Oliver T Schmidt

Aaron Towne

Tim Colonius

André V G Cavalieri

Peter Jordan

Guillaume A Brès

Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability

Coherent features of a turbulent Mach 0.9, Reynolds number 10 6 jet are educed from a high-fidelity large eddy simulation. Besides the well-known Kelvin-Helmholtz instabilities of the shear-layer, a new class of trapped acoustic waves is identified in the potential core. A global linear stability analysis based on the turbulent mean flow is conducted. The trapped acoustic waves form branches of discrete eigenvalues in the global spectrum, and the corresponding global modes accurately match the educed structures. Discrete trapped acoustic modes occur in hierarchy determined by their radial and axial order. A local dispersion relation is constructed from the global modes, and found to agree favorably with an empirical dispersion relation educed from the simulation data. The product between direct and adjoint modes is then used to isolate the trapped waves. Under certain conditions, resonance in the form of a beating occurs between trapped acoustic waves of positive and negative group velocities. This resonance explains why the trapped modes are prominently observed in the simulation and as tones in previous experimental studies. In the past, these tones were attributed to external factors. Here, we show that they are an intrinsic feature of high-subsonic jets that can be unambiguously identified by a global linear stability analysis.

Introduction

Large-scale coherent structures, or wavepackets, in turbulent jets have been subjected to intensive study, largely because of their role in the noise production of aircraft engines (Jordan & Colonius 2013). In the wake of early experimental observations by [START_REF] Mollo-Christensen | Measurements of near field pressure of subsonic jets[END_REF] and [START_REF] Crow | Orderly structure in jet turbulence[END_REF], theorists developed a number of instability wave theories to model these structures. Examples include parallel and weakly non-parallel linear stability analyses by [START_REF] Michalke | Instability of a compressible circular free jet with consideration of the influence of the jet boundary layer thickness[END_REF][START_REF] Michalke | Survey on jet instability theory[END_REF] and [START_REF] Crighton | Stability of slowly diverging jet flow[END_REF], respectively. The use of parabolized stability equations (PSE) to model wavepackets was pioneered by [START_REF] Yen | Application of parabolized stability equations to the prediction of jet instabilities[END_REF]. Later, [START_REF] Gudmundsson | Instability wave models for the near-field fluctuations of turbulent jets[END_REF] and [START_REF] Cavalieri | Wavepackets in the velocity field of turbulent jets[END_REF] found close agreement between their PSE solutions and the leading spectral proper orthogonal modes (POD) obtained from experiment. An important implication of their work is that linear analyses of the turbulent mean provide approximations to the statistically averaged coherent structures.

The capability of linear global modes to capture both the hydrodynamic near-field as well as the far-field acoustics was demonstrated by [START_REF] Nichols | Global modes and transient response of a cold supersonic jet[END_REF] for the example of a cold M = 2.5 laminar jet. Apart from the Kelvin-Helmholtz-type (K-H) shear-layer instability, the global modes contained upstream traveling acoustic waves related to modes predicted from the local analysis of [START_REF] Tam | On the three families of instability waves of high-speed jets[END_REF]. In some cases, these upstream-travelling waves may lead to resonance mechanisms. An example occurs in impinging jets, where feedback between K-H wavepackets and acoustic modes may lead to tones in the pressure field [START_REF] Tam | Theoretical model of discrete tone generation by impinging jets[END_REF]. Although some tones have also been observed in the near field of free high-subsonic jets [START_REF] Suzuki | Instability waves in a subsonic round jet detected using a near-field phased microphone array[END_REF], their relation to a feedback mechanism has not been established.

The present paper considers the global stability of turbulent jets at high-subsonic speeds, and provides a thorough comparison with experimentally validated large eddy simulation data. We first show that large-scale coherent structures are a pertinent feature of fully turbulent jets, and persist over a wide range of frequencies and azimuthal wavenumbers. Spectral POD and cross spectral density estimates are used to educe these structures from the LES data. Besides the well-known K-H instability waves in the shear-layer, acoustic waves that are trapped inside the potential core are found. In particular, we investigate a novel class of resonant acoustic modes, and show that they are a pertinent feature of high-subsonic turbulent jets. Throughout the study, emphasis is placed on the mutual comparison of the educed structures and linear global modes.

The paper is organized as follows. Our methodology is introduced in §2, including an overview of the LES database, the global stability approach, and the coherent structure eduction techniques. The identified coherent structures are discussed in §3, and compared to linear global modes in §4. A special class of resonant trapped acoustic waves is investigated in detail in §5. A dispersion relation of the latter modes is reconstructed from the global modes, and compared to the empirical dispersion relation obtained from the LES. Our findings are summarized in §6. Details on the numerical aspects can be found in appendices A-D.

Methodology

Turbulent jet database

The present study builds on the large eddy simulation (LES) of a M j = 0.9, Re ≈ 10 6 turbulent jet, performed using the compressible flow solver "Charles" developed at Cascade Technologies [START_REF] Brès | Unstructured large-eddy simulations of supersonic jets[END_REF]. By M j = U j /a j we denote the Mach number, and by Re = ρ j U j D/µ j the Reynolds number, respectively. The jet is isothermal with T j /T ∞ = 1. The subscripts j and ∞ denote jet and free-stream (ambient) quantities, a speed of sound, ρ density, D nozzle diameter, µ dynamic viscosity, T temperature, and U j the centerline jet velocity at the nozzle exit, respectively. We denote by q(x, t) = [ρ u x u r u θ T ] T (x, r, θ, t) the vector of dependent variables, with u x , u r , u θ as the cylindrical velocity components in the streamwise x, radial r, and azimuthal direction θ, respectively. All variables are non-dimensionalized by their centerline nozzle value, pressure by ρ j U 2 j , lengths by D, and time by D/U j . The frequency is represented by the dimensionless Strouhal number St = ω/(2πM j ), where ω is the angular frequency. The LES domain contains a convergent-straight nozzle. Its turbulent exit characteristics match the conditions of the tripped exit boundary layer of its experimental counterpart thanks to the introduction of synthetic turbulence combined with a wall model. The reader is referred to [START_REF] Brès | Large eddy simulation of a mach 0.9 turbulent jet[END_REF][START_REF] Brès | Large eddy simulation for jet noise: the importance of getting the boundary layer right[END_REF][START_REF] Brès | Large eddy simulation for jet noise: azimuthal decomposition and intermittency of the radiated sound[END_REF] for further details and the extensive experimental validation, and to [START_REF] Bodony | Current status of jet noise predictions using large-eddy simulation[END_REF] for an overview of the use of LES in jet noise prediction. The database used in this work consists of 10 4 snapshots, temporally separated by ∆tc ∞ /D = 0.2, and interpolated on a 656×138×128 cylindrical grid spanning x, r, θ ∈ [0, 30] × [0, 6] × [0, 2π], i.e., excluding the nozzle. Data sampled in a smaller domain but at a higher rate of ∆tc ∞ /D = 0.05 is used only to construct the frequency-wavenumber diagrams shown in figures 11, 12, 13.

The usual Reynolds decomposition

q(x, t) = q(x) + q ′ (x, t) (2.1)
of any given flow quantity q(x, t) into a long-time mean q(x) and a fluctuation part q ′ (x, t) is adopted. The visualization of the streamwise perturbation velocity u ′ x presented in figure 1 highlights the wide range of spatial and temporal scales occurring in the turbulent jet. Two lines of constant mean streamwise velocity ūx = 0.95 and 0.05 delineate the potential core and the radial extent of the jet. The same isolines at ūx = 0.95 and 0.05 are reproduced in all later figures that show the streamwise-radial plane.

Global stability analysis

In what follows, we adopt the notion that "turbulence establishes an equivalent laminar flow profile as far as large-scale modes are concerned" [START_REF] Crighton | Stability of slowly diverging jet flow[END_REF], implying that the mean Reynolds stresses are implicitly accounted for in the turbulent mean flow that we draw from the LES. The fluctuating Reynolds stresses, to the extent that they are significant at a given frequency, are assumed to act as a disorganized forcing on the large scale structures. This viewpoint has been adopted in a number of studies exploiting resolvent analysis (e.g. [START_REF] Farrell | Stochastic forcing of the linearized navier-stokes equations[END_REF][START_REF] Mckeon | A critical-layer framework for turbulent pipe flow[END_REF][START_REF] Towne | Stochastic and nonlinear forcing of wavepackets in a mach 0.9 jet[END_REF], and is supported by previous studies comparing parabolized stability equation models for the coherent wavepackets in jets to microphone and PIV data [START_REF] Gudmundsson | Instability wave models for the near-field fluctuations of turbulent jets[END_REF][START_REF] Cavalieri | Wavepackets in the velocity field of turbulent jets[END_REF]). In the present case, in advance of a full resolvent analysis of the jet, we further neglect this stochastic forcing and instead catalog the unforced behavior of the linear modes of the turbulent mean flow field. Neglecting nonlinear interaction terms in the governing equations, and assuming perturbations of the normal-mode form q ′ (x, r, θ, t) = qm (x, r)e i(mθ-ωt) ,

(2.2) yields the usual eigenvalue problem

(iω + L m ) qm = 0, (2.3)
where L m = L m (q) is the global linear stability operator based on the mean flow. In our case, L m represents the system of the linearized compressible continuity, Navier-Stokes, and energy equations for an ideal gas in cylindrical coordinates. The radial-streamwise plane is discretized by fourth-order summation by parts-finite differences developed by [START_REF] Mattsson | Summation by parts operators for finite difference approximations of second derivatives[END_REF], and the polar singularity is treated as in [START_REF] Mohseni | Numerical treatment of polar coordinate singularities[END_REF]. Nonreflecting boundary conditions are enforced using a sponge region.

