A proposed wavy shield for suppression of supersonic jet noise utilizing reflections

Abstract

In this work we propose replacing the conventional flat-surface airframe that shields the engine by a wavy surface. The basic principle is to design a wavy pattern to reflect the incoming near-field flow and acoustic perturbations into waves of a particular dominant frequency. The reflected waves will then excite the corresponding frequency of the large-scale structure in the initial region of the jet’s shear layer. By designing the frequency of the reflected waves to be the harmonic of the fundamental frequency that corresponds to the radiated peak noise, the two frequency-modes interact nonlinearly. With the appropriate phase difference, the harmonic dampens the fundamental as it extracts energy from it to amplify. The outcome is a reduction in the peak noise. To evaluate this concept, we conducted Detached Eddy Simulations for a rectangular supersonic jet with and without the wavy shield and verified our numerical results with experimental data for a free jet, as well as, for a jet with an adjacent flat surface. Results show that the proposed wavy surface reduces the jet noise as compared to that of the corresponding flat surface by as much as 4 dB.

Keywords

Jet noise shielding supersonic noise reduction rectangular jet wavy wall LES

Introduction

When a jet engine is installed on an aircraft, the jet noise is affected by the presence of solid surfaces in its vicinity. This is generally referred to as the installation noise or interaction effect. A detailed review on jet-surface interaction and shielding effect on jet noise is provided by Salehian and Mankbadi¹ The focus here is on the case where there is some distance between the jet plume and the airframe surface/wing. In several pursued aircraft design configuration concepts, the engine is mounted on top. A generic hybrid wing body design is illustrated in Figure 1(a). In such top-mounted engine design concepts, the $h / D$ parameter is around 2–3, where $h$ is the distance from the nozzle exit to the surface, and $D$ is the equivalent diameter of the jet. In such designs, the airframe provides the acoustic shielding to reduce the noise radiated to the far-field down below the aircraft.

Figure 1.

Top-mounted jet-surface interactions in hybrid wing body design. (a) Flat surface shield. (b) Wavy shield.²

The objective here is to employ a wavy profile such that it can reflect the acoustic waves at the desired frequency and can act as a passive excitation mechanism to reduce the noise more effectively when compared to the flat surface. Figure 1(b) shows the implementation of the wavy shield on a top-mounted engine configuration. On the other hand, when the engine is mounted under the wing, as illustrated in Figure 2(a), the airframe interaction tends to increase the noise radiation under the aircraft. The proposed wavy profile also has the potential to be utilized in the conventional aircraft design to reduce the unwanted installation noise. The wavy profile implementation of the engine-under-wing design is illustrated in Figure 2(b).

Figure 2.

(a) Schematics of jet flow and acoustic wave interactions with aerodynamic surfaces in a conventional configuration. (b) Wavy profile implemented in conventional engine-under-wing design.²

Several researchers have investigated the engine-airframe interactions. For the subsonic case, on the numerical side, Khavaran³ looked at the jet-surface interactions with focus on the scrubbing noise. Ramamurti et al.⁴ used Unsteady Reynolds Averaged Simulations (URANS) to study the installation geometry effects on jet noise. Paliath and Premasuthan⁵ performed Large Eddy Simulation (LES) for studying subsonic jet installation effects. A detailed review on the application of LES for jet-noise prediction is provided by Lyrintzis and Coderoni.⁶ Rego et al.⁷ used Lattice-Boltzmann method to study noise amplification effects due to jet-surface interaction. On the experimental side Head and Fisher⁸ and Brown and Ahuja⁹ observed that the noise reduction or increase due to interaction with the airframe depends on which radiation side the observer is located. In addition, it was observed that the effect varies between low and high frequencies. Bridges¹⁰ and Zaman et al.¹¹ have focused on understanding the noise-generation mechanisms in subsonic jets and the effect of the surface length and the distance between the jet and the surface (h/D). Bridges and Wernet¹² tested rectangular jets of various aspect ratios in the proximity of a flat surface in the high subsonic flow regime. These experiments reported that having an extended lip generated 2–3 dB more noise in all directions.

For the supersonic jets, which is of interest here, McLaughlin et al.¹³ carried out experimental and numerical studies on a 1.5 Mach jet at various distances from a flat surface. Measurements showed reduction of the noise on the shielded side, specifically for high frequencies. Mora et al.¹⁴ conducted comprehensive tests for a supersonic rectangular nozzle of 2:1 aspect ratio and 1.5 Mach number with and without the plate for various nozzle expansion conditions. In their study, the plate was positioned at different stand-off distances, starting where the plate touches the inner wall of the nozzle exit at $h / D = 0$ , and could be moved away from the jet up to $h / D = 3$ . A comprehensive acoustic data was collected, which is used to validate the numerical results presented in this paper.

The aforementioned experimental and numerical investigations agree that shielding plate, while increases the noise on the unshielded side, it has a beneficial effect on jet-noise reduction on the shielded side. The main objective here is to address the mechanisms involved and utilize them in a positive efficient way. The technologically relevant case where the surface is at certain distance from the jet is considered. In such geometry, there exists a reflection mechanism as the near-field flow and acoustic disturbances reflect from the flat surface. Hence, there is a potential that this mechanism can be utilized to alter the large-scale structure in the initial region of the jet in a positive way that leads to suppression of the far-field peak noise. Thus, replacing the typical flat surface shield by a wavy one is proposed here. The wavy profile should be designed in a particular way to suppress the most noise-efficient large-scale component in the initial region of the jet flow.

