Four-jet impingement: Noise characteristics and simplified acoustic model

International Journal of Heat and Fluid Flow 67 (2017) 43–58

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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff

Four-jet impingement: Noise characteristics and simplified acoustic model C. Brehm a,∗, J.A. Housman b, C.C. Kiris b, M.F. Barad b, F.V. Hutcheson c a

Mechanical Engineering Department, University of Kentucky, Lexington, KY, 40506 Aerosciences Branch, NASA Ames Research Center, Moffett Field, CA 94035 c Aeroacoustic Branch, NASA Langley Research Center, Hampton, VA 23681 b

a r t i c l e

i n f o

Article history: Received 23 September 2016 Revised 12 June 2017 Accepted 13 June 2017

Keywords: Large eddy simulation Jet impingement noise Shock shear layer interaction Noise source identification Equivalent source method

a b s t r a c t The noise generation mechanisms for four directly impinging supersonic jets are investigated employing implicit large eddy simulations with a higher-order weighted essentially non-oscillatory scheme. Although these types of impinging jet configurations have been used in many experiments, a detailed investigation of the noise generation mechanisms has not been conducted before. The flow field is highly complex and contains a wide range of temporal and spatial scales relevant for noise generation. Proper orthogonal decomposition is utilized to characterize the unsteady nature of the flow field involving unsteady shock oscillations, large coherent turbulent flow structures, and the sporadic appearance of vortical flow structures in the center of the four-jet impingement region. The causality method based on Lighthills acoustic analogy is applied to link fluctuations of flow quantities inside the source region to the acoustic pressure in the far field. It will be demonstrated that the entropy fluctuation term plays a vital role in the noise generation process. Consequently, the understanding of the noise generation mechanisms is employed to develop a simplified acoustic model of the four-jet impingement device by utilizing the equivalent source method. Finally, three linear acoustic four-jet impingement models of the four-jet impingement device are used as broadband noise sources inside an engine nacelle and the acoustic scattering results are validated against far-field acoustic experimental data. © 2017 Published by Elsevier Inc.

1. Introduction The current research study is part of NASA’s environmentally responsible aviation (ERA) project and is an integrated component of engine noise shielding investigation for the reduction of community noise. The hybrid wing body (HWB) concept is currently being studied as a promising alternative to the conventional tube-and-wing aircraft due to its large payload capacity, while offering enhanced aerodynamic efficiency. A key advantage of the HWB from a community noise perspective is that its fuselage can be used effectively to shield engine noise (Tinetti and Dunn, 2012). Within the ERA project, a series of aerodynamic and acoustic tests of a HWB design were performed at NASA Langley Research Center (LaRC). The current work is concerned with computationally modeling the broadband engine noise simulator (BENS) experiment conducted by Hutcheson et al. (2014). The main purpose of this experimental investigation was to study the effects of engine placement and vertical tail configuration on shielding of

∗

Corresponding author. E-mail address: [email protected] (C. Brehm).

http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.06.008 0142-727X/© 2017 Published by Elsevier Inc.

broadband engine exhaust noise radiation. Fig. 1a shows a CAD model of an isolated BENS which contains three so-called four-jet impingement devices (FJIDs) placed inside an engine nacelle. FJIDs are often used in aeroacoustic experiments to generate broadband noise (Hutcheson et al., 2014; Clark and Gerhold, 1999). The three FJIDs are placed inside the nacelle in off-set positions so as to avoid generation of highly directive noise, and instead to promote randomization of the noise inside the nacelle before it radiates out. Complementary to the experimental study, a computational aeroacoustic analysis (CAA) framework is seeked to study engine engine noise shielding or other noise mitigation strategies. The shielding study involves a large parameter space covering different airplane configurations and flight conditions requiring a large number of evaluations. Therefore, an efficient CAA approach will be required. The current study aims to obtain an efficient and accurate CAA approach that can be used for this purpose. An important part of developing this CAA approach is to obtain a reduced-order acoustic model of the FJID. An instantaneous flow visualization around the FJID is provided in Fig. 1b. While conducting large eddy simulations (LES) of the FJID highly interesting flow physics phenomena were observed that appear to play an important role for the noise generation process. The first step towards

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Fig. 1. (a) Three FJIDs placed inside engine nacelle (front open). (b) Instantaneous flow visualization around FJID considering Q-criterion and color contours of gauge pressure in the background. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the development of a Reduced Order Model (ROM) is to obtain an improved understanding of the noise generation process. For this reason, a large part of this paper is concerned with studying the noise generation mechanisms of the FJID. This research study is a continued effort towards gaining a more detailed understanding of jet impingement noise as previously studied in Brehm et al. (2016) for jet impingement on an inclined plate. Supersonic jet impingement problems are of great practical relevance in aeronautical applications. Examples of these types of flows that have been widely studied in the literature include under-expanded jet plumes for military and commercial aircraft, jet impingement during rocket launch, initial stage of launch abort, multi-stage rocket separation, jet-engine exhaust impingement, powered Vertical Take Off and Landing (VTOL) aircraft, etc. Although jet impingement noise is of great relevance for many applications, the noise generation processes for unsteady impinging jet flows are not well understood. The current paper focuses on two aspects: (1) noise source identification of the FJID and (2) the development of a simplified acoustic model of the FJID. To obtain an understanding of the complex fluid dynamics in the source region different post-processing strategies are used, i.e., computation of Reynolds stress terms and intermittency factor as well as application of wavelet transform, causality method and proper orthogonal decomposition. In Brehm et al. (2016), POD was previously used to identify the most energetic flow structures in a Mach 1.8 jet impinging on an inclined plate. It will be shown that the choice of the energy norm affects the coherent flow features that can be identified. While the shock motion can be effectively characterized by employing a pressure based energy norm, turbulent kinetic energy can be used to study the dynamics of shear-layers. Lighthill’s acoustic analogy is employed to study the actual noise generation process in more detail. The physical mechanisms for acoustic sound generation in subsonic and supersonic jets has been analyzed in great detail by a large number of researchers (Panda and Seasholtz, 2002; Panda et al., 2005; Tam et al., 2008; Freund, 2001; Bogey and Bailly, 2007). Following along in their steps, the causality method (Siddon, 1973; Seiner, 1974; Rackl and Siddon, 1979) is utilized in the context of Lighthill’s acoustic analogy to identify possible noise sources in the flow field. The causality method provides a way of linking the fluctuations in the unsteady flow field to acoustic pressure fluctuations in the far-field. Different components of the Lighthill’s stress tensor are cross-correlated with the acoustic pressure in the far-field. The noise emitted by the FJID is known to be broadband which is mainly attributed to noise generation from turbulent flow and its interaction with the Mach disks. The occurrence of intermittent events promoting the broadband noise characteristics has not been reported previously. Once detailed insight about the acoustic noise generation process is gained, a linear acoustic model is constructed to mimic the

FJID noise characteristics and this model is compared to available acoustic data obtained from LES of the FJID. For the linear acoustic analysis, an equivalent source method (ESM) is chosen to solve the 3D Helmholtz equation in the frequency domain. This method is computationally very efficient and it can be employed to study acoustic scattering effects, such as engine noise shielding (Tinetti and Dunn, 2012; Hutcheson et al., 2014), for a fraction of the cost required for full CFD simulations. The linear acoustic scattering analysis requires either unsteady flow data on a permeable acoustic surface encompassing the source region, or an equivalent acoustic model that can effectively generate the incident pressure field in the source region. Here, the latter route is followed by creating a linear acoustic model that is able to capture the main features of the far-field acoustic pressure field of the FJID. The current paper proceeds as follows: Section 2 provides an overview of the numerical solver and the computational setup used to perform LES of the FJID. An evaluation of the mean flow and its unsteady flow features is provided in Sections 3 and 4. In Section 5, the POD method is used to identify coherent flow features in the flow field. The causality method is introduced in Section 6 and utilized to obtain a link between the acoustic pressure field and the unsteady flow field in the source region. Section 7 presents a linear acoustic model of the FJID employing the equivalent source method. Finally, in Section 8 three linear acoustic FJID models are placed inside an isolated front-capped engine nacelle and the acoustic data is compared to experimental results. 2. Computational setup 2.1. Numerical methods In this paper, a block-structured, Immersed Boundary (IB) AMR approach is used to simulate the jet four-jet impingement problem. This methodology is capable of automatically generating, refining, and coarsening nested Cartesian volumes. LAVA’s AMR-IB method is designed to automatically generate the volume grids from a closed surface triangulation, and dynamically track important flow features as they develop. AMR is a proven methodology for multiscale problems with an extensive existing mathematical and software knowledge base (Berger and Colella, 1989; Almgren et al., 1998; Barad and Colella, 2005; Barad et al., 2009; Zhang et al., 2012). The code has been extended using data structures and interlevel operators from the high-performance Chombo AMR library (Colella et al., 20 0 0) to provide a multi-resolution capability that can coarsen and refine the grid locally as a simulation progresses. A sharp immersed-boundary representation (Mittal et al., 2008; Brehm and Fasel, 2013; Brehm et al., 2015b) and automatic grid generation requiring only a surface triangulation make it possible to model complex geometries without requiring a labor-intense