The resulting discrete generalized eigenvalue problem reads

(ωI + L m ) qm = 0, (2.4)
where I is the identity matrix, and qm = [ρ m ûx,m ûr,m ûθ,m Tm ] T (x, r) the solution vector of primitive variables. It is solved using a shift-and-invert Arnoldi algorithm. The corresponding adjoint global stability eigenvalue problem

ω * I + L † m q † m = 0, (2.5)
is defined through the definition of the adjoint operator,

L m qm , q † m E = qm , L † m q † m E
in the energy norm for a compressible gas

q, q E = q H diag T γρM 2 j , ρ, ρ, ρ, ρ γ(γ -1)T M 2 j q r dxdrdθ = q H W q,
(2.6) as derived by [START_REF] Chu | On the energy transfer to small disturbances in fluid flow (Part I)[END_REF]. Here, W is the discretized weight matrix, γ the specific heat ratio, and ( * ) indicates the scalar complex conjugate, or the conjugate transpose for vectors and matrices, respectively. For its ease of implementation, we chose the discreteadjoint framework [START_REF] Chandler | Adjoint algorithms for the navier-stokes equations in the low mach number limit[END_REF]. The adjoint eigenvectors q † m are normalized according to q † m , qm E = 1. Direct and adjoint global modes are calculated in the computational domain Ω outlined in figure 1(a) for a Reynolds number of Re = 10 5 . Details on the grid resolution, the sponge layer, and the spatial discretization can be found in appendix A. The influence of the sponge on the trapped acoustic modes is discussed in appendix B, and the effect of domain truncation is discussed in appendix C. Our choice to conduct a linear stability analysis of the turbulent mean flow is motivated, and the effect of the of Reynolds number on the global spectra is addressed in detail, in appendix D.

Coherent structure eduction techniques

As the jet is periodic in the azimuthal direction, the fluctuation field can be decomposed into Fourier modes of azimuthal wavenumber m as q ′ (x, r, θ, t) = m qm (x, r, t)e imθ , (2.7) and further, under the assumption of statistical stationarity in time, into spatio-temporal modes qm (x, r, t) = ω qm,ω (x, r)e imθ e iωt (2.8) of angular frequency ω. Spectra are estimated using Welch's method by partitioning the time series into sequences of 256 snapshots with an overlap of 50%. This partition is found to be a good compromise between a sufficiently high discrete frequency spacing, and a sufficiently high number N b = 78 of statistically independent realizations, or blocks. The power spectral density (PSD) for a given frequency and azimuthal wave number is then estimated as

Pqq (x, r) = 1 N b N b n=1 q(n) m,ω (x, r) 2 ,
(2.9)

and the cross spectral density (CSD) between any point (x, r) in the domain and a specific location (x 0 , r 0 ) as

Cqq (x, r, x 0 , r 0 ) = 1 N b N b n=1 q(n) m,ω (x, r) q * (n) m,ω (x 0 , r 0 ).
(2.10) Alternatively, frequency-domain or spectral (POD) can be applied to extract coherent structures from the column matrix of short-time Fourier realizations

Q m,ω = [ q(1) m,ω , q(2) m,ω , • • • , q(N b ) m,ω
] through the solution of the eigenvalue problem (2.11) where Q m,ω Ψ is the column matrix of orthonormal spectral POD modes whose energy is given by the eigenvalues along the diagonal of Λ. The spectral POD modes optimally represent the data in terms of the energy norm induced by the inner product q, q E (2.6) at each given frequency. As an example, CSD and spectral POD are employed to extract the coherent portion of the flow field in figure 2. Plots 2(a-d) show the first two realizations in the ensemble of short-time Fourier decompositions used for the spectral estimation.

Ψ * Q * m,ω WQ m,ω Ψ = Λ,
The K-H instability and a monochromatic high-wavenumber fluctuation in the potential core are observed in both realizations. We will show later that the wave inside the core is acoustic and propagates upstream with a negative phase velocity. In our parallel investigation, [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF] show that this latter disturbance is an acoustic wave that experiences the shear layer as an annual duct and is therefore trapped within the potential core. The CSD is particularly useful to identify spatially correlated portions of the fluctuation field in a specific region. The K-H instability, for example, is accentuated in figure 2(e,f) for a correlation location (x 0 , r 0 ) = (5, 0.5) within the shear-layer. Similarly, a location (x 0 , r 0 ) = (0.1, 0.1) close to the nozzle centerline in figure 2(g,h) highlights the acoustic wave. It also shows that the fluctuations in the core are correlated with the shear-layer instability. The K-H wave packet is identified by POD as the most energetic structure based on a global pressure 2-norm in figure 2(i,j). The comparison of the different eduction techniques leads to the following choices for the remainder of this paper. POD estimates are best suited to distill the average K-H wavepacket as it represents the most energetic coherent structure. Acoustic waves inside the potential core are most unambiguously educed via a CSD of the pressure field with respect to a point near the inlet and close to the jet axis.

Coherent features of the turbulent jet

The two families of coherent instabilities of interest for the present investigation are the Kelvin-Helmholtz wavepackets in the shear-layer, and the trapped acoustic waves in the potential core. Their frequency and azimuthal wavenumber dependence is addressed in §3.1 and §3.2, respectively.

Kelvin-Helmholtz wavepackets

In figure 3, the K-H wavepacket is identified as the most energetic coherent structure of the LES via POD for m = 0 (left column) and m = 4 (right column) over a range of frequencies. The wavepackets are characterized by a monotonic increase in wavenumber and a simultaneous decrease of their streamwise support with frequency. Their phase speed is c ph ≈ 0.8U j in all cases, and was estimated as explained below in the context of figure 8.

The K-H instability amplifies to the end of the spatially unstable region of the mean

Trapped acoustic waves

Trapped acoustic waves are extracted by use of CSD in figure 4 for the same Strouhal numbers considered in figure 3 above. The fluctuations inside the potential core are strongly correlated with the K-H wave in all cases. Inside the potential core, acoustic waves are observed for some St-m combinations, most prominently in figures 4(c,e,g,i,j). These trapped acoustic modes can be characterized by a radial order n r equal to the number of anti-nodes in the radial direction. By counting the anti-nodes, we find n r = 1 in 4(c,e), n r = 2 in 4(g,j), andn r = 3 in 4(i).

Global mode spectrum

The global linear stability eigenvalue problem (2.4) is solved as described in §2.2 for 0 m 4. For each azimuthal wavenumber m, the spectrum is shifted to 43 positions along the real axis in the interval 1.5 σ 12, or 0.24 St 1.91 correspondingly. 35 eigenvalue/eigenvector pairs are converged at each position by the standard shift-andinvert Arnoldi algorithm. The resulting spectra are presented in figure 5. All eigenvalues are stable as expected. Two common features of the spectra are of special interest for the present study. First, the discrete mode branches indicated by their leading mode (blue), and second, a branch of continuous modes from which they emerge. The structure of the spectra is similar, but with a shift towards higher frequencies with increasing m. The discrete branches are located approximately within the same frequency bands for all branches of even and odd m, respectively. The least-damped mode marked as (0, 1, 1) in figure 5(a) is found for m = 0 and St ≈ 0.37. In what follows, the modal structures of the continuous and discrete modes are investigated in §4.1 and §4.2, respectively. -0.04. This mode is mainly of K-H type with only a weak acoustic component inside the potential core. The modal shape compares well with the CSD from the LES data shown in 6(a). Here, the correlation location of the CSD is chosen such that it extracts the shear-layer instability component. A correlation location inside the potential core brings to light the trapped acoustic wave, as discussed later in the context of figure 7(a,b).

Features of the continuous branch

A remarkable agreement is found for m = 0 and St = 0.5 as depicted in figure 6(c,d). Both the LES and the modal analysis identify a global structure that combines a K-H wave train with an acoustic core mode. The regions of support and the wavenumbers of both components are accurately predicted by the global mode. A similar modal structure can be seen in the m = 2, St ≈ 1 case in figure 6(e,f). In this higher azimuthal wavenumber and frequency example, the CSD extracts the K-H wavepacket as the most energetic structure, whereas the trapped acoustic wave is dominant in the global mode.

In the spectrum, the continuous modes are unavoidably rendered discrete due to the numerical discretization. Changes in the numerical discretization scheme can freely shift their eigenvalues along the underlying continuous branch. Changing the computational domain size, as discussed in appendix C, is an example where such behavior can be observed.

The physical aspect of interest for the present study is that the continuous modes consist of two coupled wave components: a downstream propagating K-H wavepacket, and a trapped acoustic wave in the potential core.

Features of the discrete modes

The leading discrete global modes of the first three branches for m = 0, and the first branch for m = 1 are depicted in figure 7(b,d,f) and 7(h), respectively. Clearly, the discrete global modes represent the trapped acoustic mechanism. Unlike their continuous counterparts seen in figure 6(d,f) above, the discrete modes are confined within a small region x 2 close to the nozzle. Their radial order n r becomes apparent by comparing modal shapes for different branches as in figure 7(b,d,f). For higher m, the n r = 1 branch is shifted towards a higher frequency as can be deduced from 7(h) in comparison with the global spectra shown in figure 5. Note that all discrete global modes also possess a weaker K-H component, in contrast with the continuous modes which show K-H and trapped waves to be more strongly coupled. By comparison with the CSD (left column), it can be seen that the trapped acoustic modes are evidently present in the LES data. Their spatial extent and waveforms are accurately predicted by the linear global analysis. A similarly good agreement is found for all m under consideration.

The pressure profile along the jet axis shown in figure 8(a) reveals the internal structure of the modes within a given branch. It can be seen that the absolute pressure signal possesses an increasing number of anti-nodes (now with respect to the x axis) with increasing frequency. This suggests the definition of an integer streamwise wavenumber n x equal to the number of anti-nodes. Individual discrete modes can be identified by the triplet (m, n r , n x ). The latter observation also has an important physical implication. The corresponding eigenvalues are reproduced in the detail of the spectrum in figure 8(b) for comparison. Figure 8(c) depicts the phase velocity of the same modes estimated as ω/ ∂θp ∂x along the jet axis, where θ p = arg (p) is the local phase of the eigenvector pressure component. A jump discontinuity separates the pressure signal of each mode into part with a negative phase speed of c p ≈ -0.2, and a part with a positive phase speed of c p 0.8U j . The radial order n r previously addressed in the discussion of figure 7, is demonstrated in 8(d) in terms of radial velocity profiles of the leading modes of the first three branches for m = 0. In our parallel investigation, [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF] show that these waves behave like acoustic waves that experience the potential core as a cylindrical duct with a pressure release surface. Accordingly, they are effectively trapped within the potential core and their radial structure takes the form of Bessel functions. Therefore, we refer to them as duct modes.