The theory behind the proposed design of the wavy wall stems from several previous studies which showed that that introduction of excitation disturbances in initial region of the jet can result in attenuating the jet noise. Specifically, Arbey and Ffowcs Williams¹⁵ conducted experiments on a circular jet that was simultaneously excited by acoustic tones. Their observations showed that, by varying the phase between two signals at harmonically related frequencies, noise control can be achieved. In addition, Mankbadi^16–18 investigated the interactions in a turbulent jet between a large-scale structure at a given fundamental frequency and its harmonics. The harmonic component was found to dampen the fundamental by absorbing energy from it to grow. Thus, the objective here is to use wavy profile such that it would reflect the incoming near-field acoustics into reflected waves at the required harmonic frequency needed to suppress the fundamental. Thus, the wavy wall acts as a passive excitation mechanism that amplifies the appropriate harmonic at the initial region of the jet through the receptivity mechanism. Damping the fundamental, through this nonlinear interactions, can then result in reducing the peak and the total noise levels. The wavy wall profile, determined by its amplitude, wavelength, and phase shift, needs to be selected based on this theory.

The numerical approach for simulation of a supersonic rectangular jet is addressed first. The benchmark geometry and the proposed wavy wall profile is discussed, followed by the detailed discussion of the computational grid. The governing equations and the numerical procedure are discussed next including the numerical scheme, turbulence modelling, far-field acoustic prediction surface integral approach, and boundary treatment. This is followed by isolated jet results and validations against experimental data. The numerical results of the flat surface at $h / D = 3$ and comparison with experimental measurements are presented for both validation of the numerical results, as well as, understanding the mechanisms by which the shield modifies the radiated noise. Next, the wavy wall design is presented utilizing the reflection mechanism when the shield is a distance $h / D = 3$ apart and the corresponding results are examined. This is followed by the concluding remarks section.

Numerical approach

Computational domain

Geometry

To assess the value of our proposed wavy shield we compare with the base case which includes a flat plate. The convergent–divergent (C-D) rectangular nozzle ( $12.95 m m \times 25.91 m m$ ) of a supersonic jet by Mora et al.¹⁴ is considered, for which the acoustic field data for various distanced $h / D$ values were reported. The equivalent diameter of the nozzle exit is $D = 20.65$ mm. Figure 3(a) shows the 2:1 aspect ratio rectangular nozzle with a C-D profile. The nozzle’s area ratio is 1.18 with a design Mach number of 1.5, which corresponds to a Nozzle Pressure Ratio ( $NPR$ ) of 3.67. The nozzle Temperature Ratio ( $T R$ ) is chosen here is such that it resembles the experimental set-up ( $T R = T_{0} / T_{a} = 3.0$ ), where $T_{0}$ is the total temperature of the jet and $T_{a}$ is ambient temperature. The nozzle is sharply C-D in the cross-sectional minor axis plane, as illustrated in Figure 3(a).

Figure 3.

(a) Nozzle geometry (dimensions in meters). (b) Orientation of the flat plate.¹⁴

The numerical results are presented for the isolated jet, followed by the case where the jet is issuing over a flat plate as in the experiment. The thickness of the flat plate is $12.7$ mm and the plate is placed parallel the jet axis and aligned with the nozzle’s major axis, and it extends up to $x / D = 30$ downstream of the jet axis and $z / D = 10.5$ in the Major axis. This configuration is illustrated in Figure 3(b) and is explained in detail by Mora et al.¹⁴ For the simulations carried out here the flat plate is located such that the top surface of the plate is $h / D = 3$ away from the nozzle exit, which was one of the cases considered in the experiment.

In addition to the flat plate cases investigated by experimental measurements, wavy wall profiles are considered to introduce disturbances in the flow and acoustic field with the aim of enhanced noise reduction. The specification of wavy wall profile depends on several parameters such as: distance of the mean line from nozzle lip ( $h / D)$ , wavelength ( $λ$ ), and amplitude ( $A_{w}$ ). These parameters will be discussed in detail in the Wavy Wall Design and Results section. The schematics of the wavy wall design with respect to the rectangular nozzle is illustrated in Figure 4.

Figure 4.

Proposed wavy wall schematics.²

Grid

The computational grid used in the current simulations contains hexahedrally dominant cells. The entire computational domain extends to $80 D$ downstream of the nozzle exit and $10 D$ upstream of the nozzle exit, also it extends radially up to $25 D$ from both major and minor axis planes.

The grid spacing on nozzle walls is chosen such that it ensures $y^{+}$ to have a value of 30 on the wall, and to make sure the close wall calculations of boundary layer in the RANS region are accurate. the non-dimensional wall distance is calculated as: $y^{+} = (\sqrt{τ_{w} / ρ} y) / ρ$ , where $τ_{w}$ is the wall shear stress defined as: $τ_{w} = {μ (\frac{\partial u}{\partial u})}_{y = 0}$ , and $μ$ is the dynamic viscosity. The fist cell distance from the wall, $y$ , is chosen such that the mentioned $y^{+}$ criteria is satisfied on the nozzle walls. This value for $y^{+}$ is calculated considering the isentropic flow assumption along the nozzle and using the nozzle exhaust velocity $U_{j}$ . As it is illustrated in Figure 5(a), the fine grid spacing on nozzle walls are gradually increased such that the volume inside the nozzle has the maximum element sizing of $D / 50$ . Such grid spacing is kept consistent for the isolated jet. Figure 5(b) illustrates the grid spacing in major plane that has the same $y^{+}$ and expands to same maximum grid spacing as mentioned earlier.

Figure 5.

Planar cut of the computational grid near nozzle exit. (a) Minor plane. (b) Major plane.

This grid spacing is maintained and extended up to $x / D = 5$ in the jet axis direction to capture turbulent mixing near nozzle exit, and then it is gradually increased up to $D / 40$ in jet axis direction up to $x / D = 20$ . These refinement regions are illustrated by blue boxes in Figure 6. Then another refinement box is placed that is extended to $x / D = 30$ (and maintained up to $x / D = 40$ ), which gradually increases the cell size up to maximum value of $D / 30,$ shown in a similar fashion with blue box.

Figure 6.

Planar cut of the computational domain of the baseline case. (a) Minor plane. (b) Major plane.

