2.1 Numerical methods

For the current simulation of the jet-impingement problem, the unsteady compressible Navier–Stokes equations are solved in a conservative finite-difference formulation on block-structured Cartesian grids employing a sharp immersed boundary method (see Mittal et al. Reference Mittal, Dong, Bozkurttas, Najjar, Vargas and von Loebbecke2008; Brehm & Fasel Reference Brehm and Fasel2013; Brehm, Hader & Fasel Reference Brehm, Hader and Fasel2015b ) and an adaptive mesh refinement (AMR) scheme utilizing an entropy adjoint approach following Fidkowski & Roe (Reference Fidkowski and Roe2010). The solver is designed to dynamically track important flow features as they develop over time by utilizing the AMR solution procedure. To account for flow discontinuities, a sixth-order accurate shock-capturing weighted essentially non-oscillatory (WENO) scheme is utilized. The characteristic variable transformation is applied to a Rusanov-type flux vector splitting scheme. The accuracy and computational efficiency of the higher-order shock-capturing schemes of the current solver have been previously analysed in Brehm et al. (Reference Brehm, Barad, Housman and Kiris2015a ). The viscous terms are discretized with fourth-order accuracy. The fourth-order accurate explicit Runge–Kutta scheme is used for time integration. Additional information about the numerical schemes and their properties can be found in Kiris et al. (Reference Kiris, Barad, Housman, Sozer, Brehm and Moini-Yekta2014).

Figure 1.

(a) Mesh refinement regions in the computational domain. (b) Sampling lines used for postprocessing of the unsteady flow field, with greyscale contours of temperature.

2.2 Computational set-up

The mesh topology for the current simulation is schematically illustrated in figure 1(a). Grid refinement was applied in the turbulent jet, the impingement region and the wall jet where high unsteadiness and large gradients were expected. Different volume tagging surfaces were used to control the grid spacing in the impingement region (figure 1 a). Two cone sections (refinement cones 1 and 2) were used to control the volume mesh spacing in the turbulent jet, and the acoustic sampling surface used in § 4 was employed for volume tagging of the wall jet, as illustrated in figure 1(b). Within the turbulent jet, we used a grid spacing that corresponded to $D/{\rm\Delta}x=200$ grid points across the nozzle diameter. The mesh was subsequently coarsened by a factor of 2 in the second cone and in the wall jet. Overall, a total of approximately $0.6$ billion grid points were used in the current mesh. A grid refinement study was previously conducted in Brehm et al. (Reference Brehm, Sozer, Moini-Yekta, Housman, Barad, Kiris, Vu and Parlier2013) and Housman, Brehm & Kiris (Reference Housman, Brehm and Kiris2013). For the current simulation, the origin of the coordinate system is located at the intersection point of the nozzle centre axis with the impingement plate, where the $z$ -axis points towards the nozzle. The nozzle exit is located five nozzle exit diameters ( $z=5D$ ) above the origin (figure 1 b). Figure 1(b) also shows different sampling lines and sampling circles that are used throughout the paper for probing the unsteady flow field. The computational domain extends $50D$ in each coordinate direction, and characteristic boundary conditions are used to avoid wave reflections from the domain boundaries. Further, the computational mesh coarsens towards the domain boundaries, providing additional numerical dissipation to eliminate spurious reflections. The nozzle ejects an approximately ideally expanded jet which impinges on the $45^{\circ }$ inclined plate. The impingement angle is defined as the angle between the y-axis and the normal vector of the impingement plane, thus vertical impingement is obtained for $90^{\circ }$ .

The flow conditions are based on two experiments carried out by Nakanishi et al. (Reference Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2012) and Akamine et al. (Reference Akamine, Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2014) at the UT-Kashiwa hypersonic and high-temperature wind tunnel (Kashiwa wind tunnel). The experimental investigations have been accompanied by several numerical studies (see Nonomura & Fujii Reference Nonomura and Fujii2010; Nonomura et al. Reference Nonomura, Goto and Fujii2011; Nakanishi et al. Reference Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2012; Tsutsumi et al. Reference Tsutsumi, Nonomura, Fujii, Nakanishi, Teramoto and Okamoto2012, Reference Tsutsumi, Takaki, Nakanishi, Okamoto and Teramoto2014). The jet in this experiment was cold with $T=300$  K to exclude any high-temperature effects. The supersonic perfectly expanded jet reached a Mach number of $M=1.8$ at the nozzle exit. In an RANS precursor calculation, the nozzle flow was solved separately to provide boundary conditions for the LES. The boundary layer and momentum thicknesses at the nozzle exit were roughly $5.5\times 10^{-2}D$ and $8.7\times 10^{-3}D$ . For the high-fidelity simulation, the exterior of the nozzle was modelled in accordance with the experimental set-up. The simulations were performed by assuming an ideal gas with a specific heat ratio of ${\it\gamma}=1.4$ and a constant specific heat capacity of $c_{p}=1005~\text{J}~\text{kg}^{-1}~\text{K}^{-1}$ . The molecular viscosity was calculated using Sutherland’s law assuming air. For the current simulations, the flow conditions at the actual nozzle exit were $M=1.8045$ , $V_{e}=488.14~\text{m}~\text{s}^{-1}$ and $p_{e}=100\,794$  Pa. The differences from a theoretical perfectly expanded Mach $1.8$ jet were ${\rm\Delta}M/M\approx 0.25\,\%$ , ${\rm\Delta}V_{e}/V_{e}\approx 0.52\,\%$ and ${\rm\Delta}p_{e}/p_{e}\approx 0.24\,\%$ . Close to a billion grid points were used to simulate the jet flow with a Reynolds number of $Re=V_{e}D/{\it\nu}=1.6\times 10^{6}$ , where $V_{e}$ is the nozzle exit velocity, $D$ is the nozzle exit diameter and ${\it\nu}$ is the kinematic viscosity of air at the nozzle exit temperature. The physical time step was ${\rm\Delta}tV_{e}/D=2\times 10^{-3}$ and the simulation ran for about $250\,000$ time steps ( $TV_{e}/D=500$ ). Only the last $100\,000$ time steps ( $TV_{e}/D=200$ ) were used for postprocessing. For the POD analysis, only about half of the postprocessing time interval was used due to large memory requirements. The POD results for three windows with $50\,\%$ overlap showed close agreement. The analysis in this paper is mainly focused on two-dimensional cut-planes, while the noise sources are truly three-dimensional. The analysis based on two-dimensional cut-planes provides, however, a good idea about the essential features of the noise generation mechanisms. A three-dimensional analysis may provide additional insight about the acoustic noise generation mechanisms. Throughout the paper, all flow quantities (if not otherwise stated) are normalized with exit flow conditions, where the reference length scale $L_{ref}$ , pressure $p_{ref}$ and velocity $V_{ref}$ are $D$ , $p_{e}$ and $V_{e}$ respectively. The velocity components of $\boldsymbol{v}_{cart}=[u,v,w]$ refer to the components in the Cartesian reference frame. In some parts a cylindrical coordinate system is utilized with the velocity vector in the form $\boldsymbol{v}_{cyl}=[v_{r},v_{{\it\Theta}},w]$ , where $v_{r}$ and $v_{{\it\Theta}}$ refer to the radial and azimuthal velocity components.

2.3 Computational simulation strategy

It is well known that DNS is currently limited to Reynolds number regimes of up to $O(10^{4})$ due to cost constraints. Hence, DNS is not feasible for the current flow conditions and we will rely on large-eddy simulation (LES) in which molecular viscosity has little effect on the large-scale flow structures (probably less effect than the inflow shear-layer thickness or state, laminar versus turbulent, as suggested by Shur, Spalart & Strelets (Reference Shur, Spalart and Strelets2005b )). The current computational simulation strategy with respect to nozzle inflow conditions and the subgrid-scale modelling is closely aligned with the recommendations of Shur, Spalart & Strelets (Reference Shur, Spalart and Strelets2005a ) and Shur et al. (Reference Shur, Spalart and Strelets2005b ), who discuss strategies for turbulent jet flow simulations. They demonstrated that their simulation approach to predict the noise field is very accurate and computationally efficient.

For the current simulations, we are employing an implicit large-eddy simulation (ILES) approach. In contrast to classical LES, which employs an explicitly computed subgrid scale (SGS) closure, ILES relies on the inherent regularization mechanism through the truncation error of the convective fluxes, which is also referred to as the implicit SGS model (Hu & Adams Reference Hu and Adams2011) or numerical LES (Pope Reference Pope2004). The primary role of the SGS model is to remove the energy that would cascade down to the unresolved scales. Hence, the numerical scheme must be chosen carefully to provide a physically consistent implicit SGS model. In the current study, the modified sixth-order accurate WENO scheme (Hu, Wang & Adams Reference Hu, Wang and Adams2010) was used due to its superior physically motivated scale separation properties. Hu & Adams (Reference Hu and Adams2011) have demonstrated that the modified sixth-order accurate WENO scheme reproduces the Kolmogorov range of the turbulent kinetic-energy spectrum at the limit of infinite Reynolds number independent of grid resolution, while maintaining the shock-capturing capabilities of the original WENO-CU6. A detailed discussion and comparison of higher-order shock-capturing schemes within the current solver for various test problems containing turbulent flows and shocks is provided in Brehm et al. (Reference Brehm, Barad, Housman and Kiris2015a ). It is important to point out that our experience, as well as the results from other research studies (see Shur et al. Reference Shur, Spalart and Strelets2005a ,Reference Shur, Spalart and Strelets b ; Brehm et al. Reference Brehm, Sozer, Moini-Yekta, Housman, Barad, Kiris, Vu and Parlier2013), confirms that for simulations of turbulent flows away from solid boundaries, the ILES approach is a well-suited simulation strategy. Moreover, the ILES approach is very efficient in utilizing the available grid resolution for capturing the shear-layer transition process and modelling salient features of the turbulent flow. Preliminary studies and other researchers (Shur et al. Reference Shur, Spalart, Strelets and Travin2003, Reference Shur, Spalart and Strelets2005b ) have even demonstrated that the transition of the shear layer can be delayed and the proper amount of turbulence generation is not generated as well when utilizing a suboptimal explicit SGS model for these flow conditions. In addition, as pointed out in § 1.3, Nonomura & Fujii (Reference Nonomura and Fujii2011) confirmed for very similar flow conditions that the near-wall turbulent structures within the wall jet boundary layer play an insignificant role in the noise generation process.

