The connections between the vortical and near-acoustic fields of three-stream, high-speed jets are investigated for the ultimate purpose of developing linear surface-based models for the noise source. Those models would be informed by low-cost, Reynolds-averaged Navier-Stokes (RANS) computations of the flow field. The study uses two triple-stream jets, one is coaxial and the other has eccentric tertiary flow that yields noise suppression in preferred directions. Large eddy simulations (LES) validate the RANS-based models for the convective velocity *U _{c}* of the noise-generating turbulent eddies. In addition, the LES results help define a “radiator surface” on which the jet noise source model would be prescribed. The radiator surface is located near the boundary between the rotational and irrotational fields and defined as the surface on which the

*U*distribution, obtained from the space-time correlations of the pressure, matches that inferred from the RANS model. The edge of the mean vorticity field is nearly coincident with the radiator surface, which suggests a RANS-based criterion for locating this surface. The two-dimensional space-time correlations show how the asymmetry of the tertiary stream and the resulting thicker low-speed flow weakens the generation of acoustic disturbances from the vortical field.

_{c}## I. INTRODUCTION

The research effort described here targets the development of low-cost predictive models for the noise emission of complex multi-stream turbulent jets associated with the exhaust of advanced turbofan engines. For these models to be effective as design tools, they need to rely entirely on the Reynolds-averaged Navier-Stokes (RANS) solutions of the flow field. With reasonable computational resources, the RANS solutions can be obtained within hours. High-fidelity methods such as large eddy simulation (LES), coupled with surface integral methods, have evolved to the point where they can yield accurate noise predictions.^{1} However, they entail very large computational resources, long turnaround times, and enormous data sets. On the other hand, the analysis of the LES results can help in the creation of low-order models by shedding light on the physical mechanisms of noise generation as is done in this work.

The present investigation considers three-stream jets at conditions relevant to variable-cycle engines for supersonic aircraft. Henderson^{2} surveyed the acoustics of a three-stream configuration in which the core and bypass streams are internally mixed upstream of the tertiary exit; the added tertiary flow reduced high-frequency noise at broadside and peak jet noise angles. Henderson *et al.*^{3} conducted acoustic experiments and flow field simulations of the jets from three-stream nozzles with axisymmetric and offset configurations for the tertiary stream. The offset tertiary stream reduced the noise along the thick side of the jet when the core flow was operating at supersonic conditions. Henderson and Wernet^{4} investigated the mean flow field and turbulence characteristics of externally mixed, convergent three-stream nozzles using stereo particle image velocimetry. Henderson and Huff^{5} assessed the capability of the three-stream, offset duct configurations to meet the noise regulations in Chap. 14. Our research group has conducted extensive parametric studies of offset three-stream nozzle concepts and has identified promising quiet configurations that involve duct asymmetry in combination with a wedge-shaped fan flow deflector.^{6,7}

The RANS-based predictive models for multi-stream jets have focused on the acoustic analogy coupled with the methods to account for the refraction by the mean flow. Including the refraction is particularly critical for the asymmetric configurations with azimuthal directivity of the acoustic emission. The construction of the related Green's functions involves complex numerical procedures.[^{8} The application to three-stream jets with offset tertiary ducts has shown initial promise,^{3} although the asymmetry in the modeled azimuthal directivity was weaker than that in the experimental one. More recently, Papamoschou^{9} has proposed an alternative methodology in which the azimuthal influence is induced by special forms of the space-time correlations of the Lighthill stress tensor. This work has also underscored the importance of properly modeling the convective velocity *U _{c}* of the turbulent eddies, which dominate sound production.

^{9}Specifically, for the velocity ratios of the relevance to the exhaust of the turbofan engines, the sound emission is thought to be strongly influenced by the dynamics of the outer shear layer in the initial region of a multi-stream jet. Support for this observation comes from the modeling of coaxial jet noise by Tanna and Morris

^{10}and Fisher

*et al.*

^{11}A number of additional works have validated and refined this concept, including experiments on mean velocity profiles and noise source location

^{12}and near-field pressure measurements.

^{13,14}Considering the entire jet, one can generalize this observation by stating that the turbulent eddies in contact with the ambient air are the main generators of the noise. In a time-averaged sense, the action of the eddies in the outer shear layer is represented by the local peak of the Reynolds stress, resulting in the definition of the outer surface of peak stress (OSPS). The mean axial velocity on this surface is set to represent

*U*, and the axial convective Mach number, which controls the radiation efficiency, is defined accordingly. Related work by Bridges and Wernet

_{c}^{15}found that the local turbulent convection speed is roughly equal to the local mean velocity in regions with high turbulence intensity. Stuber

*et al.*

^{16}studied axisymmetric and asymmetric three-stream jets and found connections between their differences in noise generation and their convective velocities. Prasad and Morris

^{17}studied the effect of fluidic inserts on the noise of single-stream, heated supersonic jets and attributed the noise reduction enabled by the inserts to reduced convective velocities. Daniel

*et al.*

^{18}studied the noise emission from high-speed jets with temperature nonuniformity and measured the noise reduction that was connected to a reduced convective velocity.

