Noise reduction analysis of supersonic unheated jets with fluidic injection using large eddy simulations

Abstract

A set of large eddy simulations is used to perform a numerical analysis of fluidic injection as a tool for noise reduction. This technique, developed at the Pennsylvania State University, allows one to turn on and off the air injectors in order to reduce the noise during takeoff and landing without penalizing performance in other flight regimes. Numerical simulations are performed on a military-style nozzle based on the GE F400-series engines, with a design Mach number of 1.65, for overexpanded jet conditions. The numerical results are compared and validated with the outcome of experiments performed at the Pennsylvania State University. For the case chosen, the fluidic injection shows the potential of breaking down shock cells into smaller structures with different orientation and strength. This directly reduces the intensity of broadband shock associated noise, with a positive effect of reducing the overall sound pressure level by more than $1.5 dB$ along the direction of maximum sound propagation of the baseline case. The maximum noise reduction was found to be almost $3 dB$ at 55° on the azimuthal plane in between two lines of injectors.

Keywords

Jet noise computational aeroacoustics computational fluid dynamics large eddy simulation fluidic injection noise reduction unheated supersonic overexpanded jet

Introduction

Jet noise is one of the most challenging research topics in the aerospace field. The goal is to understand the noise generation mechanisms and assess nozzle designs that reduce the noise level.

The Federal Aviation Administration in the United States, as well as other aviation authorities in the rest of the world, dictates the maximum level of noise that a civil aircraft can produce. These limits are imposed to guard the health of the people living in the surroundings of airports. If an aircraft does not satisfy the requirements it cannot land or takeoff at a specific airport or may be allowed to operate only during limited time slots. Also noise is an obstacle for the development of the areas surrounding an airport. The loudness of the operating aircraft discourages people from living or working close to an airport. Last but not least is ground crew health, especially for the military field. These people work close to the jet engines and have a propensity for hearing loss caused by the noisy environment. The U.S. Department of Veterans Affairs spends about $1 billion/year for hearing loss cases.¹ During the period 1968–2006 the Navy spent $6.48 billion and the Marines spent $1.48 billion over 35 years.¹

What is jet noise? Using the definition given by Lighthill, “The jet noise is assumed to be only the noise generated by the fluid itself exiting from the nozzle.”² The jet noise, hence, does not take into account all the noise components connected with the mechanical part of the jet engine or the combustion. Jet noise is important because it is the loudest part of jet engine noise during takeoff.

The noise generation mechanisms are still not well enough understood to facilitate an effective way to drastically reduce the noise emitted. During the last decades a trial-and-error approach has been used to find possible ways to reduce the noise level. Several different designs have been studied for civil and military aircraft. Some examples are chevrons,³ beveled nozzles,^4,5 microjets,^6,7 interior corrugation, and fluidic injection.⁸ Each one of these designs has a trade-off between the noise reduction and thrust loss. Also, some of them have been found to be more effective on supersonic jets than subsonic (or vice versa).

One way to test if a new design is effective is to undertake an experimental analysis. A different and sometimes complementary approach is computational analysis. Simulations provide more detailed information than the experiments. Thanks to the ever increasing computational power, it is possible to run simulations on modern supercomputers that take into account the nozzle geometry and include the designs described above (e.g. Eastwood and Tucker,⁹ Martha et al.,¹⁰ Aikens et al.,¹¹ Gonzlez et al.,¹² Hafsteinsson et al.,¹³ Goparaju and Gaitonde,¹⁴ Dhamankar et al.,¹⁵ Brès et al.,¹⁶ Gojon and Bogey^17,18). The results are becoming more accurate and compare well with experiments.

Computational analysis can be done using different formulations. The typical and best known are Reynolds averaged Navier–Stokes (RANS), unsteady Reynolds averaged Navier–Stokes (URANS), large eddy simulation (LES), and direct numerical simulation (DNS). In order to study jet noise, RANS are not adequate because they give a time-averaged solution neglecting all the unsteady mechanisms that are directly connected to noise generation, therefore a noise model is needed. URANS can actually take into account the unsteady mechanisms, but it has some limitations on the accuracy of the solution. Both RANS and URANS use turbulence modeling that cannot capture properly the noise mechanisms since it does not solve for the turbulent scales. The most accurate way to solve the Navier–Stokes equations numerically is using DNS. DNSs require grids that can resolve all the turbulence scales down to the Kolmogorov scale. To satisfy this condition, grids with an enormous number of points are needed, which today’s supercomputers cannot handle for realistic Reynolds numbers. A more efficient way to solve the Navier–Stokes equations is the use of LES, which solves for the larger turbulent structures and models the smaller ones. Hence, LES requires less computational power than DNS. LES for jet noise for realistic Reynolds numbers can be done with current supercomputers.

Experimental analyses^19–21 show that there exists two different noise sources in jet mixing noise. The first source is the large turbulence structures, the other is the fine-scale turbulence, and they are connected with low and high emitted frequencies,^20,21 respectively. LES was introduced in computational aeroacoustics under the hypothesis that the large turbulent scales are more efficient in generating sound than the small scales.^19,22–26 Moreover, the typical LES resolution required for acoustic analysis is higher compared to the one required to accurately solve for the mean flow and turbulence intensity. Hence, the effect of the subgrid scales becomes even less important.²⁷

Solving the entire field up to the observers’ location using LES could be excessively costly in terms of computational power. The current approach is to combine the LES with surface integral aeroacoustics methods to extend the near-field solution to the far field²⁸ employing either the Kirchhoff²⁹ or the Ffowcs Williams–Hawkings (FWH) method.^30–32 Indeed, one can compute only the nonlinear near field with LES and extend the solution to the far field using these methods without the need to actually calculate the solution in the far field. This cuts dramatically the computational cost of the simulations and makes LES particularly appealing to study jet noise.

Bodony and Lele³³ published a review paper on LES used to analyze jet flows. The cases considered include both subsonic and supersonic jets. The majority of early LES runs were based on domains that do not contain the nozzle geometry. Current efforts include the geometry in order to perform more realistic flow simulations. Indeed phenomena like the boundary layer development³⁴ or the generation of shocks inside the nozzle affect the jet plume development and hence the aeroacoustics, namely the noise emitted. Therefore, simulating the nozzle geometry helps to obtain a better understanding of the noise generation mechanisms, and it allows one to analyze possible noise reduction designs. Some recent results are discussed following a chronological time line. Karabasov et al.³⁵ showed a possible way to use LES results to set the amplitude of parameters needed for a hybrid acoustic analogy based on using linear Euler equations and a RANS solution for the far field. Gao and Li³⁶ used a dispersion-relation-preserving scheme for the spatial discretization and adaptive selective damping for shock capturing to study an imperfectly expanded supersonic jet. Wang and McGuirk³⁷ developed an improved rescaling–recycling method for synthetic inlet conditions for a round subsonic nozzle, then they applied it to study an overexpanded supersonic jet from a rectangular nozzle.³⁸ Unstructured solvers were also applied to LES to perform simulations on supersonic jets.^39,40 Specifically, Mendez et al.³⁹ studied an almost perfectly expanded jet using an explicit subgrid-scale model and a shock capturing model, while Nichols and Lele⁴⁰ analyzed crackle noise on a heated jet. More recently, LES solvers have been used to analyze noise reduction designs such as beveled nozzles,¹¹ plasma actuators,^12,14 fluidic injectors with steady and flapping injectors,¹³ or chevrons.¹⁵ Dhamankar et al.¹⁵ modeled the chevrons using an immersed boundary method (IBM) that allows one to use non-body-conforming grids.

A high-order structured LES code has been developed by researchers at Purdue University to analyze some of the noise reduction designs described above. The code has an improved computational speed⁴¹ thanks to the use of a SPIKE algorithm by Polizzi and Sameh⁴² for parallelization. It can handle multiblock Cartesian and cylindrical grid topologies, including the nozzle region. The code has various additional features like a wall model,¹¹ turbulent inflow boundary conditions, and the capability to use the IBM.^43–46 The code was used to test noise reduction strategies, i.e. beveling¹¹ and chevrons.¹⁵ The present version of the code includes the possibility to simulate air injectors acting inside the nozzle.

The goal of the present research is to obtain a better understanding of the noise generation and reduction mechanisms in an unheated jet with fluidic injection by performing analyses and making comparisons with the baseline case.

The research group at the Pennsylvania State University led by P. Morris and D. McLaughlin developed a noise reduction design using injectors placed inside the nozzle. They performed several experimental^8,47–51 and numerical analyses^52–55 on different injector configurations, nozzle pressure ratios (NPRs), and total temperature ratios (TTRs). The unheated case was chosen since experimental flow-field measurements are available for comparison from Powers et al.,⁵⁰ in order to understand if our numerical simulations were reproducing properly the physics of the jet. The present work is the first step toward the analysis of heated jets that reproduce more realistic conditions and for which the fluidic injection show an enhanced noise reduction effect.⁸

An earlier version of this work was presented in Coderoni et al.⁵⁶ A preliminary analysis was based on a set of RANS simulations in order to get a picture of the flow field. The RANS simulation for the fluidic injection case was fundamental to generating a new boundary condition, in the LES code, that simulates the injectors. The present work shows only the results of the set of LES for the unheated jet.

