## Abstract

In supersonic aerospace applications, aerospike nozzles have been subject of growing interest. This study sheds light on the noise components of a cold jet exhausting an aerospike nozzle. Implicit large eddy simulations (ILES) are deployed to simulate the jet at a nozzle pressure ratio (NPR)=3. For far-field acoustic computation, the Ffowcs Williams–Hawkings (FWH) equation is applied. A mesh sensitivity study is performed and the jet instantaneous and time-averaged flow characteristics are analyzed. The annular shock structure displays short non-attached shock-cells and longer attached shock-cells. Downstream of the aerospike, a circular shock-cell structure is formed with long shock-cells. Two-point cross-correlations of data acquired at monitoring points located along the shear layers allow to identify upstream propagating waves associated to screech. Power spectral density at monitoring points in the annular shock-cell structure allows to identify its radial oscillation modes. Furthermore, a vortex sheet model is adapted to predict the annular shock-cells length and the BBSAN central frequency. High sound pressure levels (SPL) are detected at the determined BBSAN central frequencies. Finally, high SPL are obtained at the radial oscillation frequencies for the annular shock-cell structure.

## Introduction

Propulsion technologies for high-speed aircraft or rockets use nozzles through which the highly-heated flow is exhausted. However, they cannot always be operated at design conditions. In these cases, the exhausted jet is not expanded ideally leading to very intense acoustic noise. Apart from mixing noise present in subsonic configurations, other acoustic components are observed in supersonic configurations such as screech noise and broadband shock-associated noise. These noise components arise due to the shock-cell structure. In cases where flow instabilities are convected at supersonic speeds across the shear layers, Mach waves are observed. In highly-heated jets, crackle noise may also be present. All these noise components were identified in circular and rectangular nozzle jets.

Screech noise is a tonal noise component. It is due to an aeroacoustic feedback mechanism establishing between acoustic waves propagating upstream and turbulent structures being convected downstream. It has been detected by Powell [1]. First, the turbulent structures developing in the jet shear layers are convected downstream and interact with the quasi-periodic jet shock-cell structure of the jet, creating upstream propagating acoustic waves. These upstream propagating waves reach the nozzle and impinge on the trailing edge of the latter. The resonant loop is closed when the jet shear layers are excited by these upstream propagating acoustic waves. The generation mechanism for broadband shock-associated noise (BBSAN) is associated with the interactions between turbulent flow structures in the jet shear layers and the shock-cell structure. It has first been identified by Martlew [2]. Harper-Bourne and Fisher built a model allowing to compute the central frequency of BBSAN as a function of the observation angle [3].

All the aforementioned sound components are present in a cold non-ideally expanded jet. The nozzle geometry and the boundary conditions influence the intensity, the directivity, and the frequency content of the different sound components. Several geometries have been used in the industry to maximize thrust. So far, circular, rectangular, and twin-nozzles have been implemented and both the flow characteristics and their acoustic signature have been investigated. On the other hand, data on aerospike nozzle jets and the resulting aeroacoustic behavior are scarce. Aerospike nozzles have been developed in the 1950s and provide many advantages compared to conventional nozzles. They enable an altitude-adapting thrust generation and a better thrust vector control [4]. Until now, most research about aerospike nozzles has been conducted regarding thrust performance and integration within RDE [5–7]. A recent paper is providing large eddy simulations (LES) results in order to characterize the benefit of aerospike nozzle for supersonic retro-propulsion [8]. A research study is looking into the flow field characteristics but using RANS [9]. One study looks into the effect of aerospike truncation on the sound pressure levels (SPL) in the near-field [10]. No study on the jet structure and the far-field aeroacoustics of an aerospike nozzle has been performed yet.

In the present paper, the annular jet around an aerospike nozzle is simulated by means of LES. The simulation methodology is presented hereafter. The grid sensitivity study results, the near-field flow characteristics, and the resulting aeroacoustic signature are presented in the next sections.

