles (Reference Hargather and Settles2009). Scotchlite 7610 is used as the retro-reflective screen material. A Perkin Elmer Lite-Pac Xenon strobe was used as the light source. The strobe has a pulse duration of approximately 1  ${\rm\mu}$ s. The image of the light source was focused onto a 45° rod mirror using a Schneider Kreuznach Xenon 25 mm, $f$ 0.95 lens. A neutral density filter was used to reduce the intensity of the light source. A PCO.Edge sCMOS camera was used to record the images with a resolution of 2560 $\times$ 2160 pixels. A Navitar 50 mm, $f$ 0.95 lens was used as the light collection optic.

Figure 7.

Photographs of the laboratory set-up. The set-up of the imaging system (a) includes a camera, strobe light source, lens and a rod mirror. The laboratory set-up (b) shows the position of the retro-reflective screen, imaging system and probe traverse relative to the jet rig.

2.5 Description of the dataset

The shadowgraph system was used to acquire image sets at two different regions of the flow, as shown in figure 8. The upstream region covers $0<x/D_{j}<5$ and is offset in the $y$ direction to capture the shock front emission from the near-nozzle region. The downstream measurement region covers $6<x/D_{j}<13$ and is positioned to capture shocks emitted from the post-potential core region. The experiment included four image acquisition sequences: one for each imaging region for each nozzle configuration. Each sequence included 1024 images acquired at a 50 Hz frame rate.

Shock fronts appear in the shadowgraph images as discrete, rapid light-to-dark gradients as shown in the upper inset in figure 8. In many cases, as in the lower inset, shock fronts were seen to be directly attached to the jet shear layer. This agrees with the notion that the crackle source is a local event inside the jet, as opposed to a rapid steepening of high amplitude acoustic waves outside the jet shear layer.

Figure 8.

A sample snapshot from the baseline nozzle dataset illustrating the shadowgraph measurement regions. The image sets for the two regions were acquired independently.

3.1 Image preprocessing

The first step in the analysis was to remove any background gradients in the raw images. The background gradient is evident when the pixel intensity along a line is plotted as in figure 9. Similar to Fagan, L’Esperance & Zaman (Reference Fagan, L’Esperance and Zaman2014), a custom filter was created to remove the background gradient, enhance contrast and retain the sharp edge on the shadowgraph viewing area. The filter is essentially a spatially smoothed mean image that can be subtracted from each snapshot. The filter was constructed by first averaging the 1024 image set. Next, the mean image was blurred using a rotationally symmetric Gaussian lowpass filter with a size of 100 pixels and a standard deviation of 20. Finally, a binary mask was manually constructed that isolates the shadowgraph viewing area. The masked filter was applied by subtracting it from each snapshot. The result of the filtering process is shown in figure 9(c). A constant grey value offset is added within the viewing area to enhance contrast for plotting purposes. Use of the filter yields a constant mean value for the pixel intensity within the viewing area as shown in figure 9(d).

Figure 9.

Demonstration of the background gradient removal applied to the raw shadowgraph images: (a) raw image, (b) custom filter image, (c) background corrected image and (d) pixel intensity along the horizontal line in (a–c), grey is the raw intensity, black is after the filter is applied.

The shear layer edge was identified to distinguish the jet plume from the ambient region. Each background digital image is represented as an $m\times n$ matrix of intensity values, $I(m,n)$ . After background correction, the mean image is computed as the arithmetic mean of each pixel, $\overline{I}(m,n)$ . In the same way, the root-mean-square (RMS) fluctuation of the pixel intensity, $I_{RMS}$ , is calculated for each pixel. Turbulence causes strong fluctuations in the pixel intensity, so calculating $I_{RMS}$ is a simple way to visualize the development of the jet shear layer. The edge of the shear layer is found by a linear fit to a threshold trend in the $y/D$ gradient in $I_{RMS}$ . Figure 10 shows $I_{RMS}$ as a grey-scale image (black is large $I_{RMS}$ ) with the shear layer fit superimposed as a solid line. For the baseline nozzle, the shear layer fit was found to have a slope of 8.6°. For the COINs nozzle, the shear layer was found to have a slope of 8.5°.

3.2 Subimage analysis to determine shock front angles

The ambient region was gridded for subimage analysis of the shock front angles emitted by the jet. The interrogation grid is shown as open circles superimposed on the $I_{RMS}$ image in figure 10. The grid spacing is approximately $0.25D_{j}$ . In the upstream imaging region, an insignificant number of shock fronts were observed to the upper left of the grid, so that region was excluded from the analysis for expedience.

Figure 10.

The interrogation grid is shown superimposed on $I_{RMS}$ images. The solid line is the shear layer edge defined as a linear fit through the shear layer edge points shown as triangles.