The first part of the analysis is based on the baseline nozzle without injectors. The focus is on the comparison between the multifaceted geometry and a round geometry, the effect of the characteristic filter, and last to the effect of the grid. The second part of the analysis compares the solutions for a round nozzle with and without the injectors. The purpose is to identify and analyze the mechanisms that generate sound while the injectors are on, as well as the effect on the OASPL and so the effectiveness of the design.

Numerical techniques

The governing equations used in the present LES formulation are the Favre-filtered nondimensionalized Navier–Stokes equation. They are solved in the conservative form for generalized curvilinear coordinates

\frac{1}{J} \frac{\partial Q}{\partial t} = - [\frac{\partial}{\partial ξ} (\frac{F_{i} - F_{v}}{J}) + \frac{\partial}{\partial η} (\frac{G_{i} - G_{v}}{J}) + \frac{\partial}{\partial ζ} (\frac{H_{i} - H_{v}}{J})]

(1)

where t is time; ξ, η, and ζ are the generalized curvilinear coordinates of the computational space; and J is the determinant of the Jacobian of the coordinate transformation from the Cartesian physical space to computational space. The vector of the conservative variables is Q, whereas F, G, and H are the flux vectors in the generalized coordinates with subscripts i and v used to differentiate the inviscid and viscous fluxes, respectively. The implicit LES approach is used without any explicit subgrid scale modeling. It is assumed that the grid and the numerical dissipation emulate the subgrid scale effect.⁵⁷

The nondimensional quantities are defined as follows

ρ = \frac{ρ^{*}}{ρ_{r}^{*}}, u_{i} = \frac{u_{i}^{*}}{U_{r}^{*}}, p = \frac{p^{*}}{ρ_{r}^{*} U_{r}^{* 2}}, t = \frac{t^{*} U_{r}^{*}}{L_{r}^{*}}, x_{i} = \frac{x_{i}^{*}}{L_{r}^{*}}, T = \frac{T^{*}}{T_{r}^{*}}

(2)

A more complete description of the governing equations can be found in Martha.⁵⁸

The sixth-order compact finite differencing scheme of Lele⁵⁹ is used to calculate the spatial derivatives at interior points of the computational grid. Namely

α f {i - 1}_{}^{'} + f i_{}^{'} + α f {i + 1}_{}^{'} = a \frac{f_{i + 1} - f_{i - 1}}{2 h} + b \frac{f_{i + 2} - f_{i - 2}}{4 h}

(3)

with

α = 1 / 3

, a = 14/9, b = 1/9, and h being the uniform grid spacing in the computational coordinates. The scheme order is reduced to fourth order and then to third order using one-sided differencing for the first grid point away from the boundary and on the boundary, respectively. This is to avoid extending the computational stencil outside the domain.

The unresolved scales, the mesh nonuniformities, and the numerical schemes for the boundary conditions can create high wavenumber instabilities that can be suppressed using spatial filters.⁶⁰ Here the implicit tridiagonal central filtering schemes by Gaitonde and Visbal⁶⁰ are used for the internal points

α_{f} {\overset{⌣}{f}}_{i - 1} + {\overset{⌣}{f}}_{i} + α_{f} {\overset{⌣}{f}}_{i + 1} = \sum_{n = 0}^{N} \frac{a_{n}}{2} (f_{i + n} + f_{i - n})

(4)

where the coefficients are provided in Gaitonde and Visbal.⁶⁰ The sixth-order one-side-biased filters are applied on the points close to the boundaries.⁶⁰ The boundary points are left unfiltered. For a more detailed description see Martha,⁵⁸ Aikens,⁶¹ and Dhamankar.⁶²

The code can also be used to study supersonic jets thanks to the use of a fifth-order weighted essentially nonoscillatory (WENO) characteristic filter and a shock detector.^63,64 Indeed the instabilities caused by the presence of shock cells are eliminated first by detecting the shock region, then modifying the order of the compact filters in those regions and applying the characteristic filter. The filter is typically applied two times at the end of each time step to remove oscillations generated by standing shocks such as Mach disks.^11,61,63 The WENO filter formulation from Kim and Kwon⁶⁵ is used and the shock detector is described by Ducros et al⁶⁶ (see Aikens⁶¹ for details).

In order to solve the large tridiagonal systems of equations arising from the filtering process, a truncated SPIKE algorithm by Polizzi and Sameh⁴² is used. The parallelization of the code is based on a superblock-based approach. A detailed description can be found in Martha.⁵⁸

The solver can handle domains with and without the nozzle region. The cases involving a round nozzle geometry are based on the use of cylindrical grids. In order to avoid the centerline singularity, the centerline treatment proposed by Mohseni and Colonius⁶⁷ is used along the radial direction. The point skipping method proposed by Bogey et al.⁶⁸ is used along the azimuthal direction to avoid the time step restriction due to the small spacing, in that direction, near the centerline.

The time advancement is based on the standard fourth-order Runge–Kutta scheme. The solver can use different boundary conditions, the most relevant are characteristic-based inflow and outflow boundary conditions,⁶⁹ different viscous wall models,¹¹ a digital filter-based turbulent inflow boundary condition,⁴³ and a far-field radiation boundary condition.^70,71

The code includes also the IBM.⁴⁴ With this method complex solid bodies are immersed in a nonbody conforming background grid and the effect of the body on the surrounding flow is indirectly imposed. This method allows one to study different complex design shapes using the same body-conforming grid as the one used for the baseline case. Thus, it is much easier to compare different designs since the grid used is the same.

The latest feature of the code, implemented as part of this work, is the simulation of air injectors inside the nozzle. This first implementation applies a constant flow through the injector locations based on the data extracted from the RANS simulation. The injectors are not simulated entirely in the LES, only the injector exits (the hole on the nozzle internal wall) are replicated by applying the values extracted at these locations from the RANS. The injectors are first located on the grid of the nozzle internal wall. There, the injectors are imposed as simple constant supersonic inflow boundary conditions. In the present analysis, the wall model feature is used in order to reduce the number of grid points. Hence, for this case the wall model is turned off at the points where the inflow condition for the injectors is imposed.

In order to handle a multifaceted nozzle geometry, an additional modification is introduced to the inflow boundary condition to adapt the Reynolds stresses distribution on the new geometry. The radial distribution is modified in order to find the wall location at each azimuthal angle and then estimate the stresses. It has been analyzed that the error introduced following a radial line, not perfectly normal to the wall, with respect to using the normal direction from the wall is negligible (order of $10^{- 11} %$ ).

The acoustic analysis is performed through the porous FWH method for zero free-stream velocity. The results are then enhanced using the method introduced by Ikeda et al.^72,73 for the end-cap treatment. The idea is to include the quadrupole effect, on the end-cap, through an approximate quadrupole surface integral based on a frozen turbulence assumption. The details of the implementation of these methods can be found in Martha⁵⁸ and Aikens.⁶¹

Simulations setup

The present analysis is based on the experimental setup used at the Pennsylvania State University by Powers et al.⁵⁰ In the experiment a comparison between a baseline nozzle without injectors and a nozzle with injectors was performed. The unheated case was chosen because experimental flow-field measurements are available for comparison, in order to understand if our numerical simulations were reproducing properly the physics of the jet.

Figure 1 shows a sketch of the two geometries. The injectors are located to blow air inside the divergent section of the nozzle. Two injectors are placed one after the other along the same azimuthal location forming what we name as a line of injectors. In total there are three pairs of injectors equally spaced in the azimuthal direction, namely every $120 °$ .

The simulations are performed on a round and a multifaceted nozzle. The multifaceted geometry follows the original geometry used in the experiments and is defined by a sequence of long and short straight edges. The round nozzle is a simplification of the original geometry and an equivalent nozzle exit diameter is used to maintain the same exit area as the multifaceted configuration, in order to have the same nozzle area ratio.

The LES setup corresponds to the operating conditions used by Powers et al.,⁵⁰ in order to make a direct comparison of numerical and experimental results. These conditions are reported in Table 1, where the NPR defines the ratio between the total pressure at the nozzle inlet and the ambient total pressure; the injector pressure ratio is the ratio between the total pressure at the injector inlet and the ambient total pressure; the TTR is the ratio between the total temperatures at the nozzle, or the injector, inlet and the ambient. Table 2 gives a clear picture of the different simulations analyzed in the present work.

Table 1.

Simulation parameters.

Parameter	Value
Design Mach number	1.65
Equivalent nozzle exit diameter	$22.47 mm$
Nozzle pressure ratio (NPR)	3
Total temperature ratio (TTR)	1 (unheated jet)
Injector/nozzle diameter ratio	0.059
Injector pressure ratio 1 (IPR1)	2.4
Injector pressure ratio 2 (IPR2)	3.7

Table 2.

Numerical simulations.

Case	Nozzle geometry	Mesh	Characteristic filter
Baseline	Multifaceted	Coarse	Two times
Baseline	Round	Coarse	Two times
Baseline	Round	Coarse	Four times
Baseline	Round	Fine	Four times
Fluidic injection	Round	Coarse	Two times

A set of RANS simulations has been performed first to gather information and get a better understanding of the physics of the problem within a reasonable turnaround time. More details about the RANS simulations can be found in Coderoni et al.⁵⁶ The RANS results are used to setup the LES grid, determining the boundary layer thickness and the point clustering needed in the viscous region, and the mean value of the flow variables for the inflow boundary conditions of the nozzle and injectors.