Another important detail for the computational set-up is the specification of the nozzle exit boundary conditions. As pointed out in Zaman (Reference Zaman2012), the internal nozzle geometry plays a critical role in determining the nozzle exit boundary layer state. Recent studies for free jets (Bogey & Bailly Reference Bogey and Bailly2003; Lew et al. Reference Lew, Uzun, Blaisdell and Lyrintzis2004; Zaman Reference Zaman2012; Brès et al. Reference Brès, Jaunet, Le Rallic, Jordan, Colonius and Lele2015, and others) have shown that the far-field acoustic spectra are affected by the details of the nozzle exit boundary layer flow, such as boundary layer thickness and turbulent intensity. Zaman (Reference Zaman2012) showed for subsonic jet flows how the nozzle exit boundary layer state affects the emitted noise signature. He found that a noisier nozzle involved a highly disturbed laminar (or nominally laminar) boundary layer in contrast to the turbulent exit boundary layer of a more silent nozzle. Ideally, the nozzle exit flow would be modelled appropriately as a nominally laminar or turbulent boundary layer convecting unsteadiness into the jet shear layer. In the experimental set-up, however, the nozzle exit boundary layer state was not determined, making it difficult to provide the correct level of turbulence intensity. Since the available computational resources are not adequate for simulating the entire interior nozzle flow, we are relying on the fact that the shear layer will essentially transition and create a more energetic unsteadiness (ultimately a turbulent flow state) than in the nozzle boundary layer flow. For free jet flow simulations, some researchers (for example Freund Reference Freund2001; Bogey & Bailly Reference Bogey and Bailly2007) have applied unsteady forcing to seed unsteadiness in the transitional shear layer. One natural choice would be to introduce perturbations based on linear stability theory (LST), as discussed in Michalke (Reference Michalke1984), i.e. Tollmien–Schlichting waves in boundary layers and Kelvin–Helmholtz waves in shear layers. Bogey & Bailly (Reference Bogey and Bailly2007) followed a different strategy and excited the mean flow with randomized perturbations in the form of induced velocities from a vortex ring (see also Bogey et al. Reference Bogey, Bailly and Juvé2003). This was done very carefully to ensure a proper breakup of the potential core within a reasonable distance. It is cumbersome to achieve a realistic scenario with respect to frequencies and amplitudes not knowing the actual conditions in the experiment. Previous studies for jet-impingement flows, however, have demonstrated that the dominant noise sources are located in the vicinity of the impingement plate. These noise sources appear to be far less sensitive to the details of the nozzle exit boundary layer.

Hence, in the current simulation, we did not use any type of forcing based on linear stability theory and/or random noise, which on the other hand prevents accidental introduction of parasitic noise (Shur et al. Reference Shur, Spalart and Strelets2005b ). For the jet-impingement problem, an elevated acoustic background noise level was continuously generated by the impinging jet, seeding unsteadiness in the shear layer. No numerical artefacts such as a preferred frequency response that could cause some type of artificial feedback were observed in the simulations. The grid convergence study previously presented in Brehm et al. (Reference Brehm, Sozer, Moini-Yekta, Housman, Barad, Kiris, Vu and Parlier2013) and Housman et al. (Reference Housman, Brehm and Kiris2013) ensured that the relevant flow features captured with the ILES approach were sufficiently grid converged and did not contain a spurious numerical feedback mechanism. Moreover, comparison of the current grid resolution with computational set-ups of similar research studies suggested that the current grid resolution was sufficient to study the relevant flow physics associated with noise generation. For example, at similar flow conditions ( $Re_{D}=1\times 10^{6}$ ) the spatial resolution in Dauptain et al. (Reference Dauptain, Gicquel and Moreau2012) was coarser than the final mesh used here, despite the fact that the temporal and spatial resolution accuracy of the current simulation was higher. Dauptain et al. (Reference Dauptain, Gicquel and Moreau2012) obtained excellent agreement in a detailed comparison with experimental data reported by Henderson et al. (Reference Henderson, Bridges and Wernet2005). Their finest grid spacing at the nozzle lip corresponded to roughly $60$ points per nozzle diameter. The azimuthal resolution along the circumference of the nozzle was approximately $160$ points. Dauptain et al. (Reference Dauptain, Gicquel and Moreau2012) used a third-order Taylor–Galerkin method (Donea Reference Donea1984) in space with hyperviscosity at shocks on an unstructured mesh. Within the turbulent jet, our temporally fourth-order accurate and spatially sixth-order accurate solver utilized $200$ grid points per diameter in all directions on a Cartesian grid, which corresponds to $290$ to $500$ grid points along the circumference of the nozzle. (On a Cartesian grid, it is not clear how the circumferential grid spacing should be defined. The lower grid point estimate takes the diagonal spacing ( $\sqrt{3}{\rm\Delta}x$ ) within a cubic cell and the larger grid point estimate uses the actual edge length ( ${\rm\Delta}x$ ) of the cell.)

3.1 Mean flow

An overview of the mean flow is provided in figure 2(a). The time-averaged pressure distribution is visualized with coloured contours, and a solid contour line is used for the sonic line. The compression and expansion wave patterns are schematically displayed. By time averaging the unsteady flow, the gradients are significantly smoothed out. A weak Mach diamond structure downstream of the nozzle exit is present and it is caused by the mild overexpansion, as discussed in § 2. The pressure averaged over the nozzle exit area is approximately $0.5\,\%$ below the reference atmospheric pressure. For noise source identification using the causality method, it is important to keep in mind that the Mach diamond structure moves due to unsteadiness in the flow field. It was decided to use an ‘ideally’ or ‘perfectly’ expanded jet because the current study as well as prior investigations is concerned with the interaction of the turbulent jet with the plate shock and its downstream development. In the presence of a more pronounced Mach diamond structure, the local flow conditions at the impingement location vary greatly depending on the relative location in the jet structure (Henderson et al. Reference Henderson, Bridges and Wernet2005). In addition, a strong interaction of the large coherent flow structures with the Mach cone structure is expected. Hence, the complexity of the jet-impingement problem is greatly reduced by employing a perfectly expanded jet flow.

Figure 2.

Mean flow visualization in the $(y,z)$ -plane at $x=0$ . (a) Colour contours of the mean pressure field, $(p-p_{ref})/p_{ref}$ , the black solid line displays the sonic line, and schematic detailing the structure of compression and expansion waves. (b) Colour contours of normal Mach number $M_{n}$ with black lines for $M_{n}=1$ and ${\it\Gamma}^{\pm }$ marking characteristics.

The shock structures in the impingement region and in the wall jet were identified using a shock identification method based on the normal Mach number discussed in Lovely & Haimes (Reference Lovely and Haimes1999),

where $\boldsymbol{M}$ is the Mach number vector in the direction of the local flow velocity vector $\boldsymbol{v}$ and $a$ is the local speed of sound. Normal Mach number isocontours are shown in figure 2(b) (starting slightly above the plate shock). Additionally, a black solid line marks $M_{n}=1$ contour lines to highlight possible shock locations. It should be noted that the isocontours need to be interpreted carefully because only isocontours of $M_{n}=1$ , where $M_{n}$ varies from supersonic ( $M_{n}>1$ ) to subsonic ( $M_{n}<1$ ), are shock locations (Lovely & Haimes Reference Lovely and Haimes1999). The normal Mach number contours identify an impingement plate shock and a shock train downstream of the primary impingement location. The interaction of the turbulent jet with these shock structures induces significant pressure oscillations which are believed to be relevant for noise generation. Moreover, solid (black/white) lines are used to portray characteristics inside the supersonic jet denoted by ${\it\Gamma}^{\pm }$ . The characteristics were obtained with the method of characteristics applied to the linearized compressible small-disturbance potential equation utilizing the Prandtl–Glauert transformation.

For the purposes of validating the current computational approach, the time-averaged wall pressure distribution is compared with the limited available experimental data by Nakanishi et al. (Reference Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2012) and Akamine et al. (Reference Akamine, Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2014). The time-averaged pressure distribution comprises two regions, A and B, as marked in figure 3(a). The large pressure values in region A are associated with the plate shock. Tail shocks are observed in region B downstream of the plate shock, causing oscillations in the pressure distribution. The peak in the surface pressure distribution extracted at the centreline shown in figure 3(b) is slightly off centre, located at $y/D=0.1{-}0.11$ . This result corresponds well with $y/D\approx 0.1$ reported by Tsutsumi et al. (Reference Tsutsumi, Takaki, Nakanishi, Okamoto and Teramoto2014). The near-field flow measurements reported in Akamine et al. (Reference Akamine, Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2014) are only available at a few pressure sensor locations placed at the wall, shown in figure 3(b) as solid square symbols. The maximum pressure in the jet-impingement region matches the computational fluid dynamics (CFD) data by Tsutsumi et al. (Reference Tsutsumi, Takaki, Nakanishi, Okamoto and Teramoto2014), with ${\rm\Delta}p_{ave}/p_{ref}\approx 2.63$ . Assuming isentropic expansion, the stagnation pressure on the plate would be ${\rm\Delta}p_{ave}/p_{ref}\approx 1.67$ . The formation of the plate shock raises the maximum pressure on the wall by roughly $36\,\%$ . The velocity profiles in the wall jet are compared with previous hybrid RANS/LES simulations by Housman et al. (Reference Housman, Brehm and Kiris2013). Good agreement of the wall-normal velocity profiles between the two simulations is obtained on considerably different computational grids and by utilizing different simulation strategies. Neither one of the two CFD approaches compared in figure 3 employed wall-resolved LES due to the large computational resources that would be required for such a simulation. A wall-resolved LES may provide some additional insight into the relevance of near-wall unsteadiness and possible separation on the wall as observed by Honda et al. (Reference Honda, Nonomura, Fujii and Yamamoto2011) and Nagata et al. (Reference Nagata, Nonomura, Fujii and Yamamoto2013) for a lower Reynolds number.

Figure 3.