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grid generation process. The Cartesian mesh simulations presented herein are conducted by applying slip wall boundary conditions at all walls due to the high cost of resolving boundary layers with isotropic Cartesian cells. For the current FJID simulations, it expected that the boundary layer on the surface of the FJID does not play an important role. For the current implicit large eddy simulation (ILES), the unsteady compressible Navier–Stokes equations are solved in conservative finite-difference formulation on block-structured Cartesian grids. In order to study physically challenging flow fields, such as the four-jet impingement problem, which include a wide spectrum of relevant physical time and length scales, a large number of grid points and a long time integration window where relatively small time-steps are needed. In order to efficiently simulate this flow, a modified sixth-order accurate shock-capturing WENO scheme is utilized. The accuracy and computational efficiency of the higherorder shock capturing schemes within LAVA has been previously analyzed (Brehm et al., 2015a). A Rusanov-type flux vector splitting scheme is used on the Cartesian mesh. The viscous terms are discretized with second-order accuracy. The fourth-order accurate explicit Runge–Kutta scheme is used for time-integration. For more details about the LAVA AMR-IB solver the reader is referred to Kiris et al. (2016).

2.2. Geometry, grid topology and boundary conditions The geometric dimensions of the FJID are displayed in Fig. 2a and b. All dimensions are provided in terms of the nozzle exit diameter D. Close-up views of the computational mesh employed for the current simulations are shown in Fig. 3a and b. Three different grid resolutions are employed for simulating the flow around the FJID with 100, 180, and 300 million grid points and grid spacings of D/xmin = 30, 40, and 60, where D is the nozzle exit diameter and the finest mesh is concentrated around the nozzle lip. Each Cartesian block (marked with solid black lines) contains 163 grid points. Grid refinement is applied in regions where high unsteadiness and large gradients are expected, such as in the impingement region of the four jets and in the shear-layers of the deflected jets. Volume refinement boxes used to control the grid spacing in sensitive flow regions are shown in Fig. 3a and b as red solid lines. Since storing the entire volume data for each timestep requires excessively large memory, sampling planes Fig. 3c) aligned with the three coordinate directions were used to gather

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Fig. 2. Geometric dimensions of FJID: (a) (x, y)-cut-plane and (b) (y, z)-cut-plane.

the unsteady flow data at each time-step. Most of the unsteady flow analysis was applied to the data recorded in these sampling planes. The total mass-flow rate per FJID with four nozzles is m˙ = 0.0229 × 10−2 kg/s. The mass-flow rate was chosen based on values according to the experimental setup (V. Hutcheson, 2014). The streamwise velocity and pressure profiles at the exits of the four nozzles were obtained in an independent precursor simulation with an in-house unstructured solver within LAVA (Kiris et al., 2016) (see Fig. 4a and b). The simulation setup of the precursor simulation is shown in Fig. 4a, where a massflow rate of m˙ /4 (at ambient conditions T = 300 K and P = 1atm) was supplied at the pipe inlet. The inflow profiles for the unsteady simulations are provided in Fig. 4b. Note that throughout this paper all flow quantities were normalized with mean flow quantities at the nozzle exit, i.e., exit velocity, Vre f = 351.7 m/s, pressure, pre f = 3.125 × 105 Pa, temperature, Tre f = 231.8 K (averaged over the nozzle exit), and the exit diameter D = 2.1 mm. The Reynolds number based on D, Vref and Tref is approximately 75,0 0 0. 2.3. Computational simulation strategy The current computational simulation strategy follows the findings by Shur et al. (20 05a, 20 05b) with respect to nozzle inflow conditions and more importantly subgrid-scale (SGS) modeling for the jet flow-acoustic simulations. Following this simulation

Fig. 3. Computational mesh layouts in the (a) (x, y)-plane, and (b) (y, z)-plane. Refinement tag boxes that are used to control the volume grid spacings are shown with red lines. Gray-scale contours of Mach number in the background visualize unsteady flow field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 4. (a) Precursor simulation setup to obtain boundary conditions at nozzle exists of FJID and (b) streamwise velocity and pressure profiles at the nozzle exits.

approach, Shur et al. (20 05a, 20 05b) demonstrated excellent agreement between their jet noise predictions and a wide range of experimental data. Due to the high computational cost of direct numerical simulations (DNS), DNS is limited to lower Reynolds numbers (roughly of up-to O (104 )). DNS is out of reach for the current flow conditions and, therefore, a LES approach is utilized. In classical LES an explicitly computed SGS closure is employed. In contrast, the implicit large eddy simulation (ILES) approach taken here relies on the inherent regularization mechanism through the truncation error of the convective fluxes also referred to as implicit SGS model (Hu and Adams, 2011). The modified WENO-6 scheme (Hu and Adams, 2011) used here appears to provide superior physically motivated scale separation properties. It was demonstrated in Hu et al. (2010) and Hu and Adams (2011) that the modified WENO-6 scheme is able to reproduce the Kolmogorov range of the turbulent kinetic-energy spectrum at the limit of infinite Reynolds number, independent of grid resolution while retaining the shock-capturing capabilities of the original WENO scheme it is derived from. Overall, the modified WENO6 scheme seems to follow the basic physical principles of LES (see also Hu and Adams, 2011). Despite some ongoing debate on the validity of ILES, prior experience (Brehm et al., 2016), as well as other research studies (Brehm et al., 2016; Shur et al., 20 05b; 20 05a), demonstrated the suitability of the ILES approach for simulations of turbulent flows away from physical boundaries when providing sufficient grid resolution. Moreover, the ILES approach is efficient in utilizing the available grid resolution for capturing the shear-layer transition process and modeling salient features of the turbulent flow. In prior research studies (Shur et al., 20 03; 20 05b), it was shown that when utilizing an explicit SGS model the transition of the shearlayer may be delayed and that the turbulence was not captured as well. In Brehm et al. (2016), the authors demonstrated for supersonic jet impingement on an inclined plate that with the present ILES approach excellent agreement with far-field acoustic measurements by Akamine et al. (2014) could be obtained. An important detail of the computational setup is the nozzle inflow boundary conditions specification. Based on the length of the supplying pipes and the Reynolds number the flow inside the pipe can be assumed to be turbulent. For the current flow conditions, the flow can be assumed to contain fine-scale turbulent structures that will enter into the shear layer. It is understood that due to the topology change from a wall-bounded to a shearlayer flow, the flow will essentially go through a “secondary transition” process (Shur et al., 2005b) when it ejects from the nozzle exit. While employing a grid resolution to resolve the turbulence inside the nozzle is not possible with the available resources we rely on the fact that the instabilities and turbulence are stronger in the shear-layer. In the current simulations, the significant background noise (and round-off to some extent) entering the shear-layer through a receptivity process seeds unsteadiness in the shear layer and initiates the subsequent breakdown to turbulence.