An alternative to the acoustic analogy is surface-based modeling in which propagation starts from a Kircchoff surface located in the linear pressure field.^{19,20} In the frequency domain, the propagation off of the surface requires knowledge of the pressure and its normal derivative on the surface. For simple surfaces (e.g., cylindrical or conical), analytical formulations of the surface Green's function can be used to propagate to the far field. For complex surfaces, the problem can be tackled numerically by a variety of methods, including the boundary element method.^{21} In principle, the data on the Kircchoff surface would be determined by a time-resolved solution of the non-linear flow enclosed by the surface, which makes the approach costly. Considerable effort has been placed on developing simple models for the surface source, which would alleviate this cost. In several of these models for jet noise, the surface source takes the form of pressure partial fields with each partial field being an amplitude-modulated traveling wave (wavepacket) containing hydrodynamic and acoustic components. The acoustic field can then be constructed via the stochastic superposition of the pressure field emitted by each wavepacket. Such constructions can be found in Refs. 22–25 The modeling of the surface source is benefiting from the fundamental works on the wavepacket modeling of jet noise, which have made significant progress in the last decade.^{26–29} Using the source surface approach, one can predict not only propagation but also scattering from airframe surfaces,^{30,31} thus, addressing the acoustics of the propulsion-airframe integration. The surface-based models may also simplify the treatment of the azimuthal directivity from the asymmetric jets noted above. The conceptual application of this approach to multi-stream jets is depicted in Fig. 1. The source is prescribed as random partial fields on a “radiator surface” at the boundary between the inner nonlinear rotational flow field and outer linear pressure field. As will be defined in this paper, the radiator surface is a unique Kirchhoff surface on which the pressure distribution reflects the footprint of the turbulence and, in particular, the coherent structures that dominate mixing and noise generation.^{32,33} As with the acoustic analogy approach, modeling of the convective velocity *U _{c}* on the radiator surface is a critical element of the predictive scheme.

To make this model low-cost, it desired that the radiator surface and partial fields prescribed on it be based on a RANS solution of the flow field. The utility of this framework remains to be demonstrated, and the complexity of the multi-stream jet flow poses particular challenges. Therefore, before such an acoustic model can be envisioned, we need to better understand and model the relation between the inner vortical field and the linear pressure field at the edge of the jet. Of particular interest is how this relation changes when asymmetry is introduced and how well the RANS model can capture the resulting effects. To this end, we make extensive comparisons between the RANS solutions and the models derived by it and a time-resolved solution given by the LES. Our initial aim is to establish whether the RANS model has the potential to inform a differential model for acoustic emission, i.e., a model that would predict the changes from a known axisymmetric baseline. The paper is organized as follows. The nozzle characteristics and their operating conditions are outlined in Sec. II. The details of the LES and RANS simulations are explained in Sec. III. Then, their results on the mean flow fields are presented in Sec. IV. The relevant surfaces for the noise source modeling, the OSPS, and the radiator surface are studied in Secs. V and VI, respectively. Finally, the two-dimensional (2D) space-time correlations are used to investigate the noise reduction mechanisms in Sec. VII.

## II. JET FLOWS

We study the flow fields of two high-speed turbulent jets exiting from the triple-stream nozzles depicted in Fig. 2(a). The nozzles have external plugs, and the mixing of the streams is external to the nozzles. The Cartesian and polar coordinate systems used here, $x=(x,y,z)$ and $x=(x,r,\varphi )$, are illustrated in Fig. 2(b). The origin of the axial coordinate *x* is at the plug tip. Subscripts *p*, *s*, and *t* refer to the primary (inner), secondary (middle), and tertiary (outer) streams, respectively. The azimuthal angle $\varphi $ is defined relative to the downward vertical direction. Both of the nozzles have the same duct exit areas and plug dimensions. The effective (area-based) exit diameter of the primary duct is $Dp,eff=13.33$ mm, the secondary-to-primary area ratio is $As/Ap=1.44$, and the tertiary-to-primary area ratio is $At/Ap=1.06$. The diameter of the tertiary duct is $Dt=38.1$ mm. The plug diameter is 23.80 mm and its length, as measured from the primary exit plane to the plug tip, is 38.4 mm.

Nozzle AXI04U is coaxial and, thus, features uniform distributions of the secondary and tertiary duct exit widths at $Ws/Dp,eff=0.219$ and $Wt/Dp,eff=0.127$, respectively. Nozzle ECC09U has the same primary and secondary ducts as nozzle AXI04U but features a tertiary duct of variable exit width $Wt(\varphi )$, plotted in Fig. 2(c). The distribution is symmetric around the plane *z* = 0. Compared to nozzle AXI04U, the tertiary duct of nozzle ECC09U is wider in the annular segment $\u2212110\xb0<0<110\xb0$ and thinner elsewhere. On the top of the nozzle, the tertiary duct closes completely by means of a wedge-type deflector of axial length $2.1Dp,eff$ and half angle $\delta =25\xb0$. This eccentric tertiary duct causes a thickened tertiary flow on the underside of the nozzle.

The flow conditions are common for both nozzles and are listed in Table I. They represent typical exhaust conditions for a supersonic turbofan engine.^{7} In Table I, NPR is the nozzle pressure ratio, NTR is the nozzle temperature ratio, *A* is the exit cross-sectional area, $m\u0307$ is the mass flow rate, *M* is the fully expanded Mach number, and *U* is the fully expanded velocity. The Reynolds number based on the primary exit conditions and $Dp,eff$ is $1.8\xd7105$.