Resonant acoustic modes

Comparison with LES

We next examine the structure of the discrete modes in more detail by comparing the modes of the (0,1) branch with the simulation in terms of the estimated PSD defined by equation (2.9). In order to account for the small frequency spacing between individual modes, the spectral estimation is based on a larger number of 1024 snapshots in the following. This results in a smaller ensemble of N b = 19 realizations for the same overlap.

Figure 9 shows that we find a very good agreement between the global modes and the LES in terms of the location and structure of the trapped acoustic waves. The number of maximum nodes agrees precisely with our definition of the streamwise wavenumber n x , as introduced in the context of figure 8(a). The trapped acoustic waves are spatially confined in the streamwise direction. In the inlet plane, we find that the nozzle impedance associated with the LES simulation is well mimicked by the inlet sponge boundary condition. This favorable agreement was found robust with respect to changes in the sponge layer intensity, as detailed in appendix B. The length of the region over which the trapped waves have support is frequency dependent. For example, mode (0,1,1) in figure 9(a) has a streamwise spatial support of 0 x 2, whereas mode (0,1,6) shown in figure 9(a) is confined within 0 x 3.5. The observation is explained by the existence of a frequency-dependent end-condition related to the streamwise contraction of the potential core. We identify this end-condition using weakly-nonparallel stability theory in [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF], and compare the prediction to the spatial support of the global modes in §5.6.

Experimental observations

Figure 10 compares discrete mode frequencies to spectra obtained from two sets of experimental data. In a recent experimental campaign dedicated to the validation of the present LES, the near-field pressure of a M = 0.9 turbulent jet was measured in the "Bruit & Vent" jet-noise facility of the PPRIME Institute, Poitiers. In a similar experiment by Suzuki & Colonius (2006, appendix B) for M j = 0.98, T j /T ∞ = 0.84, and Re = 16 × 10 5 , the measured tones were attributed to an acoustic duct phenomenon upstream of the nozzle. Both experiments used a 6 microphone ring array located in the near-nozzle region. Spectra for the axisymmetric and first helical Fourier components are reproduced in figure 10(a) and 10(b), respectively. It can be seen that the peaks in the spectra agree well with the frequencies of the leading discrete modes found by the global analysis.

Frequency-wavenumber analysis

Our next goal is to describe the trapped acoustic instabilities in the frequencywavenumber domain. For a fixed frequency ω and azimuthal wavenumber m, the streamwise wavenumber content of the trapped modes can readily be obtained from a Fourier decomposition of pressure signal on or parallel to the jet axis. Here, we chose pm,ω (0 x 5, r 0 ), and denote by k the streamwise wavenumber. The exact length and radial location of the line segment used for the decomposition has no significant effect on the results, as long as it remains located inside the potential core. Windowing or padding of the segment is not necessary for the same reasons.

The resulting frequency-wavenumber diagram for m = 0 is shown in figure 11. The PSD is estimated by means of equation (2.9). In the positive k-plane, energy is concentrated along the line of constant slope ω/k ≈ 0.8U j ( ), corresponding to the typical downstream propagation velocity of the initial jet shear-layer instability. The curved bands of energy in the angular sector of negative phase velocities -a j c p U j -a j are the signatures of the trapped acoustic modes. These bands can be interpreted as empirical dispersion relations of the K-H and trapped acoustic modes, as they relate their frequency to their wavenumber. The lowest band originates from trapped acoustic modes of radial order n r = 1, the second of n r = 2, and so forth. Note that the shape of the energy bands implies that multiple waves with negative phase velocity can coexist at certain frequencies. Furthermore, it can be observed that the two ends of each band asymptotically approach the lines U j -a j and -a j , respectively. These two limits correspond to acoustic duct behaviour ( ) with waves propagating against the jet velocity inside the potential core, and acoustic free-stream behaviour ( ), with waves propagating upstream in the ambient flow. Details focusing on the trapped acoustic instability mechanism for higher azimuthal wavenumbers are presented in figure 12. It can be seen that the onset of the acoustic instability is shifted towards higher frequencies with increasing azimuthal wavenumber m. A comparison with the global stability spectra shown in figure 5 confirms that the the local minima (valleys) occur precisely a the frequencies of the leading global modes of each discrete branch. For m = 0 and m = 1, the PSD peaks within the n r = 1 energy bands, whereas higher radial order modes are most energetic for m > 1.

Reconstruction of the dispersion relation from global modes

We now focus on the wavenumber content of the global modes for a direct comparison with the LES. A frequency-wavenumber diagram analogous to figure 11 can be constructed from the global modes using exactly the same method as described above. For each mode, a Fourier decomposition of the pressure pm ( 0x 5, r 0 ) yields a wavenumber spectrum as depicted in figure 13(c). In the particular case of mode (0,1,2) in this example, the spectrum exhibits three distinct peaks denoted by 'k -', 'k + ', and 'K-H.'

The 'k -', 'k + ', and 'K-H' peaks are identified for all discrete modes. Connecting them in the St-k plane yields the red lines in 13(a). Analogously, the leading ( , free-stream behavior) and trailing ( , duct behavior) sections are found by connecting the dominant peaks in the spectrum of the continuous modes in negative k-plane. Continuous segments can now readily be interpreted as reconstructed dispersion relations for the trapped waves ( + • + ) and the K-H waves ( ).

A detail of the spectrum around branch (0,1) is reproduced in figure 13(b) using the same colors and symbols as in the frequency-wavenumber diagram in figure 13(a) for comparison. The concave part (red) of the lower band is directly associated with the discrete branch (0,1) modes. Analogously, the upper band at St 0.8 is associated with the (0,2) branch.

The mode located at the discrete-continuous branch intersection (+) in figure 13(a,b) separates continuous from discrete, and also continuous duct modes from continuous freestream modes. The local slope along each segment of the dispersion relation determines the group velocity of the instability wave at the corresponding wavenumber and frequency. This motivates the distinction between a k + ( ) segment of positive, and a k - ( ) segment of negative group velocity, respectively. A remarkable agreement between the reconstructed and the empirical dispersion relations is found. The discrete global modes contain both the k + and the k -wave. This observation shows that these two waves are intimately linked, and suggests a resonance phenomenon, as we will confirm in the following.

Of particular interest are the two end points of the k + segment as they mark zeros of the slope of the dispersion relation, and therefore points of zero group velocity. These points are of particular interest as they determine the long-time behavior of a system in a fixed laboratory reference frame [START_REF] Briggs | Electron-stream interaction with plasmas[END_REF]. The lower point denoted as S1 is associated with the leading mode, and the upper one denoted by S2 with the mode that marks the intersection of the discrete branch with the continuous spectrum. These points describe the transition from a mechanism that is predominantly active close to or inside the shear-layer, i.e., continuous free-stream modes, to a duct-like behavior inside the core, i.e., to continuous core modes. The anti-nodes used to determine the integer streamwise wavenumber n x , and as seen in figures 8(a) and 9, can now be interpreted in the light of the above discussion as a beating

e ik + x + e ik -x = 2 cos ∆k ± 2 x exp i k + + k - 2 x (5.1)
resulting from the superposition of the k + and the k -waves. As an example, consider mode (0,1,3) with k + = -7.1, k -= -13.0, and a resulting difference of ∆k ± = 5.9. The corresponding wavelength of the beating, λ ± = 4π/∆k ± = 2.1, can be seen to accurately match twice the distance of the maximum nodes of mode (0,1,3) in figure 8(a). The average wavenumber (k + + k -)/2 on the other hand determines the wavelength seen in instantaneous, or visualizations of the real part as in figure 7. Together with the observation that the acoustic modes are confined within a short region in the streamwise direction by end-conditions, it can be concluded at this point that a resonance between the k + and k -segments occurs. The resonance phenomenon in turn explains the high energy content along the frequency band in which k + and k -coexist. For the same reason, the corresponding modes appear as discrete eigenvalues in the spectrum.

For frequencies below the lowest-radial-order discrete branch, e.g. St 0.37 for m = 0, the potential core does not support duct modes. The absence of the duct mechanism appears to be directly related to the sudden drop of the eigenvalues at low St in the global stability spectra, as seen in figure 5 above. This becomes particularly apparent at higher azimuthal wave numbers. For m = 3 in figure 5(d), for example, all modes are heavily damped up to St ≈ 1. As soon as duct modes are supported, the discrete mode Ppp/ Ppp ∞ 1) and the isolated trapped acoustic modes ( qm (x0, r0) • q † m (x0, r0) , ). Contours of the product map correspond to 20%, 40%, 60%, and 80% of its maximum value.

branch (3,1) appears above a branch of K-H modes which now support acoustic core waves.

A caveat on the comparison between the reconstructed and the empirical dispersion relations is that the discrete nature of the trapped acoustic modes is not evident from the LES data. In the LES frequency-wavenumber diagram, the energy appears to be smoothly distributed along the individual bands. Presumably, a much longer time series would be needed to converge the statistics of individual discrete modes.

Isolation of the trapped acoustic waves

In classical local stability theory, each spatial mode is associated with a complex wavenumber given by its eigenvalue. By contrast, a single global mode of a specific complex frequency can encapsulate a variety of waves. We are interested in isolating the trapped acoustic wave component from the global modes to determine their spatial support, and for comparison with local theory.