Prior simulations suggest that the unsteadiness in the shear-layer is much less effected by the fine-scale turbulence inside the nozzle flow but more significantly by the the inflow shear-layer thickness or state (laminar versus turbulent) as suggested by Shur et al. (2005b). Inflow boundary conditions for the ILES were provided by precursor RANS simulations accounting for the internal pipe system in the experiment. Note that some researchers (Freund, 2001) apply some form of unsteady forcing to seed unsteadiness in the transitional shear layer. One natural choice would be to introduce perturbations based on linear stability theory (Michalke, 1984), i.e., Tollmien Schlichting waves in the upstream boundary layer or Kelvin–Helmholtz modes in the shear layer. It is, however, cumbersome to achieve a realistic scenario with respect to instability wave frequencies and amplitudes, which would also require a significant spatial extend for the initial disturbances to develop. Studies regarding the effect of this inflow forcing on jet noise can be found in Lew et al. (2004) and Bogey and Bailly (2003). Hence, in the present study any type of forcing based on linear stability theory and/or random noise was avoided to eliminate the dependence of the simulation on free parameters and the possibility of introducing parasitic noise and/or spurious flow structures. For the current simulations, the exiting nozzle boundary layer was modeled as turbulent in a mean sense but does not initially contain fluctuations. The unsteadiness in the shear-layer arises from numerical round-off and the elevated acoustic background noise level arising through the highly unsteady jet interactions. A rapid breakdown to turbulence of the two primary jets was observed, e.g., in the flow visualizations in Fig. 1b and 3 a and b. To identify sensitivities to the grid spacings a grid refinement study was conducted. Moreover, comparing the current grid resolution with computational setups of similar research studies (Brehm et al., 2016; Dauptain et al., 2012) suggests that the current grid resolution is sufficient to study the underlying flow physics. Sufficient grid convergence of the relevant flow physics will be demonstrated by employing a grid resolution study that is provided in Section 4.1. 3. Mean flow features An overview of some of the key flow features for this unique flow scenario are provided in Fig. 5a and b. The time-averaged flow field is visualized with color contours of gauge pressure, line contours of Mach number and a thick solid line marking the sonic line. The flow exiting the four nozzles is supersonic and the opposing jets of equal strength form a stagnation region that is bounded by four plate shocks. Due to the geometric constraints, i.e., the close distance between the four nozzles, only a small fraction of the flow is able to escape into the gaps between the nozzles of the FJID. Further, the symmetric nozzle arrangement creates a relatively stable impingement region except for the occurrence of intermittent events that occur and will be discussed in more detail below. For an unsymmetric arrangement, it is suspected that

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Fig. 5. Time-averaged flow field around the four-jet impingement device visualized by colored contours of pressure, contour lines of Mach-number and the sonic line is marked as thick, black solid line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the jet impingement region would be significantly more unsteady which is induced by violent interactions between the different jets. Two primary deflected supersonic jets form along the x-direction carrying a large fraction of the momentum flux introduced by the four nozzles, as can be seen Fig. 5a. These jets expand more rapidly and break up more quickly than common jets that are ejected from a clean nozzle. Based on the large difference in the momentum flux between the two primary jets in the x-direction (Fig. 5a) and the four diagonal jets in the (y, z)-plane (Fig. 5b), it is expected that the two jets in x-direction are the main contributors to the acoustic noise field. Although, it should be noted that the acoustic energy propagating away from the source region only accounts for a very small fraction of the total energy associated with the unsteadiness in the jets. The flow field contains complicated shock structures that are highly unsteady. The two primary jets break up quickly due to strong unsteadiness and essentially only two shock cells remain in the time-averaged flow field of the jets.

(a)

(d) sample point 3

4. Unsteady flow features 4.1. Grid resolution study The sensitivity of ILES results with respect to the grid resolution is investigated by comparing the power-spectra of pressure, Ep , at five sample points inside the source region as marked in Fig. 6a. To allow for a better comparison of the power-spectra for the different grid resolutions, the spectra were smoothed using simple averaging. A significant increase in the spectral energy can be noted for the solution on the medium grid in comparison to the coarse grid results. The difference in the power spectra, Ep , becomes significantly smaller between the medium and fine grids. The grid resolution study considering the power spectra of pressure in the near field indicate sufficient grid convergence. Further comparisons of other relevant flow quantities (not shown here) on the coarse, medium, and fine grids convey the same message. The following unsteady flow analysis is based on the fine grid solution.

(b) sample point1

(c) sample point 2

(e) sample point 4

(f) sample point 5

Fig. 6. (a) Sample point locations 1 − 6 inside the unsteady flow field. (b–f) Power spectra of unsteady pressure data extracted at sample points 1−5.

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ity fluctuations are observed in the streamwise direction u1 u1 in



 

comparison to the cross-stream component u2 u2 , which is expected for turbulence in shear-layers. Large ampli non-isotropic  tudes of u1 u1 are reached early in the shear-layer while the cross-





(a) u1 u1 

(b) u2 u2 

(c) u1 u2 

 (d) |p − a2∞ ρ |

(e) u1 u1 

(f) u2 u2 

(g) u1 u2 

  (h) |p − a2∞ ρ |

Fig. stress tensor components, Ri j /ρ =    7.  Colored contours of different Reynolds  ui u j , and the entropy fluctuation term, | p − a2∞ ρ  | . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In addition to the grid sensitivity, the different power-spectra provide some idea about the frequency characteristics in different parts of the flow field. In the shear-layer the power-spectra of pressure for sample points 1 and 5 contain broadband spectra that are relatively flat until St ≈ 1. The power-spectra of pressure in the center of the jet impingement region (sample point 4) and the end of the first shock diamond structure in the deflected jets (sample points 3) contain significantly more energy in ∞ terms of IE p = 0 E p dSt. The unsteadiness in this region is dominated by lower frequency content (St < 0.1). Sample point 2 is placed in the corner of the impingement region at the inception of the shear-layer of the primary deflected jets (in x-direction). The power-spectra for sample point 2 contains a more pronounced peak around St = 0.1 − 0.2. None of the five power spectra display any tonal noise and it can be concluded that the source region is dictated by relatively broadband spectral characteristics. It is expected that the turbulent shear layer spectra generally display broadband characteristics. In addition, the occurrence of intermittent events is observed that are also broadband in nature. 4.2. Reynolds stresses and entropy fluctuations To obtain a general overview of the unsteady flow features in the source region the Reynolds stress tensor components, Ri j / < ρ >=< ui uj > with ui = ui − ui , and the entropy fluctua-





tion term, | p − a2∞ ρ  | , based on Lighthill’s acoustic analogy are computed. Note that standard tensor notation is used for the mathematical expressions. In the above expressions, ui  refers to the mean flow velocity component and ui is the disturbance velocity component. Color contours of the Reynolds stress tensor components and the entropy fluctuation term are presented in Fig. 7a– h. The disturbance flow velocity vector was locally decomposed into three components, u1 (in direction of the mean flow), u2 (perpendicular to the mean flow inside the sampling plane), and u3 (perpendicular to the sampling plane). Thus, for the tangential component it consequently follows u2  = 0. Visualizing the Reynolds stress tensor components and the entropy fluctuation term highlights amplitudes. The four    regions   with high fluctuation   terms, u1 u1 , u2 u2 , u1 u2 , and | p − a2∞ ρ  | , display large amplitudes in different regions of the flow field and can be associated with various physical mechanisms. The two primary jets that form in x-direction break up rapidly and, therefore, the region with large unsteady fluctuations is confined within a small area around  the  FJID. Large  normal Reynolds stress tensor components, u1 u1 and u2 u2 , are identified in the jets’ shear-layers and inside the four-jet impingement region. Slightly larger veloc-



stream velocity fluctuations u1 u2 are activated later around the first Mach   disk. Large amplitudes of the Reynolds stress component u1 u2 are due to unsteady vortical flow structures inside the shear-layer. Furthermore, in the unsteady flow visualization a flapping motion of the primary and secondary jets was noted. Large amplitudes of the entropy fluctuation term are observed around the first shock diamond and inside the jet impingement region. The unsteadiness in the primary jets’ shear-layers induces an unsteady motion of the Mach diamond structure causing significant fluctuations at the Mach disk of the first shock cell. The large entropy fluctuations in the center of the jet impingement region are caused by vortical structures that are generated at the edge of the four-jet impingement region. These vortical structures gravitate towards the center of the impingement region, as will be analyzed below in more detail. Some traces of these structures can be identified in all components of the Reynolds stress tensor. To the best knowledge of the authors, the occurrence of such flow structures has not been reported previously. In order to ensure the validity of the current numerical simulations different variations of the simulation setup were tested, employing various numerical fluxes, including flux vector splitting and flux difference splitting, various higher-order WENO shock capturing schemes (Brehm et al., 2015a; Brehm, 2017), and time-integration methods. The current analysis only highlights regions with large unsteady fluctuations. It does not differentiate between sound producing and “silent” unsteadiness. Comparing the Reynolds stresses and the entropy fluctuation term in Fig. 7 with the non-normalized cross-correlation analysis results that are discussed in Section 6 provides an idea on what fraction of the unsteady flow field is producing acoustically radiating pressure waves. 4.3. Intermittency The unsteady flow visualization clearly depicts the occurrence of intermittent events that cause large pressure fluctuations away from the source region. In the present work, the intermittency of different disturbance flow quantities q is evaluated by computing the intermittency factor in the entire (x, y)-sample-plane. In this work, the intermittency factor is defined by using a binary output function I(t) in the form



I (t ) =

1 if q < 0.5q, 0 otherwise.