Stream . | NPR . | NTR . | $A/Ap$ . | $m\u0307/m\u0307p$ . |
. M | U (m/s)
. |
---|---|---|---|---|---|---|

Primary | 2.06 | 3.38 | 1.07 | 1.00 | 1.00 | 590 |

Secondary | 2.03 | 1.34 | 1.06 | 2.33 | 1.44 | 370 |

Tertiary | 1.53 | 1.24 | 0.81 | 1.31 | 1.06 | 282 |

Stream . | NPR . | NTR . | $A/Ap$ . | $m\u0307/m\u0307p$ . |
. M | U (m/s)
. |
---|---|---|---|---|---|---|

Primary | 2.06 | 3.38 | 1.07 | 1.00 | 1.00 | 590 |

Secondary | 2.03 | 1.34 | 1.06 | 2.33 | 1.44 | 370 |

Tertiary | 1.53 | 1.24 | 0.81 | 1.31 | 1.06 | 282 |

The jets AXI04U and ECC09U were part of a campaign to investigate the acoustics of coaxial and asymmetric three-stream jets.^{7,34} Small-scale experiments used helium-air mixtures to match the flow conditions shown in Table I. To demonstrate the noise suppression ability of the eccentric tertiary flow, Fig. 3 plots the far-field narrowband spectra of jets AXI04U and ECC09U in the downward polar direction of the peak emission (approximately 35° below the jet axis).^{34} The spectra are plotted versus the laboratory frequency, which is about 50 times larger than the full-scale frequency for a supersonic business jet. The eccentricity of the tertiary duct in ECC09U yields large reductions, as much as 12 dB, at a full-scale frequency in the range of 200–500 Hz. Understanding and modeling the physical mechanisms of this reduction motivates the research effort discussed in this paper.

In the presentation of the results that follow, equivalent length and velocity scales will be used to properly normalize the coordinates and flow variables. The equivalent diameter $D\u0302$ is based on the total exit cross-sectional area and has the value of 24.9 mm. The equivalent velocity is the mass-flow-rate-averaged velocity,

and has the value of 435 m/s.

## III. COMPUTATIONAL DETAILS

The computational effort encompassed the RANS solutions and LESs performed at the conditions in Table I and the Reynolds numbers listed in Sec. II. The computational fluid dynamics code is known as PARCAE^{35} and solves the unsteady three-dimensional Navier-Stokes equations on structured multiblock grids using a cell-centered finite-volume method.

The RANS computations use the Jameson-Schmidt-Turkel dissipation scheme^{36} and shear stress transport (SST) turbulence model of Menter.^{37} The solver has been used in past research on dual-stream jets, and its predictions have been validated against mean velocity measurements for dual-stream jets.^{35}

The LES computations use implicit backward three-layer second-order time integration with explicit five stage Runge-Kutta dual time stepping, residual smoothing, and multigrid techniques for convergence acceleration. The spatial discretization of the inviscid flux is based on the weighted averaged flux-difference splitting algorithm of Roe.^{38,39} The viscous flux is discretized using a second-order central difference scheme. The time-evolving jet flow is simulated using a hybrid RANS/LES approach.^{40} The Spalart-Allmaras turbulence model^{41} is used to model the turbulent viscosity near the walls, whereas in the free shear flow, the computation relies on the subtle dissipation of the upwind scheme, using the method proposed by Shur *et al.*^{39} Experimental mean velocity profiles of cold jets issuing from the nozzles of this study have been well replicated by the LES predictions; in addition, the LES-based predictions of the far-field sound pressure level spectra (in conjunction with a Ffowcs-Williams-Hawkings surface) have reproduced satisfactorily experimental spectra for the nozzles and operating conditions of this study.^{42}

The computations encompassed both the internal nozzle flow as well as the external plume. At the inlet surface of each nozzle stream, the boundary conditions specified uniform total pressure and total temperature corresponding to their perfectly expanded exit Mach number. The ambient region surrounding the nozzle flow had a characteristic boundary condition, and the downstream static pressure was set to the ambient pressure. The nozzle walls had an adiabatic, no-slip boundary condition. To aid the convergence, the RANS and LES simulations were conducted with a freestream Mach number of 0.05, equivalent to a velocity of 17 m/s.

For the RANS solutions, the mesh contained approximately 8 × 10^{6} grid points and extended to 46$D\u0302$ axially and 12$D\u0302$ radially. As the nozzles are symmetric around the *x*-*y* plane, only one half of the nozzles and jet flows were modeled to save the computational cost. The LES grids contained about 44 × 10^{6} grid points each and extended to 46$D\u0302$ axially and 23$D\u0302$ radially.

The results of nozzle AXI04U were calculated with 4100 time frames after the transient period at a time step of $\Delta t=10\u2009\u2009\mu s$, yielding a simulation time of $716D\u0302/U\u0302$. Due to limited computational resources, the simulation of jet ECC09U was moderately shorter at 3130 time frames with the same time step, resulting in a simulation time of $546D\u0302/U\u0302$. Given that nozzle AXI04U is axisymmetric, its results are averaged in the azimuthal direction whenever possible to improve their smoothness. The same treatment is not applicable to jet ECC09U.