We have observed that adjoint trapped modes have a negligible K-H component. Examples and a brief discussion of the adjoint modes used here can be found in appendix E. Hence, the map of the local product

qm (x 0 , r 0 ) • q † m (x 0 , r 0 ) (5.2)
is a way of isolating the trapped-wave mechanism from the K-H wavepackets. In a different context, such maps have been used by [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF] and [START_REF] Chomaz | Global instabilities in spatially developing flows: non-normality and nonlinearity[END_REF] to determine an upper bound for the sensitivity of eigenvalues to spatially localized feedback. Figure 14 compares the map of the direct-adjoint product with the estimated PSD from the simulation. The analysis clearly identifies the maximum nodes of the pressure PSD as the regions of highest sensitivity, and therefore closely resembles the PSD distribution of the global modes as seen in figure 9. Comparing figures 9 and 14, it becomes evident that multiplying the direct solution by the adjoint largely removes the shear-layer wavepackets. We can thus make use of the direct-adjoint product as a filter. The K-H instability constitutes , Im(∂ω/∂k). Note that in (d), the saddles S1 and S1 cannot be distinguished, and the assignment is therefore arbitrary.

a convective non-normality characterized by an upstream-downstream separation of the spatial support of the adjoint and direct modes [START_REF] Sipp | Dynamics and control of global instabilities in open-flows: a linearized approach[END_REF][START_REF] Nichols | Global modes and transient response of a cold supersonic jet[END_REF]. It is therefore widely eliminated from the product, and the trapped waves are clearly isolated. Furthermore, the match between the PSD and the isolated trapped modes suggests that the trapped instability mechanism is nearly self-adjoint.

Local linear theory

The study of the resonant trapped acoustic modes in terms of local and weaklynonparallel spatio-temporal linear theory is at the core of the parallel investigation by [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF]. Here, we revisit one of the main findings of the weakly-nonparallel theory regarding the end-condition of the resonant modes, as it allows for a direct comparison with the global modes. In particular, we are interested in finding saddle points in the complex k-plane, i.e. the points k 0 at which ∂ω/∂k = 0. In such saddle points, k 0 is a double root of the dispersion relationship. These points are relevant to provide endconditions if they are formed between a k + and a k -mode [START_REF] Huerre | Local and global instabilities in spatially developing flows[END_REF]. Defining ω 0 = ω(k 0 ) (i.e., ω 0 and k 0 satisfy the locally-parallel dispersion relationship), if ω 0 is real, a saddle point presumably corresponds to the points S1 and S2 in the reconstructed dispersion relation shown in figure 13. The reader is referred to [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF] for a detailed discussion. Under the assumption of a locally parallel flow, i.e., homogeneity in the streamwise direction, the normal mode ansatz (2.2) takes the , second saddle point location), isolated trapped modes ( ), and mode at the branch intersection ( , denoted by '+' in figure 13(a,b) for branch (0,1)). The normalized direct-adjoint product is shown along the jet axis for m = 0, and along r0 = 0.25 for m = 1 in (b). The trapped modes clearly confined between the inlet and the saddle point locations. form q ′ (r, θ, t) = qm (r)e i(kx+mθ-ωt) ,

(5.3)

where the streamwise wavenumber k ∈ C is a free parameter. The resulting eigenvalue problem for ω at some streamwise location x 0 takes the same form as in equation 2.3, but with L m = L m (q(x 0 , r), k) now as the local linear stability operator. Saddle points are identified in the complex k-plane by solving the local eigenvalue problem over a range of complex wavenumbers k, and subsequently finding the zeros of the complex derivative of ω(k) for a specific mode. In figure 15, the acoustic core modes of radial order n r = 1 are traced in the complex plane. Two saddles S1 and S2 are found for x locations close to the nozzle. Both saddles reside on the real axis, and move closer together with increasing downstream distance, as seen in figure 15(a-c). At a certain streamwise location, the two saddles collide. For locations further downstream, no saddle points with real-valued k 0 are found. Beyond this point, no k + -mechanism exists, and resonance cannot occur. This case is depicted in figure 15(d).

In figure 16, the frequency of the two saddle points is plotted over the streamwise location ( , ) for three representative branches. The (0,1) branch is considered in figure 16(a), and higher azimuthal wavenumber and radial order examples in figure 16(b) and 16(c), respectively. The isolated resonant acoustic modes ( ) and the branch intersection mode ( ) are shown in the same plots for comparison. A good agreement between the spatial support of the resonant waves as identified by the global stability analysis and the location of the S1 saddle is found in all cases. It becomes apparent that the saddle point serves as an end-condition for the trapped acoustic modes. With increasing St, resonance first occurs for the leading discrete mode of axial order n x = 1 at the lowest possible frequency that permits the interaction between two waves of opposite group velocity with k + ≈ k -. At higher frequencies, the beating between the two waves k+ > k -causes the observed modulation of the trapped modes of streamwise order n x > 1. The frequency of the highest streamwise wavenumber mode approximately coincides with the minimum frequency of the S2 saddle. Beyond this point, no resonant core modes are to be found. Apparently, the the S2 saddle provides an alternative end-condition. In contrast to the S1 saddle, S2 is associated with a k -wave which apparently has a lower reflection coefficient at the nozzle than the duct modes. This becomes apparent from an inspection of the branch intersection modes ( ). The signature of the S1 saddle is clearly visible, but the fluctuation levels between the nozzle and S1 are much lower than for the discrete modes. From the detail views depicted in figure 16(d) and 16(e), respectively, the influence of the S2 saddle can be inferred in the case of the the (0,1) and (1,1) branches, respectively. A clear beating signature is visible only between the nozzle exit and S2, whereas the fluctuation has a richer wavenumber content between S1 and S2.

The side-by-side comparison of local and global stability results in figure 16 highlights the complementary merits of both approaches. The global mode analysis reveals the connection between the different stability phenomena, identifies internal feedback, and predicts the acoustic resonance. On the other hand, the local approach provides further physical insight about the physical identity of the waves and the nature of the endconditions.

Summary and conclusions

Linear global stability modes are found to well represent large-scale coherent structures in a natural turbulent M j = 0.9 jet. Coherent structures were extracted from the LES database by means of spectral estimation techniques, and directly compared to the global modes. The favorable agreement extended over a range of frequencies and for azimuthal wavenumbers 0 m 4.

Apart from the well-known K-H wavepackets in the shear-layer, trapped acoustic waves that are confined within the potential core are observed. In the global stability spectrum, these trapped acoustic waves appear in two forms. They are either part of the continuous branch, or appear in branches of lightly damped discrete modes. The continuous acoustic waveforms can be further distinguished into two categories. Continuous duct modes are strictly confined within the potential core and propagate upstream with c p U j -a j , whereas continuous free-stream modes propagate with a phase velocity of c p -a j . This associates them with the ambient flow outside of the shear-layer.

Branches of discrete acoustic modes appear in the spectrum in a hierarchical order determined by their radial wavenumber. They possess two wave components, both of negative phase velocity, but of group velocities of opposite sign denoted as k + and k -waves, respectively. The contraction of the potential core with downstream distance imposes a frequency dependent end-condition on the trapped acoustic waves. The nozzle impedance in the inlet plane provides the second end-condition. At certain frequencies, the distance from the nozzle to the downstream end-condition becomes an integer multiple of the wavenumber difference of the two waves, and an acoustic resonance in the form a beating occurs. The dispersion relation of the trapped acoustic modes was reconstructed from the global modes, and found in remarkable agreement with an empirical frequency-wavenumber diagram extracted from the LES.

In [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF], we suggest that the upstream and downstream end-conditions for the resonance are provided by turning points originating from the nozzle impedance and the streamwise contraction of the potential core, respectively. The interpretation that the saddles in the complex k-plane act as turning points for the trapped waves is confirmed by the favorable agreement between the saddle point locations and the spatial support of the global modes. A simplified cylindrical vortex sheet model also allows us to study the Mach number dependency of the trapped acoustic mechanism. The model predicts two neutral saddle points, and therefore resonance, in the narrow range of Mach numbers 0.82 < M j < 1 for isothermal jets. For Mach numbers outside of this range, the model predictions are consistent with the findings of [START_REF] Tam | On the three families of instability waves of high-speed jets[END_REF].

The present study shows that intrinsic stability mechanisms, such as the resonant trapped acoustic modes, are brought to light by laminar global linear stability analysis of the turbulent mean flow. They appear as distinct eigenvalues in the spectrum, and the corresponding eigenvectors accurately represent the average coherent structures educed from the data. The underlying assumption of the approach is hence justified a posteriori. & Lele (2011). The latter authors found that the heads of the branches that correspond to upstream-propagating subsonic modes described by the local theory of [START_REF] Tam | On the three families of instability waves of high-speed jets[END_REF] remained fixed within the spectrum for their enlarged domain while the continuous K-H mode branches get significantly shifted. Similarly, the continuous mode branches in our calculations get shifted downwards and upwards when the domain length is decreased as can be seen in figure 19(a) and 19(b), respectively. Below a certain frequency, this behavior is not observed. In any case, the eigenvalues of the trapped acoustic modes that are the main focus of this paper are not influenced by the axial domain truncation. This observation is in accordance what [START_REF] Nichols | Global modes and transient response of a cold supersonic jet[END_REF] observed for the closely related upstream-propagating subsonic modes in their supersonic laminar jet.

Appendix D. Reynolds number effects

A crucial parameter for this study is the Reynolds number, and since we investigate the stability of a turbulent mean flow, the choice is not obvious. The turbulent mean flow is a solution to the Reynolds-averaged Navier-Stokes (RANS) equations with an appropriate turbulence model. The RANS equations can be linearized about the meanflow, as demonstrated by [START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF] and [START_REF] Meliga | Sensitivity of 2-d turbulent flow past a d-shaped cylinder using global stability[END_REF]. The results, however, are not necessarily superior to those obtained by using the linearized Navier-Stokes equations based on the molecular viscosity, as shown by [START_REF] Mettot | Quasi-laminar stability and sensitivity analyses for turbulent flows: Prediction of low-frequency unsteadiness and passive control[END_REF]. Garnaud et al. (2013b), for example, chose a molecular viscosity corresponding to Re = 1000 for their study of a turbulent jet mean flow model, noting that the choice is rather arbitrary. Other authors like [START_REF] Jeun | Input-output analysis of high-speed axisymmetric isothermal jet noise[END_REF] considered the inviscid limit by linearizing the Euler equations.

