Analysis of compressible free shear layers with finite-time Lyapunov exponents

Accepted Manuscript

Analysis of Compressible Free Shear Layers with Finite-Time Lyapunov Exponents David R Gonzalez, Datta V Gaitonde ´ PII: DOI: Reference:

S0045-7930(18)30226-3 10.1016/j.compfluid.2018.04.030 CAF 3875

To appear in:

Computers and Fluids

Received date: Revised date: Accepted date:

15 September 2017 12 March 2018 27 April 2018

Please cite this article as: David R Gonzalez, Datta V Gaitonde, Analysis of Compressible ´ Free Shear Layers with Finite-Time Lyapunov Exponents, Computers and Fluids (2018), doi: 10.1016/j.compfluid.2018.04.030

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Highlights • Finite-time Lyapunov Exponent technique provides insight into compressible flows. • FTLE structures capture distinctive turbulent and inviscid features in flow.

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• Attracting structures highlight coherent structures and dilatational consequences. • High-frequency vortex dynamics & acoustics captured by repelling features.

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• FTLE elucidates time-accurate relationship between key compressible phenomena.

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Analysis of Compressible Free Shear Layers with Finite-Time Lyapunov Exponents David R. Gonz´ alez1 NSWC IHEODTD, 4103 Fowler Rd., Bldg. 302, Ste. 107, Indian Head, MD 20640-5035.

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Datta V. Gaitonde2 The Ohio State University, 201 W 19th Ave E403, Columbus, OH, 43210.

Abstract

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Finite-time Lyapunov Exponents (FTLE) can, through suitable choice of integration parameters, capture not only convecting coherent structures in compressible turbulent flows but also propagating dilatational components. This semi-Lagrangian approach thus provides an implicit sliding decomposition technique to connect hydrodynamic and acoustic features, and furthermore facilitates a direct examination of individual time-local events such as intermittency, which are often obscured by statistical techniques. The FTLE method yields attracting (also designated stable or

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backward-integrated) and repelling (unstable or forward-integrated) structures. In this paper, we show that FTLE offers substantial additional analysis capability in compressible flows over cor-

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responding incompressible situations. Not only does the method establish a connection between attracting structures and the acoustic field, it also offers an effective means to elucidate finer details of the governing dynamics. This is achieved by leveraging repelling structures, as well as nega-

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tive FTLE magnitudes, which complement the positive values in incompressible flows. The test bed considered is a free shear layer, specifically a validated Large-Eddy Simulation of a Mach 0.9 jet, where hydrodynamic and acoustic components interact with each other in an intricate manner.

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The implications of attracting and repelling FTLE structures are found to depend on whether they arise in the core turbulent region or in regions where the acoustic energy dominates. In turbulent

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regions, attracting structures correlate well with lower frequency acoustic components propagating downstream but repelling structures isolate the higher frequency components of the principal vortex dynamics. On the other hand, away from the highly turbulent regions, attracting structures continue

to be associated with the (relatively weaker) coherent vortices while repelling structures gradually ∗ Corresponding

author Email address: [email protected] (David R. Gonz´ alez) 1 Senior Propulsion Technologist, Weapons Effects & Analysis Branch. 2 John Glenn Chair Professor, Mechanical and Aerospace Engineering.

Preprint submitted to Computers & Fluids

June 21, 2018

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capture more of the acoustic energy and show higher correlation with the Eulerian dilatation. The joint FTLE field, defined as the difference between attracting and repelling FTLE coefficients, also reveals crucial aspects of the underlying physics. In particular, it captures hydrodynamic features at time scales representative of convection, which are shown to induce and modulate acoustic events within the potential core as they relate to entrainment and vortex interactions. In compressible

time-accurate connection between different temporal events.

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flows, therefore, proper interpretation of the different components of the FTLE enables a rich,

Keywords: Lagrangian coherent structures (LCS), Finite-time Lapunov exponent (FTLE),

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Compressible flows, Hydrodynamic-acoustic connections

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Greek Symbols flow map Cauchy-Green tensor

∆θ

azimuthal coordinate grid spacing

∆r

radial coordinate grid spacing

∆t

LES time step

∆x

streamwise grid spacing

λ

Cauchy-Green tensor eigenvalues

ρ σ

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∆

density

Finite-time lyapunov exponent coefficient

τ

Symbols

Θ

c

speed of sound

Φ

D

nozzle exit diameter

h

time step between LES saved snapshots

+

N

number of grid points

Subscripts

n

snapshot time level

p0

pressure perturbation

R

radial coordinate

ReD

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Nomenclature

time after particle release azimuthal coordinate

Lagrangian particle flow map

Superscripts Van-Driest scaling

initial condition; stagnation condition

∞

freestream condition

b

backward FTLE

Reynolds number based on nozzle exit conditions

f

forward FTLE

S

arc length

j

joint FTLE; jet exit condition

T

particle integration time, temperature

Operators

t

time

L

log-transformed FTLE coefficient

X

streamwise coordinate

∇

divergence operator

x

Lagrangian particle position vector

Abbreviations

u

local particle velocity vector

DN S

Direct Numerical Simulation

EM D

Empirical Mode Decomposition

F T LE

Finite-Time Lyapunov Exponents

IM F

Intrinsic Mode Function

LCS

Lagrangian Coherent Structures

LES

Large-Eddy Simulation

RAN S

Reynolds-Averaged Navier-Stokes

V

Vortex

VI

Vortex Interaction

WP

Wave Packet

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1. Introduction The study of turbulent flows is generally performed in an Eulerian framework. Statistical postprocessing can discern the principal dynamics of the flow, and provide insights into various quantities of scientific interest. Such techniques can, however, eliminate temporal information and potentially also mask the dynamics of individual events that occur only intermittently. Lagrangian techniques

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offer an alternative viewpoint that can complement the Eulerian description. Of the various such approaches, our focus is on the finite-time Lyapunov exponent (FTLE ) approach, a summary of which is presented in Section 2.

The FTLE technique was pioneered by Haller [1] to deduce Lagrangian Coherent Structures (LCS) in incompressible flows. Depending on how the time integration is carried out, attracting

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(backward-time) and repelling (forward time) structures can be extracted. These skeletal features, i.e. material surfaces, are solely responsible for the organization of the bulk flow and provide a framework for conducting time-accurate analyses of the flow field. The primary use of FTLE has been for various incompressible flows such as atmospheric turbulence [2], vortex shedding [3], biological [4] and oceanic flows [5], low-Reynolds number wall-bounded turbulence [6] and counterflowing jets [7]. There have been relatively few applications of FTLE to compressible flows, where non-zero

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dilatational values pose challenges in interpreting the results. An initial assessment was obtained by Green [8], who performed an evaluation of the technique in a high-speed jet. The backward FTLE field was used in Ref. [9] in conjunction with dynamic mode decomposition to separate coherent

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structures related to noise generating mechanisms from those that are “silent” in an orifice-jet flow at very low Mach number (0.09). The structures so educed were shown to be superior in certain

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ways to those from the λ2 criterion [10]. More recently, Moura et al [11] used FTLE as a shock detector within a discontinuous Galerkin scheme for the solution of the flow governing equations. Our previous effort, Ref. [12], established a rigorous connection between FTLE and dilatation, and

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intermittent events were examined in this Lagrangian context. Our overall goal in this work is to further characterize and develop the FTLE method for com-

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pressible flows. Free shear layers provide a challenging problem for this analysis, since they display numerous instabilities and whose vortex dynamics is closely linked to the acoustic field. We, therefore, choose as our test bed a round perfectly expanded jet whose dynamics includes a rich interplay between convecting coherent and propagating dilatational features. The existence of coherent structures in turbulent flows [13] and their overall connection with the acoustic field have been established in numerous references. The Eulerian approach is the predominant viewpoint in more detailed analyses based on high-fidelity computations. Specifically, the near and far fields are statistically linked to each other using correlations based on Eulerian measurements or computations (see e.g., Ref.[14]). 5

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Lagrangian approaches have the potential to complement Eulerian techniques, especially in analyzing events that can be obscured through statistical processing. One example concerns intermittent events, whose importance is greatly emphasized in the literature of transition to turbulence, but is also now becoming relevant in compressible jet dynamics. A significant proportion of the acoustic field is known to arise during short periods of time [15, 16], designated “noisy” and “quiet” modes

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by Jordan et al [17]. Although the fluctuation energy is significant in both modes, the farfield noise is strongly influenced by the former. The study of intermittency with Eulerian approaches requires care since statistical processing can suppress time information. Cavalieri et al [18] note that typical statistical analyses that employ time-averaging may neglect events associated with the “noisy” mode. Lagrangian techniques can overcome this constraint because of their ability to identify and track coherent structures in a time-accurate manner, thus isolating their evolution and consequences. A

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study of intermittent events in a Lagrangian framework may be found in Ref. [12].