The LES flow field enables two-point space-time correlations throughout the domain. Considering the fluctuating flow variables $a\u2032(x,t)$ and $b\u2032(x,t)$ with zero means, their normalized space-time correlation is defined by

where $x0$ is the reference location, **x** is the displaced location, *τ* is the time separation, and the overline denotes time averaging. Equation (2) assumes stationarity in time *t*. In the analysis that follows, we will consider the space-time correlations of the pressure fluctuation $p\u2032$ with itself (*R _{pp}*), axial velocity fluctuation $u\u2032$ with itself (

*R*), as well as $u\u2032$ with $p\u2032$ (

_{uu}*R*).

_{up}The space-time correlations enable the calculation of the convective velocity by locating the time separation where the correlation peaks. Here, this calculation will be restricted to axial displacements only with $x0=(x0,r0,\varphi 0)$ and $x=(x0+\xi ,r0,\varphi 0)$. The resulting axial convective velocity *U _{c}* will be based on

*R*and

_{pp}*R*. Figure 4 shows an example of a space-time correlation. The practical implementation of the

_{uu}*U*measurement requires several space-time correlations at small axial separations

_{c}*ξ*. Because each correlation function comprises a discrete set of points, to accurately locate the maximum value of the correlation at axial separation

_{i}*ξ*, a seventh-order polynomial is fitted around the peak of the correlation curve (dashed lines in Fig. 4). The time separation

_{i}*τ*corresponds to the maximum value of the polynomial. The convective velocity for this axial separation is $Uc,i=\xi i/\tau i$, and the overall

_{i}*U*assigned to the reference point is the average of all of the $Uc,i$'s computed from the correlations whose peak values exceed 0.4.

_{c}## IV. MEAN FLOW FIELDS

### A. Mean axial velocity

Figures 5 and 6 plot the isocontours of the normalized mean axial velocity, $u\xaf/U\u0302$, on the plane of symmetry of jets AXI04U and ECC09U, respectively, and compare the RANS and LES solutions. The LES and RANS flow fields are similar with the LES predicting slightly faster spreading and, thus, moderately shorter primary potential cores. It is also noted that the wake from the plug is accentuated in the RANS solutions. For jet ECC09U, the asymmetry produced by the eccentricity of the nozzle is evident: there is a significant concentration of low-speed flow on the underside of the primary jet. The lack of tertiary flow on the upper side of jet ECC09U results in faster growth of the upper portion of the shear layer, hence, the potential core for jet ECC09U is slightly shorter than for jet AXI04U. We define the length of the primary potential core *L _{p}* as the distance from the exit of the primary nozzle (located at $x/D\u0302=\u22121.54$) to the point where the maximum mean axial velocity equals 0.9

*U*. For jet AXI04U, the LES gives $Lp/D\u0302$ = 4.5 and the RANS model gives $Lp/D\u0302$ = 6.5. For jet ECC09U, the corresponding values are $Lp/D\u0302$ = 4.2 and 6.3. As has been noted in previous studies,

_{p}^{42,43}the RANS solution has the tendency to overpredict the length of the potential core. Despite this limitation, the RANS-based noise predictions can provide useful guidance for the design of quiet propulsion systems.

^{43}

### B. Reynolds stress

We examine the distributions of the magnitude of the principal component of the Reynolds stress tensor,

where $u\u2032$ is the axial velocity fluctuation and $q\u2032$ is the transverse velocity fluctuation in the direction of the mean velocity gradient. For the LES, *g* is calculated directly from the time-resolved data. For the RANS computation, it is modeled as

where *ν _{T}* is the turbulent viscosity and $\u2207u\xaf$ is the gradient of the mean velocity. For the remainder of the report,

*g*will be loosely referred to as the “Reynolds stress.” Physically,

*g*is a measure of momentum transport by turbulence and represents a direct effect of the coherent turbulence eddies.

^{9}Therefore, the areas of high Reynolds stress may provide valuable information toward the modeling of the effects of those eddies.

Figure 7 plots the isocontours of the normalized Reynolds stress $g/U\u03022$ for jet AXI04U as predicted by the LES and RANS model. Distinct primary, secondary, and tertiary shear layers are evident near the nozzle exit. As noted in the discussion of the mean velocity profiles, the LES predicts moderately faster mixing rates. Consequently, the merging of the outer shear layers with the inner shear layer is complete by approximately $x/D\u0302=2$ for the LES and $x/D\u0302=3$ for the RANS model. The peak Reynolds stress occurs downstream of this merging. For the LES solution, the peak value of $g/U\u03022=0.010$ occurs at $x/D\u0302=4.4$; for the RANS solution, the peak value of $g/U\u03022=0.012$ occurs at $x/D\u0302=5.0$. Overall, the comparison between the RANS model and LES is satisfactory both in terms of the levels and shapes of the distributions.

The analogous plot of the Reynolds stress for jet ECC09U is shown in Fig. 8. The non-smoothness of the LES distribution is the result of the limited number of time steps of the solution. On the underside of the jet, the thicker tertiary flow slows down the spreading of the primary shear layer and results in a large suppression of the Reynolds stress. Importantly, the peak Reynolds stress shifts to a lower-speed region, compared to jet AXI04U, meaning that the most energetic eddies in contact with the ambient have a slower convection speed. On the upper side, where there is no tertiary stream, the level of the Reynolds stress is slightly higher than that in jet AXI04U. As for the axisymmetric case, the LES predicts moderately faster mixing rates than does the RANS model.