To address the Reynolds number effect, we compute spectra for different Reynolds numbers spanning two orders of magnitude, and compare them in figure 20(a-d). At the lowest Reynolds number Re = 10 3 in figure 20(a), the spectrum appears more damped and the trapped acoustic modes are not observed. This is an imported result as it indicates that crucial physical aspects can be missed when the Reynolds number is chosen too small. When the Reynolds number is increased to Re = 10 4 in figure 20(b), the trapped modes clearly appear in the spectrum. Increasing the Reynolds number further reveals that the first trapped acoustic mode branch becomes effectively Reynolds number independent for Re 5 • 10 4 . This is also observed for the higher radial mode branches, but at higher Reynolds numbers. The convergence of the frequency and amplification rate of the leading mode (0,1,1) is further detailed in 20(e) and 20(f), respectively. Notably, the Strouhal number changes only in the third significant digit over the three orders of magnitude considered. The continuous mode branches, however, get shifted up with least damped mode (0,1,1); (c,d) higher streamwise order mode (0,1,3); (e,f) higher radial order mode (0,2,1); (g,h) higher azimuthal wavenumber mode (1,1,1). The normalized pressure (✷ , -0.1 Re(pm)/ Re(pm) ∞ 0.1) is shown.

increasing Reynolds number. This result is in accordance with the findings by Garnaud et al. (2013b), who observed increased resolvent gain rates with increased Reynolds number. The determination of an appropriate efficient Reynolds number for turbulent jets remains subject of future research.

Figure 1 .

 1 Figure 1. Instantaneous streamwise perturbation velocity (✷ , -0.5 u ′ x / u ′ x ∞ 0.5) and streamwise mean velocity ( , ūx = 0.95; , ūx = 0.05) of the LES: (a) streamwise plane and computational domain Ω used for the linear stability analysis ( , solution domain; , sponge region) and (b-e) transverse planes at x = 2, 5, 10 and 15, respectively.

Figure 2 .

 2 Figure 2. Spectral decomposition and coherent feature extraction for m = 0, St ≈ 0.5 (✷ , ±0.5 of the maximum value in (a-j) and ±0.1 in (k-n)): (a-d) the first two realizations of the 256 snapshot based Fourier decomposition; (e-h,k,l) CSD using different correlation points (x0, r0); (i,j,m,n) leading POD mode estimates. The pressure and streamwise velocity component are shown in the left and right column, respectively. The CSD correlates each point of the flow-field with a location (x0, r0) marked by '+', and the POD is based on the volume weighted 2-norm.

Figure 3 .

 3 Figure 3. Spectral estimation of pressure POD modes (✷ , -0.25 Ψp/ Ψp ∞ 0.25) for different Strouhal numbers: (left column) m = 0; (right column) m = 4.

Figure 4 .

 4 Figure 4. CSD (✷ , -0.25 Cpp/ Cpp ∞ 0.25) correlating each point of the flow-field with the location (x0, r0) = (0.1, 0.1) marked as '+' in the potential core for different Strouhal numbers and (left column) m = 0, and (right column) m = 2.

Figure 6

 6 Figure6shows three representative side-by-side comparisons of coherent structures distilled from the LES via CSD (left column), and global modes of the continuous branch (right column). The eigenvalue corresponding to the mode shown in figure6(b) is located right below the least-damped mode (0,1,1) in figure5, i.e., at St ≈ 0.37 and ω i ≈

Figure 5 .

 5 Figure 5. Global stability spectra for (a) m = 0, (b) m = 1, (c) m = 2, (d) m = 3, (e) m = 4. The leading discrete modes (→) of each branch are indicated by the triplet (m, nr, nx), where m is the azimuthal, and nr and nx the the radial and streamwise integer wavenumbers, respectively. This distinction of the discrete modes is addressed in the discussion of figure 8, below. Three continuous modes (→) shown in 6(b,d,e) are also indicated.

Figure 6 .

 6 Figure 6. Comparison between (left column) CSD of the LES with location (x0, r0) marked by '+', and (right column) global modes: (a,b) m = 0, St ≈ 0.37; (c,d) m = 0, St ≈ 0.5; (e,f) m = 2, St ≈ 1. Modes are compared in terms of their normalized pressure components (✷ , -0.25 Re( pm,ω )/ Re( pm,ω ) ∞ 0.25, -0.25 Re(pm)/ Re(pm) ∞ 0.25).

Figure 7 .

 7 Figure 7. Comparison between (left column) CSD of the LES with location (x0, r0) marked by '+', and (right column) global modes: (a,b) m = 0, St ≈ 0.37, mode (0,1,1); (c,d) m = 0, St ≈ 0.87, mode (0,2,1); (e,f) m = 1, St ≈ 0.6, mode (1,1,1). Modes are compared in terms of their normalized pressure components (✷ , -0.25 Re( pm,ω )/ Re( pm,ω ) ∞ 0.25, -0.25 Re(pm)/ Re(pm) ∞ 0.25).

Figure 8 .

 8 Figure 8. Characterization of trapped acoustic modes in the potential core for m = 0: (a) pressure magnitude along the jet axis; (b) detail of the spectrum showing the (0,1)-branch (•••, modes from (a,c); •, other discrete modes; •, continuous modes); (c) local phase velocity of the pressure along the jet axis; (d) pressure magnitude profiles at x = 1. All quantities are normalized with respect to their maximum value. The discrete modes are labeled as (m,nr,nx), corresponding to their azimuthal wavenumber m, their radial order nr obtained as shown in (d), and their streamwise order obtained as shown in (a), respectively.

Figure 9 .

 9 Figure 9. Comparison between the pressure components of LES power spectral density estimate (✷ , 0 Ppp/ Ppp ∞ 1) and global mode (p m p * m , ) for branch (0,1): (a) St ≈ 0.374, mode (0,1,1); (b) St ≈ 0.382, mode (0,1,2); (c) St ≈ 0.39, mode (0,1,3); (d) St ≈ 0.397, mode (0,1,4); (e) St ≈ 0.405, mode (0,1,5); (f) St ≈ 0.41, mode (0,1,6). Contours of pm p * m correspond to 20%, 40%, 60%, and 80% of its maximum value.

Figure 10 .

 10 Figure 10. Near-nozzle exit spectrum comparing Strouhal numbers of the least-damped global modes (• • • ) to the sound pressure level (SPL) of recent "Bruit & Vent" measurements ( ), and the data of Suzuki & Colonius (2006) ( ) for Mj = 0.98, Tj/T∞ = 0.84: (a) m = 0; (b) m = 1.

Figure 11 .

 11 Figure 11. Frequency-wavenumber diagram (✷ , Ppp(0 x 5, r0)) of the LES data in the potential core region: (a) m = 0, r0 = 0; (b) m = 1, r0 = 0.1. Lines of constant propagation velocity cp = ω/k = cg = ∂ω/∂k and their corresponding values are shown in color. cp = 0.8Uj corresponds to the propagation speed of K-H instabilities.

Figure 12 .

 12 Figure 12. LES power spectral density in form of a frequency-wavenumber diagram obtained from the pressure signal along the jet axis in the potential core x ∈ [0 5]: (a) m = 0; (b) m = 1; (c) m = 2; (d) m = 3; (e) m = 4. The Strouhal numbers of the leading global modes of each discrete branch (• • •; marked by '→' in figure 5) are shown for comparison.

Figure 13 .

 13 Figure 13. Reconstruction of the dispersion relations of the trapped acoustic and K-H instabilities from a spectral analysis of the pressure signal along the jet axis in the potential core x ∈ [0 5]: (a) frequency-wavenumber diagram comparing LES (✷ , Ppp) as in figure 12(a), and global modes (•, mode (0,1,1); , , trapped-mode branches; , , continuous modes; , K-H component); (b) LST spectrum around branch (0,1) (•, mode (0,1,1); + branch intersection mode; • other (0,1) modes; •,• continuous modes); (c) wavenumber spectrum of mode (0,1,2). Colors and symbols in (a) and (b) correspond. Since all modes comprise a K-H component, it is not indicated in (b).

Figure 14 .

 14 Figure 14. Same as figure 9 but comparing LES power spectral density estimate (✷ , 0Ppp/ Ppp ∞ 1) and the isolated trapped acoustic modes ( qm (x0, r0) • q † m (x0, r0) , ). Contours of the product map correspond to 20%, 40%, 60%, and 80% of its maximum value.

Figure 15 .

 15 Figure 15. Saddle point location of the (0,1) branch in the complex wavenumber plane obtained from local spatio-temporal stability theory at different streamwise positions: (a) x = 0; (b) x = 1.5; (c) x = 2.5; (d) x = 3.8. Isolines of the real and imaginary part of the complex frequency ( , ωr; , ωi) are shown, and the locations of the the two saddle points (•, saddle S1; •, saddle S2) are obtained as the zeros of the complex derivative Re(∂ω/∂ks) = Im(∂ω/∂ks) = 0 ( , Re(∂ω/∂k);, Im(∂ω/∂k). Note that in (d), the saddles S1 and S1 cannot be distinguished, and the assignment is therefore arbitrary.

Figure 16 .

 16 Figure 16. Comparison between local spatio-temporal stability theory ( , first saddle point location;, second saddle point location), isolated trapped modes ( ), and mode at the branch intersection ( , denoted by '+' in figure13(a,b) for branch (0,1)). The normalized direct-adjoint product is shown along the jet axis for m = 0, and along r0 = 0.25 for m = 1 in (b). The trapped modes clearly confined between the inlet and the saddle point locations.

Figure 17 .

 17 Figure 17. Influence of the sponge gain ǫsponge on trapped acoustic modes and comparison with LES: (a) CSD of the LES as in 7(a); global modes with (b) ǫsponge = 50, (c) ǫsponge = 5, and (d) ǫsponge = 0.25; (e) centerline pressure profiles. The same contour levels as in figure 7 are used in (a-d), and all profiles are normalized to unit maximum amplitude in (e). The LES PSD ( Ppp, ) in (e) corresponds to the PSD along the axis in figure 9(a). The magenta line ( ) indicates the beginning of the inlet sponge region in x < 0.