LES—Grid

In the present analysis we perform the LES for the baseline case on three different grids, a fine and a coarse grid for the round nozzle geometry and a coarse grid for the multifaceted geometry. The analysis for the nozzle with fluidic injection is limited to the coarse grid for the round nozzle (see Table 2). Each grid for the LES is made of five, so-called, superblocks combined together that are not overlapping. As mentioned in the “Numerical techniques” section, the grid does not have points along the centerline in order to avoid the geometric singularity. The grids used for the baseline and the fluidic injection cases are the same and were generated using the software Pointwise. The LES grid is clustered in the boundary layer region and along the shear layer region. Inside the nozzle a wall model is used, hence a $y^{+} > 30$ is required. This helps reduce the final number of points for the grid and most important allows us to use larger time steps. From the RANS results the boundary layer thickness at the inlet of the LES grid is estimated. The boundary layer thickness along the wall normal direction is $δ_{99} = 0.189 R_{j}$ (where R_j is nozzle exit radius). A value of $δ_{99} = 0.255 R_{j}$ along the vertical direction is used in the code to take into account the angle that the inlet plane has with respect to the converging section wall.

For the fine grid, the spacing along the radial direction, near the wall, of the fine grid for the round nozzle is $0.0040 R_{j}$ ( $y^{+} = 51$ ) in order to have an adequate trade-off between the resolution in the boundary layer and the computational cost. The first 26 points have equal spacing along the radial direction to be sure that at least 19 points are in the log region and more than 40 points are placed inside the boundary layer. Once the radial spacing is set the axial and the azimuthal spacing are defined. Aikens⁶¹ suggested using cells with an aspect ratio less than 3.5 inside the boundary layer, to avoid instabilities. Hence, maximum spacings of $0.0086 R_{j}$ and $0.0120 R_{j}$ are used along the axial and azimuthal directions, respectively, inside the boundary layer. Each superblock has 512 points equally spaced along the azimuthal direction. The grid generated is kept smooth by limiting the length ratio along the i and j directions (namely the axial and radial directions). Furthermore, the aspect ratios of the cells far from the nozzle are limited by using an appropriate number of points and adjusting the point distribution without reducing the smoothness of the grid. The average grid stretching is about 0.7% in the axial direction and 1.5% in the radial direction. The spacing is uniform along the azimuthal direction. The value of the average aspect ratio is about 4. The higher values are limited to regions far from the nozzle and the mixing region. The overall number of points in the three-dimensional grid is about 130 million.

The coarse grid for the round nozzle is the same used in Coderoni et al.⁵⁶ It is extracted from the 130 million point grid and coarsened along the three directions using the same coarsening factor to obtain a total of 21 million points. In this case, the spacing along the radial direction, near the wall, is chosen to be $0.0070 R_{j}$ ( $y^{+} = 90$ ). The first 16 points near the wall have equal spacing along the radial direction to be sure that at least 10 points are in the log region and a total of 25 points are placed inside the boundary layer. In order to avoid instabilities, maximum spacings of $0.0160 R_{j}$ and $0.0220 R_{j}$ are used along the axial and azimuthal directions, respectively, inside the boundary layer. Each superblock has 288 points equally spaced along the azimuthal direction. The average grid stretching is about 1% in the axial direction and 3% in the radial direction. The spacing is uniform along the azimuthal direction.

In Figure 2 is shown a slice of the 3D grids along the azimuthal position $φ = 0 °$ . The grids’ dimensions are nondimensionalized with respect to the jet exit radius. The top half shows the coarse grid while the fine grid is shown in the bottom half.

The third grid used is the one for the multifaceted nozzle geometry, i.e. the original geometry used in the experiment. The nozzle geometry is a sequence of short and long straight sides. The grid is based on the coarse grid for the round geometry; only the diverging section of the nozzle and a small region right downstream of the nozzle are modified to follow the new geometry. The properties of the grid are the same as the coarse grid for the round nozzle. Figure 3 shows every other point of the three grids at the nozzle exit plane.

LES—Settings

The LESs are performed using the code described in the “Numerical techniques” section. The simulation parameters were obtained from the experiment and RANS settings and were then adapted to the inputs required for the LES code. The reference values are obtained using the isentropic relations based on the design Mach number. The simulation parameters are defined in Table 3. Here p_j and T_j are, respectively, the reference jet exit pressure and temperature. The settings for the coarse grids for round and multifaceted geometries are the same. Also, for all the cases $Δ t$ has been chosen in order to have a CFL number less than 1. In order to achieve that, we use five coarsening levels in the azimuthal point skipping method⁶⁸ for the fine grid and four coarsening levels for the coarse grid.

Table 3.

LES—simulation parameters.

Parameter	Magnitude
Design Mach number	1.65
Reynolds number (based on jet radius)	361,948
$p_{j} / p_{\infty}$	0.655184
$T_{j} / T_{\infty}$	0.6474587
T_j	$192.99 K$
$Δ t^{*}$ (coarse grid)	$4 \times 10^{- 3} R_{j}^{} / U_{j}^{}$
$Δ t^{*}$ (fine grid)	$2.5 \times 10^{- 3} R_{j}^{} / U_{j}^{}$

LES: large eddy simulation.

The boundary conditions applied for the LES are described in Table 4. The injectors are applied only for the fluidic injection case and they basically turn off the wall model BC at those specific points on the nozzle wall. Before the end of the domain a sponge zone is applied to have a smooth variation of the flow before it crosses the outflow boundary. The grid is interrupted along the azimuthal direction; this means that the first and last points of the grid are not directly connected. Here, a periodic boundary condition is applied at the two boundaries. The grid boundary that surrounds the centerline does not work as a common boundary. This is because the information is shared across the centerline following the centerline treatment (see the “Numerical techniques” section).

Table 4.

LES—boundary conditions.

Location	Boundary conditions
Nozzle inlet	Digital filter generated turbulent inflow
Domain end	Tam and Dong outflow
Nozzle internal wall	Wall model
Nozzle external wall	No-slip isothermal hard wall (resolved)
Outer lateral boundaries	Dong radiation
Injectors exit	Steady inflow

LES: large eddy simulation.

LES—Acoustics settings

The acoustic analysis of the baseline jet and the jet with fluidic injection is done using the FWH method (see the “Numerical techniques” section). Four acoustic data surfaces (ADSs) are used in the present analysis as shown in Figure 4. Surfaces number one and four follow the grid shape along the sides surrounding the jet, with one being tight and four being loose. The other two surfaces (numbers two and three) are telescopic surfaces and fall in between the other two surfaces. All the surfaces have an end-cap located near the beginning of the sponge zone region. In the present paper we will show only the results for the tightest surface since, as also shown in our previous analysis,⁵⁶ it gives the best results of the four surfaces. This particular surface has an initial radius of about $2 R_{j}$ and a final radius of about $7.5 R_{j}$ .

The acoustic data are sampled on the ADS every 25 iterations while using the coarse grids and every 40 iterations while using the finer grid. This means that the nondimensional time between each sample is $Δ t_{s} = Δ t_{s}^{*} U_{j}^{*} / R_{j}^{*} = 0.1$ . Hence, the maximum Strouhal number achievable, through time sampling, is

S t_{\max} = \frac{1}{Δ t_{s}} = 10

(5)

It is important to note that the Strouhal number is based on the jet diameter, while the parameters of the LES are nondimensionalized based on the jet radius.

The data are collected for a nondimensional period of $T = T^{*} U_{j}^{*} / R_{j}^{*} = 1340$ . Therefore, the minimum frequency that can be solved is

S t_{\min} = \frac{2}{T} ≃ 0.0015

(6)

In his study Mendez et al.⁷⁴ suggested that, in order to have adequate resolution of a specific frequency, at least 10 waves must be present in the record. Following this direction we assume that the minimum well-resolved Strouhal number is 0.015. It must be noted that once the acoustic data are collected the pressure history length is shorter than the original record on the ADS. This is because the sound waves have to cover the distance from the ADS to the observer locations. The effective well-resolved frequency is then slightly larger than the one estimated (St = 0.016). Furthermore, there are the cutoff frequencies due to the grid resolution as reported in Table 5. The values are extracted using the line-of-sight method described in Aikens.⁶¹ This method is based on the estimation of the cutoff frequency depending on the grid spacing near the ADS along the direction of sound propagation to the observer. Only the value for the tightest surfaces is shown, since it has been found that it gives the best acoustics results (see also Coderoni et al.⁵⁶).

Table 5.

FWH—grid frequency cutoff. The results are expressed in terms of Strouhal number for the sound propagation directions at $30 °, 60 °, 90 °$ , and $120 °$ .

Grid	Cutoff St
	$30 °$	$60 °$	$90 °$	$120 °$
Coarse	1.02	2.54	3.87	2.38
Fine	1.91	4.72	7.27	4.44

FWH: Ffowcs Williams-Hawkings method.