The grid spacing expands gradually in both major and minor directions up to $y / D = 6$ , and $z / D = 10$ and reaching the grid spacing of $D / 10$ . This conservative coarsening in axial direction up to $x / D = 40$ and in major and minor directions is chosen to have a refined box to predict acoustics. To predict the far-field acoustics, the near-field variables are captured on the Ffowcs Williams–Hawkings (FWH) surfaces, to be used for the surface integral calculations. The FWH surface used in this study is a rectangular box from the nozzle exit extending to $y / D = 6$ , and $z / D = 10$ in major and minor planes, and up to $x / D = 30$ in the jet axis direction. This near-field region is illustrated with the red box in Figure 6(a) and (b). The refined near-field acoustic region has maximum grid spacing of $D / 10$ and is to be used for FWH acoustic predictions. Sensitivity studies of the extent of the FWH up to $x / D = 40$ is investigated previously and reported by Salehian and Mankbadi.¹ Such grid spacing on FWH surface would ensure capturing acoustic waves up to Strouhal number $S t = f D / U_{j} = 0.35 - 0.4$ where $f$ is the frequency. This maximum frequency represents up to $75 %$ of the spectra (in logarithmic scale) shown in experimental results and contains most of the trend in spectral analysis of the acoustic signal, such as the peak frequency of $S t = 0.1$ observed in experimental results. From the numerical point of view, the maximum resolvable frequency is calculated based on the assumption that a minimum of 15–18 points (cells) per wavelength are required to capture the acoustic waves up to $S t = 0.35 - 0.4$ with the current numerical scheme. Such requirement has been tested for prediction of waves using second-order finite volume schemes when applied to hexahedral cells.¹⁹

The shielded cases have the same grid spacing as the baseline case inside the nozzle, as well as in the refinement boxes mentioned above in Figure 6. The only difference is in the dimensions of the near-field acoustic box, while maintaining the same grid spacing of $D / 10$ . The near-field region expands gradually in minor directions up to $z / D = 10$ like the baseline case and up to $y / D = 6$ in the reflected side, but it is bounded by the flat plate at location of the flat plate ( $y / D = - 3$ ). The near-field acoustic region is illustrated in Figure 7 (in x – y plane). No extensive grid spacing is maintained on the flat plate, when located at $h / D = 3$ , since the jet does not hit the flat plate unless in much further distance from the nozzle exit ( $x / D > 25$ ), and there is no need for boundary later grid enforcement for this case. Moreover, since one of the major objectives here is to capture the reflection of the acoustic waves from the solid boundary, and since the flat plate is at a considerable distance from the jet, boundary layer prediction on the flat plate is not considered. Figure 7 also illustrates the grid spacing in the near-field acoustic region, as well as near the flat plate. The wavy wall cases have the exact same grid spacing as mentioned for the flat plate case.

Figure 7.

The computational domain of the shielded case ( $h / D = 3$ ).

Governing equations and numerical procedure

The numerical solver and procedure are detailed in Salehian and Mankbadi²⁰ and is summarized below.

Numerical scheme

The rhoCentralFoam solver in OpenFOAM is adopted for this study. OpenFOAM is an open source Computational Fluid Dynamics (CFD) software package consisting of a set of flexible C++ modules to resolve complex fluid flows. rhoCentralFoam is an unsteady, compressible solver, that uses semi-discrete, non-staggered, Godunov-type central²¹ and upwind-central²² schemes proposed by Kurganov and Tadmor.²³ These schemes avoid the explicit need for a Riemann solver, resulting in a numerical approach that is both simple and efficient. The solver is a density based central scheme solver and solves the compressible Favre-averaged mass, momentum and energy governing equations in the Eulerian frame of reference.²⁴ The continuity, momentum, and energy equations are solved in their conservative form as

\frac{\partial ρ}{\partial t} + \nabla . [ρ u] = 0

(1)

\frac{\partial (ρ u)}{\partial t} + \nabla . [u (ρ u)] + \nabla p + \nabla . T = 0

(2)

\frac{\partial (ρ E)}{\partial t} + \nabla . [u (ρ E)] + \nabla . [u p] + \nabla . (T . u) + \nabla . j = 0

(3)

where

ρ

is the density,

u

is the fluid velocity,

p

is the pressure, and

E = e + ∥ {|u|}^{2} / 2 ∥

is the total energy per unit mass with

e

being the specific internal energy. Here,

T

is the viscous stress tensor and is represented by Newton’s Law for a non-inviscid flow as:

T = - 2 μ dev (D)

. Here,

μ

is the dynamic viscosity,

D

is the deformation gradient tensor

D = [\nabla u + {(\nabla u)}^{T}] / 2

and its deviatoric component is

dev (D) = D - (1 / 3) t r (D) I

, where

I

is a unit vector. Also,

j

is the diffusive heat flux that is represented by Fourier’s law as

j = - k \nabla T

, where

T

is temperature and

k

is the conductivity.

In addition to the above equations, the system of equations is completed with the assumption of calorically perfect gas for which $p = ρ R T$ and $e = c_{v} T = (γ - 1) R T$ , where $R$ is the specific gas constant and $γ = c_{p} / c_{v}$ is the ratio of specific heats at constant pressure and volume, respectively.

Finite Volume method is applied for expressing the differential equations. In the application of the finite volume to polyhedral cells with an arbitrary number of faces, each face is assigned to an owner cell and a neighboring cell. This is explained in detail by Salehian and Mankbadi.²⁰ The directed convective fluxes mentioned above, are interpolated using the Van Albada scheme²⁵ to provide a second order spatial discretization that, as a TVD scheme, is appropriate for capturing flow discontinuities such as shocks, and the limiter automatically provides high order stable solution. In addition, second order implicit temporal discretization²⁶ is used to ensure overall second order accuracy of the numerical simulations.