## V. OSPS

In the acoustic analogy model of Ref. 9, it was surmised that in multi-stream jets with velocity ratios of relevance to aeroengines, the turbulent eddies in direct contact with the ambient air are the principal noise generators. In a three-stream jet, these eddies are initially in the tertiary (outer) shear layer, then progressively transition to the secondary and primary shear layers as the tertiary and secondary flows become mixed with the primary flow (Fig. 1). In the context of the RANS model, the action of those eddies is represented by the statistics on the outermost peak of the Reynolds stress *g*, that is, the first peak of *g* as one approaches the jet radially from the outside toward the centerline. This results in the concept of the “OSPS,” which is thought to be important in the understanding and modeling of the multi-stream jet noise. Among the most important properties of the eddies in contact with the ambient is their convective velocity *U _{c}* and convective Mach number $Mc=Uc/a\u221e$, where $a\u221e$ is the ambient speed of sound. The convective Mach number governs the efficiency with which the eddies radiate sound to the far field; it is, thus, of paramount significance in the modeling.

The procedure for the detection of the OSPS is a modification of that described in Ref. 9. At a given axial location, the OSPS is detected by constructing rays along the direction of the mean velocity gradient, which propagate from the ambient toward the center of the jet; the first (outermost) maximum of the Reynolds stress *g* along each ray marks the location of the OSPS. This procedure is common for the RANS flow field and the time-averaged LES flow field. Figure 9 offers an example for jet ECC09U based on the RANS solution. The rays start from the low-speed region of the jet and propagate inward. They terminate at the first maximum of *g*, hence, defining the OSPS at that particular cross plane. The inner peak of the Reynolds stress is also visible in Fig. 9.

For the RANS flow field, once the OSPS has been detected, the convective velocity is modeled as the mean axial velocity on the OSPS. Denoting the radius of the OSPS as $rOSPS(x,\varphi )$, the convective velocity is expressed as

For the LES flow field, the convective velocity is determined directly by the space-time correlation of Eq. (2), as explained in the discussion of this equation. An example was shown in Fig. 4.

Three-dimensional views of the RANS- and LES-derived OSPS for the jets of this study are shown in Figs. 10 and 11, respectively. The color contours indicate the distribution of the convective Mach number *M _{c}* on the surfaces. It is evident that the LES and RANS model produce similar surfaces with moderate variations in the geometry and levels of

*M*. The LES surface for jet ECC09U is jagged because of the limited number of time steps (the corresponding surface for jet AXI04U appears smoother because it is averaged azimuthally). We discuss the general trends evident in both types of solutions; specific differences will be covered in Sec. V A. The OSPS of jet AXI04U shows a subtle convergence where the tertiary shear layer becomes mixed with the secondary shear layer, followed by a more pronounced convergence in which the outer streams become totally mixed with the primary shear. This sudden collapse is followed by a gradual convergence near the end of the primary potential core, downstream of which the OSPS diverges slowly. The peak

_{c}*M*occurs shortly downstream of the depletion of the outer streams. The asymmetry of the nozzle ECC09U has a strong effect on the shape of its OSPS. The convergence from the tertiary to secondary shear layer, as well as the stronger collapse on the primary layer, have a clear dependence on the azimuthal angle $\varphi $. Those transition points move downstream as $\varphi $ tends to zero, which is the downward direction. In addition, in the proximity of $\varphi =0\xb0$, the tertiary shear layer interacts minimally with the secondary and primary layers: it diverges until it vanishes as a result of spreading. At that point, it stops representing the outer peak of the Reynolds stress and the OSPS collapses on the primary shear layer. This creates the “bulge” visible in the downward direction of Figs. 10(b) and 11(b). Overall, the outward deflection of the OSPS on the underside of the jet causes a large reduction in the convective Mach

_{c}*M*. This is key to the noise reduction induced by the nozzle ECC09U in the downward direction as shown in Fig. 3.

_{c}### A. Comparisons of the LES and RANS results

Having discussed the detection and broad features of the OSPS, we proceed with detailed comparisons of the geometries and convective velocity distributions obtained by the RANS and LES solutions for the OSPS of jets AXI04U and ECC09U.

#### 1. Jet AXI04U

Figure 12(a) plots the radial coordinates of the OSPS of jet AXI04U as computed by the RANS model and LES. The two predictions are practically identical up to $x/D\u0302=1.7$ with the plot clearly showing the inward transition of the OSPS from the tertiary to the secondary and then to the primary shear layer. This transition occurs in the LES at about 0.8 diameters upstream than in RANS model. For $x/D\u0302>1.7$, the two surfaces are close but the LES result is shifted outward, reflecting the faster spreading of the LES jet.

The comparison of the convective velocities on the OSPS is shown in Fig. 12(b). The RANS- and LES-based trends are similar and show an increase in *U _{c}* as the most energetic eddies move from the tertiary (low speed) to the secondary (medium speed) and then to the primary (high speed) shear layer. At this point, the convective velocity peaks and starts to decline, following the decay of the mean velocity past the end of the potential core. Those three initial velocity levels are approximately $0.36U\u0302,\u20090.55U\u0302$, and $0.82U\u0302$ and correspond to $0.56Ut,\u20090.64Us$, and $0.60Up$, respectively, which are close to the typical value of $0.6Up$ in the case of single-stream jets.