While the original development of FTLE targeted strong hydrodynamic features, Gonz´ alez et al [12] demonstrated its capacity to seamlessly capture the transition of energy from the rotational, hydrodynamic structures in the turbulent region to irrotational dilatational components. Of particular note was the finding, demonstrated by application to simple acoustic constructs such as the monopole and quadrupole, that with properly tuned particle integration times, a linear combina-

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tion of the attracting and repelling FTLE components exactly reconstructs the dilatational field. In essence, the technique acts as flow decomposition in which an observation (effect) in the region

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dominated by dilatational effects can be directly attributed to the presence of either the dominant repelling or attracting flow features (cause) of the turbulence-dominated region. Since temporal

proaches.

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information is retained, FTLE complements the insights derived from statistics-based Eulerian ap-

In this work, we show that for compressible shear flows, FTLE encompasses much more informa-

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tion than was elucidated in Ref. [12]. In particular, we show that attracting and repelling structures encode different aspects of the dynamics that can be leveraged to reveal a more intricate picture of the flow. A major difference between the FTLE fields of incompressible and compressible fields is

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that only positive values arise in the former, while both positive and negative values are observed in the latter. We leverage this property for both convective as well as propagational elements of the compressible flowfield. A free shear layer in the form of a jet is again considered, but with a major difference from that of Ref. [12]: specifically, the nozzle-exit boundary layer is chosen to be turbulent (‘turbulent-exit’ case) as opposed to the prior ‘laminar-exit’ case. Features of this new database, including validation results, are presented in Section 3. The choice of a turbulent-exit condition is motivated by the known sensitivity of the shear layer development to the character of the nozzle-exit

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boundary layer, which is directly related to differences in the formation and evolution of coherent structures and their relation to the acoustic field. The results (Section 4) emphasize four primary thrusts, which identify the separate features of attracting versus repelling components as they relate to different processes in the jet. Examples of such processes of interest include vortex interactions and the partition of acoustic and hydrodynamic

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energy. Section 4.1 first explores the effects of the boundary layer at the nozzle exit by considering a turbulent profile and draws distinctions from the laminar conditions discussed in Ref. [12] in terms of both hydrodynamic (LCS FTLE ) as well as the acoustic (wave FTLE ) features. The section also discusses the differences in structures arising from the laminar-exit condition of Ref. [12] from the current turbulent-exit case, thus assimilating FTLE -based insights with Eulerian-based conclusions from the literature [19, 20, 21]. The individual properties of the attracting (§4.2) and repelling (§4.3)

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structures are then explored, where the implications of each are couched in terms of their signatures in the turbulent jet (vortical interactions) as well as in the acoustic field (wave behavior). The paper concludes after Section 4.4 which assesses the acoustic energy partition in different structures. 2. LCS Extraction via Finite-Time Lyapunov Exponents

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We first provide a summary of the FTLE algorithm for completeness and future reference. Lyapunov exponents [22] characterize the sensitivity of a system to initial conditions. The FTLE method [1, 23] quantifies the maximum rate of separation between initially-close particle trajectories

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in fluid flows. If x(τ ) denotes the position of a particle at time τ after being released at position x0 and time t0 , a flow map defining the particle trajectory between times t0 and t1 can be constructed

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as:

Φtt10 (x0 , t0 )

= x0 +

Zt1

u(x(τ ), τ )dτ.

(1)

t0

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To identify repelling and attracting surfaces, Eq. 1 can be integrated either forward (t1 > t0 ) or backward (t0 > t1 ) in time. The former yield repelling structures, while the latter generate attracting ones. The maximum degree of separation between two trajectories is characterized by the maximum

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eigenvalue of the right Cauchy-Green tensor, ∆tt10 , given by: #∗ " # " dΦtt10 (x0 , t0 ) dΦtt10 (x0 , t0 ) t1 , ∆t0 (x0 , t0 ) = dx0 dx0

(2)

where [ ]∗ denotes a matrix transpose. A measure, or coefficient, of the separation can then be developed in terms of the logarithm of the square of the largest singular value, i.e. eigenvalue, of Eq. 2: q   1 σtt01 (x0 , t0 ) = ln λmax ∆tt10 (x0 , t0 ) , 2|T | 7

(3)

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with T = t1 − t0 . High values of the FTLE indicate significant divergence in one or multiple directions. Such ‘ridges’ correlate well with material surfaces of maximum stretching [24]. We note that since σ is itself an Eulerian quantity (Lagrangian information parsed to a stationary grid), the technique is designated semi-Lagrangian. The actual calculation of FTLE fields, i.e., forward and backward flow maps, can be performed

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in several different ways. In the present work, we use the straightforward approach as presented by Shadden et al [25]. To speed up the simulations however, a parallelized approach was implemented by distributing instantaneous temporal data across processors. Each processor then simulates all particles simultaneously. At each instance in time, particle trajectories are computed in either forward or backward time over a specified interval, T , to extract repelling and attracting structures, respectively. In the current implementation, such structures are extracted from the velocity vector

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field from a large-eddy simulation (LES) as a post-processing step. Time integration of particle trajectories is implemented via a second-order trapezoidal method:   1 xn+1 = xn + u xn , tn+ 2 h,

(4)

where h is the time step between subsequent flow field snapshots. The mid-point velocity vector is computed simply as un+1/2 = (un + un+1 )/2. Linear interpolation is used to map individual

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particle positions to the LES grid at each integration step in order to determine their local velocity components. Time steps adopted in the LES are assumed to be small enough to ensure a smooth transition of the coherent structures from one window to the next (cf. Farazmand & Haller [26] for

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a discussion on the sliding time approach). The Cauchy-Green tensor, Eq. 2, is constructed via second-order centered differences and its corresponding eigenvalues are calculated with the LAPACK

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[27] software. For efficiency, the code has been parallelized with Message Passing Interface protocols by distributing the instantaneous temporal data (snapshots) between processors. A validation of the FTLE implementation on the Rayleigh-B´enard convection problem was presented previously in

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Ref. [12].

We note that significant advances have been made recently in calculating FTLE fields. For

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example, Kasten et al [28] proposed a localized approach for extracting the FTLE using the Jacobian matrix along a particle’s pathline as the measure of separation. An innovative approach that simplifies the calculations may be found in Nelson & Jacobs [29], who developed a discontinuous Galerkin scheme to compute both the forward and backward flow maps (hence, FTLE) directly from a single computation of the forward map. Subsequently, Ref. [30] extended this approach to 3-D and demonstrated time-scale differences in the evolution of structures relative to 2-D, together with the development of effective procedures to eliminate Gibbs oscillations. In the discussion below, we designate the σ (FTLE) values as σb and σf for backward (attracting) 8

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Figure 1: Experimental configuration including probe array with superimposed jet from simulations.

and forward (repelling) cases, respectively. As will become clear later, for some purposes, the value

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of σj = σf − σb , designated the joint FTLE, will be employed. Other derived quantities, including log transformed variables will be defined in context.

3. Compressible Free Shear Layer LES Database

The parameters chosen for the LES mimic the experimental observations described in Crawley et

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al [31]. Specifically, a perfectly-expanded Mach 0.9 jet is considered, issuing from a nozzle 2.54 cm in diameter at a temperature of 251 K and density of 1.4 kg/m2 into an ambient at a temperature of 272 K. The Reynolds number based on nozzle diameter is 6.3 × 105 . The experimental rig is shown

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in Fig. 1, where an instantaneous Q-criterion isolevel from the LES is superimposed for reference. The full 3-D compressible Navier-Stokes equations are solved in curvilinear coordinates. Length

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and velocity scales are non-dimensionalized by nozzle exit diameter and velocity, respectively. The numerical method used has been extensively verified and validated by comparison with numerous mean and fluctuating measurements [32, 33]. A second-order variant of the Beam-Warming time-

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stepping algorithm [34] is used to march the Navier-Stokes equations in time. For efficiency, the algorithm is couched in an approximately-factored form and has been augmented by a sub-iteration

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scheme to reduce errors associated with explicitly-specified boundary conditions, flux linearization and approximate factorization. The LES procedure follows the general approach of Grinstein et al [35]. Inviscid fluxes are treated via a third-order upwind-biased scheme [36, 37]. A key component of the algorithm is the limiter, which must be chosen carefully to minimize numerical diffusion and provide sub-grid closure. Van Leer’s harmonic limiter has been found to be very suitable for this purpose [32]. Viscous fluxes are evaluated with centered, second-order derivatives. The simulations were performed on a series of meshes on a single cylindrical domain, similar to that in prior efforts [32]. Details of the resolved grid comprised of 730 × 106 points, which exhibits 9

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good comparison with available mean and fluctuating experimental data (see below), are presented in Table 1. Grid stretching is adopted to resolve regions of high gradients while azimuthal planes Table 1: Summary of LES non-dimensional grid parameters. A constant-diameter nozzle with a length-to-diameter ratio of 1 is assumed to facilitate relaxation of the turbulent inflow conditions to an equilibrium state. Value

X ×R×Θ

30 × 12 × 2π

Nξ × Nη × Nζ Ntot Lnozz ∆x at nozzle exit ∆x+ at nozzle exit ∆x at downstream far-field ∆r+

1115 × 651 × 1005 7.295 · 108 1.0 1.0 · 10−3 71.86 2.61 3.5 · 10−5

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∆r at nozzle lip

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Parameter

at nozzle lip

∆r at centerline ∆r at far-field ∆θ at nozzle lip ∆θ+ at nozzle exit

1.26

1.0 · 10−3 0.8

3.138 · 10−3 112.98

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are distributed evenly. No-slip boundary conditions are imposed on the nozzle surfaces, along with a zero wall-normal pressure gradient. All external boundaries are treated with characteristic farfield

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conditions [38].