Figure 18 .

 18 Figure 18. Influence of the far-field boundary location on the global spectrum: the reference solution with r end = 2.25 (•, reference) is compared to an enlarged domain (•, higher domain) with r end = 3.25. The number of grid points in the radial direction is increased to Nr = 350 for the higher domain. The computed area of each spectrum is outlined in the respective color.

Figure 19 .

 19 Figure 19. Influence of the domain length on the global spectrum: the reference solution with x end = 25 (•, reference) is compared to a shortened domain with x end = 20 (•, shorter domain) in (a), and an elongated domain with x end = 30 (•, longer domain) in (b). The computed area of each spectrum is outlined in the respective color.

Figure 20 .

 20 Figure 20. Influence of the Reynolds number on the global spectrum: (a) •, Re = 10 3 ; (b) •, Re = 10 4 ; (c) •, Re = 5•10 4 ; (d) •, Re = 15•10 4 ; (e) Strouhal number, and (f) amplification rate dependence of mode (0,1,1). The reference solution with Re = 10 5 (•, reference) is reproduced in (a-d) for comparison. The computed area of each spectrum is outlined in the respective color in (a-d).

Figure 21 .

 21 Figure 21. Comparison of direct (left column) and adjoint (right column) global modes: (a,b)least damped mode (0,1,1); (c,d) higher streamwise order mode (0,1,3); (e,f) higher radial order mode (0,2,1); (g,h) higher azimuthal wavenumber mode (1,1,1). The normalized pressure (✷ , -0.1 Re(pm)/ Re(pm) ∞ 0.1) is shown.

† Email address for correspondence: oschmidt@caltech.edu

Acknowledgements

OTS gratefully acknowledges support by DFG grant no. 3114/1-1. We gratefully acknowledge support from the Office of Naval Research under contract N0014-11-1-0753 and N00014-16-1-2445. AVGC and PJ acknowledge support from the Science Without Borders program (project number A073/2013). We are grateful to S. Piantanida and R. Kari for conducting the "Bruit & Vent" measurements. The LES study was supported by NAVAIR SBIR project, under the supervision of Dr. John T. Spyropoulos. The main LES calculations were carried out on CRAY XE6 machines at DoD HPC facilities in ERDC DSRC.

Appendix

Appendix A motivates the choice of parameters for the global stability study, and gives details on the numerical discretization and the damping region. The appendices B, C, D and E address the influence of the inlet sponge layer on the trapped acoustic modes, the effect of domain truncation, the effect of the Reynolds number, and the adjoint solutions to the trapped acoustic modes, respectively. In appendices B-D, the spectrum of the production run setup with a solution domain of size x, r ∈ [0, 25] × [0, 2.25], a Reynolds number of Re = 10 5 , and sponge intensity of ǫ sponge = 5 is reproduced as a reference.

Appendix A. Computational setup

The computational domain Ω outlined in figure 1(a) is discretized by a 1150×250 points Cartesian grid. The solution domain x, r ∈ [0, 25] × [0, 2.25] is padded by a sponge region to prevent reflections from the boundaries. The resulting overall domain size is x, r ∈ [-1.2, 26.2] × [0, 2.8]. A parallel flow corresponding to the inlet profile q(0, r) is assumed in the inlet sponge layer region x < 0. The domain is constructed by first defining an equidistantly spaced mesh over x, r ∈ [-0.5, 25.5] × [0, 2.5], and second applying an exponential grid stretching of the form exp(1.5η) to the region outside of the solution domain. 0 η 1 is the normalized coordinate normal to the respective boundary, or the local sponge layer coordinate. Inside this region, the base state is assumed to be parallel, and the solution is penalized by a forcing of the form -ǫ sponge ( 6η 5 -15η 4 + 10η 3 ), where η is now normalized with respect to the stretched coordinate. A fifth-order polynomial distribution smoothly ramps the source-term from zero to ǫ sponge = 5. The support of the trapped acoustic modes that are the main subject of the linear stability study is limited to the potential core in the downstream and radial directions. The sponge has therefore a negligible influence on these modes in these directions. In absence of the nozzle in the computational domain, the inlet sponge layer plays an important role in mimicking the inlet impedance experienced by upstream traveling waves in the core, as further addressed in appendix B.

The spatial resolution used was found sufficient to resolve the increasingly fine modal structures up to St ≈ 2. In the radial direction, the resolution is chosen such that the jet boundary layer in the inlet plane, and therefore the initial shear-layer, is resolved by δ inlet /∆r = 0.08/0.010 = 8 grid points. Under the assumption that the dominant shear layer instability advects at a phase speed of c ph ≈ 0.8U j (see §4.2), a resolution of λ K-H /∆x = 0.39/0.023 ≈ 17 grid points per wavelength is achieved at St = 2, the maximum Strouhal number considered.

Appendix B. Inlet sponge layer influence

The nozzle exit plane plays a crucial role as an end-condition for the resonating trapped modes, as discussed in §5. If is therefore important that the inlet sponge layer region correctly accounts for the unknown nozzle impedance.

Figure 17 compares global mode pressure fields and pressure profiles along the centerline to the LES data. The least damped global mode (0,0,1) is chosen as an example. It can be seen that the global mode pressure fields in figure 17(b-d) are in good agreement with the LES CSD in figure 17(a) despite the wide range of sponge intensities 0.25 ǫ sponge 50. The centerline profiles reproduced in figure 17(e) allow for a more quantitative comparison. The LES PSD pressure peak location and profile shape of the trapped disturbance is accurately predicted by the global analysis for the two higher sponge gain cases. The comparison is less favorable for the ǫ sponge = 0.25. The low gain in this case allows the trapped wave to penetrate into the sponge layer, and the pressure peak gets shifted towards the inlet. An intermediate value of ǫ sponge = 5 was found adequate to prevent reflections from any of the boundaries, and to mimic the inlet impedance at the same time. The favorable agreement between the modal structures predicted by the linear analysis and those extracted from the LES seen in figures 7, 9 and 14 confirms this statement.

Appendix C. Domain size influence

We first study the effect of the radial domain extent by comparing the reference spectrum to one obtained using a 44% higher computational domain in figure 18. In the following, the lengths r end and x end denote the domain length excluding the sponge region in the radial and streamwise directions, respectively. The number of grid points is increased accordingly to keep the radial grid spacing comparable. The two spectra coincide to a large degree for all but the lowest frequencies St 0.35. At these low frequencies, an inspection of the eigenvectors shows that they get truncated in the radial direction towards the end of the domain. For the Strouhal number regime of interest, and in particular for the trapped acoustic modes, the radial domain size of the reference grid is found sufficient.

The effect of domain truncation in the streamwise direction is more critical for convection dominated flows and was addressed in detail by Garnaud et al. (2013a) and Nichols

Appendix E. Adjoint global modes

Direct and adjoint trapped acoustic modes at different radial and axial orders as well as azimuthal wavenumbers are compared in figure 21. The signature of the trapped acoustic waves inside the potential core appear evidently isolated in the adjoint modes for the reasons discussed in §5.5 above. This is a clear indication for the tapped acoustic modes being nearly self-adjoint, which is a consequence of their acoustic nature. We exploit this property in figures 14 and 16. The calculation of adjoint continuous modes is beyond the scope of this work. It is more challenging as the adjoint solution of these convective instabilities is comprised of very fine structures that are orientated against the mean-shear close to the nozzle. This is a typical observation, and explains the modes' potential for large transient growth through the Orr-mechanism.


Xiaoyan Li

Jieli Lyu

Claire Goldmann

Mathieu Kociak

Doru Constantin
email: doru.constantin@u-psud.fr

Cyrille Hamon
email: cyrille.hamon@u-psud.fr

Plasmonic Oligomers with Tunable Conductive Nanojunctions

Engineering plasmonic hot-spots is essential for applications of plasmonic nanoparticles. A particularly appealing route is to weld plasmonic nanoparticles together to form more complex structures sustaining plasmons with symmetries targeted to given applications. However, the control of the welding and subsequent hotspot characteristic is still challenging. Herein, we demonstrate an original method that connects gold particles to their neighbors by another metal of choice. We first assemble gold bipyramids in a tip-to-tip configuration, yielding short chains of variable length and grow metallic junctions in a second step. We follow the chain formation and the deposition of the second metal (i.e. silver or palladium) via UV/Vis spectroscopy and we map the plasmonic properties using electron energy loss spectroscopy. The formation of silver bridges leads to a huge redshift of the longitudinal plasmon modes into the mid-infrared region, while the addition of palladium results in a redshift accompanied by significant plasmon damping.

Noble metal nanoparticles (NPs) can support light-driven collective electron oscillations, known as localized surface plasmon resonances (LSPR), which can be tuned by the composition, shape and environment of the NPs. Plasmon hybridization can take place when NPs which sustain modes with close eigenergies enter into proximity so that the electromagnetic field of these modes overlap spatially, resulting in amplified electromagnetic fields at the junction between nanoparticles, the enhancement being more intense as the distance decreases. [START_REF] Prodan | A Hybridization model for the plasmon response of complex nanostructures[END_REF][START_REF] Halas | Plasmons in strongly coupled metallic nanostructures[END_REF][START_REF] Aizpurua | Optical properties of coupled metallic nanorods for field-enhanced spectroscopy[END_REF][START_REF] Sonnichsen | A molecular ruler based on plasmon coupling of single gold and silver nanoparticles[END_REF][START_REF] Slaughter | Toward plasmonic polymers[END_REF] These regions of intense optical activity have led to applications (ranging from biosensing to metamaterial design) that strongly depend on the detailed geometry of the metal surfaces. [START_REF] Schuller | Plasmonics for extreme light concentration and manipulation[END_REF][START_REF] Lane | SERS Nanoparticles in Medicine: From label-free detection to spectroscopic tagging[END_REF] Controlling the shape and chemical nature of the optically actives areas (or hotspots) in colloidal plasmonic nanosystems has a very wide range of applications. For instance, one can tailor plasmon coupling by bridging nanoparticles together with molecular junctions whose conductivity can be switched at will. [START_REF] Zhu | Quantum mechanical effects in plasmonic structures with subnanometre gaps[END_REF][START_REF] Pérez-González | Optical spectroscopy of conductive junctions in plasmonic cavities[END_REF] Other strategies focus on controlling the metallurgy of the junctions in which nanoparticles are immobilized on a surface or in colloidal suspension.