In order to collect all the data required for the acoustic analysis, the simulations performed several hundreds of thousands of iterations. In order to give an order of magnitude estimate of the simulation costs, examples of the total time required to perform the simulations are now discussed. Assuming the simulations run continuously $24 h/day$ , the simulations on the coarse grid would take about 14 days using 96 cores on the Rice cluster at Purdue University (that uses 10-core Intel Xeon E5 processors), while the simulation on the fine grid would take 30 days using 432 cores on the Stampede cluster at the Texas Advanced Computing Center (that uses 8-core Intel Xeon E5-2680 processors). No differences in the simulation speed were seen while moving from the baseline to the fluidic injection case and from the round to the multifaceted nozzle.

Results

In this section we present the LES results and analyze the effects of the fluidic injection on the shock cells and on the potential core length as well as the resulting noise. All the results shown are validated and compared to experimental and numerical results available in Morris et al.,⁴⁷ Powers et al.,⁵⁰ Kapusta,⁷⁵ and Dr R. W. Powers (2017, personal communication).

The LES data were sampled on the ADSs in order to calculate the acoustics in the far field using the FWH method, as described in the “Numerical techniques” section and the “LES—Acoustics settings” subsection. The spectra calculated are compared with the (lossless) acoustics results obtained by the Pennsylvania State University group. The observer is located at 100 diameters from the nozzle exit. The LES results are recorded on 13,400 samples and processed using a fast Fourier transform using windows of 1024 samples each, so that the resulting spectra are averaged over 12 records. Moreover, the spectra are averaged azimuthally over three locations, $120 °$ distant from each other. It has been chosen to perform an averaging only over three positions in order to have numerical results consistent for comparison with the fluidic injection case, even if the spectra appear to be noisy at high frequencies. The uncertainty in the experimental results is about $0.8 dB$ .

Baseline case

The analysis of the baseline case is focused on three major parts. The first part of is to verify that the differences between round and multifaceted geometries are limited. The second is to explore the effect of the characteristic filter on the solution for the round geometry. The last part is to analyze the effect of the grid on the solution. In order to perform the simulation on the finer grid we had to apply the characteristic filter four times, instead of two times as is usually done^11,61,63 for cases with standing shocks. This was done to suppress the oscillations arising from a Mach disk inside the nozzle. For that reason, in the second part of the analysis we compare the effect of applying the characteristic filter two and four times using the coarse grid for the round geometry. Table 2 shows the details of the different simulations.

Round jet on coarse grid

The nondimensional axial velocity profiles (Figure 5(a)) and the mean axial velocity along the centerline (Figure 5(b)) show good agreement between LES and experimental results. The major differences are in the shear layer region. The coarse grid simulations show a higher flow entrainment due to higher velocities than what the experiments show. The major differences along the radial location $r / D_{n o z} = 0.4$ are due to the position of the shocks right outside the nozzle. A small variation of the shock positions along the axial location can cause a noticeable difference in terms of velocity. Along the centerline the results are close to the experimental data, but the shock cell locations appear to be shifted toward the left.

Figure 5(c) shows a shear layer thicker than the experimental one. The spreading and magnitude of the root mean square (rms) velocity is higher than what the experiments show in the outer layer that surrounds the jet, and lower inside the potential core. The experimental results have a steep increase of the rms velocity along the centerline before the end of the potential core, while the increase is delayed in the LES results (see Figure 5(d)). From Dr R. W. Powers (2017, personal communication), it has been found that this can be attributed to what the group at the Pennsylvania State University calls “spatial turbulence.” If the probe volume length is not negligible compared to the jet diameter, the probe records a range of velocities that vary within the probe volume that results in an rms velocity in time even though it is only a spatial variation. The effect is that the probe shows higher values of rms velocity. Moreover, the experimental results could be affected by a nonperfect alignment along the centerline. In addition, the experiment may have a more turbulent flow entering the nozzle than what was simulated by the LES inflow boundary condition. The flow properties within the nozzle were not measured and the LES used mean and rms profiles computed from a RANS solution.

It is interesting to note that at the plane $x / D_{n o z} = 0.2$ the LES gives an explanation of the experimental point at the radial location $r / D_{n o z} = 0.2$ that originally appears to be “an outlier.” Indeed, it has been found that, at that particular location, there is an oblique shock wave that has a higher turbulence content than the flow surrounding it.

The shadowgraph comparison in Figure 6 (part 2) shows that the number of shock cells forming after the nozzle exit is in agreement with the experiment. Further downstream, the numerical results have shock cells shifted toward the left with respect to the experimental results. It must be noted that the experimental results are for a lower Reynolds number since the shadowgraph imaging was performed on a nozzle with $18.00 mm$ diameter (instead of $22.47 mm$ ) under the same operating conditions as the other experimental and numerical results presented here. A variation of about 20% of the Reynolds number is expected to have a negligible effect on the jet development since the Reynolds number is on the order of $O (10^{5})$ .³⁴

The acoustics results are presented in Figures 7 and 8 that show, respectively, the spectra for four different observer angles, namely $30 °, 60 °, 90 °$ and $120 °$ , and the overall sound pressure level (OASPL). The computational acoustics results of the $21 M$ point mesh for the round geometry with two characteristic filter applications are represented by the red line in the figures. They are in good agreement with the experimental results in the range of low–mid frequencies, while the discrepancies at the high frequencies are large (see Figure 7(c) and (d)). Experimental analysis^19,76 shows that for a nonperfectly expanded nozzle the presence of shock cells generates so-called broadband shock-associated noise (BBSAN). This noise mechanism is generated by the interaction of the shear layer with the shock cell. The BBSAN increases the OASPL (with respect to a shock-less jet) along all directions but predominantly in the range of high angles (i.e. above $80 °$ ). Also, additional experimental evidence^20,21 shows that the higher frequencies are dominant at high angles because of the sound generated by the smaller turbulent scales. Hence, the differences between the numerical and experimental results for OASPL and spectra at high angles are connected to the high frequencies and thus to the small turbulent scales and their interaction with the shock cells. By definition the LES resolves the larger scales and models the smaller scales of turbulence. Therefore, it is expected that there would be differences in the acoustic results for the high frequency range. Indeed, the LESs do not capture properly the pressure levels at high frequencies, since they do not adequately resolve the smaller turbulent scales. The grid limits the size of turbulent scales that can be resolved and adds numerical dissipation that dissipates the smallest resolved scales. The discrepancy at high frequency is also caused by additional phenomena. The screech tones can generate a self-excited jet. Also the screech tones seen in the experiment creates a cascade of harmonics that increase locally the sound pressure levels (SPLs) of the surrounding frequencies resulting in a broadband amplification at high frequencies. The screech tones and their effect on the spectra are not captured in the LES. The prediction of screech tones is a difficult problem because of their sensitivity to small details of the experimental setup^77,78 that may not be reproduced in the numerical approach. This, combined with a relatively coarse grid, may cause the simulation not to capture the screech tones.^79–81 Moreover, the location and the intensity of the shocks is sensitive to the grid fineness and affects the BBSAN levels. Hence, the OASPL difference between the numerical and experimental results at large angles is expected. In this range the maximum difference is about $3 dB$ . The region where the BBSAN peak is located is in good agreement with the experimental results. It must be noted that the experimental results for the OASPL do not take into account the screech tones. Looking at Figure 8 it can be seen that the LES shows a peak with maximum OASPL near $25 °$ . This is in agreement with the general trend for jet noise. Indeed, the large turbulent scales that convect downstream are associated with the typical emission angle of about $30 °$ . This has been found also to be the direction with the highest sound intensity levels for the LES results and it is confirmed by the spectra shown in Figure 7(a). Furthermore, a second peak can be seen at about $55 °$ , while the experimental results show a small secondary peak at about $45 °$ . This phenomenon is due to the generation of noise from the interaction between the turbulent shear layer exiting the nozzle and the shock cells downstream. Indeed, in their study Yu and Dosanjh,⁷⁶ comparing a perfectly expanded jet and an underexpanded jet, show that the presence of diamond cells may generate a small peak of the OASPL around $60 °$ . In this range of angles, the BBSAN levels from the LES appear to be slightly higher than the experimental results (e.g. see Figure 7(b)).

Overall the LES results appear to be in fairly good agreement with the experimental results. The spectra are well represented and reflect the behavior shown in the experiment except at high frequencies above the BBSAN peak for high observer angles ( $90 °, 120 °$ ). It must be noted that the maximum difference between the numerical and experimental OASPL is about $3 dB$ . Moreover, the analysis performed on other jets, using the present LES code, shows a similar trend, namely a lower level of OASPL compared to the experimental results.^11,15

Round jet and multifaceted jet

The first part of the analysis is the comparison of the results between the round and the multifaceted geometry. An interesting result is shown in Figure 9 where the Q-criterion inside the nozzle is shown for the two different geometries. The turbulent structures’ development is initially different. The multifaceted geometry shows that the turbulent structures cluster along the short edges of the nozzle, while the round nozzle has more evenly distributed turbulent structures along the azimuthal direction. At the nozzle exit, the eddies’ azimuthal distribution is similar between the two geometries, hence no strong differences should be expected in the jet development for the two cases.

In Figure 5 the comparisons of the LES and experimental mean flow and turbulence statistics are shown. It can be seen that, comparing the red and orange(online; solid thin and long dashed in print) lines, the differences between the round and multifaceted geometries are very limited in terms of axial velocity and rms velocity, as shown also in our previous analysis using RANS.⁵⁶ The major differences can be found along the centerline (see Figure 5(b) and (d)). Indeed, the locations of the shock cells are slightly different. The multifaceted geometry has shocks that are shifted in the upstream direction with respect to the results for the round geometry.