As noted earlier, the properties of the nozzle-exit boundary layer have a significant effect on the evolution of the jet and the acoustic field. Our prior FTLE results, Ref. [12], considered a laminar

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exit layer. In the present work, we consider turbulent conditions within the nozzle, imposed via the digital filtering technique [39, 40]. The specific implementation follows the work of Touber &

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Sandham [41]. Briefly, two sequences of random numbers are generated, each with unit variance and zero mean. These sequences are decorrelated by processing them with the Box-Muller algorithm [42]. Convolution coefficients and an assumed temporal correlation then provide the means of imposing

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spatio-temporal correlations typical of turbulent flows. The overall amplitude of the turbulent fluctuations are based upon a priori knowledge of turbulent Reynolds stresses using the Cholesky

decomposition assumed by Lund et al [43]. Finally, an extended strong Reynolds analogy is adopted to correlate density fluctuations to the turbulent velocity components. The scaling step in the approach requires estimates of the Reynolds stresses, which are generated by a Reynolds Averaged Navier-Stokes (RANS) precursor simulation. The digital filter results are imposed 38δ0 upstream of the nozzle exit to yield the desired nozzle-exit layer, where δ0 is the nozzle exit boundary layer

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Table 2: Reference experimental & numerical data for Mach 0.9 jets used for validation of the LES results. Diam [cm]

Current LES

2.54

Crawley, et al. [31]

2.54

Lau et al. [44] Freund [45] Bodony & Lele [46] Arakeri et al. [47]

5.51 –

ReD

T0 /T∞

105

1.0

6.2 · 105

1.0

6.2 ·

1.245 ·

106

1.0

3,600

1.0

–

88,000

1.0

2.22

5.0 · 105

thickness.

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Source

1.1

The simulation is validated by plotting centerline mean and fluctuating velocities in Fig. 2. Selected experimental and numerical reference data for Mach 0.9 jets, detailed in Table 2, are also The current simulation captures a mean potential core length in

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included in the assessment.

agreement with experiments, without any streamwise shifting often used when compensating for the unknown state of the nozzle-exit boundary layer. The properties of the exit boundary layer clearly result in simulations that match the spreading rate of published data more accurately than with a laminar exit boundary layer (see Ref. [50].) The velocity decay rate for the simulation data is slightly

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higher than the experiments of Crawley et al [31], though the experimental trend is recovered further downstream. The time-mean centerline velocity is also very similar to the data of Lau et al [44],

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which corresponds to a fit of their experimental data, given by: U = 1 − exp[1.35/(1 − x/xc )]; Uj

xc = 4.2 + 1.1Mj2 .

(5)

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In Eq. 5, xc corresponds to the approximate location of the potential core collapse while Mj is the jet exit Mach number. When compared to other numerical data, the current LES captures a very similar decay rate to the low-Reynolds number direct numerical simulation (DNS) data of Freund

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[45]. Streamwise velocity fluctuations (Fig. 2(b)) also reproduce the experimental trends fairly well. Note that the DNS of Freund [45] was conducted at a Reynolds number approximately two orders

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of magnitude lower than the current simulation. The sharp slope change at X/D ≈ 6 signals the end of the potential core. The projected directivity Fig. 2(c) displays the indicated peak at the approximately 30o shallow angle direction as expected [51]. The Lagrangian analysis was performed on a subset of the three-dimensional LES data as shown

in Fig. 3, which depicts the nozzle and contours of the axial velocity, u. Axial slices start at the nozzle exit plane (X/D = 0), and although the snapshot domain extends from the nozzle exit to X/D = 20 and R/D = 0 to 10, the FTLE analysis is conducted on the smaller domain delineated by thick lines in Fig. 3 to focus on the region of developing large-scale coherent structures and the 11

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(b) Streamwise RMS velocity fluctuations

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(a) Mean streamwise velocity

(c) Directivity

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Figure 2: Comparison of current simulations against experimental & numerical data. (−) Current LES ; () Experimental data of Crawley et al [31]; (N) Data of Lau et al [44]; (H) DNS of Freund [45]; () LES of Bodony & Lele [46]; (I) Arakeri et al [47]; Mollo-Christenen et al. (Mj = 0.9, ReD = 5.3 · 105 ) [48] and Lush (Mj = 0.88, ReD = 5 · 105 )

[49].

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Figure 3: Two-dimensional LES domain extracted for the finite-time Lyapunov exponent analyses of the jet. The Lagrangian analysis is conducted on the inset sub-domain delineated by thick lines.

genesis of radiated noise. The truncated domain extends to X/D = 10 and R/D = 6.5. Figure 3 also highlights the location of numerical probes where instantaneous LES data was tracked. The

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precise locations of these probes may be found in Ref. [12].

Propagating acoustic waves in the high-speed jets of interest are similar to the subset discussed in Green [8] where the spanwise (or in this case azimuthal) component of vorticity is dominant while

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the corresponding velocity gradient in the dilatation term is small. Hence, focus is placed here on planar FTLE analyses. A verification of this constraint used by Green is to plot the dilatation with and without the out-of-plane component for such events. Equivalents for the jet under consideration

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at an instant in time are presented in Fig. 4(a) and (b). Figure 4(a) shows the full dilatation for the near-acoustic field while (b) presents the same quantity with only the in-plane derivatives.

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The contours confirm that the criterion of out-of-plane derivative in the dilatation being small is met. Although the maximum instantaneous azimuthal velocity can be relatively large (about 45% of the in-plane velocity), such values are restricted to very small regions in the turbulent core (i.e.,

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far from the near-acoustic field) and average to zero over time. These observations are consistent with the fact that lower modes such as m = 0 (axisymmetric) and m = 1 (first helical) being dominant in the jets under study [52]. Hence, primary coherent structures in a plane have an orientation perpendicular to it – constituting the approximate heads of the hairpin vortices depicted in Fig. 1. Thus, following Green [8], these have their dominant vorticity direction perpendicular to the streamwise plane employed and justify the use of the planar FTLE.

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(b) In-plane dilatation

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(a) Three-dimensional dilatation

Figure 4: Comparison of dilatation computed with the full three-dimensional vector and in-plane components.

4. Results

4.1. Effect of turbulent-exit conditions on LCS

We first distinguish the features of the turbulent-exit FTLE field from those of the laminar-exit

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case described in Ref. [12]. The manner in which the turbulent-exit conditions are set up has been described extensively in Ref. [40]. Briefly, the upstream profile is specified using random numbers

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that are filtered to provide correlations appropriate to turbulent boundary layers, and the profile is then allowed to relax to its equilibrium state. As noted earlier, coherent structures and the related dilatational field are strongly dependent on these conditions. These results also provide an

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opportunity to summarize the choice of parameters, detailed extensively in Ref. [12], that educe convective and wave features with FTLE. These parameters are listed in Table 3: the large-scale

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Table 3: Algorithm parameters for the FTLE analyses of the turbulent jet to highlight hydrodynamic & acoustic

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features in the near-field.

Target

T

h

X/D

R/D

Nx

Ny

LCS

4.0

0.01

0 – 10

0 – 6.5

1001

1001

Acoustic

0.01

0.01

0 – 10

0 – 6.5

1001

1001

convective structures are extracted with the parameters in the LCS row, while acoustic features are obtained from the Acoustic row. The primary difference is in T , the total time of integration. The values are chosen primarily through trial and error. The larger value was obtained to highlight convective structures, and scales with the jet exit velocity and the diameter (column mode). The

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smaller smaller value emphasizes dilatational (propagating) features, and is dependent upon the frequency of the snapshots considered. In order to establish a baseline of the large structures present in the shear layer, Fig. 5 presents the attracting (σb ; frame (a)) and repelling (σf ; frame (b)) Lagrangian coherent structures (LCSs). The LCS of the jet with laminar exit conditions of Ref. [12], are shown in the frames on the right.