(1)

With the definition of the binary output function, the intermittency factor is then defined as γ = I (t ). The strategy to analyze the intermittency in the flow is based on the work by Bogey and Bailly (2007). They used the intermittency factor, where the binary output function was based on the vorticity norm |ω|, to analyze intermittent flow features at the end of the potential core of subsonic jets. In the current work, the intermittency maps were computed in the same way (see Eq. (1)) for different disturbance flow

 2 , and turbulent kinetic energy,   2

quantities, the pressure, q = p q = 0.5

 2  u

 2

+ v

+ w

, as shown in Fig. 8a and b, respec-

tively. It is assumed that for γ ≈ 0 the flow is laminar or fully turbulent and for γ ≈ 1 the signal is intermittent. The intermittency map for disturbance pressure illustrates intermittent events occurring around the Mach disk of the first shock cell in the primary jets as well as an intermittent motion of the plate shocks within the four-jet impingement region. For turbulent kinetic energy, a larger region of the flow field appears to be intermittent in

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Fig. 8. Gray-scale contours of the intermittency factor γ for (a) disturbance pressure and (b) turbulent kinetic energy.

comparison to the disturbance pressure fields. The intermittency map for turbulent kinetic energy highlights the shear layers of the primary jets and the center of the four jet impingement region. A key observation is that the intermittent event in the center of the four jet impingement region is picked up by the intermittency factor based on the turbulent kinetic energy but not the disturbance pressure. Further, the sudden expansion of the jets’ shear layers is highlighted in the intermittency map of the turbulent kinetic energy. Sample time-signals of disturbance pressure and turbulent kinetic energy are shown in Fig. 9a–c. The time-signal of (p )2 at sample point 1 in Fig. 9b is provided as an example of a time signal with low intermittency. The other two time-signals in Fig. 9a and c clearly display an intermittent behavior. This begs the question of whether temporal Fourier analysis is the right tool to analyze the unsteady character of the flow field. Discrete wavelet transforms are used in Appendix A provide an additional perspective about the highly intermittent flow field. 5. Proper orthogonal decomposition POD decomposes the flow field into a set of orthogonal basis functions that capture most of the flow energy in terms of a userdefined energy norm with the least number of modes (Lumley, 1967). In the current study, the source region is mainly composed of broadband noise sources and intermittent events and, therefore, snapshot POD is applied in the temporal domain instead of in the frequency domain (Suzuki et al., 2007). The choice of the energy norm will affect what flow physics are captured with the POD modes. The frequency information of the flow field dominated by the unsteady primary jets and shock dynamics is obtained by applying a Fourier and wavelet transform to the POD time-coefficients. Freund and Colonius (2002) have found that the effectiveness of a given representation of the initial flow data depends strongly upon the norm used and the data represented. The appropriate norm for capturing particular flow physics is not immediately clear for this particular flow configuration. In Brehm et al. (2016), it was shown that computing the most energetic p -based POD modes is

49

Fig. 10. Singular values and total energy captured by the first N POD modes.

useful for obtaining a general sense of the unsteady shock motion for supersonic jet impingement problems. In order to study the noise generation process s -based and K -based POD modes provide some idea of how noise is generated in the source region when considering that the general structure of Lighthill’s stress tensor is composed of ρ ui uj and the entropy fluctuation term s . More details about the computation of the POD modes and singular values associated with each mode energy are provided in Appendix B. 5.1. POD results In order to measure the coherence in the flow field the coherence factor ψ N is introduced. This factor provides a measure for how much energy in the pressure field is captured by the first N POD modes:



ψN =

N  m=1



λ

(m )

Ntot 



λ

(m )

,

(2)

m=1

where λ(m) are the eigenvalues of the eigenvalue problem described in Eq. (9) in Appendix B, Ntot is the total number of POD modes and N < Ntot . For jet impingement flows (Brehm et al., 2013), it is very common for a large amount of energy to be contained within a few POD modes due to the intensity of the pressure oscillations in the impingement region that are associated with shock motion. Fig. 10 shows that for the four-jet impingement problem roughly 40 POD modes are sufficient to capture 80% of the energy defined by the L2 -norm of pressure in Eq. (7). The pressure field appears to be significantly more coherent than when basing the energy norm on kinetic disturbance energy K or the entropy fluctuation term s . Fig. 11 displays the three most energetic p -based, K -based, and s -based POD modes. Depending on the choice of the energy norm, different characteristics of the unsteady flow field can be identified. The pressure field is dominated by highly-energetic pressure fluctuations in the center of the fourjet impingement region that are coupled to the shock cells inside the primary jet. Inside the jet impingement region, the dynamics in the flow field can be inferred from the p -based POD mode shapes, where all modes appear to be symmetric. The most energetic K -based POD modes highlight the shear-layers of the primary

Fig. 9. Time-signals of (a) EK at the edge of the impingement region (sample point 2), (b) (p )2 in the shear layer of the primary deflected jet (sample point 1), and (c) (p )2 in the vicinity of the Mach disk of the first shock cell.

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frequencies than for the p -based POD modes with estimated decay rates of St −2.16 , St −1.7 , and St −1.4 , respectively. The Fourier and wavelet pseudo spectra for the K -based and s -based POD modes display a broadband peak around St = 0.15 for the first two POD modes. The spectra for the s -based POD modes display similar characteristics as the power spectrum of pressure at sample point 2 in Fig. 6c. The broadband peak at St = 0.15 appears to be associated with the unsteady interaction of the shear layer of the primary jets with the Mach disk of the first shock cell. The first two axisymmetric K -based POD modes display almost identical spectra. In the POD analysis, a vortical flow structure that in its most simplistic form can be described with a traveling wave ansatz is captured as a mode pair with identical frequency characteristics. This mode pair seems to capture the interaction of flow structures in the shear layer with the Mach disk. 6. Noise source identification Fig. 11. Colored contours of the three most energetic POD modes using an energy norm based on (a–c) pressure, (d–f) turbulent kinetic energy, and (g-i) the entropy fluctuation term. The dashed lines mark the decay of the POD mode amplitudes for high frequencies. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. Fourier and DWT pseudo spectrum of POD time-coefficients of the three most energetic POD modes using an energy norm based on (a) pressure, (b) turbulent kinetic energy, and (c) entropy fluctuation term.

jets that interact with the shock cells inside the jets. The coherence of K -based POD modes quickly breaks up in streamwise direction shortly downstream of the first shock cell. The largest fluctuations are observed in the region where the jet expands rapidly and the jets’ shear layers interact with the end of the shock cells. Outside the jet impingement region, the first two K -based POD modes are axisymmetric (azimuthal wavenumber β = 0) while the third mode displays a switch in sign in the front section of the shear layer from top to bottom. All other POD modes appear to be symmetric except for some asymmetries in the center of the jet impingement region which may be related to the strong intermittency of the events occurring in this region. A coupling between the entropy fluctuations in the center of the four jet impingement region, around the jets’ shear layers, and the Mach disk of the first shock-cell can also be identified for the most energetic s -based POD modes. All POD modes show a strong coherence of the flow field in the vicinity of the FJID. Furthermore, the different POD modes display a complicated interaction/coupling between the four-jet impingement region, the shear-layers of the primary jets, and the dynamics of the shock-cell structures. Lastly, the analysis of the temporal behavior of the different POD modes remains. The time-coefficients in Eq. (11) are transformed in time using Fourier and wavelet transforms. The Fourier and wavelet spectra are shown in Fig. 12a–c. The Fourier and wavelet pseudo spectra for the p -based POD modes are relatively broad due to the intermittent unsteady nature of the flow field in the impingement region. The broadband peak for p -based POD modes occurs at around St = 0.1. It appears that the spectra for the p -based POD modes contain more high frequency content than the K -based and s -based POD modes. The spectra for the K -based and s -based POD modes seem to decay more rapidly for higher

The analysis in this section is based upon the causality idea proposed in earlier works by Siddon (1973), Seiner (1974), and Rackl and Siddon (1979). While simultaneous visualizations of the instantaneous flow and sound fields can provide an idea on specific events that generate noise (Hileman et al., 2005; Bogey and Bailly, 2003; Hileman and Samimy, 2001) the causality method is a more direct approach. The causality method relies on computing cross-correlations between flow quantities inside the source region and the radiated acoustic pressure field outside the non-linear flow regime. An attractive feature of this method is that the effects of scattering, absorption and refraction on sound radiation are automatically included by simultaneously extracting information from both the flow and acoustic fields. Starting from Lighthill’s equation an integral relation between the Lighthill stress tensor fluctuations Tr (in the direction of the source point to the far-field observer location) in the source region at xs and the acoustic pressure fluctuations at an observer location xf can be obtained. Hence, the causality relation can be written in the form