For instance, the conductivity of the junctions of gold nanoparticle dimers adsorbed on an electrode has been tuned by electrodeposition. [START_REF] Byers | From tunable core-shell nanoparticles to plasmonic drawbridges: Active control of nanoparticle optical properties[END_REF][START_REF] Hoener | Plasmonic sensing and control of single-nanoparticle electrochemistry[END_REF] Controlled melting of adjacent nanoparticles has been performed at the single particle level by an electron beam in order to connect gold nanospheres into solid networks featuring low-energy plasmon modes. [START_REF] Teulle | Multimodal plasmonics in fused colloidal networks[END_REF] Colloidal approaches include welding nanoparticle assemblies using ultrafast laser pulses. [START_REF] González-Rubio | Femtosecond laser-controlled tip-to-tip assembly and welding of gold nanorods[END_REF][START_REF] Herrmann | Threading plasmonic nanoparticle strings with light[END_REF] In another example, conductive connections were engineered by controlling the sintering of gold nanoparticle dimers within a silica shell. [START_REF] Fang | Dry sintering meets wet silverion "soldering": charge-transfer plasmon engineering of solution-assembled gold nanodimers from visible to near-infrared I and II regions[END_REF] Another group has controlled the silver thickness at the junction of asymmetric dimers linked with DNA. [START_REF] Lee | Plasmonic nanosnowmen with a conductive junction as highly tunable nanoantenna structures and sensitive, quantitative and multiplexable surface-enhanced Raman scattering probes[END_REF] Although some colloidal-based strategies have been demonstrated, engineering plasmonic assemblies with tunable conductive junctions remains challenging. Current methods rely on the controlled fusing or melting of nanoparticle assemblies but do not allow control over the metallic nature of the nanojunction. Moreover, most studies have focused on dimers rather than on larger objects, which are however more interesting for the directional control of light.

In this work, we developed a seed mediated growth procedure to deposit a metal of choice at the hotspots location in nanoparticles assemblies. We produced colloidally stable assemblies of gold bipyramids and then grew metallic patches at the nanoparticle junctions. We focused on silver and palladium due to their notable plasmonic and photocatalytic properties, respectively. UV/Vis spectroscopy and Electron Energy Loss Spectroscopy (EELS) were used to describe the evolution of the collective plasmonic response of the nanostructures. Upon bridging the nanoparticles we observed a strong variation of the optical properties, which can be tuned by the nature of the metallic junction.

Elongated chain-like structures comprising separated nanoparticles are often referred to as plasmonic "polymers", in direct analogy with their molecular counterparts. [START_REF] Slaughter | Toward plasmonic polymers[END_REF][START_REF] Liu | Step-growth polymerization of inorganic nanoparticles[END_REF][START_REF] Guerrero-Martinez | Molecular thinking for nanoplasmonic design[END_REF] In waterbased synthetic procedures, such objects are generally impractical for further use, due to the fast self-assembly kinetics (which leads to precipitation) and/or to surface passivation by thiolated ligands. A recent study provided an alternative to dithiolated linkers by showing that sulfate ions can mediate end-to-end attachment of gold nanorods. [START_REF] Abtahi | Sulfate-mediated end-to-end assembly of gold nanorods[END_REF] We built upon this study to assemble gold bipyramids (GBPs) oligomers and refined the process to better control the chain formation kinetics (Figure 1). The preparation and characterization of GBPs assemblies are described in the Supporting Information. In short, oligomers were obtained by introducing 1 mM MgSO4 into the GBPs suspension (Figure 1A). We monitored the evolution of the optical properties by means of timeresolved UV/Vis spectroscopy (Figure 1B). The intensity of the longitudinal surface plasmon resonance (LSPR) band at 696 nm intensity drops over 20h while the transversal surface plasmon resonance (TSPR) band at 510 nm did not change noticeably. Simultaneously, a new plasmon band appears which gradually increases in intensity and red-shifts with respect to the LSPR band of isolated GBPs. Resulting assemblies were analyzed by Transmission Electron Microscopy (TEM) and showed a progressive elongation of the chains over time and a preferential end-to-end arrangement (Figure 1D-E and Figure S2). This structuration leads to coupling of the dipolar modes in the chain-like assembly similar to the trend observed in chains of plasmonic nanoparticles with other shapes, such as spheres and rods. [START_REF] Halas | Plasmons in strongly coupled metallic nanostructures[END_REF] We tuned the chain formation kinetics by adjusting the CTAB concentration in the micromolar range (Figure S3). However, without additional intervention the reaction progresses until flocculation and is impractical for further use. Fortunately, we found that the self-assembly process can be stopped at the desired stage simply by adding excess surfactant to the reaction mixture. The extinction spectra of the oligomer solutions did not evolve over 1 month after supplementing the reaction mixture with a 10 mM hexadecyltrimethylammonium chloride (CTAC) solution, demonstrating that the system is locked in a stable state (Figure 1C).

Adding surfactant after 5h of reaction yields dispersions with a predominance of the monomers plasmon band over the oligomer one. In comparison, almost no monomers remained in suspension after 20h (Figure 2B and Figure S5). This experiment shows that quenching the reaction at different time intervals allows controlling the number of GBPs per chain. Moreover, the assemblies are sturdy: they can be centrifuged and re-dispersed without affecting their optical properties (Figure S6). Overall, this strategy presents two advantages: (i) Tip-to-tip assembly simply results from a combination of salt and surfactants (with no need for stronglybinding molecules). (ii) The reaction can be frozen at any stage of the process, yielding colloidally-stable chains of the desired length.

We then explored how engineering conductive nanojunctions could control the optical properties of the nanochains by seed mediated growth. We bridged the gap between GBPs tips by injecting the assemblies as seeds in a growth solution containing a metal precursor and a mild reducing agent (ascorbic acid). In order to build nanojunction, it is important to introduce a relatively low quantity of metal (e.g. 0.2 molar equivalent Metal/Au) to avoid the formation of core-shell structures (Figure S7). Silver, palladium and platinum have been deposited in this way (Figure 2). In the following, we denote by "oligomers" and "{Ag, Pd, Pt or Au}-oligomers" the assemblies before and after the creation of the conductive nanojunctions, respectively. The formation of connected oligomers is demonstrated by the TEM images, in which the GBPs are initially distinct (Figure 2A) but form connected solid objects after the overgrowth reaction (Figure 2B-D, Figure S8). A clear evolution of the optical properties is also observed by UV/Vis spectroscopy (Figure 2B). In addition to the LSPR band corresponding to the monomers (dashed line), the suspensions exhibit two plasmon bands corresponding to the oligomers. In the case of Pd-and Pt-oligomers, the LSPR bands significantly broadens due to the interband transitions in palladium and platinum (Figure 2B-C). [START_REF] Khurgin | Reflecting upon the losses in plasmonics and metamaterials[END_REF][START_REF] Langhammer | Plasmonic properties of supported Pt and Pd nanostructures[END_REF][START_REF] Garcia | Surface plasmons in metallic nanoparticles: fundamentals and applications[END_REF] In contrast to palladium and platinum, silver displays a sharp plasmonic response in the visible-IR spectral range that is preserved when deposited on gold. In the case of silver, the LSPR band of the monomers at 696 nm shifts towards lower energies due to the deposition of thin Ag layers (Figure 2D). [START_REF] Fernandes | Acoustic vibrations of Au nano-bipyramids and their modification under Ag deposition: a perspective for the development of nanobalances[END_REF] The plasmon band of the oligomers shifted out of the visible range after connecting the NPs together. We performed electrodynamical simulation by the Boundary Element Method (BEM) using the MNPBEM17 toolbox [START_REF] Hohenester | Simulating electron energy loss spectroscopy with the MNPBEM toolbox[END_REF][START_REF] Hohenester | MNPBEM -A Matlab toolbox for the simulation of plasmonic nanoparticles[END_REF] and the LSPR of continuous oligomers of different length was shifted systematically beyond 1200 nm (Figure S9). This suggests that the LSPR of the Ag-oligomers was shifted outside the experimentally accessible spectral range. Although UV/Vis measurements demonstrate a clear transition of the optical properties after connecting the particles together, it only provides information on large particle ensembles, and does not give access to the IR range relevant for long assemblies. Therefore, we used scanning transmission electron microscopy-electron energy loss spectroscopy (STEM-EELS) in order to study in more detail the collective plasmonic properties of such structures at the single particle level. Indeed, STEM-EELS is an effective technique for interrogating the plasmonic response of individual and assembled nanoparticles, including one-dimensional chains. [START_REF] Teulle | Multimodal plasmonics in fused colloidal networks[END_REF][START_REF] Kociak | Mapping plasmons at the nanometer scale in an electron microscope[END_REF][START_REF] Nelayah | Mapping surface plasmons on a single metallic nanoparticle[END_REF][START_REF] Barrow | Mapping bright and dark modes in gold nanoparticle chains using electron energy loss spectroscopy[END_REF][START_REF] Mayer | Direct observation of plasmon band formation and delocalization in quasi-infinite nanoparticle chains[END_REF][START_REF] Koh | Electron Energy-Loss Spectroscopy (EELS) of surface plasmons in single silver nanoparticles and dimers: influence of beam damage and mapping of dark modes[END_REF] Hyperspectral measurements with a spatial resolution of a few nanometers enable the direct observation of plasmon modes. The nature of each plasmon mode can be determined by its spatial distribution and low loss energies: transverse (T) and longitudinal (L) plasmon modes can thus be characterized. The notation LN, where N-1 stands for the number of charge nodes along the assembly direction is convenient to describe modes from the lowest to highest energy.