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Significant differences are immediately evident between the two flow fields. First, the potential core in the current turbulent jet (left), denoted by the solid black line, spans almost 6 nozzle diameters downstream of the nozzle exit plane as opposed to approximately 4.5 diameters with the laminar boundary layer (right). Coherent structures, vividly displayed in the backward-integrated FTLE fields (σb ), are very compact, resulting in a more gradual initial jet spreading rate. In contrast, the laminar exit boundary layer is much thinner and more susceptible to inviscid, Kelvin-Helmholtz-type

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instabilities. As a result, the coherent structures formed are larger (right frame of Fig. 5(a)), to yield a more rapid jet spreading rate. These relative differences in initial spreading rate between turbulent and laminar-exit conditions are consistent with the results of Bogey et al [53]. At downstream locations, however, the spreading rate of the turbulent jet increases more rapidly than the laminarexit case, and has the larger width of the two jets.

An important observation is that attracting structures have much broader regions of negative

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FTLE magnitudes than repelling structures – note especially the large negative σb downstream of the potential core collapse in both cases. Incompressible flows do not display such negative values,

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which are associated with the non-solenoidality of the flow. The broader span of such negative magnitude regions in the turbulent-exit jet indicate much stronger interactions beyond the collapse of the potential core. More importantly, even though the coherent structures in the turbulent-exit case

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are finer than those in the laminar-exit case, the attracting LCSs demonstrate strong interactions between leading and trailing vortices in the shear layer closer to the nozzle. This phenomenon can

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be discerned in Fig. 6, where the turbulent-exit jet (frame (a)) displays strong negative ridges much closer to the nozzle than the laminar-exit case (frame (b)). The stronger interactions in the former case are associated with the increased turbulent kinetic energy and such negative ridges in the σb

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field are shown later to designate the onset of vortex pairing events. Turning to the repelling structures, Fig. 5(b) shows that these maintain a similar character in

both cases, although the turbulent-exit simulation displays a steeper inclination angle to account for the smaller initial jet spreading rate. As discussed later, these structures complement the attracting LCS and play an important role in generating the near acoustic field.

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(a) Backward-time integrated (attracting), σb

(b) Forward-time integrated (repelling), σf

Figure 5: Candidate Lagrangian Coherent Structures present in Mach 0.9 jets. Left and right frames correspond to turbulent and laminar exit conditions, respectively

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(b) Laminar-exit jet

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(a) Turbulent-exit jet

Figure 6: Attracting convective LCSs in the vicinity of the nozzle for the Mach 0.9 jets. Turbulent exit conditions induce strong interactions between leading & trailing vortices closer to the nozzle exit than the laminar case. This is evidenced by the strong, negative ridges in the σb fields.

4.2. Dynamics of Attracting Structures

We now examine the features of attracting structures, σb , in the context of the known phe-

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nomenology of the effect of nozzle-exit properties. Primary features captured by the σb field using the wave FTLE (Acoustic row in Table 3) are highlighted in Fig. 7. Since the range of values is

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relatively large, and both negative and positive values are observed, a new quantity is introduced,

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L:



σtt01 . L σtt01 = log10 |σtt01 | |σtt01 |

(6)

The subscript b or f will be used on L when σb or σf are used for σ, respectively. For subsequent

1 and ○) 2 are marked at 30 and 90 degree angles in Fig. 7(a). These reference, two vectors (items ○

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directions are the focus of acoustic studies because of they encapsulate the effects of large structures and fine-scale turbulence respectively [51]. The Lb results clearly capture distinct waves in each direction. Results with the corresponding laminar-exit condition, taken from Gonz´ alez et al [12],

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are shown in Fig 7(b). Features associated with the shallow angle radiation are clearly evident, but those associated with the sideline radiation are muted. Large vortical structures present in laminar transitional jets induce higher noise levels at shallow-angles, leading to the increased acoustic activity captured in Fig. 7(b). 3 is clearly evident in Fig. 8, which is the same as the One source of the sideline radiation ○ instantaneous visualization shown in Fig. 7(a) but zoomed into the vicinity of the nozzle. The turbulent boundary layer interacts with the entrained flow surrounding the nozzle (seen in the light 17

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(a) Turbulent-exit

(b) Laminar-exit

Figure 7: Wave phenomena induced by the attracting (σb ) FTLE structures for the two exit conditions. Frame (a) 1 and ○, 2 as are several structures, ○3 ○ 8 whose implications are shows two wave propagation vectors are marked ○

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discussed in the text.

Figure 8: Wave & convective structures captured by the Lb field in the vicinity of the nozzle exit plane in the turbulent nozzle-exit simulation. Induction of the sideline-radiating wave components is clearly evident adjacent the shear layer, as is the turbulence in the potential core and the flow entrainment from ambient.

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green contours around [X/D, Y /D] = [0, 0.5−1.0]). Wave-like entities (red cellular contours adjacent to the entrained flow) are generated at the interface of the jet and the ambient fluid and travel up towards the top of the nozzle lip. Some of these structures split into two just downstream of the initial roll-up evident in the shear layer (at approximately X/D = 0.5): one travelling towards the top of the nozzle lip; the other propagating towards the sideline direction. The sideline-travelling

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waves give rise to intermittent features which largely engulf probe P1 in the near-field array. A 4 and relatively regular structures ○ 5 present within more detailed view of the exit turbulence ○ the potential core (discussed in subsequent sections) is also afforded in the visualization of Fig. 8. 1 vector Though difficult to discern from a still frame, the radiating waves in the vicinity of the ○ (Fig. 7) depict both downstream- (30o ) and sideline-radiating (90o ) components.

Various interface structures of interest are also noteworthy in this set of figures, an example

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6 in Fig. 7(a). These correspond to either entrainment or expulsion events of which is shown as ○ from the shear layer, such as the upstream wave-like disturbances shown which are intermittently entrained into the shear layer. Some of these features are completely entrained while others continue propagating along the edge of the shear layer, eventually transitioning into aft-radiating waves, i.e. noise. Wave sources are observed to be more active near the end of the potential core. This behavior represents the Lagrangian counterpart of the Eulerian analysis of Ref. [54], indicating this

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region as being closely associated with the emitted aft-propagating noise radiation. Other sources 7 and ○, 8 are located further downstream of aft-propagating waves, some marked in Fig. 7(a) as ○

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from the potential core collapse region.

In addition to the sideline- and aft-propagating noise components, the turbulent jet also shows 4 in Figs. 7(a) and 8). Turbulent flucevidence of unsteady features within the potential core (item ○

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tuations inside the potential core, originating from upstream due to the turbulent flow specification, persistently interact with the shear layer and coalesce into relatively regular structures further down-

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5 in Fig. 7(a), are continually modulated as they propagate towards stream. The features, labeled ○ the end of the potential core. This space-time modulation is the Lagrangian manifestation essential to the wavepacket ansatz in recreating farfield-propagated noise [55]. Very recently, Unnikrishnan &

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Gaitonde [56] extracted the acoustic component by directly decomposing a Large-Eddy Simulation of a similar high-speed jet. The acoustic mode has the form of a highly coherent wavepacket in the potential core and extends up to 10 diameters downstream of the nozzle. They argue that these structures are induced by the “dominant periodic nature from the developing shear layer” and are highly modulated by vortex intrusions into the potential core. The basic features of this intricate dy5 essentially representing the FTLE namics are captured by the current Lagrangian analysis, with ○ manifestation of the wavepacket. However, as will be shown in Section 4.3, a proper combination of

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attracting and repelling structures through the joint FTLE highlights this phenomenon even more vividly. In addition to these acoustic effects, attracting structures are also prominent in describing the core vortex dynamics. The role of vortex roll-up and pairing in the development of jets has been discussed in numerous efforts cited earlier, which also highlight the major influence of the state of

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the boundary layer exiting the nozzle. For example, Bogey et al [53] and Bogey & Marsden [57] note that as exit conditions shift from laminar towards turbulent, azimuthal modes in the shear layer become more prominent and peak frequencies increase. This, thus, indicates the presence of smaller structures in turbulent jets, and farfield noise is reduced. The reduced roll-ups and pairings leads to longer potential cores, lower velocity fluctuation levels, and slower jet spreading rates [53].