Turbulence modelling

In this study, the $k - ω$ SST DES turbulence model is adopted, where the URANS models are employed only in the boundary layer, while the LES treatment is applied everywhere else. Therefore, the computational cost is more efficient compared to the full LES that requires extensive near wall treatment. For the current simulations, a statistically steady solution is achieved with the $k - ω$ SST RANS model first, then the DES simulations are carried out using the RANS results as an initial solution.

The URANS $k - ω$ SST turbulence model relies on solving two transport equations for the turbulence kinetic energy, $k$ , and turbulence specific dissipation rate, $ω$ , as explained by Wilcox.²⁷ The DES formulation of the $k - ω$ SST²⁸,²⁹ model is achieved such that in the LES regions of the grid, the solution would reduce to a Smagorinski-like sub-grid model,³⁰ as described by Strelets,²⁹ such that the eddy viscosity is proportional to the magnitude of the strain tensor, and to the square of the grid spacing. Therefore, the only term of the RANS model that is different in the DES mode is the dissipative term of the $k$ transport equation

The FWH surface integral formulation

Far-field acoustic results are obtained using the FWH surface integral technique.³¹ The FWH equation is an inhomogeneous wave equation derived by manipulating the continuity equation and the Navier-Stokes equations. If we assume that the control surface contains all acoustic sources, the volume integrals outside this surface can be dropped. The Farassat 1 A formulation of the FWH equations developed by Brentner and Farassat³² is utilized such that the far-field acoustic, can be represented as

p^{'} (x, t) = {p^{'}}_{T} (x, t) + {p^{'}}_{L} (x, t) + {p^{'}}_{Q} (x, t)

(4)

Details of the implementation of the formulations in OpenFOAM using the dynamic libraries are explained in Epikhin et al.³³ For a non-moving control surface, the surface integral equations are simplified to

4 π {p^{'}}_{T} (x, t) = \int_{f = 0}^{} {[\frac{ρ_{o} ({\dot{U}}_{n})}{r}]}_{ret} d S

(5)

4 π {p^{'}}_{L} (x, t) = \frac{1}{c} \int_{f = 0}^{} {[\frac{{\dot{L}}_{r}}{r}]}_{ret} d S + \int_{f = 0}^{} {[\frac{L_{r}}{r^{2}}]}_{ret} d S

(6)

U_{i} = (\frac{ρ}{ρ_{o}}) u_{i}

(7)

L_{i} = p_{i j} {\hat{n}}_{j} + ρ u_{i} u_{n}

(8)

where all the terms can simplified to,

U_{n} = U_{i} . {\hat{n}}_{i}

{\dot{U}}_{n} = \frac{\partial U_{n}}{\partial t}

L_{r} = L_{i} . {\hat{r}}_{i}

, and,

{\dot{L}}_{r} = \frac{\partial L_{r}}{\partial t}

Here, r is the distance between source and observer. ${\dot{L}}_{r}$ , and ${\dot{U}}_{n}$ represent the source time derivatives. The subscripts r or n denote a dot product of the vector with the unit vector in the radiation direction $\hat{r}$ , or the unit vector in the surface normal direction $\hat{n}$ respectively. The term $p_{i j}$ is the compressive pressure tensor, which is resulted from ambient pressure subtracted from the static pressure. The term “ret” refers to retarded time. The term, $f = 0$ , represents closed surface integration on the control surface. The last term in equation (4), ${p^{'}}_{Q}$ , is the volume integral which represent quadrupole (volume) sources in the region. The contribution of the volume integrals becomes very small when the source surface encloses the source region.³⁴ Hence this term is ignored in the computations presented here, since the FWH is at a considerable distance from the sources.

Boundary conditions

At the nozzle inlet, a total pressure condition of $3.67 MPa$ is specified and the jet was expected to be ideally expanded with a $NPR$ value of 3.67. Temperature at the inlet of the nozzle is prescribed to $900 K$ to ensure the TR = 3.0. where ambient pressure is $p_{a} = 101325 P a$ , and has a temperature value of $T_{a} = 300 K$ .

Advective far-field condition was imposed on the rest of the domain boundaries, which corresponds to “waveTransmisive” boundary conditions in OpenFOAM. This non-reflecting condition is based on the same idea of non-reflecting boundary condition as mentioned by Poinsot and Lele³⁵ without full inter-field coupling. In summary, the non-reflecting boundary condition ensures that the material derivative of any variable is kept as zero at the outlet boundary condition, as shown in the following equation. Here, $ϕ$ is a generic representation of any variable flux, and $u_{adv}$ is the advection speed which is defined as the sum of the component of the velocity normal to the boundary and the speed of sound ( $u_{adv} = u . \hat{n} + c$ ).

\frac{D ϕ}{D t} = \frac{\partial ϕ}{\partial t} + u_{adv} . \frac{\partial ϕ}{\partial x}

(9)

The nozzle inner walls are prescribed as adiabatic no-slip condition, so the RANS simulations near the wall can predict the boundary layer with the specified $y^{+}$ , as mentioned previously. On the other hand, on the flat plate adiabatic slip conditions are imposed. Since the flat plate is only to reflect the acoustic wave, the no-penetration rule is enforced by imposing $\partial p / \partial n = 0$ for pressure, and zero normal velocity $u . \hat{n} = 0$ .

Isolated jet results and validations

First a grid sensitivity study is carried out to validate the numerical results for the isolated jet case. Four refinement zones are depicted in Figure 8. The fine grid refinement regions have the grid spacing as described previously. However, the coarse grid has bigger cell size as mentioned in Table 1.

Figure 8.

The Refinement zones in the computational domain.

Table 1.

Grid spacing of different refinement zones in the computational domain.