## Methodology

### Geometry Parameters.

Compressible implicit LES of an annular supersonic jet with a truncated aerospike nozzle at a constant nozzle pressure ratio NPR =3 and a temperature ratio TR =1 are performed. The outer and inner diameters of the axisymmetric nozzle are respectively *D*_{out} = 154 mm and *D*_{in} = 128.6 mm. The aerospike body has a length *L* = 177 mm. The truncated aerospike nozzle is shown in Fig. 1. Some performance parameters were studied in Ref. [11]. Previous investigations showed that aerospike truncation leads to reduced SPL in the near-field [10], but also to thrust decrease due to the higher base drag of the bluff body [12].

The equivalent diameter of the annular geometry is *D*_{eq} = 84.7 mm. This corresponds to the diameter that the jet would have if it were round. The ideally expanded equivalent jet has a velocity *u*_{j} = 399 m/s and a Mach number *M*_{j} = *u*_{j}/*c*_{j} = 1.36. In the rest of the paper, all geometric parameters and velocities will be scaled with the equivalent diameter *D*_{eq} and the velocity *u*_{j}, respectively. The equivalent diameter of the ideally expanded jet is *D*_{j} = 47.4 mm. Using this, the viscosity being *ν* = 1.789 × 10^{−5} Pa · s, the corresponding Reynolds number is Re_{j} = *u*_{j}*D*_{j}/*ν* = 1.06 × 10^{6}. The nozzle lip has a width of 12.7 mm.

### Flow Solver and Numerical Parameters.

Large eddy simulations are carried out using an in-house finite volume-based compressible flow solver. It has been validated for a wide range of configurations and operating conditions [13–20]. Good agreements of flow and acoustic fields between the computational fluid dynamics prediction with the solver and experimental data are achieved. An explicit four-stage Runge–Kutta algorithm is used for time integration and a second-order central difference scheme is used for spatial discretization. An artificial dissipation scheme after Jameson has been used and validated for different configurations to capture shock waves to avoid Gibbs-like oscillations near shocks and to relax sub-grid-scale turbulent energy [13,21].

The axisymmetric computational domain is represented in Fig. 2. The flow direction is the *z*-direction. It is of total length of 65 *D*_{eq}. At the far-field outlet (*z*/*D*_{eq} = 57), the domain has a diameter of 44 *D*_{eq}. At *z*/*D*_{eq} = 0, corresponding to the outlet of the annular duct, the domain has a diameter 27 *D*_{eq}. At the co-flow inlet *z*/*D*_{eq} = −8, the domain has a diameter 29 *D*_{eq}. The domain size has been chosen in accordance with previous studies of supersonic jets [13,19,20].

A total pressure *p*_{t} = 304,000 Pa and total temperature *T*_{t} = 293 K are applied at the inlet plane. At the outlet plane and radial far-field, the pressure is set at *p* = 101,325 Pa, the temperature at *T* = 293 *K*, and the velocity at *u* = 0 m/s. At the co-flow inlet plane (*z*/*D*_{eq} = −8), a secondary flow with Mach number of 0.05 is applied. Weak adiabatic no-slip boundary conditions are applied in the annular chamber and at the aerospike body. Characteristic boundary conditions are used to avoid reflections [22]. Moreover, the grid is stretched outside of the flow region at a rate of 5% to further dissipate the acoustic waves propagating and make the far-field boundaries reflection-free. Lower stretching rates lead to reflected waves at the far-field boundaries and higher stretching rates lead to reflections from the grid [23,24]. A similar strategy has been applied in previous studies [13–20].

To compute the acoustic far-field, the Ffowcs Williams–Hawkings equation (FWH) is solved. It requires data acquisition on a permeable surface enclosing the acoustic sources. The surface is open in the downstream to avoid pseudo-sound due to eddies crossing the surface [25]. The corresponding domain enclosing the source region has a length of 42 *D*_{eq} and a diameter of 7.5 *D*_{eq} at the annular duct outlet (*z*/*D*_{eq} = 0) and 15 *D*_{eq} at *z*/*D*_{eq} = 40. The surface is shown as a thin white line in Fig. 2 with the indicated dimensions. The simulations are performed for a total physical time of 65 ms (∼306 *D*_{eq}/*u*_{j}) with a transient of 15 ms. Data for the far-field acoustic analysis are exported on around 400,000 nodes for a physical time of 50 ms.