P SD p (x f , f ) = −

π f2 ra2∞



V

CSDTr ,p (x f , xs , f )dV,

(3)

where the auto-correlation function of p is transformed to the power spectral density P SD p and the cross-correlation function between Tr and p turns into the cross-spectral density CSDTr ,p . The contributions from the surface integrals in Eq. (14) are negligible in this work. Although, the surface contribution is not considered in Eq. (3), surface scattering effects are still implicit accounted for in CSDTr ,p . Further details about the causality method and how it is applied here is provided in Appendix C (see also Brehm et al., 2016). Fig. 13 shows contours of the peak normalized cross-correlation function C p0 ,p of the acoustic pressure at different sample points (solid circles) with the unsteady pressure in the (x, y)-sampling plane (superscript 0 denotes the normalization). The dashed lines mark the time-delay of the cross-correlation. An approximate location of the noise source can be obtained by tracing the contour lines of time-delay back to the region with high fluctuation intensity, where the normalized cross-correlation function has small values. The lines of time-delay for the first two sample point locations in Fig. 13a and b can be roughly traced back to the four-jet impingement region and roughly the region around the Mach disk of the first shock cell. In the present flow field, the normalized cross-correlation functions, C p0 ,V  and C p0 ,s , did not display large r

peak values inside the source region for the first two observer points similar to C p0 ,p shown in Fig. 13a and b. For the highly broadband time characteristics of the flow in the source region, the

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Fig. 13. Normalized cross-correlation of acoustic pressure, C p ,p , at different sample points marked with solid circles (•) and pressure on the sampling plane.

Fig. 14. Non-normalized cross-correlation of acoustic pressure at the first observer point (marked in Fig. 13a) with pressure fluctuations and components of the Lighthill’s stress tensor in the (x,y)-sampling plane.

normalized cross-correlation functions is not able to identify coherent flow features that may play an important role in the noise generation process. Some regions in the turbulent jets are highlighted for the last sample point furthest away from the FJID in Fig. 13c. These regions are located in the part of the flow field with lower fluctuation intensities where the primary jets are substantially broken up. Panda (2005) pointed out that simply utilizing normalized correlations may lead to an incorrect interpretation of the noise source distribution because it does not provide information about the fluctuation intensity. Hence, in order to determine the details of the acoustic source distribution non-normalized crosscorrelation functions are computed. Fig. 14a–c show the nonnormalized cross-correlation functions of acoustic pressure at the first observer location (as in Fig. 13a) with pressure fluctuations and components of the Lighthill’s stress tensor in the (x, y)sampling plane. C p ,p displays large amplitudes in the four-jet impingement region and around the Mach disk of the first shock cell. The large values in C p ,p indicate that a significant amount of shock noise appears to be generated at the Mach disk of the first shock cell. The cross-correlation function C p ,V  highlights the jets’ shearr layers and the unsteady Mach disks. Furthermore, large values of C p ,s can be observed in the center of the four-jet impingement region. Both cross-correlation functions, C p ,V  and C p ,s , display simir lar amplitudes over a large spatial extent and highlight three dominant regions that are important for the noise generation process, i.e., the four-jet impingement region, the shear layer of the primary jets, and the Mach disk of the first shock cell. The cross-correlation functions were also computed for the other two observer locations in Fig. 13b and c. It was noted, however, that the spatial distributions of the cross-correlation functions are fairly insensitive to the observer point locations used here. 7. Development of a reduced-order linear acoustic model Based on the findings in the previous analysis of the unsteady flow field, a simplified linear acoustic model of the FJID is constructed in the following discussion. The model is based on utilizing an in-house linear acoustic scattering code that was developed for this purpose and is part of the LAVA framework (Kiris et al., 2016). The scattering code is able to perform acoustic calculations with far less computational resources than when conducting di-

rect noise computations resolving the relevant scales. This is especially advantageous when propagating sound waves to observer locations that are far away from the source region. The linear acoustic scattering code is based on solving the linear Helmholtz boundary value problem in the frequency domain using a highly scalable equivalent source method (see Ochmann, 1995). Details about the equivalent source method are provided in Appendix D. To solve the boundary value problem, NM monopole sources are placed inside the scattering surface that contains NS > NM surface elements (here, triangles). Prior grid resolution studies confirmed that the edge lengths for the surface triangulation t must be chosen such that the highest frequency acoustic wave of interest is resolved with λ = 5t . Utilizing the free-space Green’s function for a monopole source the overall acoustic pressure field is obtained following the superposition principle. The strength of the monopoles is obtained by solving an over-determined system of equations that accounts for the boundary conditions at each element of the scattering surface (see Appendix D for details). It is assumed that the incident pressure field is either available on a radiating surface where the unsteady data is extracted from high-fidelity CFD simulations (permeable surfaces are used that encompass the source region) or an equivalent acoustic model can be used to effectively mimic the noise being generated in the source region. One key issues of utilizing a permeable acoustic surface is that when enclosing the source region (containing two rapidly expanding jets) with a closed surface it is not straightforward to avoid or to remove pollution from hydrodynamic pressure fluctuations. In the present work, the latter route was taken by creating a simplified acoustic model that is able to capture the main features of the acoustic sound field. In this approach, the understanding of the noise generation mechanisms in the FJID obtained from the previous analyses can be directly tested by comparison with available CFD data in the acoustic near-field. It must be ensured that the three FJID noise source models are temporally uncorrelated when deploying the three models inside the engine nacelle (see Fig. 1) to simulate the BENS experiment (see Section 8). This is achieved by applying random phase relations between the three FJIDs models. In a prior research study (Tinetti and Dunn, 2012), the FJID was modeled as a broadband monopole noise source. This simple model does not account for scattering effect of the housing assuming uni-directional directivity pattern. In this section, a modified FJID acoustic model is discussed. The previous analysis of the ILES simulations provides valuable information about the noise generation process in the FJID. Three important noise source regions were identified, i.e., the four-jet impingement region, the shear-layers of the primary jets and the end of the first shock diamond structure. Due to the particular nature of the primary jets, the conventional jet noise models do not apply here (or would need to be extended), especially when considering the intermittency that was observed in the center of the jet impingement region and around the Mach disk. Moreover, it was noted that the surface of the FJID itself provides a scattering field that must be accounted for. To model the acoustic characteristics of the FJID a simple acoustic model as shown in Fig. 15 was used. Three monopole sources, denoted by S0 , S1 and S2 , were placed in the vicinity of the FJID scattering surface. The monopole source S0 is meant to capture the dy-

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Fig. 17. Phase plots in sampling plane for (a) CFD, and (b–d) different acoustic models. In the CFD solution, the region with significant hydrodynamic pressure fluctuations is shaded and dashed lines mark the dominant wave fronts that can be observed in the phase plots.

Fig. 15. Preliminary linear acoustic model containing three sources, S0 , S1 and S2 , placed in the vicinity of the FJID scattering surface.

namics in the high-pressure four-jet impingement region. The objective of S1 and S2 is to primarily model the unsteadiness around the Mach disk of the first shock cell and account indirectly for the accumulative noise generation in the shear-layers of the primary jets. It is clear that this model is highly simplified and does not account for some basic features of the noise generation process. For example, if Mach radiation from large-scale coherent flow structures is a key noise generation mechanism a single monopole placed on the center-line of the primary jet would not be able to reproduce the actual directivity pattern. Furthermore, the jet mean flow causes refractions of the acoustic waves. Nevertheless, it appears that this model is able to capture the key features of the acoustic pressure field. The current study focuses on results for a single frequency St ≈ 0.2 (or f = 31.5 kHz). This frequency corresponds to the peak frequency (roughly 1.8kHz for full scale of the HWB) for turbomachinery noise (Hutcheson et al., 2014). Since the FJID produces a broadband noise field it may be important to consider the characteristics of broadband noise sources in the linear acoustic model. In the current investigation, monochromatic and broadband (white) noise sources are considered. The assumption of white noise is justified because the objective is to model broadband noise in one-third octave-bands (as in the experimental study by Hutcheson et al., 2014) and the CFD results show that the amplitude is fairly constant within the one-third octave-band centered around St = 0.2. When comparing monochromatic noise and broadband noise it is important to establish an equivalent monochromatic noise source amplitude. Considering the root-mean-square pressure field of the signal a simple relationship between the source strengths of single and multiple frequency signals can be obtained as

A2mono =

N  n=1

A2bb,n .