	Zone 1	Zone 2	Zone 3	Zone 4	# of Cells
Coarse grid	$D / 25$	$D / 20$	$D / 15$	$D / 5$	$8.57 \times 10^{6}$
Fine grid	$D / 50$	$D / 40$	$D / 30$	$D / 10$	$51.03 \times 10^{6}$

In Figure 9, the time averaged axial velocity component is compared with the data available in the literature. Red line represents the numerical LES simulation results presented by Viswanath et al.³⁶ for a nozzle with $NPR = 3.67$ and $T R = 3.0$ , which is the exact same operating condition as the current work. It can be observed that numerical from the fine grid show very good agreement with the results published by Viswanath et al.³⁶ While the coarse grid results do not accurately predict the shocks, completely fail to predict the dissipation of the jet. The inability to capture the dissipation of the jet in the coarse grid results is mainly due to the nature of turbulence modeling. As described earlier, accurate prediction of the dissipation rate in DES formulations depends on the grid spacing.

Figure 9.

Time averaged center line velocity. (Isolated jet).

The grid spacing similar to the fine grid carried out here was tested by Liu et al.³⁷ and showed successful predictions of the behavior of a circular heated jet ( $M = 1.5$ ) with a second order of accuracy numerical scheme. The grid sensitivity was carried out to confirm that the fine grid can accurately predict the shocks, and the dissipation trend of the jet in a similar manner and provide evidence on necessity of the described grid spacing here for the rectangular heated jet.

The Turbulent Kinetic Energy (TKE) is illustrated in Figure 10. TKE here is normalized with respect to the jet velocity squared ( $TKE / U_{j}^{2}$ ). The isolated jet case is compared with the experimental results for the ideally expanded heated jet ( $NPR = 3.67, T R = 2.0$ ) reported by Baier et al.³⁸ These experimental results are chosen for validation, since these results have the closest operating conditions to the current numerical simulations among all experimental results available in the literature for this nozzle geometry, at this time. The numerical results exhibit similar structure of turbulence.

Figure 10.

TKE normalized by jet velocity squared. (Isolated jet).

The numerical shadowgraph is calculated and compared with the shadowgraph results of the experiment reported by Mora et al.¹⁴ in Figure 11. Looking at the results for the nozzle without the plate, the Mach waves propagating downstream of the jet seem to be the main sources of noise in the far-field. Mora et al.¹⁴ mentioned existence of crackle noise, specifically for heated jets. Crackle³⁹ is characterized by intermittent positive pressure fluctuations radiating downstream at an angle associated with the peak jet noise. Such waves are somewhat different from Mach waves which are long, straight and have about equal angles.⁴⁰

Figure 11.

Instantaneous numerical shadowgraph. (b) Instantaneous Schillerian (Mora et al.¹⁴).

To be able to investigate the effect of flat plate on radiated noise in far-field, acoustic spectra is presented at the maximum radiation angle of $^{152} °$ . Similarly, another probe angle is measured from the upstream of the jet axis, but on the shielded side. The acoustic results are calculated and compared with experimental data presented by Mora et al.¹⁴ The location of the probes, the reflected side, and the shielded side are illustrated in the schematics shown in Figure 12.

Figure 12.

Schematics of the microphone probe locations.

For the spectral data presented here, 4 sequences of 1024 samples are collected at a sampling frequency of $204.8 kHz$ . Fast Fourier Transform (FFT) was applied to obtain the narrowband noise spectrum. The frequency is normalized to obtain Sound Pressure Level (SPL) ( $d B rel 20 μ P a$ ), as a function of Strouhal number, as explained in the previous sections. Figure 13 compares SPL spectra between the reflected side and the shielded (point $A$ and $A'$ , $ψ =^{152} °$ ) for the free jet case. Generally, the results shown in both figures show favorable agreement with the corresponding experimental data, both in terms of the predicted level of acoustics, as well as the trend of spectra.

Figure 13.

Acoustic spectra at $ψ = 152^{o} .$ (a) Reflected ( $A$ ). (b) Shielded side ( $A'$ ).

Jet-Flat plate results and validations

Following up with the baseline and wall jet cases mentioned earlier, the main objective is to investigate the effect of distance of the flat plate from the jet axis on the flow field and acoustics of the jet and compare with the baseline.

Figure 14(a) illustrates the Mach number contour for the case where plate is placed at ( $h / D = 3$ ). The bulk of flow field is not bounded by the plate, and the jet spreads out and the jet flow barely starts sweeping on the plate after $x / D = 30$ . The fluctuating component of pressure, shown in Figure 14(b) and denoted by $p'$ , exhibits the acoustic waves reflecting from the flat plate. The dominant downstream travelling waves hit the plate at around $x / D = 5$ on the flat plate and reflect into the jet flow.

Figure 14

Instantaneous low field and acoustics of the flat plate at $h / D = 3$ .

These reflections have an impact on the turbulence structure of the jet. The effect of the location of flat plate on TKE is shown in Figure 15. The reflections from the plate interact with the jet plume and energize the shear layer on the plate side. Hence, causing the asymmetry in the TKE structure for the ( $h / D = 3$ ) case when compared to the isolated jet seen in Figure 10.

Figure 15

Turbulent Kinetic Energy normalized by jet velocity squared. (Flat plate at $h / D = 3$ ).

Figure 16 compares SPL spectra between the reflected side and the shielded side (point $A$ and $A'$ , $ψ =^{152} °$ ) for the ( $h / D = 3$ ) case. As seen earlier for the isolated jet, the acoustic spectra results show favorable agreement with the corresponding experimental data, both in terms of the predicted level of acoustics, as well as the trend of spectra.

Figure 16.

Acoustic spectra at $ψ = 152^{o} .$ (a) Reflected ( $A$ ). (b) Shielded side ( $A ’$ ).

To visualize the effect of flat plate, the acoustic results for the isolated jet and flat plate are plotted together in Figure 17. The acoustic data from numerical investigations suggest that, although the flat plate design provides the considerable acoustic shielding effect in the shielded direction. However, the noise levels increase in the reflected side. It can be concluded that, the noise increase in the reflected side is mainly due to interaction of the reflected waves with the jet flow and energizing the noise sources in the shear layer.

Figure 17.