## Grid Convergence Study

A convergence study has been performed for the cold annular supersonic jet. Three different grids of 119 blocks have been used with 110, 140, and 170 millions of nodes respectively. The grids are axisymmetric and the resolution in the radial direction is kept constant for all the grids. The mesh size in the direct vicinity of the nozzle up to a streamwise distance of *z*/*D*_{eq} = 5 is kept similar. The average cell size in this region is shown in Table 1. To assess the mesh sensitivity, the grid is stretched in the flow direction (*z*-direction) while the resolution in radial and azimuthal directions is kept constant for all three meshes. The grid is differently stretched in the region between *z*/*D*_{eq} = 5 and the far downstream right-hand-side disk of the open FWH surface located at a streamwise distance *z*/*D*_{eq} = 40. The stretching parameters and dimensionless length of the largest cell within the FWH surface in the flow direction are given in Table 1. In the annular duct, the mesh is designed to fulfill a dimensionless distance to the wall of *y*^{+} = 1. Outside of the annular duct, the mesh stretching is kept below 1% within the FWH surface to guarantee a sufficiently high resolution of the flow structures relevant for noise generation mechanisms. Outside of the FWH surface, the mesh stretching is increased to 5% in order to dissipate the acoustic waves and thus avoid reflections at the far-field boundaries. In the radial direction, the mesh has a minimal resolution of 16 grid points per wavelength for frequencies of 20, 000 Hz. Thus, acoustic waves of high frequencies propagate undisturbed toward the FWH surface and are dissipated in the buffer zones situated beyond the FWH surface.

Mesh size | 110M | 140M | 170M |
---|---|---|---|

Cell size z/D_{eq} < 5 | 9.4 · 10^{−3} | 8.3 · 10^{−3} | 7.1 · 10^{−3} |

Stretching parameter | 0.7% | 0.5% | 0.3% |

Cell size z/D_{eq} ∼ 40 (FWH) | 0.38 | 0.21 | 0.16 |

Time-step Δt u_{j}/D_{eq} | 1.8 · 10^{−4} | 1.4 · 10^{−4} | 1.2 · 10^{−4} |

Mesh size | 110M | 140M | 170M |
---|---|---|---|

Cell size z/D_{eq} < 5 | 9.4 · 10^{−3} | 8.3 · 10^{−3} | 7.1 · 10^{−3} |

Stretching parameter | 0.7% | 0.5% | 0.3% |

Cell size z/D_{eq} ∼ 40 (FWH) | 0.38 | 0.21 | 0.16 |

Time-step Δt u_{j}/D_{eq} | 1.8 · 10^{−4} | 1.4 · 10^{−4} | 1.2 · 10^{−4} |

Due to the over-expanded conditions, shock waves are generated at both the annular duct exit (at *z*/*D*_{eq} ∼ 0.5) and downstream of the aerospike body (at *z*/*D*_{eq} ∼ 2). No difference in terms of flow field solution obtained with the three different grid resolutions is observed in the direct vicinity of the annular duct. Figure 3 shows the effect of mesh refinement on the time-averaged axial velocity along the jet centerline in the flow direction downstream of *z*/*D*_{eq} ∼ 2. The shock-cell length is defined as the distance between two successive minima and is reported in Fig. 4(a). Ten, nine, and eight shock-cells are visible for the grids with 170, 140, and 110 million cells, respectively. Moreover, the coarsest grid provides shorter shock-cells compared to the two finer grid resolutions. Figure 3 shows a lower change in velocity across the shocks for the coarsest grid compared to the two other grids. To better characterize this effect, the axial velocity change across the expansions for each shock-cells is shown in Fig. 4(b). It shows that the two finest grids behave more similarly than the two coarsest grids. These findings regarding the shock-cell length and the axial velocity change across the shocks suggest that the obtained results for 140 and 170 million nodes grids are similar enough. The 170 million nodes grid is sufficient to properly analyze the flow field properties. It is worth to note that Fig. 4(a) displays a shock-cell length decay of 3% which matches with André’s results [26]. These were also obtained with a secondary flow of *M* = 0.05.