(a) Contours of the time-averaged gauge pressure on the impingement plane and (b) comparison of the pressure distribution with experimental data. The experimental data are digitized from Tsutsumi et al. (Reference Tsutsumi, Takaki, Nakanishi, Okamoto and Teramoto2014). (c) Comparison of wall-normal velocity profiles in the wall jet with the hybrid RANS/LES simulation by Housman et al. (Reference Housman, Brehm and Kiris2013). The sonic line is shown as a dashed line.

3.2 Unsteady flow visualization

Figure 4.

Instantaneous flow visualization of a turbulent jet impinging on a $45^{\circ }$ inclined plate in the $(y,z)$ -cut-plane: (a) passive trace particle visualization coloured by vorticity magnitude, (b) isocontour of $Q$ -criterion,  $QD/V_{ref}=1.2\times 10^{5}$ , and normalized gauge-pressure contours and (c) greyscale contours of dilatation.

The following discussion is concerned with characterizing the turbulent jet upstream of the impingement location, the jet–plate interaction and the particularities of the unsteady wall jet containing tail shocks. A wide range of relevant temporal and spatial scales can be observed from the passive trace particle visualization (coloured by vorticity) and the $Q$ -criterion isocontour surface (with $QD/V_{ref}=1.2\times 10^{5}$ ) in figure 4(a,b). The unsteady plate and tail shocks can be identified in the dilatation contours in figure 4(c). Fine-grained turbulence and small turbulent eddies organized in large-scale coherent flow structures are visible. Due to the short plate-to-nozzle distance, large levels of vorticity are mainly concentrated in the jet and wall jet shear layers. A general idea about the wave-propagation pattern can be obtained in the normalized gauge-pressure contours in the $(y,z)$ -cut-plane in figure 4(b). At least three distinct wave directivities are apparent. The wave fronts emitted by the turbulent jet shear layer (marked with arrows and labelled ‘(1)’) have been carefully studied in the literature for free jets. Tam et al. (Reference Tam, Viswanathan, Ahuja and Panda2008) provided a simple model for the generation of these acoustic waves. The large-scale coherent flow structures may be regarded as a wavy wall moving with supersonic speed past the ambient air, thus emitting Mach waves.

The flow visualizations in figure 4(b,c) clearly depict Mach radiation from large-scale coherent flow structures in the turbulent jet upstream of the impingement location. It is assumed that noise generation from fine-scale turbulence is also present for the current setting. However, the wave fronts are diffused in figure 4(b,c) because the generated noise is nearly omnidirectional, as noted by Tam et al. (Reference Tam, Viswanathan, Ahuja and Panda2008). The two additional types of wave-propagation patterns marked in figure 4(b) have been previously observed by other researchers (see Nonomura et al. Reference Nonomura, Goto and Fujii2011; Nakanishi et al. Reference Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2012; Tsutsumi et al. Reference Tsutsumi, Takaki, Nakanishi, Okamoto and Teramoto2014). The wave fronts ‘(2)’ appear to be emitted from the jet-impingement region and the wave fronts ‘(3)’ seem to originate from the unsteadiness in the wall jet. The relevant physical mechanisms governing the generation of these acoustic waves are, however, not well understood at this point.

3.3 Unsteady flow features: from nozzle exit to impingement point

The vortical flow structures shown in figure 4(b) are initially obliquely aligned to the jet axis. The initial low-amplitude disturbances downstream of the nozzle lip rapidly reach large amplitudes where they saturate and higher modes are nonlinearly generated (or through secondary, tertiary, etc. instability mechanisms). The exact nature of the transition process is not analysed here but some estimates are provided below. Further breakdown of the jet into small-scale structures can be observed in figure 4(b), where the flow field in the shear-layer region is assumed to be ‘fully turbulent’ approximately $1D$ downstream of the nozzle exit. The appearance of small-scale turbulent eddies in the instantaneous flow visualization in figure 4(a–c) indicates that the current simulation approach is able to capture the ‘second transition’ of the jet, as it was referred to by Shur et al. (Reference Shur, Spalart and Strelets2005a ,Reference Shur, Spalart and Strelets b ). Additional evidence is provided by analysing the power spectra of different disturbance flow quantities in figure 5(a–c). Ideally, the state of the boundary layer at the nozzle exit would contain resolved turbulence which enters the jet flow in the shear layer (resolution requirements are out of reach for the current Reynolds number). Due to the strong shear-layer instability mechanism, the flow is quickly transitioning in the shear layer. This seems acceptable considering that the instabilities and turbulence are much stronger in the shear layer than in the boundary layer.

Figure 5.

Power spectra of flow quantities (a) $p^{\prime }=(p-\overline{p})/p_{ref}$ , (b) $v_{r}^{\prime }=(v_{r}-\overline{v}_{r})/V_{ref}$ and (c) $w^{\prime }=(w-\overline{w})/V_{ref}$ extracted at sample points on the shear-layer line in figure 1(b). It should be noted that the spectra were smoothed to ease the comparison of the data.

Figure 5(a–c) displays the spectral energy $E$ of normalized disturbance pressure $p^{\prime }=(p-\langle p\rangle )/p_{ref}$ and disturbance velocities $v_{r}^{\prime }=(v_{r}-\langle v_{r}\rangle )/V_{ref}$ and $w^{\prime }=(w-\langle w\rangle )/V_{ref}$ at four different locations in the shear layer of the jet. The point probes are located at four vertical positions, i.e. $z/D=4.75$ , $4.50$ , $4.25$ and $2.00$ , on the shear-layer line right below the lip of the nozzle exit displayed in figure 1(b). Two dashed lines have been added which correspond to the theoretically expected asymptotic behaviour of a turbulent flow in the inertial subrange with $E\sim St^{-5/3}$ and the dissipation range with $E\sim St^{-7}$ of the von Kármán energy spectrum (Pope Reference Pope2000). The initial high-frequency peak at $St\approx 30$ decays quickly in the downstream direction. The cause of this high-frequency peak, whether it is numerical or physical, is not clear at this point. It can be disregarded since it did not seem to play a major role in the current study. This conjecture was scrutinized by comparing the current results with other (Tsutsumi et al. Reference Tsutsumi, Takaki, Nakanishi, Okamoto and Teramoto2014) and lower-resolution simulations (Brehm et al. Reference Brehm, Sozer, Moini-Yekta, Housman, Barad, Kiris, Vu and Parlier2013).

According to Michalke (Reference Michalke1984), the largest amplification rates in the jet shear layer are expected for frequencies around $St=O(1)$ (and $St\approx 2{-}3$ with a phase speed of $v_{ph}/V_{ref}\approx 0.6{-}0.7$ for the helical mode with $k_{{\it\phi}}=1$ ) due to the aforementioned Kelvin–Helmholtz (K–H) type instability mechanism. The dominant frequencies in the initial shear layer observed in the current simulations are within the range of the most amplified frequencies in a round free jet estimated by linear stability theory (see Michalke Reference Michalke1984). By computing the shear-layer thickness of the mean flow and considering the stability analysis results by Michalke (Reference Michalke1984), the region of unstable K–H modes contains frequencies up to approximately $St\approx 4{-}6$ (for $k_{{\it\phi}}=1$ ) for $z/D=4{-}5$ . For the current Mach number, it is expected that the helical mode with $k_{{\it\phi}}=1$ is more amplified than the axisymmetric mode with $k_{{\it\phi}}=0$ . The initial oblique alignment of the $Q$ -criterion isosurface in figure 4(b), slightly downstream of the nozzle lip, supports this conjecture. The space–time resolution in the present simulation was based on the estimated relevant frequency range that is needed to resolve the initial shear-layer breakdown. Indeed, a comparison of the spectra in figure 5(a–c) for different grid resolutions (Brehm et al. Reference Brehm, Sozer, Moini-Yekta, Housman, Barad, Kiris, Vu and Parlier2013) indicates that the present ILES is able to capture the initial growth of the primary instability waves correctly as well as the subsequent breakdown of the flow to turbulence.

The disturbance flow energy spectra for $1<z/D<4$ are basically identical to the spectra at $z/D=2$ in terms of their asymptotic behaviour. For the current grid resolution, the inertial subrange in the von Kármán energy spectrum following the asymptotic behaviour with $f^{-5/3}$ seems to be well established. Hence, from this location ( $z/D\approx 4$ ) forward (with $z/D$ decreasing towards the impingement plate) the flow in the shear layer may be considered to be fully turbulent. In the dissipation range with the $f^{-7}$ asymptotic behaviour, most of the turbulent kinetic energy is supposed to be dissipated by the physical action of the molecular viscosity. For the current not fully resolved simulations (not DNS), it is assumed that most of the dissipation is actually introduced by the numerical scheme acting as a filter to the unresolved scales and not by the physical action of the molecular viscosity. Comparison of the spectra in figure 5(a–c) with spectra from lower-grid-resolution simulations in Brehm et al. (Reference Brehm, Sozer, Moini-Yekta, Housman, Barad, Kiris, Vu and Parlier2013) suggests that the energy spectra are resolved for up to roughly $St=6$ , well into the frequency range relevant here. The simulation accuracy is further validated in § 4 by comparing the acoustic noise signature in the far field with acoustic measurements by Akamine et al. (Reference Akamine, Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2014). Assuming a normal-mode representation of turbulent fluctuations based on LST, one could assume that the disturbance flow spectra are identical for all flow quantities. It is well known, and indeed Panda et al. (Reference Panda, Seasholtz and Elam2005) demonstrated, however, that the normal-mode assumption is not necessarily a good assumption for fully turbulent jet flows. An obvious difference in the spectra is that $E_{w^{\prime }}$ contains more energy in the lower-frequency range ( $St<1$ ) than $E_{v_{r}^{\prime }}$ and $E_{p^{\prime }}$ .

Figure 6.

Spectral energy distribution versus Strouhal frequency, $St$ , and azimuthal wavenumber, $k_{{\it\phi}}$ , for different jet-axis locations $z/D=4.5$ , $3.0$ and $1.0$ ((a–c), (d–f) and (g–i)) and flow quantities $p^{\prime }$ , $v_{r}^{\prime }$ and $w^{\prime }$ ((a,d,g), (b,e,h) and (c,f,i)).