Acoustic spectra at $ψ =^{152} ° .$ (a) Reflected ( $A$ ). (b) Shielded side ( $A'$ ).

On the other hand, the noise reduction due to the acoustics shielding, may benefit from further noise reduction, especially at the peak frequency region. Hence, modifications in the shielding plate profile is suggested next, to improve the noise reduction of the shielding wall in both directions.

Wavy wall design and results

Theoretical estimation of wavy wall profile parameters

The objective here is to introduce disturbances to reduce the noise. We consider here the technologically-relevant case of $h / D = 3$ where the distance from the engine is such that there is some significant reflections of the flow and acoustic disturbances. The proposed wavy shield design has a sinusoidal profile in the form of

y_{w} = A_{w} \sin [k (\frac{x - x_{0}}{D}) - ϕ]

(10)

where,

k = 2 π / λ

is the wave number, with the wavelength of

λ

, and

A

is the amplitude. Here

x_{0}

is where max reflection occurs along the line

h = 3 D

, and

ϕ

degree is the phase shift between the peak reflected waves and the fundamental frequency mode at the nozzle exit shear layer. These profile parameters are estimated here.

We start by identifying the dominant frequency and wavelength of the acoustic waves for the isolated case at the peak radiation angle. Therefore, the acoustic waves along the two radiation angles of $ψ =^{136} °$ , and $^{152} °$ measured from upstream of the jet axis are investigated. These radiation paths along with the horizontal line denoting the shielding plate are illustrated in Figure 18(a) on the acoustic pressure field. The acoustic pressure wave signal along the radiation angles of $ψ =^{136} °$ and $^{152} °$ are shown to have a dominant wavelength of approximately around $4.5 D - 5 D$ . This corresponds to a Strouhal number consistent with the peak frequency observed previously in acoustic spectra (Figure 13), which shows the peak frequency occurring at $S t = 0.12 \sim 0.13$ . Since Strouhal number is defined as $S t = f D / U_{j}$ , for a wavelength defined as $λ = c / f$ , One can easily calculate the corresponding wavelength as

λ = [\frac{1}{M S t}] D

(11)

Figure 18.

(a) Acoustic pressure field, and the measure line illustrations.

Here, $M$ is the isentropic jet exhaust Mach number. For the given peak frequency, the wavelength would have a value of around $4.5 D - 5 D$ . This calculation is consistent with the wavelengths observed in Figure 18(b) and (c).

The phase lag of the wavy wall profile is taken to be equal to π, since Arbey and Ffowcs Williams¹⁵ noticed that this phase lag results in maximum suppression of noise. In addition, Mankbadi^17,18 and Raman and Rice⁴¹ have also found this phase lag value cause the most effective reduction of the fundamental component in the jet flow. The starting point of phase lag employed in the wavy wall profile here, is measured based on the location of maximum magnitude of waves, which happens around $x = 5 D$ along the line $h = 3 D$ as shown in Figure 18(d) and (e).

To determine the amplitude $A_{w}$ of the wavy profile, we are first guided by Arbey and Ffowcs Williams¹⁵ which showed the noise is suppressed when the initial amplitude of the harmonic at the nozzle exit is half that of the fundamental. For the isolated jet case, we show in Figure 19 the velocity fluctuations along the shear layer, normalized by the isentropic jet exit velocity.

Figure 19.

RMS of velocity fluctuation along the shear layer. (Isolated jet).

The Root Mean Square (RMS) of velocity fluctuations shows the magnitude of fluctuations near the nozzle exit is ∼2% of the jet exhaust velocity $U_{j}$ , shown by the red arrow in the figure. Thus, we want our reflected excitation signal to be about $0.01 U_{j}$ at the nozzle shear layer. Since the value of the momentum thickness at the nozzle exit is about $0.1 D$ , and with accounting for decay during transmission, we estimate the amplitude of the wavy wall profile needed to be $A_{w} ≅ 0.01 D$ .

Arbey and Ffowcs Williams,¹⁵ and Mankbadi^16–18 have concluded that the introduction of a harmonic can effectively reduce the fundamental. With the fundamental for the isolated jet in the current study around $S t = 0.1$ , then we need to introduce a harmonic at $S t = 0.2$ which gives $λ = 2.5 D$ . Figure 20 illustrates the fundamental frequency of the waves in the peak radiation angle and its first harmonic for the isolated jet.

Figure 20.

Fundamental frequency of the waves in the maximum radiation angle. (Isolated jet).

With the arguments presented here, three cases of the wavy wall profiles are simulated in this study, which are defined in Table 2. It should be noted that $π$ phase shift is considered in the wavy wall profile of all these three cases. We need to emphasize that our focus is on the nonlinear harmonic-fundamental interactions, but we will still compare with the linear superposition mechanism by considering the case when the wavy wall wavelength corresponds to that of the fundamental frequency, but with the phase shift.

Table 2.

Parameters of the wavy wall cases.

	$h / D$	$λ$	$A_{w}$
Case 1 - Fundamental	3	5 $D$	0.5 $D$
Case 2 - Fundamental	3	5 $D$	0.05 $D$
Case 3 - Harmonic	3	2.5 $D$	0.05 $D$

The near-Field results

The numerical simulations are carried out for the three wavy wall cases, and the results are compared with the corresponding flat plate result for $h / D = 3$ . The instantaneous fluctuating component of the pressure field for the four simulations are shown in Figure 21. We focus here on the radiated near-field acoustics. We note that for the wavy wall case 1, which has a large amplitude, the reflections were excessive and shock waves occurred. This actually led to intensifying the noise instead of reducing it. But, for the cases 2 & 3, where the amplitude was selected based on the theoretical estimations, reduction of the intensity of the near-field acoustic waves can be observed. This can be easily seen by focusing on the waves in the peak emission direction in the region around ( $x / D = 15$ , $y / D = 6$ ) . Careful comparison of case 2 to case 3 qualitatively shows that case 3 (harmonic excitation) is more effective than case 2 (fundamental excitation) in reducing noise.