Figure 5 shows the normalized time-averaged axial velocity (Figs. 5(a), 5(b), and 5(c)) and the normalized axial velocity fluctuations (Figs. 5(d), 5(e), and 5(c)) at three axial locations *z*/*D*_{eq} = 3, *z*/*D*_{eq} = 5, and *z*/*D*_{eq} = 8 in the jet. The axial velocity profiles are very similar for all resolutions. Small discrepancies are noticeable for the velocity fluctuations. The 140 million cells mesh is showing slightly higher fluctuation levels (with 8%) at *z*/*D*_{eq} = 3, as compared to the other grid resolutions.

Overall, the results are similar for the three grids. The 170 millions nodes mesh estimates sharper shocks and a longer potential core as compared to the two coarser resolutions. The results in the following sections will be discussed based on the simulation with the finest mesh of 170 million nodes.

## Near-Field Flow Results

The present section focuses on the flow characteristics of the annular jet of the aerospike.

Figure 5 helps characterize the jet structure behind the truncated aerospike body. Close to the aerospike, the jet still displays a slight annular character. Figure 5(a) shows two velocity peaks corresponding to an annular jet structure at *z*/*D*_{eq} = 3. This annular feature of the jet is confirmed by Fig. 5(d). The peaks at *r*/*D*_{eq} = ±0.65 correspond to the outer annular shear layers. The peak at lower intensity at *r*/*D*_{eq} = 0 corresponds to the merged inner annular shear layers. Figures 5(c) and 5(f) at an axial distance *z*/*D*_{eq} = 8 show a Gaussian profile for the velocity and two peaks in the axial velocity fluctuation data, respectively. This suggests that the jet has lost its annular character and fully circular at the downstream location of *z*/*D*_{eq} = 8.

The time-averaged Mach number field normalized by *M*_{j} is presented in Fig. 6(a). Note that the domain is axisymmetric and that contour profiles in other section planes provide the same result.

Since there is no area change in the annular duct, the flow has a Mach number *M* = 1 before exhausting the aerospike nozzle. It is first expanded and highly accelerated before being further exhausted in the ambient free-field generating shock-cell structures. At the annular duct outlet (at *z*/*D*_{eq} ∼ 0), the jet is first expanded radially inwards and outwards. The first “cell” observed is an expansion cell. The generated expansion fan is reflected on the inner and outer annular shear layers of the annular jet. After this expansion, the jet undergoes a shock again. In the downstream around the aerospike body (between *z*/*D*_{eq} ∼ 0.5 and *z*/*D*_{eq} ∼ 2), several shock-cells bent toward the axial direction are observed. After reaching the aerospike tip at a distance *z*/*D*_{eq} = 2, the jet undergoes another expansion and a second shock-cell structure arises.

The supersonic jet displays two distinct shock-cell structures: an annular and a circular one. The annular structure close to the aerospike body is further divided into two separate regions. The first part of the annular jet is not attached to the aerospike body and displays short shock-cells (up to a distance *z*/*D*_{eq} = 0.85). There, the Mach number change across the shock is Δ*M*/*M*_{j} = 0.3. In the radial inwards direction, a whole separation region where the velocity drops is formed. The second part of the annular jet is reattaching to the aerospike body and displays longer shock-cells compared to the non-attached annular jet. A model for the prediction of the annular shock-cell length is proposed in the following section. It corresponds to the region between *z*/*D*_{eq} = 0.85 and *z*/*D*_{eq} = 2. The shocks in this region are weaker than in the non-attached shock-cell structure. The average Mach number change across the shocks is Δ*M*/*M*_{j} = 0.03.