The acoustics results presented in Figures 7 and 8 allow us to complete the comparison of the two geometries. It can be seen that the multifaceted geometry shows similar acoustic levels at medium and high frequencies with respect to the round nozzle, while the lower frequencies show a lower SPL. This turns into having a jet be slightly quieter. The maximum variation of OASPL is about $2 dB$ only at $10 °$ and immediately drops to $1 dB$ for slightly larger angles, where the lower frequencies are dominant. The difference in OASPL, then, reduces to less than $1 dB$ as we move to larger observer angles. Since the differences between the two geometries are limited, the rest of the analysis is performed using only the round geometry.

Effect of the characteristic filter

The second part of the present study is based on the analysis of the effect of applying the characteristic filter two or four times every time step. To assess the effect of this approach we compare the results obtained using the coarser grid.

Applying the filter four times instead of two appears to add a dissipative effect to the calculations. It can be seen in Figure 5(b), where some of the fine-scale structure oscillations are smoothened out in the axial range $x / D_{n o z} \in [0, 2]$ . This also affects the shock cells downstream shifting them more upstream with respect to the case with the filter applied two times. This can also be seen in the shadowgraph comparison in Figure 6.

The axial rms velocity (see Figure 5(c) and (d)) shows that applying the filter four times slightly decreases the turbulence levels exiting the nozzle. The two simulations have a potential core length of about $8 D_{n o z}$ .

The acoustics results in Figures 7 and 8 show, as expected, that applying the characteristic filter more times adds dissipation to the solution, resulting in lower sound levels. The effect is spread over all the frequencies as shown by the spectra. At small observer angles the lower frequency part of the spectra are reduced, while at large observer angles the high frequencies are reduced by the two additional applications of the filter. This generates a distorted OASPL graph that is reduced about $2 dB$ along most of the observer angles with respect to the case with the characteristic filter applied two times. Between $50 °$ and $70 °$ OASPL is slightly increased. This is connected to a louder BBSAN peak region (see Figure 7(b)). This could be related to the shock cell locations that are shifted upstream with respect to the case with the characteristic filter applied two times.

Effect of the grid

The last part of the analysis is focused on the effect of the grid on the solution. The nondimensional axial velocity (Figure 5(a)) shows that the fine grid results follow the experimental trend more closely. The profiles for the fine grid in the outer region of the shear layer show a lower entrainment of the outer flow compared to the coarser grid results. In this region the level of agreement with the experimental results is increased. It seems (from Figure 5(b)) that the fine grid results follow more closely the experimental results toward the end of the potential core. The length of the potential core is longer (about 9 diameters) than in the results with the coarse grid. Also the shock cells are shifted further downstream with respect to the coarse grid results and show more details, along the axial range $x / D_{n o z} \in [0, 2]$ , as the grid refinement counteracts the dissipative effects of applying the characteristic filter four times (see Figure 6).

The axial rms velocity levels get lower as the grid is refined (see Figure 5(c) and (d)). The results are more in agreement with experimental results mostly in the outer shear layer region, as seen in the nondimensional axial velocity profiles. On the other hand, along the centerline the results have a reduced intensity for the rms velocity.

This set of findings shows that the fine grid case results in a jet with a shear layer that grows slower than the coarse grid cases. This leads to a longer potential core and lower jet decay and spread rate.

Figure 10 shows the contours of the Mach number and the TKE along the z = 0 plane and compare the LES results for the round geometry. The Mach number contours show that the magnitude is extremely close between the simulations and the major difference is in the shift of the shock cell locations, as seen earlier in Figure 6. Indeed, the LES on the fine grid shows shock cells that are located further downstream than the results for the coarse grid simulations and are more in agreement with the experimental shadowgraph. Furthermore, the LESs on a coarse grid do not predict the formation of a small Mach disk inside the nozzle. The TKE contours show that the LES solutions on a coarse grid have higher levels of turbulence exiting the nozzle, as well as a thicker layer of turbulence in the shear layer than the case with the fine grid.

The acoustics results in Figures 7 and 8 show, as described before, that applying the characteristic filter four times produces lower sound levels than applying it two times. The refinement of the grid appears to mitigate the filter effect at high frequencies and at the BBSAN peak locations. Indeed, using a finer grid improves the resolution of the smallest turbulent scales and so the quality of the spectra at high frequencies. This can be seen in Figure 7 especially for the observer angle $120 °$ , where the effect of the higher frequencies is dominant. The spectra peak connected to the BBSAN show higher levels for $30 °, 90 °$ , and $120 °$ , but lower levels at $60 °$ , with respect to the results shown for the coarse grid case with the characteristic filter applied four times. The solution in these ranges gets slightly closer to the experimental results thanks to the refinement of the grid because the position and geometry of the shock cells is improved. The lower frequencies have lower spectra levels for most of the angles. This can be connected to the lower turbulence levels in the shear layer that generate weaker large scales that convect downstream with respect to the coarser grid cases. The grid is still not fine enough to properly capture the screech tones shown in the experimental results. Looking at the OASPL in Figure 8, comparing the green and blue(online; dashed-dot and dotted in print) lines, the effect of refining the grid can be clearly seen. At observer angles smaller than $30 °$ or higher than $90 °$ the finer grid shows higher OASPL than the coarse grid with the same characteristic filter settings. The effect is limited, less than $1 dB$ , for shallow angles while it is larger, up to $2 dB$ , at wider angles. Therefore, it can be linked to a better resolution of the smaller turbulent scales and their contribution to the high-frequency noise. For the observer angles from $30 °$ to $90 °$ the finer grid has reduced OASPL levels by about $1 dB$ . Even if the results appear to become more distant from the experimental ones, the OASPL trend shows more realistic features. The main peak located at $60 °$ is now reduced almost at the same level as the peak at about $25 °$ , namely the physical direction of maximum sound propagation. This happens because the shock cell locations and geometry are modified. Therefore, to summarize: the excessive filtering shifts the shock cells upstream increasing the SPL levels of the BBSAN peak and so the OASPL between $30 °$ and $90 °$ ; grid refinement shifts the shock cells downstream, changing the BBSAN to levels closer to the experimental ones and reducing the OASPL along that range of observer angles.

Round jet with fluidic injection

In this section we focus on the analysis of the LES results of the nozzle with injectors. The experimental results are shown along two azimuthal planes: one aligned with the injectors ( $φ = 0 °$ ) and one in between two lines of injectors ( $φ = 60 °$ ). For sake of brevity, only the comparison of the numerical and experimental results along the plane aligned with the injectors is shown on different x-planes. The numerical results are averaged along three azimuthal positions.

The nondimensional axial velocity profiles (Figure 11(a)) show that the LES follows the trend of the experimental results. The numerical results underpredict the axial velocity near the nozzle exit and overpredict it as one moves downstream.

Figure 11(b) shows the mean axial velocity along the centerline. The LES seems in rough agreement with the experimental results for $φ = 60 °$ , up to $x / D_{n o z} = 2.5$ ; however, in our LES the magnitude of the oscillations of the velocity is much larger than in the experiments. Also, it should be pointed out that the experimental velocity measurements are available at fewer points, thus some of the peaks may have been missed. The potential core is longer compared to the experimental results. The discrepancy between the experimental results at different azimuthal orientations of the nozzle indicates an imperfect alignment of the probe on the centerline. This error leads to a scattering in the experimental data, so that it is hard to tell the overall level of agreement of the LES with the experiment.

Figure 11(c) shows the scaled rms velocity profiles. The LES results have high rms velocity levels exiting the nozzle downstream of the injectors compared to the experimental results. Similar to what was seen for the rms velocity of the baseline case, the position of the shocks near the nozzle exit, located at the axial location $x / D_{n o z} = 0.2$ , cause a small peak of axial rms velocity at the radial locations $r / D_{n o z} = 0.15$ .

Figure 11(d) shows a similar trend to the baseline case for the scaled axial rms velocity along the centerline. The shock cells generate high peaks of scaled axial rms velocity in the region just downstream of the nozzle exit. The LES results are in good agreement with the experimental results for the plane in line with the injectors ( $φ = 0 °$ ) up to an axial location of about $x / D_{n o z} = 2$ , then the increase of the rms velocity deviates from that of the experiment and is delayed in the LES. This agrees with the longer potential core described before. The differences between the numerical and experimental results can be connected to the “spatial turbulence” effect due to the probe size, as seen in the baseline case, to the misalignment of the probe with respect to the centerline, and possible differences in the inflow conditions within the nozzle.

A set of contour plots is shown in order to better visualize the results in the flow field. The effect of the injectors on the distribution and geometry of the shock cells in the potential core and the wave propagation in the near field are shown by the contours of the velocity dilatation in Figure 12. The large shocks of the baseline nozzle are broken down into numerous smaller shocks with different orientations.

Figure 13 shows the Mach number and TKE contour plots. Comparing the contours with Figure 10, it can be seen that (looking at the Mach number contours) the shock structures break down into smaller shocks with reduced intensity. This effect on the region inside the nozzle and right outside it affects the intensity of the downstream shocks, which is reduced with respect to the baseline case. The TKE levels are in accordance with what is seen for the baseline case: high levels of TKE at the nozzle exit that then dissipate moving further downstream. Furthermore, the turbulent kinetic energy (TKE) is high past the two injectors, due to the mixing effect, and then decays faster than in the regions without injectors.