The semi-Lagrangian methodology highlights these phenomena in a complementary manner. We

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continue to focus on attracting structures to highlight the dynamics of the region close to the jet nozzle exit. Figures 9 and 10 examine vortex dynamics: the former takes a broader view and connects such dynamics to acoustic events, while the latter isolates specific pairing events for greater clarity. In these figures, V is employed to identify individual vortex structures, while VI is employed to highlight interactions between vortical structures. Figure 9 (0 ≤ X/D ≤ 4) shows convective LCSs on the left set of frames and wave FTLE (Lb ) contours at the same instant on the right set. The

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convective LCSs capture several vortex interactions that strongly influence the organization of the shear layer, potential core, and near-field. Four such events are identified, VI1 through VI4. The

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ridges formed by strong negative σb , described previously in Fig. 5, are a prelude to the onset of vortex interactions. For example, VI2 in Fig. 9(a) outlines the interaction between a vortex roller (X/D ≈ 2) grazing the mean potential core boundary (solid black line) and a preceding vortex

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located to its northwest. Between these two structures lies a very strong negative σb ridge. As their interaction evolves and the two vortices convect downstream, this negative ridge decreases in size

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until the two vortices merge into a much larger structure (Fig. 9(b)). This structure subsequently convects in the shear layer until it encounters a different vortex downstream of it in Fig. 9(c) (indicated by the re-emergence of negative ridges surrounding VI2 ). VI1 is an example of a vortex

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pairing/interaction similar to that experienced by VI2 by Fig. 9(c), thereby demonstrating the cyclical nature of such events. Note, however, that the evolution of VI2 when it reaches the end

of the domain shown is different than that of VI1 as a result of the natural variability of similar successive events in the flow. Such interactions between vortical structures are demonstrated in more detail in Fig. 10. Three vortices (V#) are identified at an arbitrary time, τ0 (Fig.10(a)). Vortex V1 straddles the low-speed side of the shear layer in a counter-clockwise rotating motion and is the furthest downstream of

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(b) τ = τ0 + 0.8

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(a) τ = τ0

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(c) τ = τ0 + 1.6 Figure 9: Vortex pairing, as captured by the attracting FTLE LCSs. Convective structures educed by long time integration are shown on the left while the Wave FTLE at the same instance are depicted on the right.

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(b) τ = τ0 + 0.3

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(a) τ = τ0

(d) τ = τ0 + 0.9

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(c) τ = τ0 + 0.6

(e) τ = τ0 + 1.2

(f) τ = τ0 + 1.5

Figure 10: Evolution of individual vortex pairing events depicted by the convective attracting FTLE. Three vortices initially identified coalesce into a single entity within approximately T = 1.

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the structures identified. V2 and V3 begin interacting at approximately τ0 , inducing a counterclockwise rotation as the high-speed jet pulls V3 closer to the jet axis while V2 rolls up above it (frame (b)). These two vortices merge into a larger structure (V2’) which then starts interacting with V1 until all three features coalesce into a single, large-scale vortex after approximately one characteristic time unit. In relatively high-speed flows, such vortex merging events have not been

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educed through traditional Eulerian means, thus indicating the power of the Lagrangian technique. Note that the vortex pairing/interactions described above, along with VI4 in Fig. 9, can take place very close to the nozzle. Such structures are, initially, relatively small in size (on the order of the exit boundary layer thickness), making a visual representation of such events difficult. Nonetheless, their signature, and that of the larger structures downstream, on the potential core and near-field can be discerned by inspecting the wave FTLE contours on the right in Fig. 9.

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The region inside the time-mean potential core also displays coherent features. Between the vortex interactions identified in the convective LCSs (also evident in the wave FTLE on the right) lie the aforementioned wavepacket-like structures, here identified as WP. Their spatio-temporal structure and coherence evolves continuously as they propagate through the potential core and interact with the vortical structures in the shear layers. At the Mach number under consideration, these features occur in a region that encompasses “trapped” acoustic waves deduced by Towne et

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al [58] and Schmidt et al [59], who postulate that these acoustic waves resonate as a result of repeated reflections between the nozzle exit and the streamwise-contracting potential core. The modulation

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of structures can also be related to the results of Unnikrishnan & Gaitonde [56] who demonstrate the perturbation of the acoustic wavepacket due to vortex intrusion. While the FTLE are scalar fields, the time evolution of the LCSs, VIs, and WPs provides some supporting evidence to these findings.

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In particular, we observe that VIs tend to force the motion of the paired vortices towards the jet centerline (cf. Fig. 10) as a result of their counter-clockwise rotation. Though the FTLE show rather

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compact features, these contours only outline the dominant regions towards which fluid particles are converging within the vortex. The physical structure itself encompasses a broader extent than the

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FTLE ridges, protruding into the potential core during pairing events. 4.3. Dynamics of Repelling Structures We now highlight the role of repelling LCSs, which complement attracting structures, in estab-

lishing the shear layer dynamics. Indeed, a direct connection between FTLE and the dilatational

field can only be made if both types of structures are considered together. In the monopole-induced field discussed in Ref. [12], each FTLE coefficient captured a specific portion of dilatational waves: σf targeted the positive (increasing pressure) stroke while σb honed in on the attracting (negative) phase. This fact, combined with the association established above between negative σb ridges and 23

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vortex interactions/pairings, suggests there should be a correlation also with the repelling LCSs in this process. LCSs are frequently identified from FTLE by extracting surfaces equal to a certain percentage of the maximum Lagrangian coefficient magnitude [4]. Considering only the maximum (i.e. positive) contours, a composite view of both attracting and repelling structures can be obtained by using the reconstruction: (7)

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σj = σf − σb ,

where σj is denoted the joint FTLE field. The variable seeks to effectively combine the attracting and repelling structures so that the key properties of each are elicited in a single variable. Thus, it demonstrates the generation and modulation of acoustic events within the potential core as they relate to entrainment and vortex interactions. In purely acoustic flow fields, this linear combination fully reconstructs the dilatation [12] if the wave FTLE parameters are used. We now explore the

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implication of σf in a highly-compressible flow with strong hydrodynamic components.

Application of Eq. 7 to the convective LCS fields exposes a more complete view of the role of attracting and repelling structures in the vortex interaction process. This is elucidated in Fig. 11, where the forward (σf ), backward (σb ), and joint FTLE (σj ) contours for the turbulent jet are plotted in the vicinity of the nozzle at the same instant in time. As noted previously, the attracting

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LCSs (highlighted in blue in the σj field, Fig. 11(c)) are very compact. More importantly, they are intricately intertwined within the repelling structures (σf ), whose ridges are located precisely within the peak negative ridges in the σb field (Fig. 11(b); several identified in frame (c)). Therefore,

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σf ridges are an integral aspect of the vortex interaction/pairing events denoted by VI1 and VI2, appearing in both the leading and trailing vortex pairs. The joint FTLE coefficient (Eq. 7) also provides an additional perspective on the near-field

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acoustics. For example, it is not immediately apparent that the log-transformed acoustic σf (denoted by Lf ) contours in Fig. 12(a) capture as much of the near-field acoustics as the Lb contours

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in Fig. 12(b). This is merely a consequence of the log-scaling chosen to highlight the acoustic components. In reality, wave dynamics are also encoded in the Lf . For example, wave-like features are also captured in the frame of Fig. 12(a), particularly within the potential core at approximately

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X/D = 2. Comparing this frame to the Lb analog (Fig. 12(b)), it is evident that this feature in the

repelling structures must interact with the attracting analog to form a complete entity. Applying Eq. 7 to the acoustic coefficients highlights this in great detail, demonstrating the manner in which each FTLE coefficient represents a particular portion of the wavepacket within the potential core (Fig. 11(c)). Moreover, the contributions of high-frequency acoustic components in the near-field are also highlighted, where the sideline radiation is primarily enhanced. The space-time modulation of such wavepackets is also best captured by the joint Lagrangian 24

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(b) σb

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(a) σf

(c) σj

Figure 11: Construction of the Joint FTLE (σj , (c)) from the forward (a) and backward (b) coefficients. The joint field provides a holistic view of the convective LCSs in a turbulent jet. Note how regions of high negative σb correspond

to ridges in σf , implying a strong influence of repelling structures in promoting vortex interactions.

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σ

σb (b) Lb = log10 (|σb |) |σ |

(a) Lf = log10 (|σf |) |σf |

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f

σ

(c) Lj = log10 (|σj |) |σj | j

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Figure 12: Log-transformed (a) forward, (b) backward & (c) joint FTLE fields in the turbulent jet. Note the highly coherent wavepackets in the joint FTLE field, resulting from the linear combination of the two Lagrangian coefficients. High-frequency waves near the nozzle and prevalent events adjacent to the shear layer are also exposed. Aft-traveling waves are present, but masked by the contours.