Behind the aerospike body, a longer shock-cell structure is observed with around 10 shock-cells (see also Fig. 3). It extends from the aerospike tip at *z*/*D*_{eq} = 2 to *z*/*D*_{eq} = 10. The jet is still annular for the two first shock-cells due to the aerospike body truncation. The following shock-cells (from *z*/*D*_{eq} ∼ 4.5) merge to build a circular shock-cell structure. The observed shock-cell length decreases with increasing axial distance from the aerospike body. This length decay is predicted by the model proposed by Tam for circular jets [27]. It will be applied and compared to the observed shock-cell length from the LES in the next section. Finally, the shocks are stronger in the circular jet than in the attached annular jet but weaker than in the non-attached annular jet. The average Mach number change across the shocks for the first five shock-cells is Δ*M*/*M*_{j} = 0.1.

The normalized pressure fluctuations *p*_{rms}/*p*_{0} are presented in Fig. 6(b). High fluctuation levels indicate strong dynamic effects in jets such as flapping. These dynamic effects are affecting the acoustic signature of the investigated device in the near- and far-field. Therefore, it is crucial to identify the regions where such high fluctuation levels are present and to additionally determine the frequency associated with these dynamic effects.

The aforementioned distinct two parts of the jet, an annular and a circular one, are visible in Fig. 6(b). The highest fluctuation levels are observed in the annular jet structure. The annular shock structure emphasized in Fig. 6(a) is distinguishable as well in Fig. 6(b). It indicates that the shock-cells around the aerospike body are oscillating. The shock between the first and the second shock-cell displays the highest fluctuation levels at around *p*_{rms}/*p*_{0} ∼ 0.2. Such high levels are also observed at the location of the separation bubble. Marginally lower fluctuation levels are observed at the location between the expansion cell and the first shock-cell. This implies that the position of the first shock is more fixed than the position of the second shock further downstream. Consequently, it would translate into a stiffer motion of the first shock-cell compared to the second one. Power spectral density (PSD) for the points with the highest fluctuation levels are shown in the last section (see points *P*_{1}, *P*_{2}, *P*_{3}, and *P*_{4} in Fig. 8). Furthermore, high fluctuation levels are observed in the shear layer emerging right after the annular duct outer lip. Cross-correlations along this annular shear layer line are computed in the last section of this paper (see line *L*_{1} in Fig. 8). Slightly lower fluctuations levels are observed further downstream in the vicinity of the aerospike nozzle for the third and fourth shock-cells for which the flow is reattaching. Two lobes of lower pressure fluctuations are observed above |*x*|/*D*_{eq} > 1 at an axial distance *z*/*D*_{eq} ∼ 1.3. They correspond to emerging acoustic waves generated in the annular shock-cell structure.

Further downstream, high fluctuation levels are observed at the expansion location after the aerospike body (around *z*/*D*_{eq} ∼ 2.15). A recirculation bubble is formed behind the aerospike bluff body and displays fluctuation levels of around *p*_{rms}/*p*_{0} ∼ 0.12. The first shock-cells in the circular jet (between *z*/*D*_{eq} ∼ 2.15 and *z*/*D*_{eq} ∼ 5) exhibit slightly lower fluctuation levels than the smaller circular shock-cells further downstream. These higher fluctuation levels can be linked to the interaction between turbulent flow structures in the jet shear layers and the shock-cell structure. This mechanism is associated with the generation of screech noise and BBSAN [1–3]. Hence, the downstream part (from *z*/*D*_{eq} > 4.5) of the circular jet might contribute to a larger extent to sound propagation in the far-field than the upstream part.

### Shock-Cell Structure for an Annular Jet.

The length of the shock-cells in the potential core is related to the frequencies of BBSAN and screech [28]. Therefore, the correct computation of the shock-cell length is crucial for far-field noise prediction. Several models have been developed for shock-cell length prediction. Tam and Tanna [29] as well as Harper-Bourne and Fisher [3] developed models to predict the shock-cell length in round jets. These models have been adapted and validated for rectangular jets since [30]. However, it has not been extended to annular jets yet. Let us adapt the model developed in by Tam and Tanna [29].