One can distinguish the fundamental dipolar mode (L1), its harmonics (N≥2) as well as bright (odd N) and dark plasmon modes (even N). [START_REF] Barrow | Mapping bright and dark modes in gold nanoparticle chains using electron energy loss spectroscopy[END_REF][START_REF] Willingham | Energy transport in metal nanoparticle chains via sub-radiant plasmon modes[END_REF][START_REF] Barrow | Electron energy loss spectroscopy investigation into symmetry in gold trimer and tetramer plasmonic nanoparticle structures[END_REF] Moreover, plasmon mapping reveals the high symmetry of the modes across the nanostructures and the spatial distribution of the modes as well as the number of nodes further confirms the identification of LN≥2 modes. This technique complements bulk UV/Vis spectroscopy measurements by studying in more detail the collective plasmonic properties of assembled species at the single particle level.

In this work, we used a NION-Hermes 200 S monochromated Cs-corrected STEM fitted with an NION-IRIS EELS spectrometer and a Hamamatsu Kuros camera. The microscope was operated at 60 keV with a typical 20 meV spectral resolution and 0.5 nm probe size. Typical spectral-images were acquired with 5 ms dwell time and 15000 pixels. The dwell time was set such that the zero-loss peak was not saturated and the spectra could therefore be re-aligned after acquisition. From the spectral images, either spectra or filtered map could then be extracted. In this work, spectra were obtained by summing around 200 pixels in a region of interest and filtered maps were typically integrated over 200 meV energy range. The sample were prepared by drop-casting the NP on a standard silicon nitride film grid. Herein, we used STEM-EELS to follow the transition of the collective modes from GBPs oligomers to Ag-oligomers and compared the results with BEM modelling (Figure 3). Single bipyramids exhibit two energy loss peaks, corresponding to the L1 and T modes centered at 1.63 eV and 2.23 eV respectively (Figure 3b), in agreement with the EELS maps (Figure 3a).

As the number of particles in the chain increases, the L1 mode shifts to lower energies and new modes appear. This evolution of the plasmonic properties is well reproduced by the BEM model (Figure 3c), even though in the simulation we considered perfectly aligned oligomers and did not account for the particles/Si3N4-membrane interaction. In the case of dimers, we observed two LN peaks, one at 1.44 eV (L1) and 1.72 eV (L2) that are at lower and higher energies respectively than the L1 mode of the monomers. EELS maps confirm this assignment as the L1 and L2 modes are distributed around the ends of the oligomers and at their center, respectively. This observation is consistent with the trends predicted by the plasmon hybridization theory, in which the L1 mode of the dimers correspond to the bonding mode while the L2 mode correspond to the antibonding mode. For trimers, the L1 and L2 mode are shifted to lower energies and a new L3 mode is detected. From BEM simulations, we would expect the L2 mode to have a higher loss probability around the central position (Figure S10). In the experimental EELS map, the L2 mode is localized only at one particle junction, presumably because of the lower symmetry of the nanostructure compared to the BEM model. By increasing further the number of particles in the chain, the L1 mode energy converges asymptotically to a finite value due to the so-called infinite chain limit effect [START_REF] Willingham | Energy transport in metal nanoparticle chains via sub-radiant plasmon modes[END_REF][START_REF] Barrow | Surface plasmon resonances in strongly coupled gold nanosphere chains from monomer to hexamer[END_REF][START_REF] Citrin | Plasmon polaritons in finite-length metal-nanoparticle chains: the role of chain length unravelled[END_REF] and the addition of new modes within a finite energy range results in a broad plasmon band. [START_REF] Mayer | Direct observation of plasmon band formation and delocalization in quasi-infinite nanoparticle chains[END_REF] Those results are consistent with previous STEM-EELS studies describing the collective modes of spherical NPs chains, which observe as many L modes as nanoparticles in the chain. [START_REF] Barrow | Mapping bright and dark modes in gold nanoparticle chains using electron energy loss spectroscopy[END_REF][START_REF] Mayer | Direct observation of plasmon band formation and delocalization in quasi-infinite nanoparticle chains[END_REF][START_REF] Barrow | Electron energy loss spectroscopy investigation into symmetry in gold trimer and tetramer plasmonic nanoparticle structures[END_REF] Bridging the oligomers with silver has a dramatic influence on the position of the energy loss peaks (Figure 4A-B). In the case of hexamers (n = 6), the L1 mode energy is as low as 0.55 eV (2250-nm equivalent free-space photon wavelength). We attribute the large spectral shifts to the transition from capacitive to conductive coupling, which allows charge transfer between neighboring NPs. [START_REF] Halas | Plasmons in strongly coupled metallic nanostructures[END_REF][START_REF] Zhu | Quantum mechanical effects in plasmonic structures with subnanometre gaps[END_REF][START_REF] Kadkhodazadeh | Coexistence of classical and quantum plasmonics in large plasmonic structures with subnanometer gaps[END_REF][START_REF] Alber | Multipole surface plasmon resonances in conductively coupled metal nanowire dimers[END_REF] The BEM calculation reproduces this evolution by modelling a continuous gold nanostructure (Figure 4C and Figure S11). In a control experiment, NPs in oligomers were partially fused by the electron beam induced welding to create Au-oligomers.

Although Au-oligomers and Ag-oligomers were produced by two different approach, we found similar spectral and spatial distribution of the plasmon modes (Figure S12). Previous studies have shown that low-energy surface plasmon modes in Au-oligomers was resulting from the conductive junction, [START_REF] Zhu | Quantum mechanical effects in plasmonic structures with subnanometre gaps[END_REF][START_REF] Teulle | Multimodal plasmonics in fused colloidal networks[END_REF][START_REF] Su | Optical characterization of gold nanoblock dimers: from capacitive coupling to charge transfer plasmons and rod modes[END_REF][START_REF] Wen | Charge transfer plasmons: optical frequency conductances and tunable infrared resonances[END_REF] an interpretation that can be applied to the case of Ag-oligomers. This experiment confirms that a conducting silver bridge was deposited at the gap of Agoligomers by our approach. Note that the L1 mode does not systematically shift toward lower energies as the number of particles in the chain increases, but also depends on the symmetry of the assembly and can even shift toward higher energies in the case of oligomers that are not straight (see Figure 4 and Figure S13). The connected oligomers share similarities in terms of number, nature and energy of modes with their unconnected counterparts. For instance, the L2 modes are occurring at 1.70 eV and 1.72 eV in the case of dimers and connected dimers respectively. In the case of trimers, a similar loss peak energy is observed for the L3 modes at 1.85 eV and 1.84 eV before and after connection, respectively. This suggests that the higher L modes in separated and linked structures have similar physical properties, in agreement with a previous study that used templated nanostructures. [START_REF] Osberg | Systematic study of antibonding modes in gold nanorod dimers and trimers[END_REF] We also performed STEM-EELS on Pdoligomers, whose L1 mode is systematically positioned between those of oligomers and Agoligomers (Figure S14). In this case, the width of the plasmon peaks is large compared to the ones observed for Ag-oligomers due to damping by interband transitions in Pd.

In summary, we connected colloidal nanoparticles by a seed-mediated growth approach that presents three original opportunities to control the properties of plasmonic dispersion. (1) The self-assembly of gold bipyramids is controlled at will in order to modulate the optical response from the visible to the NIR range. (2) By depositing silver conductive junction, plasmon are shifted towards the mid-infrared window. (3) Catalysts such as palladium and platinum can also be deposited at the hotspots. Although the plasmonic response is damped, such a configuration is ideal for photocatalysis because hot charge carriers would be generated in the vicinity of the catalyst. [START_REF] Besteiro | Understanding hotelectron generation and plasmon relaxation in metal nanocrystals: quantum and classical mechanisms[END_REF] Similarly, our method could be applied to deposit semiconductor materials at the hotspots location to form double Schottky barriers. Overall, this benchtop method is simple, scalable and yields high-aspect-ratio nanostructures with optical properties that can be tailored across a very wide spectral range, from the visible to the mid-infrared region. We believe that this work provide a leap forward for all plasmonic application that can benefit from a precise design of the hotspots in NPs oligomers. connected oligomers, STEM-EELS characterization of Au-connected oligomers obtained by electron beam induced welding, STEM-EELS characterization of bended nanostructures, STEM/EELS of Pd-connected oligomers. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 1 :

 1 Figure 1: Chain formation and colloidal stabilization. A) Scheme illustrating the tip-to-tip

Figure 2 :

 2 Figure 2: Tunable conductive nanojunctions in plasmonic oligomers engineered by seed-

Figure 3 :

 3 Figure 3: STEM-EELS characterization of oligomers and corresponding theoretical

Figure 4 :

 4 Figure 4: STEM-EELS characterization of Ag-oligomers and corresponding theoretical

Acknowledgements

The CNRS is acknowledged for funding and support. Jieli Lyu acknowledges financial support by the China Scholarship Council (CSC). The present work has benefited from the electronic microscopy facility of Imagerie-Gif, (http://www.i2bc.paris-saclay.fr), member of IBiSA (http://www.ibisa.net), supported by "France-BioImaging" (ANR-10-INBS-04-01), and the Labex "Saclay Plant Science" (ANR-11-IDEX-0003-02). This work has received support from the National Agency for Research under the program of future investment TEMPOS-CHROMATEM with the Reference No. ANR-10-EQPX-50.

Supporting Information

Material and method sections, procedures for GBPs preparation, self-assembly, details on modeling, TEM images of the GBPs and corresponding UV-vis spectra, details on how to control chain formation kinetics, UV-Vis spectra of the GBPs diluted in water, UV-Vis spectra of stabilized oligomers after one month, UV-Vis spectra of the oligomers before and after centrifugation, additional TEM images of the oligomers, BEM modelling of oligomers and