(4)

In this expression, it is assumed that the continuous broadband spectrum is modeled by summing over a discrete frequency interval, Stn ∈ [Stlow , Sthigh ], while a random phase relation between the different harmonics is assumed. The broadband solution was obtained by superimposing the individual narrowband solutions at the N equally spaced frequencies in the interval assuming random phase relations. Fig. 16a–d show the SPL levels for the CFD data and three sample linear acoustic models. All results shown here were obtained for a frequency of St = 0.2. The SPL levels from the CFD data were computed in one-third octave-bands with a center frequency of St = 0.2. By comparing the linear acoustic results with acoustic data extracted from the CFD runs the amplitudes of the sources can later be adjusted. For the linear acoustic model, two solutions are shown for the narrowband solution (St = 0.2) on the top and the broadband solution (St ∈ [0.179, 0.225]) on the bottom. The narrowband and broadband solutions are almost identical, which is not surprising considering the small variation in the interval with St/(0.5St) > 8. Consequently, from this point forward the narrowband solutions are utilized. Utilizing narrowband solutions significantly reduces the computational cost since the broadband solution requires N narrowband solutions in the frequency band St ∈ [0.179, 0.225] that are superimposed. For lower frequencies and different noise source configurations, this assumption needs to be re-evaluated because the broadband results can be significantly different from narrowband results. The wave propagation patterns are visualized by plotting the phase of the complex Fourier coefficients in the sampling plane. Fig. 17a–d show the phase plots for CFD and three linear acoustic models. Considering the similarities in the phase plots between the CFD data and the different acoustic models, it seems possible to come up with amplitudes for the current monopole distribution such that a fair match with the CFD data can be obtained. The phase relations that have to be imposed among the noise sources to produce a realistic sound field is unclear at this point. To decorrelate the sources the overall solution is obtained as an average (black line) of solutions with random phase relationships between the sources (gray lines) as shown in Fig. 18 for S0 = S1 = S2 , 2S0 = S1 = S2 , and S0 = S1 + S2. It was, however, assumed that S1 and S2 have a zero phase relation. This assumption may not necessarily hold true in the case where the flapping motion of the

Fig. 16. Sound pressure level in sampling plane for (a) CFD, and (b–d) different acoustic models.

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Fig. 18. Different acoustic models with randomized phases between S0 and S1 = S2 . The distance between the two monopoles S1 and S2 is d/D = 5.7. Different realizations of the random phase relations between S0 and S1, 2 are shown with gray lines and the averaged solution with a solid black line.

Fig. 19. (a) BENS linear acoustic scattering simulation setup and (b) equivalent monopole sources inside engine nacelle. The engine nacelle was capped in front according to the experimental setup.

jets on either side of the FJID are out-of-phase. The POD analysis and the causality method, however, confirm the in-phase assumption where a symmetry between both sides was identified with respect to the large-scale motion inside the source region. Furthermore, the time-delay in the cross-correlation functions to obtain a peak correlation between the left and right side of the FJID is τ = 0. Multiple variations of the monopole amplitudes and monopole distances (data sweep not shown here) were tried in order to determine the best combination. Fig. 18a–c show some of the best models that were tested. Considering the different data sets, it appears that the linear acoustic models using S0 = S1 = S2 and 2S0 = S1 = S2 (both with d/D = 5.7) obtain the closest match with the CFD data. Note that the pressure levels in the free-jet region shown in the reduced-order model in Fig. 18b–d underpredict the levels shown in the CFD data (shaded region in Fig. 17a) due to the presence of significant hydrodynamic pressure fluctuations in the unsteady jets. Hence, a meaningful comparison of the SPL values for ∈ [0, 20°] is not possible at this point. 8. Application of linear acoustic FJID model to BENS experiment The final part of this work is concerned with placing three instances of the linear acoustic FJID model developed in Section 7 in an engine nacelle as discussed in Hutcheson et al. (2014). Fig. 19a schematically illustrates the computational setup used in this study, where only the no-flow case (M∞ = 0) is considered and the engine inlet was capped. The setup is similar to the one discussed in Tinetti and Dunn (2012) but includes some additional geometry, such as plug and bracket. Three linear acoustic FJID models are enclosed in the front capped engine nacelle as shown in Fig. 19a. For the current analysis, NS = 1 × 106 triangles and NM = 60 × 103 monopole sources are placed inside the scattering surfaces as displayed in Fig. 19b. For the ESM, 400 processors were utilized for 20 min of wall clock time. The sensitivities of the acoustic scattering analysis with respect to various modeling parameters were studied by varying (1) the monopole offset distance to the scattering surface, (2) the number of equivalent monopole sources, (3) the number of triangles on the scattering surface, and (4) the number of sample sets with different phases between the three FJID acoustic models.

Fig. 20. Comparison of sound pressure level for acoustic model and experimental data.

The scattering solution procedure proceeds in the following three steps: 1. Obtain linear acoustic scattering solution employing the ESM separately for each of the three FJIDs 2. Apply random phases (RP) to each of the three solutions, superimpose solutions and compute phase-averaged solution (here, NRP = 10 0 0). 3. Extract sound pressure levels at sampling points from the phase-averaged solution and compare to experimental data. Step 2 is needed to ensure that the three FJID models are statistically uncorrelated. The NRP random phases were generated considering conventional Latin hypercube sampling. It is essentially the same approach as discussed before in Section 7. Fig. 20a–c provide a comparison of the experimental data with acoustic scattering results. Three types of linear acoustic FJID models were utilized: (a) S0 without accounting for the scattering effects from the FJID surface, (b) S0 with FJID scattering surface, and (c) S0 = S1 + S2 with FJID scattering surface. For the second and third models, fair comparison with experimental data is achieved. The results demonstrate that accounting for the scattering effects from the FJID surface appears to be important to match the experimental data. It must also be pointed out, however, that when including the FJID models inside the engine nacelle the sensitivity with respect to the particularities of the noise generation process discussed above are significantly reduced. This is not surprising for this particular application since the acoustic pressure waves can only escape through a very narrow gap in the engine exit plane. Finally, it should be noted that the computational cost for obtaining an ESM solution for the BENS experiment is significantly lower than conducting direct noise computations with LES. To put the computational cost for scattering calculations into perspective, a fairly well-resolved LES of the isolated FJID requires roughly 300 million grid points and approximately 1 million timesteps. 9. Summary and conclusions A key focus of this work was to provide insight into the noise generation mechanisms of the FJID that was used in the BENS experiment as a broadband noise source. To understand the noise generation process a detailed analysis of the flow features in the

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source region was conducted. The different components of the Lighthill stress tensor, i.e. the Reynolds stress tensor components and the entropy fluctuation term, showed large fluctuation amplitudes inside the jets’ shear layers around the first Mach disk, and inside the four jet impingement region. The large fluctuations inside the four jet impingement region are associated with the sporadic appearance of vortical flow structures that originate at the inception of the secondary jets. In the intermittency analysis the jet impingement region, the region around the first Mach disk, and the shear layers of the primary jets, displayed large intermittencies. POD was used to identify the most energetic coherent flow features in the source region. The most energetic p -based POD modes display large amplitudes in the center of the four-jet impingement region and they appear to be associated with the appearance of flow structures that intermittently originate at the inception of the four diagonally ejected secondary jets. The most energetic K -based POD modes mainly capture the dynamics in the shear layers and the interaction of the shear layers with the shock cells. The most energetic s -based POD modes also display a strong coherence between the unsteadiness inside the impingement region and the unsteady motion of the first shock diamond structure of the two primary jets. The frequency spectra of the POD time coefficients are generally broad and no distinct narrowband peaks were identified. The wavelet analysis showed that broadband noise is introduced through the occurrence of intermittent events and the interaction of the shear-layers with the Mach disks. The sudden shifts in frequency in the wavelet spectra were traced back to the appearance of flow structures in the four jet impingement region. By employing the causality method a link between different components of Lighthill’s stress tensor and the acoustic pressure in the far field was established. In addition to the noise generation in the shear layer that is common for jet noise, this analysis identified two key noise generation mechanisms: (1) shear-layer interaction with the first Mach diamond structure, and (2) large entropy fluctuations within the impingement region that is coupled to unsteady shock motion inside the two primary jets. 9.1. Simplified four jet impingement acoustic model The second part of the paper was concerned with constructing a reduced-order linear acoustic model that is able to capture the main characteristics of the FJID acoustic pressure field. Based on the noise source analysis, it appears that the acoustic pressure field of the FJID can be modeled reasonably well with a simple linear acoustic model considering three monopole sources while accounting for scattering effects of the FJID surface. The single monopole source in the center of the FJID is used to capture the dynamics in the four-jet impingement region. The two monopole sources, separated by a distance d, model the unsteadiness around the Mach disk of the first shock cell and it can partially account for the accumulative noise generated in the shear-layers of the primary jets. It is expected that this model cannot fully account for all acoustic phenomena, such as refraction effects from the free jets, noise generation within the free shear-layer through small and large-scale turbulent structures, etc. However, by modeling the most dominant noise sources, it was possible to match the acoustic data extracted from the direct noise simulations reasonably well. The linear acoustic analysis demonstrated that it is important to include the scattering surface of the FJID itself to be able to match the direct noise simulations data. 9.2. Computational predictions of broadband engine noise simulator In a first step towards modeling the Broadband Engine Noise Simulator (BENS) experiment studied by Hutcheson et al. (2014), three linear acoustic FJID models were placed inside a front capped