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(a) τ = τ0 + 0.3 (from Fig. 10)

(b) τ = τ0 + 1.5 (from Fig. 10)

Figure 13: Wavepackets encoded in the joint FTLE field, demonstrating their spatial coherence within the potential

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core. One such packet is identified at the two instances selected. A complete view of the structures’ evolution throughout the jet is masked by the turbulence field induced by the shear layer as FTLE provides a composite view of the entire flow.

field, as evident in Fig. 13. Here, the Lj field at the time instances in Figs. 10(b) and 10(f) are shown, highlighting the evolution of the wavepacket within the ovals. From the broad span at the initial

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timeframe, the wavepacket compresses its axial span into a more compact structure. Numerous other lobes of the wavepacket are also captured within the potential core. In addition to these in-core dynamics, numerous interface structures are continuously ejected or

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entrained from the ambient throughout the time period analyzed. Many of these also correspond to wave sources which can be identified in the Lb contours from the backward FTLE coefficients. However, the joint field ultimately masks the lower frequency acoustic components captured in

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the log-transformed backward FTLE, as can be appreciated by comparing Figs. 12(b) and (c). In particular, the small wave source adjacent to probe P3 (X/D ≈ 3.25) in the Lb is completely masked

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in the joint Lj contours. This wave source is intimately tied to the interface structure shown in Fig. 12(c) but is not reflected in the unified field. The same is true of the broad wave downstream of the wave source.

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These findings make it clear that each of the FTLE coefficients provides insight into unique

and specific aspects of the underlying acoustic fields when a very small integration time step, T , is adopted. The forward field (repelling structures) is strongly biased towards high-frequency dynamics. As the red contours of Fig. 11(c) suggest, the broad repelling structures are composed of numerous filament sub-structures. On the other hand, the attracting FTLE field emphasizes the lower-frequency spectrum, owing to its more direct connection to large-scale coherent structures. Finally, while the joint FTLE also demonstrates a bias towards high-frequency structures, its strength 27

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Figure 14: Instantaneous view of dilatation (top) and time-rate-of-change of pressure (bottom) in the turbulent jet. Vorticity magnitude contours are used to highlight the shear layer. Superimposed are several rays along which the evolution of FTLE coefficients is explored. These include rays at 30- and 90-degrees relative to the jet axis and the

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time-mean boundaries of the potential core (PC) and shear layer (SL).

ultimately lies in providing details related to the shear layer-turbulence interactions that spawn and affect the evolution of wavepackets. In Section 4.4, we further explore the energy partition

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between the attracting and repelling flow features, focusing on the jet near-acoustic field to further characterize their contributions.

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4.4. Acoustic Energy Partition in Lagrangian Coherent Structures A quantitative assessment of the information contained in attracting and repelling structures can

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be made by examining their magnitudes along discrete points in space and time and comparing them with the Eulerian fields. The primary variable of interest is the dilatation, whose value increasingly approaches the rate of change of pressure fluctations in regions where viscous effects and turbulence

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are minimal, as follows from the linearized continuity equation: ∇ · u0 = −

1 ∂p0 , ρ0 c20 ∂t

(8)

with ρ0 c20 being the acoustic impedence. Figure 14 shows the dilatation (top) and pressure timerate-of-change (∂p/∂t; bottom) for the jet at a particular instant in time. As anticipated, the two variables share very similar characteristics outside of the turbulent core. In Ref. [12], it was shown in the context of a monochromatic acoustic field that the dilatational field could be completely 28

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reconstructed by considering the Lagrangian analysis over a single integration time step. This is consistent with the fact that the dilatational operator embedded in the Cauchy-Green tensor (see Eq. 2) forms a core of the FTLE algorithm. A key question, alluded to in the previous section, concerns the effectiveness of the reconstruction of Eq. 7 for complex flows such as the turbulent jet under consideration, where viscous and turbulent

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effects are important. We address this by considering FTLE coefficients along two representative rays highlighted in Fig. 14 along angles of 30 and 90 degrees relative to the jet axis and extending about six nozzle diameters in the radial direction. These are known to be approximately aligned with sound associated with larger and smaller scales respectively [51]. Time-mean boundaries of the potential core (PC) and shear layer (SL) are also displayed in the figure for reference.

Figures 15 and 16 show the evolution of both Eulerian flow field parameters, ∇ · V and ∂p/∂t,

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and the reconstruction obtained by the linear combination (Eq. 7) of the wave FTLE coefficients along the 30o degree ray at a representative time instant. The variable S corresponds to the distance along each ray, beginning at [X/D, Y /D] = [0, 0] in the 30-degree ray and [0.25, 0] in the sideline ray. The FTLE reconstruction replicates the true dilatation in regions of relatively weak viscous effects, even inside the potential core. Only in the shear layer region does the correspondence fail. Thus, in these regions, the correspondence of wave FTLE results to dilatation breaks down. Convective

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FTLE, however, are applicable here and continue to correspond to the coherent structure dynamics. Several other important points also emerge from the results in Figs. 15 and 16. First, dilatation

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and FTLE regain their correspondence at a very short distance away from the turbulent shear layer. In the 30-degree direction, where the data traverses a broader span of the shear layer, dilatation and FTLE converge to the same magnitudes approximately one nozzle diameter beyond the potential

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core-shear-layer boundary (S/D ≈ 0.75; see Fig. 15(b)). Along the sideline direction (Fig. 16(b)), this correspondence is recovered after only half a nozzle diameter from the onset of the initial

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divergence of the two fields, which occurs at approximately S/D = 0.4. Finally, both the Eulerian- and Lagrangian-derived dilatation curves mimic the behavior of the pressure time-rate-of-change towards the ‘farfield’, as expected in linear flows from Eq. 8. In these

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regions (Figs. 15(c) and 16(c)), the evolution of dilatation is identical to that of ∂p/∂t. These findings scope the validity of the dilatation reconstruction from the wave FTLE in a realistic flow field with

broadband frequency content and validates the effort to provide a more detailed characterization of the contribution of each Lagrangian component to the structure of the near acoustic field. As described in Section 4.2, each FTLE component displays a bias towards specific aspects of the two-source theory of Tam et al [51]. Hence, it is insightful to process the solution into its constituent modes and to then relate these to modes extracted from the dilatation field. Although various

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(b) Near Nozzle

(c) Farfield

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(a) Full Length

Figure 15: Evolution of Eulerian (∇ · V , ∂p/∂t) & Lagrangian parameters along the 30o aft-propagating direction. The latter are shown as the reconstruction in Eq. 7 computed with the Acoustic parameters of Table 3.

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(b) Near Nozzle

(c) Farfield

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Figure 16: Evolution of Eulerian (∇ · V , ∂p/∂t) & Lagrangian parameters along the sideline direction. The ‘FTLE’ curve corresponds to the reconstruction in Eq. 7 computed with the Acoustic parameters of Table 3.

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Figure 17: Empirical mode decomposition of dilatation, FTLE coefficients, and Lagrangian dilatation at the 30o LES farfield probe. EMD extracts nearly identical IMFs in both dilatational fields.

Fourier-based techniques are available, extraction of different scales using these can be tedious, since various user choices are required. We therefore adopt a data-driven approach that automatically yields an orthogonal set of component modes. The Empirical Mode Decomposition (EMD) [60]

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proves an ideal analysis tool for the current unsteady field – see for example Agostini et al [61]. The procedure is discussed in detail in their work, with emphasis on its use to separate scales.

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Briefly, the signal extrema are fitted with cubic splines, and a recursive sifting procedure is enforced that subtracts the mean of the splines from the original signal. Upon convergence, the signal has effectively been split into component intrinsic mode functions (IMFs), of successively lower

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frequency bands. The IMFs are non-lossy and because of their orthogonality property, allow for a straightforward comparison of modal features of each signal

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For concreteness, we focus on signals at the ‘farfield’ end points of the 30- and 90-degree directional rays of Fig. 14, located at [X/D, Y /D] = [10, 6] and [0.25, 6]. The dilatational and FTLE results are shown in the top plots of both Figs. 17 and 19. At both the aft and sideline locations, the

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forward-integrated FTLE coefficient (σf ) captures the bulk of the dilatational energy. The signal itself has a non-zero mean and rarely achieves negative magnitudes. This non-zero mean results from the relatively close proximity of the probes to the jet, where hydrodynamic effects are still present. In contrast, the backward coefficient (σb ) displays oscillatory behavior around zero, with

a relatively small amplitude. The IMFs of the Lagrangian dilatation correlate well with those extracted from the Eulerian counterpart at this location. Note that the σf signal also replicates most of the dilatational trends, though there are significant differences in the intrinsic mode functions as

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Figure 18: Auto- and cross-correlations of LES dilatation against the Lagrangian coefficients and reconstruction at

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the 30o ‘farfield’ position.

a result of the missing information captured in σb . Because the scalar FTLEs tend primarily to