The LES data were sampled and processed in the same way as for the baseline case. The spectra are averaged azimuthally over three locations, $120 °$ distant from each other, and shown for two locations, the azimuthal planes at $φ = 0 °$ and $φ = 60 °$ . The results shown are obtained using the tighter ADS described in the “LES—Acoustics settings” subsection (i.e. what called surface one in Figure 4).

Figures 14 and 15 show the spectra on the two planes for four different observer angles, namely $30 °, 60 °, 90 °$ , and $120 °$ , while Figure 16(a) shows the corresponding OASPL. As done for the baseline case, the SPL and OASPL graphs are analyzed together in order to give a more complete description of the results. As with the baseline case, the spectra agree well at $30 °$ and $60 °$ observer angles, but show more discrepancies at $90 °$ and $120 °$ in the high frequency range. The reasons for the discrepancy have already been discussed in the analysis of the baseline case. In addition to that, the results at $φ = 60 °$ , for observers at $90 °$ and $120 °$ , show two peaks that could be connected to the shedding generated by the constant air injection. The numerical spectra show better agreement with the experimental results compared to the baseline case. This is because for the baseline case screech is present in the experiments but not captured in the computation, whereas for the injection case there is no screech in both the experiments and the LES. The presence of screech in the experimental baseline case could cause jet self-excitement and broadband amplification resulting in higher spectra differences in the mid to high frequencies for the baseline case. The LES results for the azimuthal plane $φ = 60 °$ appear to agree better with the experimental results than the ones for the azimuthal plane $φ = 0 °$ . At this location ( $φ = 0 °$ ), indeed, the spectra levels appear to be lower than the ones shown by the experimental spectra.

It must be noted that the benefit of using the injectors is to reduce the BBSAN. Comparing Figure 7(b) with Figures 14(b) and 15(b) it can be clearly seen that the injectors reduce the BBSAN peak at the azimuthal location $φ = 0 °$ and almost completely remove the BBSAN effect at $φ = 60 °$ . A reduction of the BBSAN due to the injectors is shown also in all the other spectra. Indeed the injectors break down the shock cells into smaller and weaker shocks altering their locations and orientations as well. Moreover the jet plume is deformed by the injectors. The overall effect is to create a nonaxisymmetric plume with weaker shock cells that interact differently with the shear layer, directly affecting the mechanisms that generate the BBSAN. This becomes an acoustic benefit due to a reduction of the BBSAN.

Looking at Figure 16(a) it can be seen that the LES, in the azimuthal plane $φ = 0 °$ , now shows a peak with maximum OASPL at $60 °$ , while the peak around $25 °$ has a lower value of OASPL. The direction of maximum sound propagation for the baseline case shows more than $1.5 dB$ of reduction (see Figure 16(b)). The experimental data show a similar behavior, but with the peak located at about $45 °$ instead of $60 °$ , analogous to what is seen comparing experimental and LES results in the baseline case. This means that the principal direction of the sound propagation is modified by the injectors, while the maximum value of the OASPL is reduced by about $1 dB$ . The OASPL levels are about $2 - 3 dB$ lower than the experimental results up to $90 °$ , then the difference increases up to $5 - 6 dB$ . This trend is similar to what is found in the baseline case comparing experimental and LES results, even though the difference for large observer angles was smaller for the baseline case.

In the azimuthal plane $φ = 60 °$ , the LES results show a peak around $25 ° - 30 °$ that is consistent with the experimental data for that location. It shows, also, another peak at about $70 °$ that is located where the experimental results show the lowest levels of OASPL. For the observer angles up to $90 °$ the LES results differ less than $1.5 dB$ with respect to the experimental data. Thus, the difference is similar to what is seen for the other azimuthal plane.

Since the OASPL for the baseline jet and the jet with injectors shows lower noise levels with respect to the experimental results, a useful comparison is to analyze the trend of ΔOASPL, i.e. the difference of the OASPL of the case with injectors versus the OASPL of the baseline case. Figure 16(b) shows a noise reduction where the ΔOASPL is negative and a noise increase where the variation has positive values. The LES results show a global reduction of noise for all observer locations, while the experiment depicts an increase of noise around $50 °$ in the azimuthal plane $φ = 0 °$ , due to additional high frequency phenomena caused by the shedding generated by the injectors. The LES results on this plane follow closely the experimental results up to $100 °$ with a difference for the ΔOASPL that is generally less than $1 dB$ , except around $50 °$ where the difference is less than $1.5 dB$ . Beyond $100 °$ the LES diverges from the experimental results. In the azimuthal plane $φ = 60 °$ the LES ΔOASPL differs less than $1 dB$ with respect to the experimental results except in the range of observer angles between $65 °$ and $90 °$ where the difference is less than $2 dB$ . The LES results show that, along the principal direction of sound propagation for the baseline nozzle ( $25 °$ ), the injectors can reduce the OASPL by more than $1.5 dB$ . The maximum noise reduction happens at about $55 °$ in the azimuthal plane $φ = 60 °$ , where the OASPL is reduced by almost $3 dB$ .

The last point of this analysis is the change in the thrust due to the injectors. Since the jet operates in overexpanded conditions the thrust equation is

T = \int_{A_{e}} ρ_{e} u_{e}^{2} d A + \int_{A_{e}} (p_{e} - p_{a}) d A

(7)

It has been found that using the injectors in this configuration reduces the jet thrust by about 0.7%. The specific thrust variation is also estimated dividing the thrust by the respective mass flow rate at the nozzle exit. The use of the injectors causes a reduction of specific thrust by about 2.4%. This is caused by the injectors that blow additional mass into the nozzle, increasing the mass flow rate by about 2% with respect to the baseline case.

Conclusions

The goal of the present research is to study computationally the noise reduction effectiveness of fluidic injectors for an unheated supersonic overexpanded jet. The LESs of the baseline nozzle and the nozzle with injectors are performed and validated with the results of experiments performed at the Pennsylvania State University.

The analysis is divided in different parts: a general comparison between experimental and numerical results; a comparison between multifaceted and round geometries; examination of the effect of the characteristic filter; investigation of the effect of grid refinement; evaluation of the noise reduction effectiveness of the fluidic injectors.

The LESs are performed using a high-order structured LES code developed at Purdue University. The code structure makes it very suitable for parallelization for Cartesian and cylindrical grid topologies. Radiation boundary conditions on the outer boundaries and a sponge zone at the end of the domain are used in order to avoid any spurious reflections of the acoustic waves. A wall model is used inside the nozzle in order to reduce the computational costs. The inflow boundary conditions for the jet and the injectors are based on the RANS results obtained previously. The jet inflow is combined with a digital filter approach in order to generate a turbulent inflow.

The acoustic results are obtained through the porous FWH method combined with the formulation introduced by Ikeda et al.,^72,73 for the end-cap treatment. Four ADSs are used. For the sake of brevity only the tightest surface is used to perform the acoustic analysis since it is found that it gives more accurate results.

The LES results of the flow field for the baseline case are in good agreement with experimental results, except for the rms velocity near the centerline, where the experimental results show higher values than the LES outcome. The trend of the SPL follows the experimental spectra well, without capturing the screech tones. However, at larger observer angles, the numerical results underestimate the levels of the high frequencies in the region past the BBSAN peak. For the nozzle with injectors, the LES results show less agreement with the experimental flow-field results, especially along the centerline, where the experimental outcome has a different trend depending upon the azimuthal orientation of the probe with the jet. On the other hand, the acoustic spectra for the fluidic injection case appear more in agreement with the experimental results than for the baseline case. This seems to be due to the elimination of screech tones when using injectors. Indeed, the jet self-excitement mechanism, which is not captured by the LES in the baseline case, is no longer present when the injectors are turned on, hence reducing the discrepancies between numerical and experimental acoustic results.

A comparison between a round axisymmetric nozzle and a multifaceted nozzle was performed to show that the two geometries produce similar flows and acoustic fields. This result was used to simplify the setup of the other LES (e.g. the use of a cylindrical grid).

The characteristic filter used for shock capturing is found to be dissipative and applying it more than two times affects the solution. The higher dissipation modifies the resolution of the shock cells, reduces the turbulence levels exiting the nozzle, and so affects the noise imprint of the jet. The OASPL, indeed, is reduced at low and high angles and the main direction of sound propagation is modified from about $30 °$ to about $60 °$ .

Refining the grid increases the resolution of the small turbulent scales and makes the locations of the shock cells more accurate. The quality of the results is still limited by the additional applications of the characteristic filter. The jet has a slower spreading, with a longer potential core and lower TKE levels, and makes less noise than the coarse grid cases. The higher grid resolution improves the acoustic levels mostly at large observer angles where the high frequencies are dominant.