Figure 21.

Instantaneous acoustic pressure. (a) Flat plate. (b) Case 1. (c) Case 2. (d) Case 3.

To elaborate the effect of wavy wall on the acoustic waves, the SPL is shown in Figure 22 in a similar manner for all 4 cases. The figure, again, shows that the high-amplitude wavy wall case 1 over intensifies the reflected waves and increases the sound sources. On the other hand, both case 2, and 3, with the amplitude properly designed, clearly show that SPL is effectively reduced. Focusing on the peak radiation direction, we can see that case 3 is more effective than case 2 in reducing the noise. We note that case 3 is designed based on the nonlinear interaction theory to excite the harmonic mode, while case 2 is designed based on linear superposition to excite the fundamental mode. Further quantitative studies of the noise reduction are given next.

Figure 22.

Near-field sound pressure level. (a) Flat plate. (b) Case 1. (c) Case 2. (d) Case 3.

Acoustic spectra and overall sound pressure level

In order to provide a quantitative assessment of the noise reduction of the wavy wall, the SPL spectra is shown in Figure 23 at the peak radiation direction (152^°) for the unshielded (reflected), and shielded side (point $A$ and $A'$ ) for the flat plate, case 2, and 3. Since, FAA regulations mainly enforce limitations on the perceived noise on the ground, the main interest here is on the shielded side where the noise reach the ground in the case of supersonic engine mounted on top. Figure 23 shows that both wavy walls of case 2, and 3 effectively reduce the noise on both sides of the shield, but case 3 (harmonic) is more effective than case 2 (fundamental) over a wide range of frequencies, particular on the shielded side.

Figure 23.

Acoustic spectra at $ψ = 152^{o}$ (a) reflected ( $A$ ), and (b) shielded side ( $A ’$ ).

To provide a clearer picture of the effectiveness of the shielding device as an acoustic reduction mechanism, the noise change is presented in Figure 24 in the form of $Δ OASPL$ , which is defined as

Δ OASPL = OASP L_{(Shielded)} - OASP L_{(Isolated)}

(12)

Figure 24.

OASPL reduction effectiveness. (a) reflected side, (b) shielded side.

For the unshielded (reflected) side the flat plat increases the noise as compared to the isolated case. But, the wavy walls actually reduce the noise by about 2 dB in this unshielded side, as shown in Figure 24(a). On the shielded side, while the flat plat does reduce the Overall Sound Pressure Level (OASPL), the wavy walls reduce the noise more than that of the flat plate by 2 dB and 3 dB for the case 2 and 3, respectively.

Another important aspect of the shield is its effect on the peak frequency noise, which may be a more important factor than the OASPL. Figure 25 compares the acoustic spectra at the 152^° radiation direction (point $A$ and $A'$ ), for the isolated jet (no plate), flat plate, and the most effective wavy wall (case 3 - harmonic). The peak level is effectively reduced by the harmonic wavy wall, when compared with the flat plat or to the isolated jet. To provide a clear quantitative measure of the peak noise attenuation of the two shielding devices, Figure 26, compares the peak SPL at the far-field observer probe location on the shielded sided ( $A'$ ). The SPL level is averaged across the peak noise frequency range ( $0.08 < S t < 0.15$ ) for each case to provide a consistent comparison. Figure 26 shows that the wavy wall (harmonic) reduce the peak noise level by 3.7 dB compared to the flat plate. This is equivalent to about 42% more peak noise reduction compared with the flat plate shield.

Figure 25.

Acoustic spectra at $ψ = 152^{o}$ (a) reflected ( $A$ ), and (b) shielded side ( $A ’$ ).

Figure 26.

Peak noise reduction effectiveness.

The mechanism of noise reduction via utilization of the wavy wall reflections

To explain the mechanism responsible for the noise reductions observed by the harmonic wavy wall profile, the effect of the reflected waves on the source of the noise is investigated. Figure 27 shows the RMS of pressure fluctuations in the jet for the isolated jet, flat plate, and the wavy wall cases. The contours of pressure fluctuations show that the reflection from the flat plate intensify the fluctuations in the lower shear layer of the jet causing an asymmetrical turbulence structure in the jet (Figure 27(b)). Figure 27(c) shows that the manipulated reflection by the wavy produce two specific effects: I) In the top shear layers it reduces the noise source as compared to that of the isolated case and to that of the flat plate case. II) It reduces the fluctuations in the bottom shear layer as compared to that of the flat plate case.

Figure 27.

RMS of pressure fluctuations in source region.

The RMS of pressure fluctuations along the top and bottom shear layers are compared in Figure 28 along the line where the mean flow gradient is maximum. Focusing on the top shear layer (Figure 28(a)), the figure shows that the wavy wall reduces the RMS as compared to the isolated case or to that of the flat-plate case. This reduction is caused by the fundamental-subharmonic interaction mechanism imposed by the wavy wall that excites the harmonic of the fundamental frequency. Turning to the bottom shear layer (Figure 28(b)), although both flat and wavy shields increase the fluctuations relative to the isolated case, the RMS pressure fluctuations in the wavy wall case is smaller than that of the flat plate case. In addition, in the wavy case the fluctuations downstream are less than that of the isolated case. Thus, the noise reduction mechanism can be explained by the following statements: I) The wavy wall reduces the fluctuations in the top shear layer relative to the isolated case and relative to the flat plate case, and II) the intensifications of the fluctuations in the lower shear caused by the presence of the flat plate is significantly reduced when the wavy wall is used instead. These two modifications imposed by the wavy wall result in reducing noise source, and consequently lead to reduction in the SPL spectra.

Figure 28.

RMS of pressure fluctuations along the top and bottom shear layer.