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positive magnitudes, peaks in σb signal are seemingly out of phase with dilatation, leading to IMFs that least resemble those of the Eulerian field. Cross correlations of dilatation with the Lagrangian variables also highlight the effective contri-

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given by:

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butions of the repelling structures. Figure 18, in particular, shows the normalized cross correlations 0 Rxy =p

Rxy , Rxx (0)Ryy (0)

(9)

at the 30o farfield. Here, Rxy corresponds to the unscaled cross correlation, Rxx/yy (0) are the autocorrelations of each signal at zero lag, and τd in Fig. 18 is the non-dimensional lag. The repelling

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structures and dilatation at τd = 0 display near-perfect correlation, emphasizing the similarity between the two signals. The phase lag in the attracting structures is clearly seen in Fig. 18, where the peak correlation with dilatation is offset by approximately 0.5 characteristic times. However, the

cross correlation between the Eulerian and Lagrangian dilatations is identical to the auto-correlation of the Eulerian field, further validating the Lagrangian acoustic field analysis technique. In the sideline direction, σf and the Lagrangian dilatation (Fig. 19) data share an even more striking resemblance to one another. This further confirms that unstable manifolds given by the

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Figure 19: Empirical mode decomposition of 90o farfield LES probe data. The high-frequency components show noticeable differences between the various EMD modes of the different parameters.

forward FTLE coefficients are, indeed, primarily associated with high-frequency components in the

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flow. Contributions from the large-scale coherent structures within the shear layer, represented by the σb , are nearly an order of magnitude smaller than the forward-integrated component. As sug-

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gested by the data in Fig. 16, the 90-degree ‘farfield’ position is still at a relatively small distance away from the jet; hence, hydrodynamic components impose a stronger influence than at the downstream ‘farfield’ location, leading to the differences between the Eulerian and Lagrangian dilatations

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and their IMFs. Correlations of the Lagrangian parameters with dilatation are noticeably different along the sideline direction (Fig. 20) as compared to the aft-radiating direction (Fig. 18), an expected feature associated with the dual noise source theory of jets [51]. Thus, σf shares many similarities

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with dilatation.

To further emphasize the importance of the repelling structures on farfield noise, spectral analyses

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of the farfield signals, including their intrinsic mode functions, were conducted to expose the energy partition inherent in each component. As expected from the signals of Figs. 17 and 19, repelling structures constitute the bulk of the radiated energy at these locations, as can be appreciated from Fig. 21 for the aft-direction (corresponding results for the sideline direction are similar but not shown). respectively. In fact, the signal from the repelling structures always contains more energy than that propagated by the attracting structures.

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Figure 20: Auto- and cross-correlations of LES dilatation against the Lagrangian coefficients and reconstruction at

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the 90o ‘farfield’ position.

Figure 21: Fourier transform of dilatational and FTLE parameters at the 30o far-field probe.

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5. Conclusions The features of the FTLE method have been examined to elucidate the dynamics of compressible turbulent flows. A free shear layer LES comprised of a Mach 0.9 free jet is employed, where hydrodynamic features of the type that exist in incompressible flows interact closely with acoustic (dilatational) aspects. As shown in previous work, these can be educed by suitably changing the

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time of integration. In this work, we have placed particular focus on inferring the individual and combined contributions of attracting and repelling components of FTLE fields. A more detailed assessment indicates that the two types of structures manifest different phenomena in the turbulent core and the near acoustic field. In general, attracting structures are more representative of largescale features identified in experiments (low-frequency phenomena). Vortex pairing events can be

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very clearly visualized in a manner that is difficult to do in direct fashion with Eulerian methods. Repelling structures are more biased towards relatively higher frequency phenomena. Although they capture a larger portion of the acoustic energy and also correlate better with the Eulerian acoustic near field, they also complement attracting structures in the highly vortical region. A joint FTLE comprised of the difference between the two fields highlights shear layer-turbulence interactions. The different fields thus clarify the manner in which diverse events induce and modulate wavepackets

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present within the potential core. This includes entrainment and ejection phenomena that evolve as acoustic energy, which then propagates as farfield noise. Overall, for compressible flows, the FTLE approach is shown to complement statistics-based processing commonly employed in such flows,

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Acknowledgements

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because of its ability to retain the time-localization information in the events of interest.

The authors are greatly endebted to Dr. M. Crawley for providing the particle image velocimetry

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data required to extract the experimental first and second moment data for the high-speed jet under study. The authors would like to acknowledge the support of the Office of Naval Research under an Independent Applied Research project through the N-STAR program, monitored at NSWC

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IHEODTD by Dr. K. Clark. Support from a SMART Fellowship while the first author was a graduate student at The Ohio State University is also greatly appreciated, along with many fruitful discussions and comments provided by our colleagues. Partial funding for this effort was also provided to the second author by the Office of Naval Research under a task monitored by Dr. K. Millsaps. Computational resources were supported by a grant of HPC time from the DOD Supercomputing Resource Centers (DSRC) at AFRL, NAVO, and ERDC and the Ohio Supercomputer

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References [1] G. Haller, Distinguished material surfaces and coherent structures in three-dimensional fluid flows, Physica D 149 (2001) 248–277. [2] F. Lekien, S. D. Ross, The computation of finite-time Lyapunov exponents on unstructured

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meshes and for non-Eucliden manifolds, Chaos 20 (2010) 017505. [3] B. M. Cardwell, K. Mohseni, A Lagrangian view of vortex shedding and reattachment behavior in the wake of a 2D airfoil, in: 37th AIAA Fluid Dynamics Conference, Miami, FL, 2007, AIAA Paper 2007–4231.

[4] M. A. Green, C. W. Rowley, A. J. Smits, Using hyperbolic Lagrangian coherent structures to

AN US

investigate vortices in bioinspired fluid flows, Chaos 20 (2009) 017510.

[5] G. Haller, F. J. Beron–Vera, Coherent Lagrangian vortices: the black holes of turbulence, Journal of Fluid Mechanics 731 (2013) R4.

[6] M. A. Green, C. W. Rowley, G. Haller, Detection of Lagrangian coherent structures in threedimensional turbulence, Journal of Fluid Mechanics 572 (2007) 111–120.

M

[7] D. Wu, J. Li, Z. Liu, Y. Xiong, C. Zheng, P. R. Medwell, Eulerian and lagrangian stagnation plane behavior of moderate reynolds number round opposed-jets flows, Computers & Fluids

ED

133 (2016) 116–128. doi:http://dx.doi.org/10.1016/j.compfluid.2016.05.001. [8] M. A. Green, Eulerian and Lagrangian methods for coherent structure analysis in both computational and experimental data, in: 51st AIAA Aerospace Sciences Meeting, Dallas, TX, 2013,

PT

AIAA Paper 2013–0629.

[9] V. Nair, E. Alenius, S. Boij, G. Efraimsson, Inspecting sound sources in an orifice-

CE

jet flow using lagrangian coherent structures, Computers & Fluids 140 (2016) 397–405. doi:http://dx.doi.org/10.1016/j.compfluid.2016.09.001.

AC

[10] J. Jeong, F. Hussain, On the identification of a vortex, Journal of Fluid Mechanics 285 (1995) 69–94.

[11] R. C. Moura, A. F. C. Silva, E. D. V. Bigarella, A. L. Fazenda, M. A. Ortega, Lyapunov exponents and adaptive mesh refinement for high-speed flows using a discontinuous Galerkin scheme, Journal of Computational Physics 319 (2016) 9–27. [12] D. R. Gonz´ alez, R. L. Speth, D. V. Gaitonde, M. J. Lewis, Finite-time lyapunov exponent-based analysis for compressible flows, Chaos 20 (8) (2016) 083112. doi:10.1063/1.4961066. 37

ACCEPTED MANUSCRIPT

[13] E. Mollo-Christensen, Jet noise and shear flow instability seen from an experimenter’s viewpoint (similarity laws for jet instability as suggested by experiments), Journal of Applied Mechanics 34 (1967) 1–7. [14] C. Bogey, C. Bailly, Downstream subsonic jet noise: link with vortical structures intruding into

CR IP T

the jet core, C.R. Mechanique 330 (2002) 527–533. [15] D. Juv´e, M. Sunyach, G. Comte-Bellot, Intermittency of the noise emission in subsonic cold jets, Journal of Sound and Vibration 71 (3) (1980) 319–332.

[16] M. Kearney-Fischer, A. Sinha, M. Samimy, Intermittent nature of subsonic jet noise, AIAA Journal 51 (5) (2013) 1142–1155.

AN US

[17] P. Jordan, M. Schlegel, O. Stalnov, B. R. Noack, C. E. Tinney, Identifying noisy and quiet modes in a jet, in: 13th AIAA/CEAS Aeroacoustics Conference, Rome, Italy, 2007, AIAA Paper 2007–3602.