As the turbulent jet evolves, the unsteady flow goes through significant changes with respect to frequency and azimuthal wavenumber content. In the next step, the three-dimensional nature of the flow field was analysed by applying a double Fourier transform in time and azimuthal direction ${\it\phi}$ . To perform these transforms, the unsteady flow data were recorded on $20$ circles centred around the jet axis with a radius $r/D=0.5$ , as shown in figure 1(b). Each circle contained $200$ equispaced sample points. Figure 6 displays the spectral energy content for the disturbance flow quantities at three vertical positions $z/D=4.5$ , $3.0$ and $1.0$ . It should be noted that $St$ is plotted on a logarithmic scale, and for this analysis the radial velocity, $v_{r}^{\prime }$ , was used. The characteristics of the unsteady flow field between the nozzle exit and the impingement plate can be roughly divided into three regions: (i) transitional shear layer ( $5\geqslant z/D>4$ ), (ii) turbulent jet flow ( $4\geqslant z/D>1.5$ ) and (iii) impingement region ( $z/D\leqslant 1.5$ ). These three regions can be characterized by their particular frequency and wavenumber spectra in figure 6. The transitional jet flow (figure 6 a–c) is dominated by frequencies $St=O(1)$ and azimuthal wavenumbers up to roughly $k_{{\it\phi}}=10$ . As one may expect for a Mach $1.8$ cold jet (Michalke Reference Michalke1984), the oblique modes $k_{{\it\phi}}>0$ appear to be initially more amplified than the axisymmetric mode $k_{{\it\phi}}=0$ . All spectra also display a region with large amplitudes for low frequencies ( $0.1<St<0.8$ ) and low azimuthal wavenumbers. Further downstream, the spectra broaden, in both frequency and azimuthal wavenumber space. The spectral energy distribution for $z/D=3$ is ‘typical’ for the spectral energy distributions of the turbulent jet in region (ii). Distinct differences can be observed between the spectra of disturbance pressure and disturbance velocities with respect to the azimuthal wavenumber content. The energy spectra of the disturbance velocities are dominated by a wider range of azimuthal wavenumbers (up to $k_{{\it\phi}}\approx 20$ ) in comparison to the energy spectrum of the disturbance pressure (up to $k_{{\it\phi}}\approx 10$ ). The power spectrum for $w^{\prime }$ contains more low-frequency content at low azimuthal wavenumbers than $E_{p^{\prime }}$ and $E_{v_{r}^{\prime }}$ . With decreasing distance to the plate, the spectral energy shifts towards lower frequencies for all flow quantities, whereby the differences in the spectral wavenumber content remain. Slightly upstream of the impingement location (figure 6 g–i), where the turbulent jet interacts with the oblique plate shock, all three energy spectra, $E_{p^{\prime }}$ , $E_{v_{r}^{\prime }}$ and $E_{w^{\prime }}$ , are dominated by low frequencies with $St<1$ and lower azimuthal wavenumbers than upstream. The $E_{v_{r}^{\prime }}$ spectrum peaks at larger azimuthal wavenumbers than the $E_{p^{\prime }}$ and $E_{w^{\prime }}$ spectra. The shift towards lower azimuthal wavenumbers at $z/D=1.0$ may be partially explained by the fact that the circle cuts through the impingement region. It shows that the effects of the inclined plate can be felt over a significant distance upstream of the impingement point. The differences in the Fourier spectra are very subtle, and we will demonstrate that the employment of auto-correlation functions is advantageous to distinguish particularities in the unsteady time signals for the different flow quantities.

In the following temporal correlation analysis, three important aspects for characterizing the unsteady flow field between the nozzle exit ( $z/D=5$ ) and the impingement location ( $z/D=0$ on the centreline) are investigated: (i) temporal (and derived streamwise) length scales of the disturbance flow quantities, (ii) disparity in scales between the shear layer, jet axis and in the vicinity of the plate shock, and (iii) the relationship between different disturbance flow quantities. To understand the noise generation mechanisms, it is important to identify the relevant scales in the different regions of the flow field as well as how the flow quantities are interconnected in the process. Due to the nonlinear nature of the Navier–Stokes equations, it may not be expected that individual flow quantities can be isolated in this process because a coupling between these quantities will probably occur. However, in the following analysis, it will be demonstrated that the dominant temporal scales in the disturbance velocities and pressure can be different in the vicinity of the plate shock.

The dominant scales in the flow field were analysed by computing the temporal auto-correlation function of the different disturbance flow quantities, $\boldsymbol{q}^{\prime }(\boldsymbol{x},t)$ , where $q_{n}^{\prime }(\boldsymbol{x},t)$ is the $n$ th component of the primitive variable vector $\boldsymbol{q}^{\prime }=[p^{\prime },v_{r}^{\prime },v_{{\it\Theta}}^{\prime },w^{\prime },T^{\prime }]^{\text{T}}$ recorded at location $\boldsymbol{x}$ . For a free jet, Tam et al. (Reference Tam, Viswanathan, Ahuja and Panda2008) employed a similar strategy to identify the coherence time scale that is associated with far-field acoustic noise in the upstream and sideline directions due to small-scale turbulence and in the downstream direction due to Mach radiation. The normalized auto-correlation function, $C_{q_{n}^{\prime },q_{n}^{\prime }}^{0}$ , is given by

Throughout this paper, $\langle \,\rangle$ denotes time averaging and the superscript $0$ is used for normalized correlation functions. The auto-correlation function is merely the Fourier transform of spectra of the disturbance flow quantities, $q_{n}^{\prime }$ . Furthermore, according to Tam et al. (Reference Tam, Viswanathan, Ahuja and Panda2008), a coherence time scale can be associated with the shape of the auto-correlation function. In this work, the integral temporal scales, $T_{q_{n}^{\prime }}$ in figure 7(a,b), were obtained by integrating the auto-correlation functions in figure 8. The plots in figure 8 present the auto-correlation functions at particular streamwise locations upstream of the impingement location. Assuming Taylor’s hypothesis, this coherence time scale can also be converted into a physical length scale using an appropriate propagation speed.

Figure 7.

Integral temporal scales along (a) the shear-layer line and (b) the centreline. The smoothed data are shown with thick lines and the original data with thin lines.

Figure 8.

Auto-correlation functions of disturbance pressure, $v_{r}^{\prime }$ velocity and $w^{\prime }$ velocity for sample points in the shear layer (a–d) and on the centreline (e–g) of the jet: (a) $z/D=4.50$ , (b) $z/D=2$ , (c) $z/D=0.20$ , (d) $z/D=0.25$ , (e) $z/D=2$ , (f)  $z/D=0.46$ , (g) $z/D=0.15$ .

The auto-correlation functions for all disturbance flow quantities are very similar close to the nozzle exit, displaying deep undershoots. The deep negative peaks in the auto-correlation function for all flow quantities are typical features (or imprints) of large coherent flow structures in the transitional jet flow, as explained in Tam et al. (Reference Tam, Viswanathan, Ahuja and Panda2008). Further downstream, due to the randomization of the time signals, the initial deep undershoot disappears as the jet shear layer progresses further through the ‘second transition’ process and ultimately breaks down to turbulence. Upstream of the impingement region, the temporal integral scale $T_{w^{\prime }}$ is always larger than $T_{v_{r}^{\prime }}$ and $T_{p^{\prime }}$ in the shear layer. Once the turbulent jet flow reaches the jet-impingement region ( $z/D=0.20$ and $-0.25$ ), the interaction of the unsteady jet with the plate shock causes significant changes in the physical scales. The integral temporal scales $T_{p^{\prime }}$ and $T_{v_{r}^{\prime }}$ increase in the vicinity of the plate shock while $T_{w^{\prime }}$ decreases; $T_{v_{r}^{\prime }}$ approaches the value of $T_{w^{\prime }}$ below the plate shock. The large disparity in the temporal scales for the time signals of the velocity and pressure fields below the plate shock suggests that in this region, the pressure and velocity fields are dictated by widely different physical scales, possibly indicating anisotropies in the flow or even decoupling of flow quantities. A likely explanation of the disparity in scales between the velocity and pressure fields is provided in § 3.4. It will be shown in § 3.4 that the wall jet contains coherent flow structures as well as a stationary mode mostly perceivable in the pressure field spatially extending from the impingement region to the tail shocks in the wall jet.

Next, the differences between the unsteadiness in the shear-layer region and the centre of the turbulent jet are analysed. In § 5, we utilize this information to explain why stronger normalized correlations are obtained in the turbulent jet core than in the shear layer. Based on prior research studies for free jets (by, for example, Panda & Seasholtz Reference Panda and Seasholtz2002; Panda et al. Reference Panda, Seasholtz and Elam2005; Bogey & Bailly Reference Bogey and Bailly2007; Tam et al. Reference Tam, Viswanathan, Ahuja and Panda2008), the flow features in the jet centre and inside the shear layer can be expected to be different. These differences are aggravated for the impinging jet analysed here because the potential core length of an equivalent free jet ( ${\rm\Delta}z_{c}/D\approx 8$ ) is larger than the actual plate-to-nozzle distance. Therefore, the turbulence originating in the shear-layer region cannot penetrate into the centre of the jet before impinging on the inclined plate. The integral temporal scales for all flow quantities in the jet core are very similar until the jet reaches the impingement region. Hence, it appears that the disturbance flow quantities are closely related to each other. Assuming a nearly constant propagation speed of the dominant flow structures at the centreline, it seems that the integral scales along the centreline scale perfectly with the momentum thickness, ${\it\theta}$ , of the jet.

At $z/D=2.0$ , the auto-correlation functions in the centre, $y/D=0$ , and in the shear layer, $y/D=0.5$ , display some differences. The disturbance flow field in the jet centre appears to be more coherent, which can be noted by larger undershoots right next to the peak. The increased temporal integral scales $T_{v_{r}^{\prime }}$ and $T_{p^{\prime }}$ in the jet core indicate larger temporal coherence for $v_{r}^{\prime }$ and $p^{\prime }$ in the centre of the jet. The integral temporal scales rise significantly around the location of the plate shock ( $z/D=0.40$ ). Right above the impingement plate, $z/D=0.15$ , $C_{v_{r}^{\prime },v_{r}^{\prime }}^{0}$ displays a significantly smaller width of the peak, and thus a smaller integral time scale, than $C_{p^{\prime },p^{\prime }}^{0}$ . It should be noted that we repeated the analysis of the disturbance flow field along a line with a ${\rm\Delta}y/D=0.1$ offset to centreline and obtained nearly identical results. In this way any possible artefacts that may only occur just in the centre of the jet could be excluded. The key finding of this discussion is that the integral temporal scales and relationships between the different disturbance flow quantities are widely different along the jet centreline and the shear-layer line.