The Fundamental-Harmonic Interaction: It was shown above that the wavy wall designed using the wavelength of the harmonic is more effective than the one with the wavelength of the fundamental. The former is utilizing the nonlinear mode-mode interaction, while the latter applies linear superposition. Since the noise generation is a nonlinear process, it is not surprising that the “harmonic” wavy wall based on utilizing the nonlinear interaction is more effective. Therefore, we further examine here the fundamental and harmonic modes to further explain and verify the nonlinear mechanism associated with the harmonic-based wavy wall.

We utilize Fourier transform to obtain the fundamental and harmonic components of the noise-efficient large-scale structure in the initial region of the jet. In the following figures, the magnitude of the FFT of the pressure oscillations, $|\hat{p}|$ , is taken as some what a representation of the level of the noise source.

First, the flat plat is compared with the isolated case in Figure 29, which shows the fundamental frequency component, $S t = 0.1$ , corresponding to the peak noise in the spectra. The figure shows that for the isolated case, the source is equally concentrated around both the top and bottom shear layers. But when the flat plate is introduced, while it shields the bottom radiation, it has intensified the source in both the top and bottom shear layers due to the reflections of the acoustic waves. This increase in the sound source counterbalances the shielding effect of the plate on the bottom side radiation, and results in intensifying the radiated sound in the non-shielded reflected side.

Figure 29.

Magnitude of pressure oscillation FFT ( $S t = 0.1$ ). (a) Isolated jet, (b) Flat plate.

The wavy wall is designed in a way that it reflects the incoming disturbance occurring at a wide range of frequencies into waves focused on the needed harmonic wavelength such that it will excite the harmonic mode at the nozzle exit’s shear layer. Then, through the fundamental-harmonic nonlinear interaction, the fundamental is damped because the harmonic extracts energy from it to grow. This reduction in the fundamental is what leads to the reduction of the peak radiated noise.

This nonlinear exchange of energy can be seen in Figures 30 and Raman and Rice,⁴¹ in which the magnitude of pressure oscillations at $S t = 0.1$ (fundamental frequency) and $S t = 0.2$ (harmonic frequency) are shown for the harmonic wavy wall (case 3) and compared with the isolated jet case. In Figure 30, $S t = 0.1$ , When the harmonic wavy wall is introduced, two main mechanisms can be observed: I) The $S t = 0.1$ source in the top shear layer is substantially reduced relative to that of the isolated case and relative that of the plate. II) The enhancement of the bottom shear layer source at $S t = 0.1$ is reduced when compared to that of the flat plate (which was shown in Figure 29(b)). This is because the harmonic nonlinearly extracts energy from this fundamental component. The growth of the harmonic is verified in Figure 31, which shows that the harmonic component ( $S t = 0.2$ ) is excited by the presence of the wavy wall and that it grows by extracting energy from the fundamental. Since the higher frequency components are less efficient than the lower frequency ones in radiating sound, the total radiated sound, as well as the spectra peak sound, are reduced as was shown earlier in Figures 24 and 26. In summary, these figures illustrate that the wavy wall profile can provide effective noise reduction, compared to the original flat plate shield by manipulating the reflected waves. The manipulation of the reflected waves introduces fundamental-harmonic nonlinear interactions, which is used as a control device for jet noise reduction.

Figure 30.

Magnitude of pressure oscillation FFT ( $S t = 0.1$ ). (a) Isolated jet, (b) Wavy wall (Harmonic).

Figure 31.

Magnitude of pressure oscillation FFT ( $S t = 0.2$ ). (a) Isolated jet, (b) Wavy wall (Harmonic).

Concluding remarks

This work explores the possibility of modifying a flat airframe surface that shrouds the engine by a wavy surface to reduce the source of supersonic jet noise. To assess the benefit of this idea, we used a hybrid Large-Eddy Simulation to calculate both the flow and acoustic fields of a rectangular supersonic jet. The effect of the proposed wavy surface on the radiated noise was compared to that of the isolated jet case, as well as, to that of the conventual flat surface at a distance $h / D = 3$ from the jet exit, where $D$ is the equivalent diameter of the jet.

Our predicted flow and acoustics agree well with the experimental data for both the isolated jet as well as when a flat surface is included. The latter case is examined to understand why a flat shield has some benefit on the shielded side but enhances the noise on the other side. It was concluded that the flat surface reflects the near-field impinging flow and acoustics disturbances. The reflected waves, being at various frequencies, amplify various flow disturbances in the initial region of the jet and enhances the jet noise sources.

This reflection mechanism has led us to propose a new modification to the shielding airframe surface by utilizing this reflection to effectively reduce the noise source rather than enhancing it. This is done by estimating the Strouhal number and wavelength of the dominant radiated noise, which is labelled here as the fundamental frequency. The wavy wall is then designed with the appropriate wavelength, phase shift, and amplitude in such a way that the reflected waves from the wavy wall propagate back to the jet at a particular frequency. By choosing this frequency to correspond to the harmonic frequency, it interacts nonlinearly with the fundamental frequency responsible for the peak noise and reduces it as well as corresponding radiated far-field noise.

The computational results have verified this concept as it shows that the noise source in the initial region of the jet associated with the noise-efficient large-scale structure has been reduced when the flat surface is replaced with the wavy surface. On the shielded side, the results show that this design achieves as much as 3 dB reduction of OASPL and as much as 4 dB reduction of peak noise, relative to what the flat surface achieves.

On the non-shielded (reflected) side, a flat surface increases the noise. On the other hand, since the wavy surface reduces the source itself, it also reduces the noise of the reflected side. The wavy surface reduces the OASPL by 2.2 dB and as much as 4 dB in the peak noise compared to the flat surface. Thus, the wavy wall surface can be beneficial to engines mounted on the top of the airframe or under the wing.

Footnotes

Authors' note

Saman Salehian is now affiliated with Department of Aerospace Science Engineering, Tuskegee University, Tuskegee, AL, USA.

Acknowledgements

This research is carried out using high performance computing cluster, VEGA, provided by Embry-Riddle Aeronautical University.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Reda R Mankbadi

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