*θ*and takes the form: $p(r,z)=\u2211n=1\u221e\varphi n(r)\psi n(z)$, with

*r*the radius and

*z*the axial distance along the flow. The eigenvalues functions are of the general form [31]

*J*

_{0}and

*Y*

_{0}are the Bessel functions of the first kind and second kind of zeroth order,

*A*

_{n}and

*B*

_{n}are scalars yet to be determined,

*β*

_{n}corresponds to the roots of a boundary condition equation. In the case of a circular jet, the term $BnY0(2\beta nrDj)$ disappears since pressure is a continuous field and

*Y*

_{0}is singular when

*r*goes toward 0. In the circular case, only the scalars

*A*

_{n}and

*β*

_{n}need to be computed. In the case of an annular jet, the radius corresponding to the boundary condition at the inner shear layer is not equal to zero. Thus,

*B*

_{n}is an additional scalar yet to be determined.

In the vicinity of the aerospike nozzle system, the annular jet exhibits shock-cell structures with different features. A separation bubble is initiated downstream of the annular exit in the presence of adverse pressure gradients on the aerospike nozzle. Thus, the first shock-cell structure at the outlet of the annular duct emerges in an annular jet separated from the aerospike body, extended up to an axial distance 0.85 *D*_{eq}. It displays short shock-cells. The second structure attached to the aerospike is stretching from an axial distance 0.85 *D*_{eq} to the tip of the aerospike body.

*r*

_{inner}=

*r*

_{i}. The non-attached part of the jet is modeled by three steps since three cells are recognizable (see Fig. 6(a)). The first step corresponds to the expansion cell directly at the annular duct outlet. The two following steps correspond to the two non-attached shock-cells in the annular jet. The attached part of the annular jet is modeled by a single long step because the aerospike curvature in this region is low. Thus, the aerospike geometry is replaced by an equivalent aerospike with wall-perpendicular direction corresponding to the radial direction. This simplifies the boundary condition equations which require the computation of the derivative with respect to the wall-normal direction. In the non-attached annular jet, the pressure perturbation at the inner and outer shear layers is zero. This leads to the following condition at the boundaries:

*ϕ*

_{n}(

*r*) = 0 at

*r*= {

*r*

_{o},

*r*

_{i}}. This condition translates to

*r*

_{o}and

*r*

_{i}are the radii where the axial velocity

*w*is equal to half the maximal velocity at a given axial distance

*z*. For a fixed axial distance

*z*, it translates to the condition

*w*(

*z*,

*r*

_{o}) = max (

*w*(

*z*))/2 or

*w*(

*z*,

*r*

_{i}) = max (

*w*(

*z*))/2 with

*r*

_{o}>

*r*

_{i}. For each step of the equivalent aerospike and hence for each shock-cell, the radii

*r*

_{o}and

*r*

_{i}are averaged over the whole sheet length. The drawn dashed line for the non-attached jet (see Fig. 7) shows that the inner shear layer radius

*r*

_{i}is strictly larger than the aerospike nozzle radius over the shock-cell extension domain. In the attached jet part, the dashed line is crossing the aerospike body since the flow is attached to the aerospike surface. The change of bluff body radius in the attached flow region compared to the shock-cell length is smaller. This justifies the choice of using one single step to simplify the boundary condition equations in the attached flow region. For an attached flow, the velocity at the wall is zero and hence the derivative of the pressure along a normal direction to the body is zero. This translates into $d\varphi ndr=0$ for

*r*=

*r*

_{i}and

*ϕ*

_{n}(

*r*

_{o}) = 0. Assuming an aerospike wall parallel to the flow direction as for the red dashed line in Fig. 7, this leads to

*β*

_{n}are computed, the length of the shock-cell is given by the wavelength $\lambda 1=\pi DjMj2\u22121/\beta 1$ corresponding to the axial first mode

*ψ*

_{1}. The obtained results for the four annular shock-cells are summarized in Table 2. The observed values for the first two shock-cells in the non-attached part of the jet from the LES are in very good agreement with the computed values using the adapted vortex sheet model. The undertaken discretization of the aerospike body in Fig. 7 provides matching shock-cell lengths. The length for the two last shock-cells in the attached part of the jet is slightly lower than the simulated values. This might be linked to the choice of simplifying the aerospike body with a single step in this region (see Fig. 7).