Figure 9.

Space–time plots of the streamwise disturbance velocity along (a) the shear-layer line and (b) the jet axis, as shown figure 1(b). (c) Instantaneous snapshot of the shock indicator function in the symmetry plane displaying weak shocks in the core of the jet.

The space–time (or $(z,t)$ ) plots in figure 9(a,b) for the streamwise disturbance velocity are used to determine the propagation speeds in the core, $v_{p,c}$ , and shear layer, $v_{p,s}$ , of the jet. The widely different patterns in the space–time plots are another indicator for the differences in temporal scales in the core and the shear layer of the jet. The following discussion is aimed at obtaining an understanding of the unsteady nature of the flow in the jet shear layer and the jet core. The propagation speed of the most dominant flow features can be estimated by following a peak (light grey region) or valley (black region) in the $(z,t)$ plots. Dashed lines have been added which visualize the propagation speed of the most dominant flow features, the backward travelling acoustic wave with $v_{a}^{-}=1-a_{ref}/V_{ref}=0.44$ and the jet exit velocity, $V_{e}=V_{ref}$ . The most dominant flow features inside the shear layer propagate significantly faster than along the jet axis. The coherent flow structures inside the shear layer propagate supersonically with approximately $v_{p,s}=0.69V_{ref}$ , where the speed of sound is approximately $a=0.56V_{ref}$ . The estimated propagation speed of roughly $v_{p,c}=0.34V_{ref}$ in the centre of the jet is shown as a dashed line in figure 9(b). In flow visualizations of a planar jet by Shariff & Manning (Reference Shariff and Manning2013) employing ray tracing, fine-scale features inside the jet can be identified to travel significantly slower than the coherent flow structures in the shear layer. Based on the propagation speed of the dominant flow features in the centre of the jet ( $v_{p,c}<v_{a}^{-}$ ) in figure 9(b), it is, however, unlikely that these flow features are associated with (backward travelling) acoustic waves. Although it is theoretically possible that an interference pattern of acoustic waves travels faster than the speed of sound, it is more likely that small shock structures propagate inside the jet. In figure 9(c), the shock identification method of Lovely & Haimes (Reference Lovely and Haimes1999) was used to visualize possible shock locations inside the jet. Equation (3.1) was extended to account for unsteadiness (the movement of the shocks). Greyscale contours of the normal Mach number $M_{n}$ are shown in figure 9(c). Small shock structures appear to originate at the edge of the shear layer and propagate to the centre of the jet. Estimation of the speeds of these shock structures utilizing simple shock relations produces velocities that are very close to the previously estimated propagation speed of roughly $v_{p,c}=0.34V_{ref}$ .

3.4 Unsteady flow features: wall jet

Figure 10.

Space–time plots of (a) disturbance pressure and (b) streamwise velocity fluctuations extracted along streamline 1 in figure 1(b). The vertical lines mark the approximate locations of the tail shocks.

The role of the large-scale coherent flow structures in the wall jet is explored here. Upstream of jet impingement, Mach radiation is clearly emitted by supersonically convecting large-scale structures, as can be seen in figure 4(b,c). Previously, it was assumed that the interaction of these flow structures with the plate shock and the solid wall defines their predominant role in the noise generation process. The flow visualizations in figure 4(a–c) indicate, however, that large structures persist in the wall jet past the impingement region. It remains to be shown whether these flow structures propagate downstream supersonically relative to the ambient medium and, thereby, are also able to emit Mach radiation. Further, the interaction of these flow structures with the tail shocks may provide an additional mechanism for the generation of sound waves. These two-noise sources are assumed to be different with respect to their particular wave-propagation pattern. While the acoustic waves generated by coherent flow structures are mainly directional, the unsteady shock motion generates more omnidirectional wave patterns.

Figure 11.

The FFT amplitude of $p^{\prime }$ along sampling line 3. The vertical lines mark the approximate locations of the tail shocks.

Figure 10 shows space–time ( $s,t$ ) plots of disturbance pressure and streamwise velocity fluctuations along a streamline in the wall jet with arclength $s$ (see figure 1 b). In the $(s,t)$ plots of disturbance pressure, the imprints from the tail shocks are clearly visible and the mean tail shock positions are marked with vertical dashed lines. For the streamlines further away from the wall (not shown), these distinct features are progressively less pronounced. The tail shock interaction with the unsteady flow is less visible in the $(s,t)$ plots for streamwise disturbance velocity. The velocity field is less affected by the presence of the tail shocks. Figure 10(a,b) can provide some possible clues about the expected frequency content of the emitted acoustic waves from the shear-layer shock train interaction. The dominant spacings of the streaks in the streamwise disturbance velocity pattern that can be attributed to dominant flow features are fairly constant, with ${\rm\Delta}tV_{ref}/D\approx 2.7$ . This would lead to an interaction frequency of $St\approx 0.38$ for the shear-layer (single) tail shock interaction. Further, the spacings of the tail shocks introduce another frequency scale considering that the tail shocks are being distorted by the unsteady shear layer. Assuming a propagation speed of the coherent flow structures in the wall jet of $v_{p,w}=0.74$ and approximating the distances $L_{1,2}=0.93$ (between tail shocks 1 and 2) and $L_{2,3}=0.77$ (between tail shocks 2 and 3), this interaction frequency can be estimated as $St\approx 0.8$ –1. This is clearly a simplified view because the shear-layer shock train interactions are more complicated and nonlinear in nature, providing a broader frequency content. A better estimate of the frequency content associated with the tail shock oscillations is obtained by applying a Fourier transform to the pressure field in the wall jet. Figure 11 shows the fast Fourier transform (FFT) amplitude of $p^{\prime }$ along sampling line 3. Large Fourier amplitudes are shown around the tail shocks (marked with vertical dashed lines). As expected, the unsteady tail shocks contain a wide spectrum of frequencies. The largest amplitudes are, however, obtained for low Strouhal numbers.

The space–time plots provide mainly a qualitative description of the flow features. A more quantitative analysis of the unsteady flow features in the wall jet is obtained by computing the frequency–wavenumber spectra. The spectra contain information about the wave phase speeds ( $v_{p}={\it\omega}/k$ ) and modal characteristics inside the unsteady wall jet. The frequency–wavenumber spectra for disturbance pressure and streamwise disturbance velocity are displayed in figure 12(a,b). The plots were generated by placing sample points on a streamline following the wall jet (in figure 1 b) and applying a double Fourier transform in time and streamwise direction to the disturbance flow quantities. To reduce spectral leakage, a windowing function was utilized similar to the one used in Freund (Reference Freund2001). The windowing function follows as

where $s_{s}$ and $s_{e}$ define the first and last points on the sampling line, and $s_{1}$ and $s_{2}$ were chosen so that the solution was ramped over the first and last $5\,\%$ of the total length of the sampling line. Two distinct regions can be identified in the spectra for disturbance pressure. Region ‘I’ contains the propagating part of the solution. Region ‘II’ is associated with a non-propagating ( $\text{d}s/\text{d}t=0$ ) part of the solution, and may be linked to the unsteady shock structure above the wall that is quasistationary in space (standing wave). While region II extends over a large area in the frequency–wavenumber spectra for the disturbance pressure, it is less present in the disturbance velocity spectra. In § 3.3, we showed that the auto-correlation functions of the disturbance pressure and velocity fields are significantly different in the impingement region. Region II is associated with the unsteady shock structure and, therefore, becomes more visible in pressure.

Figure 12.

Frequency–wavenumber spectra for (a) disturbance pressure and (b) disturbance velocity fluctuations extracted along streamline 1 in figure 1(b).

The main motivation for computing the frequency–wavenumber spectra is that we can extract information about which parts of the solution are radiating and non-radiating. The wave equation has the important property that the part of the solution with ${\it\omega}>0$ and a real wavenumber $k$ (in the propagation direction) can only be acoustically radiating if the frequency and the wavenumber follow the inequality $|k|<{\it\omega}/a_{\infty }$ . Hence, only the fraction in the spectrum above the dashed line following the equation ${\it\omega}D/a_{\infty }=k_{r}D$ can be part of the acoustically radiating solution. A large portion of the spectra are evanescent pressure waves that are negligible a few wavelengths away from the source region without an associated sound field. The analysis seems to support the fact that the wall jet contains coherent flow structures that travel supersonically and thereby emit Mach radiation. The current analysis purely accounts for sound generation in the mean streamline propagation direction. It is likely, however, that the unsteady shock structures mainly generate sound radiation in directions that are not aligned with the mean streamline. The fact that a large fraction of the frequency–wavenumber pairs associated with the unsteady shock structure are located in the non-radiating part of the spectrum does not downplay their importance for the noise generation process originating from the wall jet. From the analysis of the noise generation mechanisms in underexpanded supersonic jets, it is known that sound waves emanate from the terminal locations of shock waves in the shear layer. In contrast to Mach radiation originating from large-scale coherent structures convecting supersonically, which has a dominant directivity, these sound waves appear to be more omnidirectional, spreading out almost spherically from the point of generation. Thus, only a very small fraction of the shock noise (in the direction of the wavenumber vector) can be captured in the current frequency–wavenumber analysis.