engine nacelle. By employing this preliminary model, fair agreement with experimental data was obtained. The CAA results, however, also demonstrate that the acoustic predictions are not very sensitive to some of the details of the linear acoustic FJID model. This may be explained by considering that the pressure waves can only escape through a narrow opening at the engine exit. An important aspect for the BENS acoustic analysis is that the current approach for modeling the acoustics of the BENS experiment provides a truly predictive capability because it is solely based on the CFD/CAA predictive capabilities of the four-jet impingement problem. The amplitudes of the reduced-order acoustic model were determined from CFD results of a single FJID. Appendix A. Discrete wavelet analysis Wavelet transforms (Chui, 2002; Daubechies, 1992; Gurley and Kareem, 1999) offer a different perspective on analyzing the unsteady flow field. While the location information, i.e., here the location in time, is generally lost in the Fourier transform, wavelet transforms provide the advantage that they can capture both frequency and location information. By employing wavelet transforms the occurrence of intermittent (or sporadic) events is scrutinized. A discrete wavelet transform (DWT) using Ricker wavelets was applied to the time-signals recorded at the sample points in Fig. 6a. The resolution of the wavelets is mapped to a scale that can be interpreted as a pseudo-frequency, here denoted by

˜ = St

Stc . aTs

(5)

˜ , is computed from the samIn Eq. (5), the pseudo-frequency, St pling period, Ts , the center frequency, Stc , and the scale a of the wavelet (Abry, 1997; Indrusiak and Möller, 2011). In this context, it is assumed that the wavelet can be associated with a harmonic time-signal of frequency Stc . For the current results, the center frequency is given by the frequency that maximizes the Fourier transform of the wavelet modulus. Fig. 21a–f present colored contours of wavelet transformed time-signals of pressure recorded at sample points 1 − 6 in Fig. 6a. The pseudo frequency and the amplitude associated with these peaks significantly varies over time and does, therefore, not display the typical signature that would be associated with a perfectly harmonic time-signal. The DWT of the time-signal at sample point 1 contains a large amount of high-frequency content that is typical for high-Reynolds number shear-layers. The pressure timesignal at this sample point is less intermittent as it has previously been noted in Section 4.3. The Mach disk motion of the first shock cell is dominated by lower frequencies as can be seen in the DWT of the time-signal at sample point 3. The appearance of intermittent events cause a sudden shift in the pseudo frequency and can clearly be identified in the DWTs. Sample point 4 in the center of the jet impingement region captures the formation of sudden pressure fluctuations that affects the time-signals at all sample points. An advantage of DWT analysis is that it provides the location in time when shifts in frequency occur. The occurrence of intermittent events is further scrutinized by visualizing the flow field at instances in time when these events are identified in the DWT contours in Fig. 21a–f. Fig. 22 displays instantaneous snapshots of gauge-pressure contours and contour lines of dilatation at t/tre f = 80, 105, and 124. The flapping motion of the primary jets causes the first shock cell to move towards sample point 3 inducing large pressure fluctuations with higher frequency content at t/tre f = 80 in the wavelet spectrum. In unsteady flow visualizations it was noticed that instantaneous flow structures are generated at the edge of the jet impingement region and they gravitate towards the center. At t/tre f = 105, the formation of such a structure can be detected at

C. Brehm et al. / International Journal of Heat and Fluid Flow 67 (2017) 43–58

(a) sample point 1

(b) sample point 2

(c) sample point 3

(d) sample point 4

(e) sample point 5

(f) sample point 6

55

˜ , t ) of pressure recorded at sample points 1−6 (a-f) versus non-dimensional time and pseudo Strouhal freFig. 21. Colored contours of the wavelet pseudo-spectrum A(St quency in log-scale. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 22. Instantaneous snapshots of gauge-pressure contours and contour lines of dilatation at t/tre f = 80, 105, and 124. The arrows mark the point probe locations where the signal was recorded.

sample point 2 introducing low frequency content. Large pressure fluctuations are intermittently being generated in the center of the four-jet impingement region that can be identified by significant frequency shifts in the wavelet pseudo-spectrum. The large amplitudes with low frequency content in the DWT pseudo spectrum at t/tre f = 124 for sample point 4 is associated with a flow structure being located right above the sensor location. In the more conventional approach of studying unsteady flow fields, the time-signals are analyzed by employing Fourier transforms that are not able to reveal this intermittent time behavior. FFT data can be compared with the DWT data by computing a DWT pseudo spectrum which allows for a straightforward interpretation of the pseudo frequency. Fig. 23 shows a comparison of DWT with FFT data. In order to compute an estimate power spectrum from the DWT coefficients the approach by Gurley and Kareem (1999) was followed. In the time interval t ∈ [t1 , t2 ], the DWT pseudo spectrum was simply computed with

ADW T

 

˜ =α St



 t2 t1





˜ , t dt A2 St t2 − t1

,

(6)

where α is a constant scaling factor to obtain amplitudes that are comparable to FFT amplitudes. Overall the integral DWT amplitudes ADWT match the general trends of the FFT coefficients very well. Hence, the definition of the DWT pseudo Strouhal number appears to be consistent with the FFT results. This allows for a straightforward interpretation of the frequency characteristics over

Fig. 23. Integral DWT amplitude defined by Eq. (6) and magnitude of FFT coefficients at different sample points 1−6.

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time as displayed in Fig. 21. An advantage of the smooth integral DWT amplitude curves is that it can be used for a clearer comparison between different data sets. For intermittent flow fields the DWT provides information about when intermittent events occur while in addition the frequency information in the sense of FFT remains. For more details about DWT and its relation to FFT the reader is referred to the literature (Chui, 2002; Daubechies, 1992; Gurley and Kareem, 1999).

For problems where the spatial dimension, m, is much greater than the number of available time-steps, N, it is more efficient to employ the “snapshot” method (Sirovich, 1987) for POD. Hence, in the current analysis the snapshot method is used. For more details about the POD snapshot method used here, see Chatterjee (20 0 0) for a basic introduction to POD, and Rowley (2001) and Freund and Colonius (2002) on details on how it can be applied to fluid dynamics problems. In the current POD analysis, it is assumed  (x, t ) = ( p , u , v , w , s ), is that a vector of flow quantities, here q defined over a region of interest and additional weighting fac (x, t ) in the  weigh the contribution of the components of q tors ω vector norm,

|q| =





Nq 



ω

2 k qk

dx,

(7)

k=1

. By employing POD, where Nq is the number of components of q the series expansion converges optimally with respect to the L2 norm defined in Eq. (7). Kinetic disturbance energy, the entropy  can be used fluctuation term, pressure or other combinations of q  = in the energy norm by choosing the weighting vector as ω (0, 1, 1, 1, 0 ), (0, 0, 0, 0, 1), and (1, 0, 0, 0, 0), respectively. The POD eigenmodes are computed as

ψ (n) (x ) =

The current theoretical framework is based on the acoustic analogy employed in Lighthill’s Eq. (13). In the current paper, flowacoustic interaction is not explicitly accounted for and the noise sources shall be interpreted in terms of Lighthill’s sources. The starting point for the causality method is Lighthill’s equation defined as