The trend of noise reduction due to the use of the injectors is well captured. The results show that the injectors break down the shocks into smaller structures affecting the shape and the intensity of the shocks downstream of the nozzle exit. This causes a reduction of the interaction between the shear layer and the shock cells. Hence, the BBSAN is reduced with respect to the baseline nozzle case, so that a positive reduction of the OASPL at most of the observer locations is obtained. The LES results show a reduction by more than $1.5 dB$ along the principal direction of the sound propagation (i.e. about $25 °$ ). This induces a consequent change in the loudest angular location (i.e. about $60 °$ ) on the azimuthal plane in line with the injectors. The maximum noise reduction of about $3 dB$ happens in the azimuthal plane $φ = 60 °$ , in between the injectors, at observer angle of about $θ = 55 °$ . The thrust reduction due to activating the injectors is about 0.7%.

Fluidic injection appears to be a promising noise reduction design thanks to its effectiveness and to the possibility of turning on the injectors only when needed without affecting aircraft performance during other times. Our future work is focused on extending the analysis to the effect of fluidic injection on heated jets.

Footnotes

Acknowledgements

The authors wish to thank Nitin Dhamankar and Chandra Sekhar Martha of Intel Corp., Kurt Aikens of Houghton College, Ali Uzun of NASA Langley Research Center, and Yingchong Situ of Google for their prior implementation of the LES solver used for the simulations presented here as well as for answering several questions about the code. The support of Shanmukeswar Vankayala of Purdue University in order to understand the code, while implementing new features and setting up the simulations is greatly appreciated. A special thanks goes to Philip Morris and Ching-Wen Kuo of the Pennsylvania State University for providing the nozzle geometries used in the computational analysis. Fundamental was the help of Russell Powers of the U.S. Navy to better understand the experiment in order to make a fair comparison with the numerical results, as well as the acoustic data he provided us. The Rigel cluster was used on some of the presented simulations. The Rice and Conte clusters of the Rosen Center for Advanced Computing (RCAC) were utilized for some of the presented simulations. The Stampede cluster at the Texas Advanced Computing Center (TACC) was utilized under allocation TGASC040044N. The discussions with Luigi Morino, during the analysis of the simulations, have been valuable. The authors wish also to thank the Embry-Riddle Aeronautical University for the doctoral scholarship awarded to Marco Coderoni.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported in part through computational resources provided by Information Technology at Embry-Riddle Aeronautical University, Daytona Beach, Florida. The research was supported also through computational resources provided by Information Technology at Purdue (ITAP), West Lafayette, Indiana. Resources for the simulations with the finer grid were provided by the Extreme Science and Engineering Discovery Environment (XSEDE),⁸² which is supported by National Science Foundation grant number ACI-1548562.

Appendix

References

Doychak

Department of Navy Jet Noise Reduction Project overview. In: Partners in Environmental Technology Technical Symposium and Workshop, Washington, DC, 2010.

Lighthill

MJ.

Jet noise. AIAA J 1963; 1: 1507–1517.

Bridges

Brown

Parametric testing of chevrons on single flow hot jets. NASA technical memorandum 213107, 2004.

Viswanathan

Czech

MJ.

Adaptation of the beveled nozzle for high-speed jet noise reduction. AIAA J 2011; 49: 932–944.

Powers

Senft

McLaughlin

DK.

Acoustic measurements of scale models of military style supersonic beveled nozzle jets. In: 17th AIAA/CEAS aeroacoustics conference, Portland, Oregon, 5–8 June 2011, paper no. 2011-2702. AIAA.

Krothapalli

Greska

Arakeri

High speed jet noise reduction using microjets. In: 8th AIAA/CEAS aeroacoustics conference and exhibit, Breckenridge, Colorado, 17–19 June 2002, paper no. 2002-2450. AIAA.

Greska

Supersonic jet noise and its reduction using microjet injection. PhD Thesis, FAMUFSU College of Engineering, The Florida State University, Tallahassee, Florida, 2005.

Powers

Kuo

McLaughlin

DK.

Experimental comparison of supersonic jets exhausting from military style nozzles with interior corrugations and fluidic inserts. In: 19th AIAA/CEAS aeroacoustics conference, Berlin, Germany, 27–29 May 2013, paper no. 2013-2186. AIAA.

Eastwood

Tucker

PG.

Hybrid les-rans of complex geometry jets. Int J Aeroacoust 2011; 10: 659–684.

10.

Martha

Situ

Louis

et al. Development and application of an efficient, multiblock 3-D large eddy simulation tool for jet noise. In: 50th AIAA aerospace sciences meeting, Nashville, TN, 9–12 January 2012, paper no. 2012-0833. AIAA.

11.

Aikens

Blaisdell

Lyrintzis

AS.

Analysis of converging-diverging beveled nozzle jets using large eddy simulation with a wall model. In: 53rd AIAA aerospace sciences meeting, Kissimmee, Florida, 5–9 January 2015, paper no. 2015-0509. AIAA.

12.

Gonzlez

Gaitonde

Lewis

MJ.

Large-eddy simulations of plasma-based asymmetric control of supersonic round jets. Int J Comput Fluid Dyn 2015; 29: 240–256.

13.

Hafsteinsson

Eriksson

Andersson

et al . Noise control of supersonic jet with steady and flapping fluidic injection. AIAA J 2015; 53: 3251–3272.

14.

Goparaju

Gaitonde

DV.

Large-eddy simulation of plasma-based active control on imperfectly expanded jets. J Fluids Eng 2016; 138: 071101-1–071101-12.

15.

Dhamankar

Blaisdell

Lyrintzis

AS.

Analysis of turbulent jet flow and associated noise with round and chevron nozzles using large eddy simulation. In: 22nd AIAA/CEAS aeroacoustics conference, Lyon, France, 30 May–1 June 2016, paper no. 2016-3045. AIAA.

16.

Brès

Ham

Nichols

et al . Unstructured large-eddy simulations of supersonic jets. AIAA J 2017; 55: 1164–1184.

17.

Gojon

Bogey

Flow structure oscillations and tone production in underexpanded impinging round jets. AIAA J 2017; 55: 1792–1805.

18.

Gojon

Bogey

Flow features near plate impinged by ideally expanded and underexpanded round jets. AIAA J 2017; 56: 445–457.

19.

Tam

CKW.

Jet noise generated by large-scale coherent motion, aeroacoustics of flight vehicles: theory and practice. In: Hubbard HH (ed.) NASA Langley Research Center, Aeroacoustics of Flight Vehicles: Theory and Practice vol. 1, pp. 311–390

20.

Tam

CKW

Golebiowski

Seiner

JM.

On the two components of turbulent mixing noise from supersonic jets. In: 22nd AIAA/CEAS aeroacoustics conference, State College, PA, 6–8 May 1996, paper no. 96-1716. AIAA.

21.

Tam

CKW

Viswanathan

Ahuja

et al . The sources of jet noise: experimental evidence. J Fluid Mech 2008; 615: 253–292.

22.

Crighton

DG.

Acoustics as a branch of fluid mechanics. J Fluid Mech 1981; 106: 261–298.

23.

Zaman

KBMQ.

Far-field noise of a subsonic jet under controlled excitation. J Fluid Mech 1985; 152: 83–112.

24.

Zaman

KBMQ.

Flowfield and near and far soundfield of a subsonic jet. J Sound Vib 1986; 106: 1–16.

25.

Hussain

AKMF.

Coherent structure and turbulence. J Fluid Mech 1986; 173: 303–356.

26.

Mankbadi

Hayer

Povinelli

LA.

Structure of supersonic jet flow and its radiated sound. AIAA J 1994; 32: 897–906.

27.

Mankbadi

RR.

Review of computational aeroacoustics in propulsion systems. J Propul Power 1999; 15: 504–512.

28.

Lyrintzis

AS.

Surface integral acoustics methods: from the (CFD) near-field to the (acoustic) far-field. Int J Aeroacoust 2003; 2: 95–128.

29.

Lyrintzis

AS.

Review: the use of Kirchhoffs method in computational aeroacoustics. J Fluids Eng 1994; 116: 665–676.

30.

Ffowcs Williams

Hawkings

DL.

Sound generation by turbulence and surfaces in arbitrary motion. Phil Trans R Soc Lond Ser A Math Phys Sci 1969; 264: 321–342.

31.

di Francescantonio

A new boundary integral formulation for the prediction of sound radiation. J Sound Vib 1997; 202: 491–509.

32.

Brentner

Farassat

Analytical comparison of the acoustic analogy and Kirchhoff formulation for moving surfaces. AIAA J 1998; 36: 1379–1386.

33.

Bodony

Lele

SK.

Current status of jet noise predictions using large-eddy simulation. AIAA J 2008; 46: 364–380.

34.

Bogey

Bailly

Large eddy simulations of transitional round jets: influence of the Reynolds number on flow development and energy dissipation. Phys Fluids 2006; 18: 065101.

35.

Karabasov

Afsar

Hynes

et al . Jet noise: acoustic analogy informed by large eddy simulation. AIAA J 2010; 48: 1312–1325.

36.

Gao

Large eddy simulation of supersonic jet noise from a circular nozzle. Int J Aeroacoust 2011; 10: 465–474.

37.

Wang

McGuirk

Large eddy simulation of high speed nozzle flows – assessment and validation of synthetic turbulence inlet conditions. In: 20th AIAA computational fluid dynamics conference, fluid dynamics and co-located conferences, Honolulu, Hawaii, 27–30 June 2011, paper no. 2011-3555. AIAA.

38.

Wang

Mcguirk

JJ.

Large eddy simulation of supersonic jet plumes from rectangular con-di nozzles. Int J Heat Fluid Flow 2013; 43: 62.