3.5 Proper orthogonal decomposition

Application of POD results in a decomposition of the flow field into a set of basis functions which represents most of the flow energy, as defined by a user-defined energy norm, with the least number of modes (Lumley Reference Lumley, Yaglom and Tatarsky1967). For many fluid dynamics applications, POD has been used to identify the most energetic flow structures. At this point, it must be noted, however, that it is not guaranteed that existing POD modes have dynamical significance. Therefore, the POD analysis results need to be carefully interpreted. For example, Freund & Colonius (Reference Freund and Colonius2002) applied POD to a Mach 0.9 turbulent jet. They alluded to the fact that the radiated sound from turbulence amounts to a very small fraction of the total energy. Hence, a POD approach in the temporal domain may not be able to capture the physics of the radiated sound waves. Furthermore, beforehand it is not necessarily clear which energy norm should be used to capture the relevant flow physics. Freund & Colonius (Reference Freund and Colonius2002) found that the efficiency of representing the flow data depends strongly upon the norm used and which data are represented. When analysing the acoustic field, it may be advantageous to follow an approach similar to the one described in Suzuki et al. (Reference Suzuki, Bodony, Ryu and Lele2007). Instead of using snapshots in time, Suzuki et al. (Reference Suzuki, Bodony, Ryu and Lele2007) used Fourier coefficients as snapshots for the POD. Suzuki et al. (Reference Suzuki, Bodony, Ryu and Lele2007) argued that this approach is preferable for acoustics because the acoustic waves contain much less energy than the turbulent jet or the unsteady shock structures. In the current study, POD is applied in the temporal domain since we are mainly interested in analysing the coherence in the source region. The spectral energy content of the different POD modes can be retained from the time coefficients. The spectra can be compared with the far-field noise signature, and thus we can infer information about the relative importance of the most energetic flow structures with respect to the sound field produced. The reader is referred to appendix A for details about the current implementation of the POD method.

4.1 Far-field acoustic noise predictions

First, our computational results are compared with far-field acoustic noise measurements by Nakanishi et al. (Reference Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2012). The acoustic data comparison was performed at six microphone locations, $40D$ away from the impingement location, as shown in figure 16. Here, the Ffowcs-Williams and Hawking (FW–H) integral method was utilized to propagate the unsteady flow data in the near field from the permeable surfaces (acoustic control surface) to the six microphone locations in the far field. The FW–H method was chosen over the Kirchhoff method because the FW–H method is less sensitive to nonlinearities at the control surface. A more detailed discussion of this approach and the acoustic control surface selection is provided in Housman et al. (Reference Housman, Brehm and Kiris2013). To study surface scattering effects, we also computed the noise field with a linear acoustic scattering code based on the equivalent source method presented in Kiris et al. (Reference Kiris, Barad, Housman, Sozer, Brehm and Moini-Yekta2014) and Brehm et al. (Reference Brehm, Housman, Kiris and Hutcheson2015c ). It was found that the results were insensitive to surface scattering effects provided that the correct location for the acoustic surface was chosen. The different permeable surfaces and the far-field sample locations are shown in figure 16(a). The far-field predictions are based on flow field information on the selected FW–H surfaces. Hence, it is important to construct an FW–H surface suitable for this purpose. It is crucial to avoid a large contribution from hydrodynamic pressure fluctuations on the permeable surfaces. The placement of the FW–H surface is based on a widely used practice discussed in Liu et al. (Reference Liu, Corrigan, Kailasanath, Heeb, Munday and Gutmark2014). Considering the FW–H equation, it can be shown (see Lyrintzis Reference Lyrintzis2003) that when the surfaces are placed in the near-field region where $\hat{p}^{\prime }\sim 1/r(x)$ the far-field predictions will converge (here, $r(x)$ is the local radius on the FW–H surface and $\hat{p}^{\prime }$ is the pressure fluctuation intensity).

Figure 16.

(a) The permeable surfaces used in the FW–H method and the far-field sample locations. (b) The overall acoustic sound pressure levels (OASPLs) extracted at $r=40D$ .

In figure 16(b), the overall acoustic sound pressure level (OASPL) is compared with the experimental data considering four different permeable FW–H surfaces displayed in figure 16(a). The temporal Fourier transforms were computed using five windows with $T\approx 120D/V_{ref}$ and 50 % overlap. Hanning windowing was applied to reduce spectral leakage and accumulated noise in the spectra. Two distinct local maxima can be identified in the OASPL distribution. The first maximum occurs at roughly $75^{\circ }$ and may be associated with noise generated from the deflected wall jet. The second local maximum occurs at roughly $120^{\circ }$ and is mainly associated with pressure waves originating in the jet-impingement region. Surface 1 is clearly too small to capture all of the important dynamics in the wall jet region, thus the first maximum at $75^{\circ }$ is underpredicted. The addition of a vertical extension to surface 1 down to the impingement plane (denoted by surface $1+2$ ) results in a large increase in the OASPL at low observation angles ${\it\Theta}$ . This is not surprising and has been observed in many other studies, such as Liu et al. (Reference Liu, Kailasanath, Boris, Heeb, Munday and Gutmark2012, Reference Liu, Corrigan, Kailasanath, Heeb, Munday and Gutmark2014). Surface 2 cuts directly through the deflected turbulent jet flow and is, thereby, contaminated by significant hydrodynamic pressure fluctuations. The significantly larger FW–H surfaces 3 and $3+4$ provide similar OASPLs. These results capture the overall trends of the OASPL with respect to the observation angle, albeit at slightly higher noise levels than in experiments. The absolute values are predicted within an acceptable range of $\pm 3$  dB.

Figure 17.

Power spectral density (PSD) versus Strouhal number at six probe locations with $r=40D$ and observation angles of (a) ${\it\Theta}=60^{\circ }$ , (b) ${\it\Theta}=75^{\circ }$ , (c)  ${\it\Theta}=90^{\circ }$ , (d) ${\it\Theta}=105^{\circ }$ , (e) ${\it\Theta}=120^{\circ }$ and (f) ${\it\Theta}=135^{\circ }$ .

In addition to the OASPL, a comparison of the power spectral density (PSD) at the same six far-field point probes is presented in figure 17(a–f). The PSD is defined as

where $p_{a}=20~{\rm\mu}\text{Pa}$ is the audible reference pressure, $f_{min}=1/T$ is the lowest frequency in the sample, $T$ is the length of the time interval and $\hat{p}$ is the pressure Fourier amplitude at the sensor location. The FW–H surface $3+4$ in figure 16(b) was used for the present results. The data were smoothed to remove large oscillations. The CFD results compare well with the experimental far-field measurements for all observation angles. For smaller observation angles ( ${\it\Theta}=75^{\circ }$ and $90^{\circ }$ ), the maximum in the SPL distributions occurs at smaller Strouhal numbers than for larger observation angles ( ${\it\Theta}=105^{\circ },120^{\circ }$ and $135^{\circ }$ ). For the two most dominant propagation directions, i.e. ${\it\Theta}=75^{\circ }$ and ${\it\Theta}=120^{\circ }$ , the maxima occur at two distinct frequencies, $St\approx 0.2$ and $St\approx 0.5$ respectively.

Although the universal spectra of Tam et al. (Reference Tam, Golebiowski and Seiner1996) were only validated for free jets, it appears that these spectra can be employed to match the data at  $75^{\circ }$  and  $120^{\circ }$ . It should be noted that for these two spectra the main contribution comes from the F-spectrum at low frequencies. The G-spectrum is, however, required to obtain the correct decay for high frequencies. The best match of the spectrum at $75^{\circ }$ was obtained when using different peak frequencies for the F-spectrum, $St_{F}=0.22$ , and the G-spectrum, $St_{G}=0.55$ . It must also be noted that the local peak in the spectrum at $90^{\circ }$ cannot be recovered using these two spectra, similarly to peaks in the spectra associated with shock noise presented in Tam et al. (Reference Tam, Golebiowski and Seiner1996).

The validated results provide great confidence that our computational simulation approach is able to capture the key physical mechanisms responsible for noise generation in the nonlinear flow region enclosed by the FW–H surfaces. The current predictions of the far-field noise signature exceed the prediction accuracy provided by previous high-fidelity simulations of the same flow field by Tsutsumi et al. (Reference Tsutsumi, Takaki, Nakanishi, Okamoto and Teramoto2014). The improved prediction accuracy of our results may be attributed to increased grid resolution and the use of numerical schemes with lower numerical dissipation.

Figure 18.

The magnitude of the Fourier coefficients (in SPLs) for four octave-banded frequencies, $St=0.16$ , $0.33$ , $0.66$ and $1.31$ (a–d).

4.2 Near-field acoustic noise analysis

The noise-propagation pattern in the near field was analysed by computing temporal Fourier transforms of the unsteady pressure field in the $(y,z)$ -plane with $x=0$ . The magnitude (in SPLs) and phase (in radians) of the complex Fourier coefficients are shown in figures 18 and 19, which portray a general idea about the noise source distribution and the wave-propagation pattern. The apparent wave fronts in the phase plots and fluctuation intensity in the SPL contours clearly point towards two pre-eminent noise source regions: the jet-impingement region and the wall jet. One needs to be very careful when interpreting SPL and phase plots. Phase plots, for example, are insensitive to the wave amplitude. Although clear wave fronts may emerge in the phase plots, they can be associated with low-amplitude waves. Further, phase plots do not discriminate against acoustic and, for example, evanescent or hydrodynamic instability waves. Tracing the wave fronts back to possible noise sources may also be problematic considering that the radiated sound experiences refraction due to high gradients in the velocity and density fields. Lastly, an important well-known issue of noise source identification is that measurement of the acoustic field outside the source region is not sufficient to uniquely determine the acoustic sources (Rienstra & Hirschberg Reference Rienstra and Hirschberg2004). This issue is addressed in more detail in the following section when introducing the causality approach. In the context of noise source identification, the spectral analysis of the acoustic near field provides information that will be explored to guide the placement of observer locations for the causality approach in § 5. In order to obtain a sensible cause and effect relationship between the source region and the two distinct maxima observed in the OASPL in figure 16(b), the observer locations (also referred to as field points) must be placed in regions where we can clearly separate the physical mechanisms responsible for the two predominant noise-propagation directions.