For each instantaneous shadowgraph image, subimages centred on each of the interrogation grid points were extracted and filtered as shown in figure 11. Subimage $k$ is denoted $I_{k}^{\prime }(p,q)$ , where $p$ and $q$ are the respective horizontal and vertical pixel indices centred on the $k$ th interrogation point. Canchero et al. (Reference Canchero, Tinney, Murray and Ruf2016) evaluate the sensitivity of the angle search routine and find that the ratio of interrogation size to integral length scale along the peak wave front angle need only exceed 5.5 to eliminate error associated with the subimage size. Within the limits of the optical system, the integral length scale associated with the shock front becomes shorter as the shock becomes more distinct. The observed streaks shown in figures 8 and 11 are on the order of 5–6 pixels in width. Therefore, a subimage size of $128\times 128$  pixels was selected, which is well above the necessary size to yield well resolved local wave front angles. Each $I_{k}^{\prime }$ is normalized and a two-dimensional radial Tukey window is applied to reduce edge effects in subsequent analysis. Figure 11(c) shows a single normalized and filtered subimage, $\hat{I} _{k}^{\prime }$ , before it is passed to the angle-finding routine.

Figure 11.

Image processing sequence showing the selection, normalization and filtering of a $128\times 128$ pixels subregion of the parent image: (a) subimage shown on interrogation grid, (b) extracted subimage, (c) normalized and filtered subimage ready for input to angle-finding routine.

The angle of the shock front in the subimage is evident from inspection. For example, in figure 11(c) the wave front appears as the dark line passing through the centre from the top left to bottom right of the image. An autocorrelation is performed to reveal the dominant angle of the wave in the subimage. The two-dimensional autocorrelation, $R_{k}(p,q)$ , is computed by performing a two-dimensional Fourier transform, applying a high-pass filter in the form of a circular spatial mask, multiplying the result by its complex conjugate, and inverse Fourier transforming with appropriate normalization. The Fourier transform of $\hat{I} _{k}^{\prime }$ is shown in figure 12(a) with the 6-pixel-radius circular mask applied to reject low frequency features in the image. The mask is the small black circle in the centre of the figure. Figure 12(b) shows $R_{k}(p,q)$ . The strong correlation region appears as a bright streak passing from the top left to bottom right of the image through the central maximum. The normal to this correlation streak gives the dominant wave angle for interrogation point $k$ .

Figure 12.

Analysis of the subimage: (a) two-dimensional Fourier transform of the subimage, (b) autocorrelation with the corresponding wave front angle measured and indicated.

Figure 13.

Shock front angle-finding method: (a) Radon transform of the subimage autocorrelation, (b) normalized Radon transform amplitude along $p^{\prime }=0$ showing the central peak at 54°.

The shock front propagation angle is determined from the integral Radon transform of $R_{k}$ . This transform yields a function, $\mathscr{R}_{k}({\it\theta},p^{\prime })$ , which is the projection of $R_{k}$ onto a line normal to the projection angle, ${\it\theta}$ , as shown in figure 13(a).

(3.1) $$\begin{eqnarray}\mathscr{R}_{k}({\it\theta},p^{\prime })=\int _{-\infty }^{\infty }R_{k}(p^{\prime }\cos {\it\theta}-q^{\prime }\sin {\it\theta},p^{\prime }\sin {\it\theta}+q^{\prime }\cos {\it\theta})\,\text{d}q^{\prime }.\end{eqnarray}$$

Note that $(p^{\prime },q^{\prime })$ is rotated counter-clockwise from $(p,q)$ by angle ${\it\theta}$ . The transform was computed for $0^{\circ }>{\it\theta}>179^{\circ }$ in $0.5^{\circ }$ increments. $\mathscr{R}$ is defined with its centre of rotation at $(p,q)=(0,0)$ , which is also where $R_{k}$ has its maximum. As a result, the maxima in $\mathscr{R}$ will occur along $p^{\prime }=0$ . Therefore, the search for the maximum projection angle reduces to finding the maximum on $\mathscr{R}_{k}({\it\theta},0)$ as shown in figure 13(b). In this case, the maximum occurs at $54^{\circ }$ as shown in figure 12(b), indicating the local shock front angle.

Two tests were performed to validate the shock front angles for each $\mathscr{R}_{k}$ . First, if multiple maxima were found in $\mathscr{R}_{k}({\it\theta},0)$ , the result was retained only if the second highest peak was less than 0.7 of the dominant peak value. Second, the result was validated against the span of the wave front: if a wave front extends across the entire subimage, the width of the correlation peak should be significant. Therefore, the result was retained only if the correlation coefficient remained above 0.2 for more than 10 pixels. The threshold of 10 pixels was determined after manually evaluating $\mathscr{R}_{k}$ for a number of subimages both with and without a shock front present. Given the strict validation criteria, the uncertainty in the reported maximum is $\pm 0.25^{\circ }$  based on the discretization of ${\it\theta}$ into $0.5^{\circ }$ increments.