^{15}There are moderate quantitative differences between the RANS and LES results with the RANS results predicting a peak value of

*U*that is about 14% higher than that predicted by the LES. These peaks of

_{c}*U*also take place at slightly different locations, $x/D\u0302=3.6$ for the RANS results and $x/D\u0302=2.0$ for LES, which is explained by the difference in transition to the primary stream in each OSPS.

_{c}#### 2. Jet ECC09U

Because of the eccentricity of nozzle ECC09U, the resulting OSPS shape is dependent on the azimuthal angle $\varphi $. For brevity, we only show comparisons for $\varphi =0\xb0$ and $\varphi =180\xb0$. The radial coordinate results for $\varphi =0\xb0$ are plotted in Fig. 13(a). There is a reasonable agreement between the RANS and LES predictions, both capturing the collapse of the OSPS near $x/D\u0302=4.3$, where the outer shear layer vanishes and the OSPS transitions to the primary shear layer. The axial location of this transition is earlier in the LES than it is in the RANS solution, which is consistent with the faster spreading of the LES flow, as is also shown for jet AXI04U. Downstream of this transition, the curves have similar trends with the LES-based OSPS showing a faster spreading and, therefore, an outward shift. Past $x/D\u0302=13$, the LES-based OSPS loses accuracy as a result of the lack of convergence of the statistics. Figure 13(b) compares the convective velocities obtained by the RANS model and two-point correlations on the LES. The noise on the LES-based *U _{c}* at $x/D\u0302<\u22121$ is not considered physical but a result of the numerical difficulty in locating the OSPS and performing two-point correlations very close to the tertiary nozzle lip. Overall, the LES and RANS curves are similar and show a slightly decaying

*U*where the OSPS occurs on the outer shear layer. Near $x/D\u0302=4.3$, the collapse of the OSPS to the primary shear layer causes the convective velocity to rise suddenly. The LES predicts a peak

_{c}*U*value that is about 9% lower than that obtained from the RANS solution.

_{c}The corresponding results for $\varphi =180\xb0$ are shown in Fig. 14. The radial coordinates show similar trends with an overall faster spreading of the LES jet. Because the tertiary stream is deflected away from the top of the nozzle, the OSPS follows the secondary shear layer, which is quickly merged with the primary shear layer. This transition occurs near $x/D\u0302=0.7$ for the LES and around $x/D\u0302=1.2$ for the RANS model. Downstream of this transition, the LES result shows a more rapid spreading rate. Despite the location discrepancy shown in Fig. 14(a), the RANS- and LES-based convective velocities plotted in Fig. 14(b) are still in overall agreement. Similar to jet AXI04U, there is a stepped increment in the convective velocity as the shear layers mix. In this case, because the tertiary flow is deflected such that there are only primary and secondary flows at the top of the jet, only one sudden rise is shown. The fact that the LES predicts the transition from the secondary to the primary shear layer upstream from the RANS model naturally leads to an earlier rise of the corresponding convective velocity. After that, the lower LES-based *U _{c}* is explained by the faster spreading of the OSPS.

Comparing the *U _{c}* distribution on the underside of jet ECC09U [Fig. 13(b)] with that of jet AXI04U [Fig. 12(b)], we note a substantial reduction in the region $0\u2264x/D\u0302\u22644$. This region influences the middle and high frequencies, which are of particular relevance to aircraft noise. The peak convective Mach number in that region is reduced from 1.10 to 0.57 in the LES solution and from 1.19 to 0.48 in the RANS solution. This reduction occurs because the outermost eddies are shifted to a lower velocity regime. The resulting decrease in the radiation efficiency is evident by the large reduction in the sound pressure level shown in Fig. 3 at the mid and high frequencies. Although there are discrepancies on the order of 10% between the RANS and LES models in the prediction of

*U*, the RANS model captures well the changes in

_{c}*M*and their spatial extent and is, thus, expected to provide useful guidance in a differential noise prediction model.

_{c}## VI. RADIATOR SURFACE

The radiator surface is a surface close to the jet axis on and outside of which the propagation of the pressure perturbation is governed by the homogeneous linear wave equation. It is on this surface that the noise sources could be modeled in the form of linear partial fields.^{25} This model would be informed by turbulence statistics of the vortical field computed by the RANS solution. As we move away from this surface, the hydrodynamic information is lost rapidly. One of the most important elements of a surface-based source model is the convective velocity *U _{c}*. This section provides a specific definition for the radiator surface, evaluates it for the jets of this study, and offers a practical criterion for locating it based on the RANS solution.