The SPL contours are shown in figure 18(a–d) for four octave-banded frequencies $St=0.16$ , $0.33$ , $0.66$ and $1.31$ (with centre frequency $St_{0}=0.041$ ). For higher frequencies, the sharp decay in the SPL can be attributed to the coarsening grid far away from the source region. The largest amplitudes in the near field are observed for $St=0.33$ and $St=0.65$ . Arrows have been added to roughly identify distinct near-field source regions that may be associated with different noise generation mechanisms. It is highly unlikely that distinct noise sources have similar directional dependence (Tam et al. Reference Tam, Viswanathan, Ahuja and Panda2008). Nevertheless, it can be expected that some of the regions in the SPL plots are partially overlapping in certain areas. Based on the shape of the SPL contours, at least three distinct regions of noise generation can be identified, i.e. the shear layer of the turbulent jet (1), the jet-impingement region (2) and the wall jet (3). For high frequencies, Mach radiation from the turbulent jet (1) upstream of the impingement plane propagates downstream, and some radiation originating from the wall jet (3) can be identified. The unsteadiness at the inception of the shear layer is dominated by high frequencies, and the SPL contours spread away from the shear layer. In § 3.3, it was shown that further downstream lower frequencies become increasingly more dominant. For $St=0.33$ and $St=0.66$ , the SPL contours display two distinct maxima expanding out from the impingement region (2) and the unsteadiness in the wall jet (3). The pre-eminent noise sources are suspected to be located in these two regions. For the smallest octave-band frequency shown here, $St=0.16$ , the second maximum in the SPL contours, associated with the impingement region, almost disappears and the noise from the wall jet region becomes the dominant noise contributor.

Figure 19.

Phase plots for four frequencies, $St=0.16$ , $0.33$ , $0.66$ and $1.31$ (a–d).

Phase plots are used to unveil the directivity pattern of the acoustic pressure waves outside the source region. The phase plots in figure 19(a–d) are obtained at the centre frequencies that were used for the SPL plots. The phase angles are displayed in radians, varying from $-{\rm\pi}$ to ${\rm\pi}$ with a black–white–black colour map. Different types of waves, such as acoustic, vorticity, entropy, etc., can be identified in the phase plots. Dashed lines and arrows have been added to highlight the directivity of the wave fronts. Although refraction is captured in this analysis, it is not straightforward to determine the actual acoustic noise source regions from these plots. It is generally not clear whether the phase contours in the source regions are associated with purely acoustic pressure waves or other waves, such as vorticity waves, entropy waves, etc. For low Strouhal numbers ( $St=0.16$ and $St=0.33$ ), the phase contour lines continuously extend in a wide arc into the wall jet. Thus, it can be conjectured that the noise generated in the impingement region appears to be phase-locked to the noise originating from the wall jet for these frequencies. In the intermediate frequency range ( $St=0.33$ and $St=0.66$ ), black dashed lines mark wedges that may be associated with different acoustic sources. In figure 19(d), wave fronts within the unsteady jet shear layer are clearly visible for $St=1.31$ . These wave fronts are almost perpendicular to the sonic line of the mean flow, implying that these waves propagate downstream with the mean flow inside the shear layer. It must be noted that the instability waves in the transitional jet close to the nozzle lip are not clearly distinguishable from the acoustic sound waves once they reach larger amplitudes. These vortical flow structures with a dominant frequency around $St=O(1)$ generate acoustic waves by travelling supersonic past the ambient air. The acoustic speed of the ambient fluid is close to the phase speed of the dominant flow structures. The plate and tail shock oscillations on the impingement plate cannot easily be identified in the phase plot because they do not behave like a travelling wave and do not contain a pronounced propagation direction. The POD analysis in § 3.5 was able to capture the dynamics of the shocks.

5.1 Normalized cross-correlations

The normalized cross-correlation functions $C^{0}$ (where the superscript $0$ denotes the normalization) provide valuable information about the interplay of vortical flow structures with the unsteady shocks and solid surface, which leads to significant hydrodynamic pressure fluctuations and the generation of acoustic waves. In order to differentiate between shear and self-noise, the stress tensor is decomposed into its first- and higher-order contributions,

where ${\it\rho}=\langle {\it\rho}\rangle +{\it\rho}^{\prime }$ and the fluctuating part of the velocity component in the observer direction is $V_{r}^{\prime }=V_{r}-\langle V_{r}\rangle$ . It should be noted that approximation of Lighthill’s stress tensor in the form ${\it\rho}u_{i}u_{j}\approx \langle {\it\rho}\rangle u_{i}u_{j}$ by neglecting density fluctuations is reasonable for low-speed unheated flows. For the current high-speed turbulent jet flow impinging on an inclined plate, density fluctuations are inherent to the physics. Thus, all terms in Lighthill’s stress tensor contain a non-negligible dependence on density. The flow quantities in the source region, such as ${\it\rho}^{\prime }$ , $V_{r}^{\prime }$ and $p^{\prime }$ , are normalized with ${\it\rho}_{ref}$ , $V_{ref}$ and ${\it\rho}_{ref}V_{ref}^{2}$ respectively. The far-field acoustic pressure is normalized with ${\it\rho}_{\infty }a_{\infty }^{2}$ , as is common in the traditional acoustic analogy approach, where ${\it\rho}_{\infty }$ and $a_{\infty }$ are the ambient density and speed of sound outside the source region. Before computing the cross-correlation functions, the signals were bandpass filtered with $St_{low}=0.05$ and $St_{high}=4$ .

Figure 20.

Colour contours of peak normalized cross-correlation with the observer point in the ${\it\Theta}=120^{\circ }$ direction and contour lines of time delay with solid lines for ${\it\tau}>0$ and dashed lines for ${\it\tau}<0$ : (a) $C_{p^{\prime },p^{\prime }}^{0}$ , (b) $C_{p^{\prime },V_{r}^{\prime }}^{0}$ , (c) $C_{p^{\prime },s^{\prime }}^{0}$ .

In order to impart a clear interpretation of the results, we computed the peak normalized cross-correlation functions for the different components of Lighthill’s stress tensor in the entire $(y,z)$ -plane (see figure 20) instead of limiting ourselves to sample points and lines (as in previous works). The employment of the spatial distribution of the peak normalized cross-correlation function eases the way for an improved interpretation of the results. It is important to keep in mind that regions with large values of the peak normalized cross-correlation function do not inevitably identify acoustic source regions. They highlight parts of the flow field where the unsteady flow features, irrespective of their amplitudes, strongly correlate with the acoustic pressure at the observer location. This information can be utilized to obtain a physical understanding of how the noise is being generated. In this context, it seems clear that the large-scale coherent flow structures convecting with the turbulent jet, before interacting with the shock structures or the solid surface, are not able to directly generate noise that is detectable upstream of the jet flow direction, for example at an observation angle of ${\it\Theta}=120^{\circ }$ .

The peak normalized cross-correlation functions for the first-order and entropy fluctuation terms are visualized in figure 20(a–c). Solid ( ${\it\tau}>0$ ) and dashed ( ${\it\tau}<0$ ) contour lines of the time delay ${\it\tau}$ are used to portray the propagation direction of the fluctuating flow quantities that are highly correlated with the acoustic pressure. For the cross-correlation study, two field points were placed outside the LES region with a radial distance to the impingement location of $r/D=5$ and observation angles of ${\it\Theta}=75^{\circ }$ and $120^{\circ }$ relative to the jet axis. The field points were placed in a region where the grid resolution is still sufficient to support the relevant frequency range. A solid circle ( $\bullet$ ) is used to mark the position of the observer location in the flow field (see figures 20 and 21). The chosen observer angles correspond to the two angles where the local maxima in the OASPL were observed in § 4. Further, the two observer locations were placed in the two distinct regions with large SPLs in figure 18(a–d) of the acoustic near field (see § 4.2), where the two acoustic noise signatures associated with noise sources in the impingement and wall jet regions can be virtually separated. Although the two point probes were placed much closer to the source region than the far-field sensors, the spectra display very similar characteristics to the far-field spectra. Nakanishi et al. (Reference Nakanishi, Okamoto, Teramoto, Okunuki and Tsutsumi2012) also confirmed experimentally that the acoustic spectra at $r/D=15$ and $r/D=40$ are similar to each other.

Figure 21.

Colour contours of peak normalized cross-correlation with the observer point in the ${\it\Theta}=75^{\circ }$ direction and contour lines of time delay with solid lines for ${\it\tau}>0$ and dashed lines for ${\it\tau}<0$ : (a) $C_{p^{\prime },p^{\prime }}^{0}$ , (b) $C_{p^{\prime },V_{r}^{\prime }}^{0}$ , (c) $C_{p^{\prime },s^{\prime }}^{0}$ .

Away from the immediate source region, the distribution of the peak normalized cross-correlation function $C_{p^{\prime },p^{\prime }}^{0}$ as shown in figure 20(a) provides a depiction of how the acoustic waves propagate away from the source region. As one may expect, the acoustic pressure correlates well in a large area sufficiently far away from the source region, and the distances between the time-delay contour lines are fairly constant, implying wave propagation with $a_{\infty }$ . The region where the noise is generated can be estimated by tracing the contour lines of time delay back into the source region. Dashed lines have been added to mark the area (or wedge) with large correlations outside the source region. Interestingly, the regions with high fluctuation intensities (in the source region) obtain small values of the normalized cross-correlation function. Although large fluctuations are generated in the source region, a large portion of this unsteadiness is not acoustically radiating.

For all three normalized cross-correlation functions displayed in figure 20(a–c), large correlations are obtained inside the turbulent jet. The alignment of the time-delay contour lines provides information about the propagation direction of the highly correlated flow features. A small illustration has been added in figure 20(b), highlighting the propagation direction. The highlighted region is assumed to propagate in the direction of decreasing time delays normal to the contour lines of time delay. The abrupt changes in the peak correlation functions $C_{p^{\prime },p^{\prime }}^{0}$ and $C_{p^{\prime },s^{\prime }}^{0}$ follow the directions of the characteristics that are visualized in figure 20(a). The abrupt changes of regions of large $C_{p^{\prime },p^{\prime }}^{0}$ and $C_{p^{\prime },s^{\prime }}^{0}$ (aligned with characteristics) are probably associated with an interaction of the flow features inside the jet with the weak Mach diamond structure. It should be noted that the particular shapes of these regions are not sufficiently statistically converged to allow for a more detailed interpretation.