∂ 2 Ti j ∂ρ − a2∞ ∇ 2 ρ = , ∂t ∂ xi ∂ x j

Appendix B. Computation of POD modes

2

Appendix C. Causality method

N   (x, ti ) , ri(n ) q

(8)

i=0

 (x, ti ) is where the subscript i indicates the time-step (i = 0 to N), q the vector of flow quantities at time-step i, and ri(n ) are individual elements of r (n ) . The right eigenvectors, r (n ) , are solutions to the algebraic eigenvalue problem:

Cr (n ) = λ(n )r (n ) ,

where the right-hand-side term contains the Lighthill’s stress tensor Ti j = ρ ui u j + δi j ( p − a2∞ ρ ) neglecting the effect of viscosity, a∞ is the speed of sound, ui is the unsteady velocity field, and p is the unsteady pressure field. In the current discussion, it is assumed that viscous effects are negligible for the noise generation as well as feedback from the acoustic field to the source. It is important to remember that Lighthill’s equation is simply a reformulation of the mass and momentum equations and is valid for every solution that obeys these equations. Lighthill’s equation cannot distinguish between radiating and non-radiating disturbances. Theoretically, the relationship a∞ < ω/α can be used to separate the radiating and non-radiating parts, where ω is the angular frequency and α is the spatial wavenumber. In the current approach, this relationship is not utilized explicitly. A slightly different route to identify the radiating part of the solution is taken here. The basic idea is to multiply the Lighthill’s stress tensor with the acoustic pressure fluctuations in the far field which essentially acts as a filter operation, because sampling points located in the far field only sense the radiated part of disturbances (Panda et al., 2005). The most relevant parts of the causality approach are presented below. The right-hand side of Eq. (13) can be regarded as a forcing term to the wave equation. Acoustically, the source terms represent a distribution of quadrupoles embedded in an ambient medium, whereby flow discontinuities are not explicitly contemplated. The desired solution at a far-field point is derived in terms of an integral equation after manipulating Eq. (13) and utilizing the timedomain free-space Green’s function.

 xi x j Ti j (xs , t  ) ∂2 1 dV + 4π 4π a2∞ r 2 ∂ t 2 V |x f − xs |  xj ∂ (δi j p + ρ ui u j ) + ni dS. 4π r ∂ t S | x f − xs |

p (x f , t ) ≈

(9)

where the correlation tensor C is defined in index notation by

Ci j =

1 

N+1



 (x, ti ), q  (x, t j ) . q

(10) 

Nq

, q ] = ( k=0 Here, the brackets denote the inner product [q ωk q2k )dx (see Eq. (7)). The eigenfunctions are normalized with their  ( n ) ( n )   ,ψ  corresponding eigenvalues ψ = δ  λ(n ) , and the timenn

coefficients are computed as

a(n ) (t ) =



1

λ (n )



 (n ) (x ) .  (x, t ), ψ q

(11)

The flow field can be reconstructed/recovered from the eigenmodes and time-coefficients:

 (x, t ) ≈ q

I 

 (n ) (x ) . a(n ) (t )ψ

(12)

n=0

The POD modes are orthogonal to each other and sorted by their respective energy norm (see Eq. (7)) contents, whereby the drop– off in mode energy toward the higher mode numbers is typically significant. Therefore, a small number of modes, I, will suffice for capturing a large percentage of the flow’s energy defined in Eq. (7). This is particularly the case when the flow dynamics are governed by large energetic structures of organized (or coherent) fluid motion.

(13)

∂ ∂t

 S

ρ ui n dS |x f − xs | i (14)

Following the analysis in Proudman (1952), as more recently revisited by Panda et al. (2005), the scalar component of the stress tensor Tr is introduced that represent longitudinal quadrupoles. The scalar quantity is the fluctuating stress tensor component Tr in the direction of the source point to the far-field observer location, i.e., r = x f − xs . The acoustic intensity at a far-field point is obtained by multiplying Eq. (14) with the far-field acoustic pressure and computing the acoustic intensity,







  1 ∂2   Tr (xs , t + τ )dV, p (x f , t ) 4π a20 r V ∂ t 2  1 ∂2 = CT ,p (xs , x f , τ )dV, 4π a20 r V ∂τ 2 r

p , p (x f , τ ) =

(15)

where the correlation function CTr ,p is evaluated after shifting the near field data Tr by the propagation time or time delay τ . Eq. (15) links the near field dynamics to the far-field acoustic pressure fluctuations. Applying the cross-correlation is extremely powerful since it naturally separates the hydrodynamic and acoustic fluctuations (Panda et al., 2005). To avoid the computation of the second derivatives in time, a temporal Fourier transform is applied to

C. Brehm et al. / International Journal of Heat and Fluid Flow 67 (2017) 43–58

Eq. (15) and find

P SD p (x f , f ) = −

π f2 ra2∞

 V

CSDTr ,p (x f , xs , f )dV,

(16)

where the auto-correlation function of p is transformed to the power spectral density P SD p and the cross-correlation function between Tr and p turns into the cross-spectral density CSDTr ,p . Appendix D. Equivalent source method An outline of the Helmholtz boundary value problem (BVP) solver that is also referred to as Equivalent Source Method (ESM) is described here. At each frequency the following radiating/scattering Helmholtz BVP is posed:

∂ P  , ∀x ∈ ∂ ; (17) ∇ 2 P + k2 P = 0, ∀x ∈ ; = −ˆıkρ∞ c∞ uˆ · n ∂n    ∂P lim R + ıˆkP  = 0 (Sommerfeld radiation condition ), ∂R R=|x|→∞ where P is the acoustic pressure, k = 2π f /c∞ is the wavenumber, ρ ∞ is the free-stream density and c∞ is the free-stream sound  = 0 is enspeed. On perfectly reflecting scattering surfaces, uˆ · n  is the Fourier transform of u · n  forced. For radiating surfaces, uˆ · n (time-averaged velocity has been subtracted out) obtained from the CFD simulation. An alternative Dirichlet boundary condition for the acoustic pressure on radiating surfaces is also available. The Helmholtz BVP is solved by considering a set of NM monopole sources that are placed inside radiating and/or scattering surfaces, where the location of the monopoles is xm . The incident pressure field may originate from radiating surface data extracted from direct noise computation, such as LES, or some type of acoustic source model. The scattering surface represents the geometry of interest for the scattering calculations, such as FJID and engine nacelle surfaces. It is assumed that the number of surface elements (here triangles) NS used to discretize the radiating/scattering surface is less than the number of monopole source, i.e., NM < NS . Using the free-space Green’s function,

G(x, xm ) =

1 e−ˆık|x−xm | , 4π |x − xm |

(18)

an approximation to the acoustic pressure field, p , is constructed such that:



p (x, t ) =

NM 



am G(x, xm ) eıˆωt ,

(19)

m=1

where ω is the angular frequency and {am }m=1,NM are complexvalued coefficients. The ultimate objective is to solve for the complex coefficients {am }m=1,NM . By constructing p as a linear combination of free-space Green’s functions, the wave equation and the Sommerfeld radiation condition in Eq. (17) are satisfied exactly in the entire domain. The remaining part of the Helmholtz BVP is to determine {am }m=1,NM such that the boundary condition on the radiating/scattering surfaces is satisfied. In order to approximately satisfy the boundary condition, the following linear overdetermined system of equations for each surface triangle NS is constructed to determine the unknown coefficients {am }m=1,NM .



   ∂ G(xn , xm )  (xn ) for n = 1, · · · , Ns . am = −ˆıkρ∞ c∞ uˆ · n ∂ n m=1 NM 

(20) In the current implementation the boundary conditions are applied at each centroid of the surface triangles. The solution

57

of the overdetermined system is obtained by applying a parallel implementation of the complex-valued conjugate-gradient (CG) algorithm to the normal equations A∗ Ax = A∗ b where A = ∂ G(xn , xm )/∂ n, x = (a1 , · · · , aM )T , b = −ˆıkρ∞ c∞ ([uˆ · n ](x1 ), · · · , [uˆ ·  ](xN ))T , and A∗ is the Hermitian transpose of A. n N Once the coefficients {am }mM=1 are found, the solution can be evaluated at any observer location x ∈ . For a large number of observer locations (e.g., evaluation on a plane), a parallel version of the solution evaluation has been implemented. Once the solution is evaluated at each observer location for all frequencies, it can be transformed back into the time domain, which has also been implemented. For the ESM, we commonly utilize a ratio of NS /NM = 3 − 15 for the number of surface triangles NS to the number of monopole sources NM . 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