### A. Definition

It is desirable that the convective velocity distribution on the radiator surface matches that of the underlying eddies that dominate the noise emission. It is then sensible to look for a connection between the convective velocity distributions on the OSPS and at the edge of the jet. The first issue to address is whether the LES-based convective velocity should be based on the space-time correlations of the axial velocity fluctuation $u\u2032$ or the pressure fluctuation $p\u2032$. Due to their physical associations, we designate $u\u2032$-based space-time correlations for the inner vortical field, where the turbulent structures affect the velocity of the flow directly, and $p\u2032$-based correlations for the region near and beyond the edge of the jet, where we seek the pressure imprint of the vortical eddies. This choice is supported by earlier works, which found that the space-time correlations of $u\u2032$ capture the extent of the turbulence eddies but lose their effects (their “footprint”) away from them. On the other hand, the correlations based on $p\u2032$ capture the footprint of the turbulence events better and, thus, show a physically meaningful transition from the hydrodynamic to acoustic fields.^{44} Accordingly, the specific definition of the radiator used in the present work is the surface near the edge of the jet where the *R _{pp}*-based

*U*matches the

_{c}*R*-based convective velocity on the OSPS at the same axial and azimuthal locations.

_{uu}### B. Evaluation for the jets of this study

Figure 15 displays the isocontours of the *R _{pp}*-based

*U*, normalized by the equivalent velocity $U\u0302$, on the meridional planes of jets AXI04U and ECC09U at $\varphi =0\xb0$ and $\varphi $=180°, respectively. The result for jet AXI04U has been averaged in the azimuthal direction. At a given axial location,

_{c}*U*has a radial trend whereby it decreases outside the OSPS, reaches a minimum, and then rises sharply. The sharp rise is associated with the transition from the hydrodynamic to the acoustic fields. Previous studies have shown similar trends for single- and multi-stream jets.

_{c}^{15,16,45}To achieve the aforementioned property of the radiator surface, we search for a surface near the edge of the jet where the

*U*distribution matches that on the OSPS. The results are the white lines plotted in Fig. 15. They track the hydrodynamic-acoustic transition of the

_{c}*U*maps very closely. The smoothness of the

_{c}*U*-match lines and their proximity to the hydrodynamic/acoustic boundary in the

_{c}*R*-based

_{pp}*U*suggest that the

_{c}*U*information on the OSPS is transmitted to the jet rotational/irrotational boundary. It is notable that a highly distorted OSPS, such as that of jet ECC09U, yields a smooth radiator surface. This is even more apparent in the three-dimensional renderings of Fig. 16, which overlays the LES-derived OSPS with the radiator surface for jets AXI04U and ECC09U. The results provide encouragement that there is a surface with the desired properties of the radiator surface on which the RANS-derived convective velocity (on the OSPS) would inform the definition of the partial fields for the noise source modeling.

_{c}It is instructive to examine the effect of the eccentricity of the tertiary stream on the pressure distribution on the radiator surface. To this end, Fig. 17 plots the axial distribution of the root mean square of $p\u2032,\u2009p\u2032rms$, on the radiator surfaces of jets AXI04U and ECC09U at $\varphi =0\xb0$ (downward direction). The eccentricity reduces the pressure level by a factor of about 2, which is consistent with the reduction in the Reynolds stress that we see when comparing Figs. 7(a) and 8(a). This reduction and the decline in the radiation efficiency resulting from the lower convective Mach number are factors that contribute to the reduction in the far-field sound pressure level that is shown in Fig. 3.

### C. Approximation based on the mean flow

A predictive approach based on the RANS model alone would not have the benefit of the space-time correlations to locate the radiator surface. We, thus, search for a criterion based on the mean flow field that would yield an approximate representation of the radiator surface. The main attribute of the radiator surface is that it is placed at the boundary between the rotational and irrotational fields. It is, therefore, relevant to study the mean vorticity distribution as a means of developing the desired criterion. Figure 18 plots isocontours of the normalized mean vorticity magnitude $|\omega \xaf|D\u0302/U\u0302$ on the meridional plane of jet AXI04U. The magnitude has a wide dynamic range and reaches peak values of approximately $|\omega \xaf|D\u0302/U\u0302=20$ in the shear layers near the nozzle exit. To accentuate the vorticity distribution near the jet edge, a smaller dynamic range has been applied so that the core vortical region appears saturated. The radiator surface is included in Fig. 18. It is observed that the radiator surface follows the outer edge of the mean vorticity field. The same observation holds for jet ECC09U and is not shown here for brevity.

Although a fixed threshold of $|\omega \xaf|D\u0302/U\u0302$ may work well around the potential core of the jet, it will fail downstream as the magnitude of the mean vorticity decays together with the maximum mean velocity of the jet. To account for this, we consider a criterion based on the local mean vorticity. Specifically, we seek the surface defined by

where $|\omega \xaf|OM$ is the outermost maximum of $|\omega \xaf|$ at a given axial and azimuthal location, $rOM$ is the radial location of this maximum, and $\kappa <1$ is a threshold. The search procedure for $|\omega \xaf|OM$ is similar to the detection of the OSPS exemplified in Fig. 9. Then, as Eq. (6) indicates, the threshold is applied as one approaches the jet from the ambient toward the centerline. It was found that the threshold $\kappa =0.125$ works satisfactorily for both jets of this study as illustrated in Figs. 19 and 20. Figures 19 and 20 indicate an accurate representation of the radiator surface using the criterion of Eq. (6). Although this is based on only two jets, it builds confidence that a RANS-based criterion for locating the radiator surface is achievable.