For the observer point in the ${\it\Theta}=120^{\circ }$ direction, the largest correlation values are realized in the centre of the jet for $V_{r}^{\prime }$ . The region of large $C_{p^{\prime },V_{r}^{\prime }}^{0}$ stretches over a significant streamwise extent of roughly $2D$ . Large correlation values are detected even downstream of the plate shock after an abrupt reduction of $C_{p^{\prime },V_{r}^{\prime }}^{0}$ in the vicinity of the plate shock. For a free turbulent jet, Bogey & Bailly (Reference Bogey and Bailly2007) observed large values of the correlation function at the end of the potential core region. They used this and additional flow visualization to conjecture that the noise generation mechanism for the downstream jet-noise component is linked to periodic and intermittent intrusion of vortical structures into the jet core. We are following a similar train of thought here. The current data support the presumption that large coherent structures with a coherence length comparable to the diameter of the jet are indirectly detectable (through normalized cross-correlation) in the jet centre. The obvious conjecture is that these flow structures interact with the solid wall as well as the plate shock, causing large fluctuations of Lighthill’s stress tensor and pressure fluctuations on the impingement plate.

Significantly lower correlations are observed towards the edges of the jet in the shear layer. The radial dependence of the peak normalized cross-correlations with the velocity fluctuations agrees with similar observations from other studies considering free jets (see, for example, Panda et al. Reference Panda, Seasholtz and Elam2005; Bogey & Bailly Reference Bogey and Bailly2007). Various explanations for the radial dependence of the cross-correlation function are plausible. From prior studies by Panda & Seasholtz (Reference Panda and Seasholtz2002) and Tsutsumi et al. (Reference Tsutsumi, Takaki, Nakanishi, Okamoto and Teramoto2014) considering a free jet with the same flow conditions, it is known that the potential core length extends over roughly $8D$ downstream of the nozzle exit. Because the plate-to-nozzle distance is only $5D$ , the turbulence initiated in the shear layer does not contaminate the entire jet cross-section before jet impingement. Therefore, the shear layer containing highly energetic vortical flow structures consists of a richer frequency spectrum (see also § 3) than the jet centre, resulting in a strong decorrelation between the time signals in the jet shear layer and the acoustic far field. In an analysis utilizing correlation functions, it is common to point towards regions with large correlation values. However, an interesting observation here is that for the entropy fluctuation term, distinct regions of very low correlation appear at the edge of the source region. None of the other cross-correlation functions share this distinguishing feature.

For the second observer location in the ${\it\Theta}=75^{\circ }$ direction, the cross-correlation function $C_{p^{\prime },p^{\prime }}^{0}$ depicted in figure 21(a) clearly indicates that the noise is mainly originating from the wall jet. Large values of $C_{p^{\prime },V_{r}^{\prime }}^{0}$ are obtained in the wall jet downstream of the plate shock inside the supersonic flow region. Further, strong correlations with the entropy fluctuation term are obtained at the edge of the shear layer of the wall jet in figure 21(c). The large values of $C_{p^{\prime },s^{\prime }}^{0}$ inside the shear layer of the wall jet in figure 21(c) could be linked to coherent flow structures that propagate in the direction of the sonic line. This, however, begs the question of why we did not observe similarly large values of $C_{p^{\prime },s^{\prime }}^{0}$ inside the turbulent jet for the first observer point. The strong correlations suggest that these flow features are involved in the generation of noise predominantly observed in the ${\it\Theta}=75^{\circ }$ direction. We assume that these strong correlations are not associated with the unsteady shock structures in the wall jet due to the fact that $C_{p^{\prime },p^{\prime }}^{0}$ did not display similarly large correlations in this region.

A simple derivation can be used to describe how the entropy fluctuation term can be physically interpreted. We start with a well-known thermodynamic relation

where $\text{D}/\text{D}t$ represents the material time derivative and $s$ is the entropy. Writing equation (5.2) in terms of the disturbance quantities, subtracting $\text{D}{\it\rho}^{\prime }/\text{D}t$ from the right- and left-hand sides and introducing the so-called excess density as ${\it\rho}_{ex}={\it\rho}^{\prime }-p^{\prime }/a_{\infty }^{2}$ leads to

where the first right-hand side term vanishes for constant heat capacity, as assumed here. It follows that sound is produced though the entropy fluctuation term by spatial density variations (last term) and due to non-isentropic processes (second term), such as large gradients in the shear layer and discontinuities across shock waves. The entropy fluctuations in the shear layer may be related to large gradients in this region. The impingement zone also contains large gradients as well as flow discontinuities that may spawn entropy fluctuations.

Figure 22.

Coloured contours of peak normalized cross-correlation $C_{p^{\prime },{\it\rho}^{\prime }(V_{r}^{\prime })^{2}}^{0}$ and contour lines of time delay with solid lines for ${\it\tau}>0$ and dashed lines for ${\it\tau}<0$ : (a) $120^{\circ }$ , (b) $75^{\circ }$ .

The (normalized) cross-correlation analysis for second-order terms (not shown) demonstrated that the second-order disturbance flow quantities correlate significantly less with acoustic pressure than the first-order disturbance flow quantities. This observation agrees with what was reported for free jets in prior research studies by Panda et al. (Reference Panda, Seasholtz and Elam2005) and Bogey & Bailly (Reference Bogey and Bailly2007). The normalized cross-correlation results for the third-order terms are displayed in figure 22. Correlations of the third-order term are not negligible, which is consistent with observations for free jets by Panda et al. (Reference Panda, Seasholtz and Elam2005). The magnitudes of the peak cross-correlations are, however, not as large as those of the first-order terms. The spatial distribution of $C_{p^{\prime },{\it\rho}^{\prime }(V_{r}^{\prime })^{2}}^{0}$ is similar to what was observed for the first-order terms. For the first observation angle of ${\it\Theta}=120^{\circ }$ , larger magnitudes of the cross-correlation function are noted in the potential core region of the jet upstream of the plate shock. For the second observation angle of ${\it\Theta}=75^{\circ }$ , no regions with large cross-correlations are clearly detectable inside the source region.

5.2 Non-normalized cross-correlations

Based on the causality relation in (B 6), the sound-producing regions in the source region are identified by computing the non-normalized cross-spectral density (or cross-correlation) function between the acoustic pressure in the far field and different parts of Lighthill’s stress tensor. It should be noted that when simply computing Reynolds stresses only regions with large fluctuations are highlighted, but there is no differentiation between sound-producing and ‘silent’ unsteadiness. The following analysis utilizing non-normalized cross-correlation functions, on the other hand, extracts the part of the solution in the source region that is correlated with the acoustic pressure. The first- and higher-order contributions of the source term (integrand of (B 6)) are visualized in figure 23(a–f) considering the field point in the ${\it\Theta}=120^{\circ }$ observation direction (marked with ▪). All first-order terms display similar maximum amplitudes, whereas their spatial distributions are distinctly different. Large amplitudes of $C_{p^{\prime },V_{r}^{\prime }}$ are observed in the region where the shear layer impinges on the wall as well as in the shear layer itself. The cross-correlation function of the entropy fluctuation term peaks where the shear layer interacts with the plate shock. Interestingly, while $C_{p^{\prime },V_{r}^{\prime }}$ peaks in the vicinity of the sonic line, $C_{p^{\prime },s^{\prime }}$ peaks closer to the centreline of the turbulent jet. A possible explanation is that $C_{p^{\prime },V_{r}^{\prime }}$ captures the interaction of the impinging shear layer with the solid wall and $C_{p^{\prime },s^{\prime }}$ captures the interaction with the plate shock. Figure 23(c) displays the source contribution from the pressure term. Although the contribution of the surface integral in (B 5) is visualized in the entire $(y,z)$ -plane, it must be noted that only the pressure fluctuations on the solid wall are relevant for the acoustic intensity in the far field (left-hand side of (B 5)). Indeed, $C_{p^{\prime },p^{\prime }}$ reaches its maximum on the wall just below the shear-layer impingement point. The higher-order terms shown in figure 23(d–f) reach significantly lower values in comparison to the first-order terms. For all three higher-order terms, the peak amplitudes are obtained right above the impingement plate below the oblique shock. The term $C_{p^{\prime },(V_{r}^{\prime })^{2}}$ shows similar features to $C_{p^{\prime },V_{r}^{\prime }}$ , while the amplitudes in the shear layer are lower. When including density fluctuations, the third-order term $C_{p^{\prime },{\it\rho}^{\prime }(V_{r}^{\prime })^{2}}$ peaks at around the same location as $C_{p^{\prime },p^{\prime }}$ .

Figure 23.

Colour contours of peak non-normalized cross-correlation for $120^{\circ }$ observation angle: (a) $C_{p^{\prime },V_{r}^{\prime }}$ , (b) $C_{p^{\prime },s^{\prime }}$ , (c) $C_{p^{\prime },p^{\prime }}$ , (d) $C_{p^{\prime },(V_{r}^{\prime })^{2}}$ , (e) $C_{p^{\prime },u^{\prime }v^{\prime }}$ , (f) $C_{p^{\prime },{\it\rho}^{\prime }(V_{r}^{\prime })^{2}}$ .

For the sample point in the $75^{\circ }$ observation direction, the distributions of the different non-normalized peak cross-correlation functions (not shown here) look almost identical to the results for the sample point in the $120^{\circ }$ observation direction. Although the non-normalized cross-correlation function contains information about the linear dependence between two time signals, regions with large fluctuation amplitudes are weighted more strongly than regions with small fluctuation amplitudes, such as the potential core region inside the turbulent jet. This explains why the non-normalized cross-correlation function is less sensitive to the observer location.

An expected finding is that the distributions of the normalized and non-normalized cross-correlation functions are widely different. For example, $C_{p^{\prime },V_{r}^{\prime }}^{0}$ displays large amplitudes inside the jet axis (for the first field point), whereas $C_{p^{\prime },V_{r}^{\prime }}$ peaks in the shear layer and in the region where the shear layer impinges on the plate. This study demonstrates that the normalized cross-correlation function provides information about how the noise is physically generated, but it does not necessarily highlight the actual source locations. In contrast, the non-normalized cross-correlation function retains information about fluctuation amplitudes in the source region and, therefore, it can be used to detect the noise source distribution. Considering the large fluctuation amplitudes and the low normalized cross-correlation in the impingement region, it is evident that this part of the unsteady flow field is not very efficient in generating sound waves. This is not surprising because only a small fraction of the unsteadiness is contributing to the acoustic field.