[18] A. V. G. Cavalieri, P. Jordan, Y. Gervais, M. Wei, J. B. Freund, Intermittent sound generation and its control in a free-shear flow, Physics of Fluids 22 (2010) 115113.

M

[19] K. B. M. Q. Zaman, Effects of initial boundary-layer state on subsonic jet noise, AIAA Journal 50 (8) (2012) 1784–1795.

ED

[20] C. Bogey, C. Bailly, Effects of inflow conditions and forcing on subsonic jet flows and noise, AIAA Journal 43 (5) (2005) 1000–1007. [21] R. L. Speth, D. V. Gaitonde, Near field pressure and associated coherent structures of excited

PT

jets, in: Proc. of ASME 2014 Fluids Engineering Division Summer Meeting, Chicago, IL, 2014, FEDSM2014–21186.

CE

[22] A. M. Lyapunov, The general problem of the stability of motion (translated into english by A. T. Fuller), International Journal of Control 55 (1992) 531–773.

AC

[23] S. C. Shadden, Lagrangian Coherent Structures, in: R. Grigoriev (Ed.), Transport and Mixing in Laminar Flows: From Microfluidics to Oceanic Currents, Wiley-VCH, 2012, pp. 59–89.

[24] S. C. Shadden, F. Lekien, J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D 212 (2005) 271–304. [25] S.

C.

Shadden,

J.

O.

Dabiri,

J.

dabiri.caltech.edu/software.html (2006). 38

E.

Marsden,

LCS

Matlab

Toolkit

v2,

ACCEPTED MANUSCRIPT

[26] M. Farazmand, G. Haller, Attracting and repelling Lagrangian coherent structures from a single computation, Chaos 23 (2013) 023101. [27] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd Edition, Society

CR IP T

for Industrial and Applied Mathematics, Philadelphia, PA, 1999. [28] J. Kasten, C. Petz, I. Hotz, B. R. Noack, H. C. Hege, Localized finite-time Lyapunov exponent for unsteady flow analysis, in: Proc. Vision, Modeling, and Visualization (VMV’09), 2009, pp. 265–276, vol. 1.

[29] D. A. Nelson, G. B. Jacobs, DG-FTLE: Lagrangian coherent structures with high-order

AN US

discontinuous-Galerkin methods, Journal of Computational Physics 295 (2015) 65–86.

[30] D. A. Nelson, G. B. Jacobs, High-order visualization of three-dimensional lagrangian coherent

structures

with

DG-FTLE,

Computers

&

Fluids

139

(2016)

197–215.

doi:http://dx.doi.org/10.1016/j.compfluid.2016.07.007.

[31] M. Crawley, A. Sinha, M. Samimy, Near-field and acoustic far-field response of a high-speed jet

M

to excitation, AIAA Journal 53 (7) (2015) 1894–1909.

[32] D. V. Gaitonde, M. Samimy, Coherent structures in plasma-actuator controlled supersonic jets:

ED

Axisymmetric and mixed azimuthal modes, Physics of Fluids 23 (2011) 095104. [33] R. L. Speth, D. V. Gaitonde, Parametric study of a Mach 1.3 cold jet excited by the flapping

PT

mode using plasma actuators, Computers & Fluids 84 (2013) 16–34. [34] R. Beam, R. Warming, An implicit factored scheme for the compressible Navier-Stokes equa-

CE

tions, AIAA Journal 16 (4) (1978) 393–402. [35] F. F. Grinstein, L. G. Margolin, W. J. Rider, Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics, Cambridge University Press, 2007.

AC

[36] P. Roe, Approximate riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics 43 (1981) 357–372.

[37] B. van Leer, Towards the ultimate conservation difference scheme V, a second-order sequel to Godunov’s method, Journal of Computational Physics 32 (1979) 101–136. [38] J. Blazek, Computational Fluid Dynamics, Elsevier, 2001.

39

ACCEPTED MANUSCRIPT

[39] M. Klein, A. Sadiki, J. Janicka, A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations, Journal of Computational Physics 186 (2003) 652–665. [40] M. C. Adler, D. R. Gonz´ alez, C. Stack, D. V. Gaitonde, Synthetic generation of turbulence for

CR IP T

direct simulation from modeled statistics, Computers & FluidsSubmitted. [41] E. Touber, N. D. Sandham, Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble, Theoretical and Computational Fluid Dynamics 23 (2009) 79–107.

[42] G. E. P. Box, M. E. Muller, A note on the generation of random normal deviates, The Annals

AN US

of Mathematical Statistics 29 (2) (1958) 610–611.

[43] T. S. Lund, X. Wu, K. D. Squires, Generation of turbulent inflow data for spatially-developing boundary layer simulations, Journal of Computational Physics 140 (1998) 233–258. [44] J. C. Lau, P. J. Morris, M. J. Fisher, Measurements in subsonic and supersonic free jets using laser velocimeter, Journal of Fluid Mechanics 93 (1) (1979) 1–27.

M

[45] J. B. Freund, Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9, Journal of Fluid Mechanics 438 (2001) 277–305.

ED

[46] D. Bodony, S. Lele, On using large-eddy simulation for the prediction of noise from cold and heated turbulent jets, Physics of Fluids 17 (8) (2005) 085103. [47] V. H. Arakeri, A. Krothapalli, V. Siddavaram, M. B. Alkislar, L. M. Lourenco, On the use of

PT

microjets to suppres turbulence in a Mach 0.9 axisymmetric jet, Journal of Fluid Mechanics 490 (2003) 75–98.

CE

[48] E. Mollo-Christensen, M. A. Kolpin, J. R. Martucelli, Experiments on jet flows and jet noise far-field spectra and directivity patterns, Journal of Fluid Mechanics 18 (1964) 285–301.

AC

[49] P. A. Lush, Measurements of subsonic jet noise and comparison with theory, Journal of Fluid Mechanics 46 (3) (1971) 477–500.

[50] D. R. Gonz´ alez, R. L. Speth, M. C. Adler, L. P. Riley, D. V. Gaitonde, M. J. Lewis, Lagrangian characterization of a subsonic turbulent jet near-field, in: 46th AIAA Fluid Dynamics Conference, Washington, DC, 2016, AIAA Paper 2016–4094. [51] C. K. W. Tam, K. Viswanathan, K. K. Ahuja, J. Panda, The sources of jet noise: experimental evidence, Journal of Fluid Mechanics 615 (2008) 253–292. 40

ACCEPTED MANUSCRIPT

[52] J. W. Hall, J. Pinier, A. Hall, M. Glauser, Two-point correlations of the near and far-field pressure in a transonic jet, in: 2nd Joint U.S.-European Fluids Engineering Summer Meeting, Miami, FL, 2006, FEDSM2006–98458. [53] C. Bogey, O. Marsden, C. Bailly, A computational study of the effects of nozzle-exit turbulence

Conference, Portland, OR, 2011, AIAA Paper 2011–2837.

CR IP T

level on the flow and acoustic fields of a subsonic jet, in: 17th AIAA/CEAS Aeroacoustics

[54] C. Bogey, C. Bailly, An analysis of the correlations between the turbulent flow and the sound pressure fields of subsonic jets, Journal of Fluid Mechanics 583 (2007) 71–97.

[55] P. Jordan, T. Colonius, Wave packets and turbulent jet noise, Annual Review of Fluid Mechanics

AN US

45 (2013) 173–195.

[56] S. Unnikrishnan, D. V. Gaitonde, Acoustic, hydrodynamic and thermal modes in a supersonic cold jet, Journal of Fluid Mechanics 800 (2016) 387–432.

[57] C. Bogey, O. Marsden, Influence of nozzle-exit boundary-layer profile on high-subsonic jets, in: 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, 2014, AIAA Paper 2014–2600.

M

[58] A. Towne, A. V. G. Cavalieri, P. Jordan, T. Colonius, V. Jaunet, O. T. Schmidt, G. A. Br`es, Trapped acoustic waves in the potential core of subsonic jets, in: 22nd AIAA/CEAS Aeroa-

ED

coustics Conference, Lyon, France, 2016, AIAA Paper 2016–2809. [59] O. T. Schmidt, A. Towne, T. Colonius, P. Jordan, V. Jaunet, A. V. G. Cavalieri, G. A. Br`es, Super- and multi-directive acoustic radiation by linear global modes of a turbulent jet, in: 22nd

PT

AIAA/CEAS Aeroacoustics Conference, Lyon, France, 2016, AIAA Paper 2016–2808. [60] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung,

CE

H. H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. Lond. A 454 (1998) 903–995.

AC

[61] L. Agostini, M. Leschziner, D. Gaitonde, Skewness-induced asymmetric modulation of smallscale turbulence by large-scale structures, Physics of Fluids 28 (1) (2016) 015110.

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