## VII. 2D SPACE-TIME CORRELATIONS

The connection between the inner vortical field and the edge of the jet is further investigated using 2D space-time correlations of the LES data. The focus is on the interaction between the turbulent eddies near the inner (high-speed) shear layer and the rest of the domain with emphasis on events near the radiator surface. For the reasons given in Sec. VI, we consider the correlation *R _{up}* between $u\u2032$ in the high-speed turbulent region and $p\u2032$ elsewhere. The formulation of Eq. (2) is used with the reference point $x0=(x0,r0,0)$ and displaced point $x=(x,r,0)$.

On the meridional plane $\varphi =0\xb0$, the reference point is placed at $(x0,r0)=(2.0,0.3)D\u0302$. This point is on the OSPS of jet AXI04U and near the middle of the high-speed shear layer of jet ECC09U. The resulting space-time correlation $R\u0302up$ is plotted in Fig. 21 for jets AXI04U and ECC09U at three time separations. The evolution of $R\u0302up$ for jet AXI04U shows two main lobes of opposite signs traveling downstream at a speed slightly faster than $0.6U\u0302$. At zero time separation (*τ* = 0), the lobes show a strong correlation pattern radiating from the vortical region to the radiator surface and then on to the near-acoustic field. For nonzero time separations ($\tau =\xb11.92D\u0302/U\u0302$), the correlations remain strong in the near-acoustic field but weaken inside the vortical region. The correlation peaks near the radiator surface represent the footprint of large turbulent structures that pass through the reference point and dominate the surrounding linear field.^{46} However, inside the vortical field, those large eddies coexist with smaller scales that become uncorrelated quickly and, thus, decrease the values of the two-point correlations for $\tau \u22600$. The fact that the peaks of the correlation linked to the linear field follow well the location of the radiator surface is further confirmation of its appropriate placement in Sec. VI.

In comparison with jet AXI04U, jet ECC09U shows much lower values of correlations at all time separations. At the zero time separation, the peak correlation of jet ECC09U in the near-acoustic field is $Rup=\u22120.13$ versus $Rup=\u22120.21$ for jet AXI04U. At the nonzero time separation, the correlations for jet ECC09U become even weaker. The thickened low-speed flow of jet ECC09U not only suppresses the turbulence level of the inner shear layer, as evidenced in Fig. 8, but also weakens the correlation between the inner shear layer and the emitted acoustic field. The reduced correlation can be attributed to the lower radiation efficiency of the eddies in the inner shear layer.

## VIII. CONCLUDING REMARKS

This computational study explored connections between the vortical and near-acoustic fields of multi-stream jets whose understanding will aid in the noise source modeling of these complex flows. The ultimate goal is the development of linear, surface-based models that would be informed by low-cost RANS solutions. The study used two triple-stream jets, one coaxial and the other with an eccentric tertiary flow that yields noise suppression in preferred directions. The jets exhausted at the conditions simulating the takeoff set point of a supersonic turbofan engine. An essential requirement for the model is accurate representation of the convective velocity *U _{c}* of the noise-generating turbulent eddies.

LES-based correlations were used to assess the key assumptions in the RANS-based model. The direct evaluation of the Reynolds stress and convective velocity *U _{c}* from the LES show a reasonable agreement with the RANS-based modeled values. This suggests the validity of modeling the convective velocity of the noise-generating turbulent as the mean axial velocity on the OSPS. The LES results also help define a radiator surface on which the jet noise source model would be prescribed. The radiator surface is located at the boundary between the rotational and irrotational field and defined as the surface near the jet edge on which the

*U*distribution, obtained from the space-time correlations of the pressure fluctuations, matches the convective velocity based on the axial velocity fluctuations on the OSPS. A criterion based on the mean vorticity is formulated that accurately approximates the shape of this surface. The connection between the inner vortical field and edge of the jet is also investigated through 2D space-time correlations of the velocity and pressure fields. The correlations shed light on the noise generation from the high-speed region of the jet and show how the asymmetry of the tertiary stream and the resulting thicker low-speed flow weaken the radiation efficiency of the high-speed eddies.

_{c}One of the most important findings of this study is that there is a surface in the linear near field of a complex multi-stream jet, the radiator surface, on which the pressure disturbances convect at the same speed as the vortical eddies that are considered to be the main noise generators. This convective speed can be modeled as the mean flow velocity on the OSPS and, therefore, can be informed by a RANS solution. The nozzle asymmetry causes changes in the geometry of the OSPS and convective velocity that are captured by the RANS model with a reasonable degree of accuracy. Despite the irregular shape of the OSPS, the radiator surface is relatively smooth and its geometry can be approximated using a mean-vorticity criterion and, therefore, it can also be based on the RANS solution. These results suggest that the RANS solution has the potential to predict two of the most important elements in the proposed modeling—the geometry of the radiator surface and the convective velocity distribution on it. A complete model will require additional information, including the length and time scales. It is hoped that these can also be based on the RANS model, and this is the topic of the current work. The ultimate validation of this framework will entail the accurate prediction of the changes in the acoustics induced by the mean flow distortion, a task that remains to be demonstrated.

## ACKNOWLEDGMENTS

This work was partially funded by National Aeronautics and Space Administration (NASA) Phase II SBIR Contract No. 80NSSC19C0089, under technical monitor Dr. Brenda Henderson. Spectral Energies, LLC was the prime contractor. A.A. has also received support from a Balsells Fellowship.