39.

Mendez

Shoeybi

Sharma

et al . Large-eddy simulations of perfectly expanded supersonic jets using an unstructured solver. AIAA J 2012; 50: 1103–1118.

40.

Nichols

Lele

SK.

Large eddy simulation of crackle noise in hot supersonic jets. J Acoust Soc Am 2013; 134: 4128–4128.

41.

Situ

Martha

Louis

et al . Petascale large eddy simulation of jet engine noise based on the truncated spike algorithm. Parallel Comput 2014; 40: 496–511.

42.

Polizzi

Sameh

A parallel hybrid banded system solver: the spike algorithm. Parallel Comput 2006; 32: 177–194.

43.

Dhamankar

Martha

Situ

et al. Digital filter-based turbulent inflow generation for jet aeroacoustics on non-uniform structured grids. In: 52nd AIAA SciTech aerospace sciences meeting, National Harbor, Maryland, 13–17 January 2014, paper no. 2014-1401. AIAA.

44.

Dhamankar

Blaisdell

Lyrintzis

Implementation of a sharp immersed boundary method in a 3-D multi-block large eddy simulation tool for jet aeroacoustics. In: 53rd AIAA SciTech aerospace sciences meeting, Kissimmee, Florida, 5–9 January 2015, paper no. 2015-0504. AIAA.

45.

Dhamankar

Blaisdell

Lyrintzis

AS.

A simple extension of digital filter-based turbulent inflow to non-uniform structured grids. Aerosp Sci Technol 2016; 56: 29–33.

46.

Dhamankar

Blaisdell

Lyrintzis

AS.

Overview of turbulent inflow boundary conditions for large-eddy simulations. AIAA J 2018; 56: 1317–1334. https://doi.org/10.2514/1.J055528

47.

Morris

McLaughlin

Kuo

CW.

Noise reduction in supersonic jets by nozzle fluidic inserts. J Sound Vib 2013; 332: 3992–4003.

48.

Powers

McLaughlin

Morris

et al. Supersonic jet noise reduction by nozzle fluidic inserts with simulated forward flight. In: 20th AIAA/CEAS aeroacoustics conference, Atlanta, GA, 16–20 June 2014, paper no. 2014-2474. AIAA.

49.

Morris

McLaughlin

Powers

et al . Prediction, experiments and optimization of high-speed jet noise reduction using fluidic inserts. In: 50th AIAA/ASME/SAE/ASEE joint propulsion conference, Cleveland, Ohio, 28–30 July 2014, paper no. 2014-3737. AIAA Propulsion and Energy Forum.

50.

Powers

Hromisin

McLaughlin

et al. Mean velocity and turbulence measurements of supersonic jets with fluidic inserts. In: 54th AIAA aerospace sciences meeting, San Diego, California, 4–8 January 2016, paper no. 2016-0001. AIAA SciTech.

51.

Morgan

McLaughlin

Morris

PJ.

Experimental results for supersonic jet noise reduction using nozzle fluidic inserts. In: 23rd AIAA/CEAS aeroacoustics conference, Denver, Colorado, 5–9 June 2017, paper no. 2017-3518. AIAA AVIATION Forum.

52.

Sikarwar

Morris

PJ.

Optimization of blowing in a convergent-divergent nozzle for noise reduction. In: 20th AIAA/CEAS aeroacoustics conference, Atlanta, Georgia, 16–20 June 2014, paper no. 2014-2473. AIAA AVIATION Forum.

53.

Sikarwar

Morris

PJ.

The use of an adjoint method for optimization of blowing in a convergent-divergent nozzle. Int J Aeroacoust 2015; 14: 327–351.

54.

Kapusta

Powers

Morris

et al . Numerical simulations for supersonic jet noise reduction using fluidic inserts. In: 54th AIAA aerospace sciences meeting, San Diego, California, 4–8 January 2016, paper no. 2016-0758. AIAA SciTech.

55.

Morgan

Morris

McLaughlin

et al . Further development of supersonic jet noise reduction using nozzle fluidic inserts. In: 55th AIAA aerospace sciences meeting, Grapevine, Texas, 9–13 January 2017, paper no. 2017-0683. AIAA SciTech Forum.

56.

Coderoni

Lyrintzis

Blaisdell

GA.

Aeroacoustics of supersonic jets with fluidic injection. In: 23rd AIAA/CEAS aeroacoustics conference, Denver, Colorado, 5–9 June 2017, paper no. 2017-3210. AIAA AVIATION Forum.

57.

Uzun

Blaisdell

Lyrintzis

AS.

Coupling of integral acoustics methods with LES for jet noise prediction. Int J Aeroacoust 2004; 3: 297–346.

58.

Martha

CS.

Toward high-fidelity subsonic jet noise prediction using petascale supercomputers. PhD Thesis, School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, 2013.

59.

Lele

Compact finite difference schemes with spectral-like resolution. J Comput Phys 1992; 103: 16–42.

60.

Gaitonde

Visbal

Pade-type higher-order boundary filters for the Navier-Stokes equations. AIAA J 2000; 38: 2103–2112.

61.

Aikens

KM.

High-fidelity large eddy simulation for supersonic jet noise prediction. PhD Thesis, School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, 2014.

62.

Dhamankar

NS.

An immersed boundary method for efficient computational studies of nozzles designed to reduce jet noise. PhD Thesis, School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, 2016.

63.

Blaisdell

Lyrintzis

AS.

High-order shock capturing schemes for turbulence calculations. Int J Numer Methods Fluids 2010; 62: 473–498.

64.

Aikens

Blaisdell

et al . Numerical investigation of 3-D supersonic jet flows using large-eddy simulation. Int J Aeroacoust 2012; 11: 783–812.

65.

Kim

Kwon

JH.

A low dissipative and dispersive scheme with a high order weno dissipation for unsteady flow analyses. In: 34th AIAA fluid dynamics conference and exhibit, Portland, Oregon, June 28–July 1, 2004, paper no. 2004-2705. AIAA.

66.

Ducros

Ferrand

Nicoud

et al . Large eddy simulation of the shock/turbulence interaction. J Comput Phys 1999; 152: 517–549.

67.

Mohseni

Colonius

Numerical treatment of polar coordinate singularities. J Comput Phys 2000; 157: 787–795.

68.

Bogey

Cacqueray

Baily

Finite differences for coarse azimuthal discretization and for reduction of effective resolution near origin of cylindrical flow equations. J Comput Phys 2011; 230: 1134–1146.

69.

Kim

Lee

Generalized characteristic boundary conditions for computational aeroacoustics. AIAA J 2000; 38: 2040–2049.

70.

Tam

CKW

Dong

Radiation and outflow boundary conditions for direct computation of acoustic and flow disturbances in a nonuniform mean flow. J Comput Acoust 1996; 4: 175–201.

71.

Bogey

Bailly

Three-dimensional non-reflective boundary conditions for acoustic simulations: far field formulation and validation test cases. Acta Acust 2002; 88: 463–471.

72.

Ikeda

Enomoto

Yamamoto

et al . On the modification of the Ffowcs Williams-Hawkings integration for jet noise prediction. In: 19th AIAA/CEAS aeroacoustics conference, Berlin, Germany, 27–29 May 2013, paper no. 2013-2277. AIAA.

73.

Ikeda

Enomoto

Yamamoto

et al . Quadrupole corrections for the permeable-surface Ffowcs Williams-Hawkings equation. AIAA J 2017; 55: 2307–2320.

74.

Mendez

Shoeybi

Sharma

et al . Post-processing of large-eddy simulations for jet noise predictions. Center for Turbulence Research, Annual Research Briefs, Stanford University, 2009.

75.

Kapusta

Simulations of the flow generated by fluidic inserts for supersonic jet noise reduction based on steady RANS simulations. Master Thesis, Department of Aerospace Engineering, The Pennsylvania State University, Old Main, State College, PA, 2015.

76.

Dosanjh

DS.

Noise field of a supersonic Mach 1.5 cold model jet. J Acoust Soc Am 1972; 51: 1400–1410.

77.

Tanna

An experimental study of jet noise part II: shock associated noise. J Sound Vib 1977; 50: 429–444.

78.

Norum

TD.

Screech suppression in supersonic jets. AIAA J 1983; 21: 235–240.

79.

Shur

Spalart

Strelets

Les-based noise prediction for shocked jets in static and flight conditions. In: 16th AIAA/CEAS aeroacoustics conference, Stockholm, Sweden, 7–9 June 2010, paper no. 2010-3840. AIAA.

80.

Brès

Bose

Ham

et al . Large eddy simulations of supersonic rectangular jets from sinuous exhaust system. In: 23rd AIAA/CEAS aeroacoustics conference, Denver, Colorado, 5–9 June 2017, paper no. 2017-3207. AIAA AVIATION Forum.

81.

Markesteijn

Semiletov

Karabasov

et al . Supersonic jet noise: an investigation into noise generation mechanisms using large eddy simulation and high-resolution PIV data. In: 23rd AIAA/CEAS aeroacoustics conference, Denver, Colorado, 5–9 June 2017, paper no. 2017-3029. AIAA AVIATION Forum.

82.

Towns

Cockerill

Dahan

et al . XSEDE: Accelerating scientific discovery. Comput Sci Eng 2014; 16: 62–74.