L/D_{j} | 1 | 2 | 3 | 4 |
---|---|---|---|---|

LES | 0.47 | 0.41 | 0.80 | 0.83 |

Adapted vortex sheet model | 0.47 | 0.41 | 0.77 | 0.77 |

L/D_{j} | 1 | 2 | 3 | 4 |
---|---|---|---|---|

LES | 0.47 | 0.41 | 0.80 | 0.83 |

Adapted vortex sheet model | 0.47 | 0.41 | 0.77 | 0.77 |

Similarly, the shock-cell length in the circular jet is compared. The results are in agreement with Tam’s model for circular jets. The shock-cell structure after the aerospike body not being perfectly circular, the first shock-cell is shorter than the following ones. The results are reported in Table 3 with the shock-cell decay of 3% observed in Ref. [26]. This decay is due to the presence of the co-flow of Mach number *M* = 0.05.

## Noise Generation and Acoustic Results

### Noise Generation Mechanisms.

The dynamic behavior of the jet plays a fundamental role in the noise generation mechanisms. The PSD for the radial velocity at four monitoring points in the annular jet structure allows to characterize the annular jet dynamics. These points are located in the annular jet region where the pressure fluctuations are the highest (see Fig. 6(b)). Figure 10(a) shows the PSD for a point corresponding to the middle position of the separation bubble located at (*r*/*D*_{eq}, *z*/*D*_{eq}) = (0.67, 0.62) (point *P*_{1}). Figure 10(b) shows the PSD for a point located at the beginning of the first shock-cell after the first shock in the annular jet at (*r*/*D*_{eq}, *z*/*D*_{eq}) = (0.76, 0.44) (point *P*_{2}). Figure 10(c) shows the PSD for a point situated between the two first shock-cells in the annular jet at (*r*/*D*_{eq}, *z*/*D*_{eq}) = (0.7, 0.3) (point *P*_{3}). Figure 10(d) shows the PSD for a point situated right above the second shock-cell in the annular jet at (*r*/*D*_{eq}, *z*/*D*_{eq}) = (0.76, 0.74) (point *P*_{4}). The four observation points are shown in black in Fig. 8.

Figure 10(a) shows two main peaks for the radial velocity of the point *P*_{1} located at the separation bubble. The frequencies associated with these peaks are St = 0.54 and St = 1.21. Secondary peaks are observed close to the frequencies of the main peaks and frequencies around St = 0.68 and St = 2.20. These results suggest that the separation bubble trapped between the annular shock-cell and the aerospike body has three oscillation modes. The peak at St = 0.54 is associated with a coupling with an azimuthal jet mode and the screech frequency identified by two-point cross-correlation further in this section (see Fig. 11(c)). Figure 10(c) for point *P*_{3} shows a single peak of high amplitude with a central frequency St = 1.21 corresponding to the oscillation of the annular shock-cell structure. The peak has a slight broadband character. Figure 10(d) for point *P*_{4} located above the second shock-cell displays two main peaks at 1.21 and 1.36. The latter peak is a sub-peak which is also visible in Fig. 10(a) and could explain the broadband character of the main peak at St = 1.21 in Fig. 10(c). These observations suggest that the annular shock-cell structure is oscillating at the same frequency as the separation bubble (St = 1.21). Analogously to screech which corresponds to a flapping motion of the shock-cell structure [14,32], radial oscillations of the shock-cell structure might lead to noise generation at oscillation frequencies.

The main radial velocity oscillation frequency for the first shock-cell is St = 2.59 as shown in Fig. 10(b). Secondary peaks are observed at St ∼ 1.3. The first shock-cell is flapping radially at a higher frequency compared to the second shock-cell. The frequency is roughly double of the frequency observed for the broadband peaks between St ∼ 1.21 and St ∼ 1.36. Hence, it can correspond to a sub-harmonic of the shock-cell main oscillation mode. A peak at the same Strouhal number St = 2.59 is also observed in Fig. 10(d) but at a lower amplitudes than the main peaks at the lower frequencies. The motion of the first shock-cell is stiffer and as a result, the frequency associated to this motion is higher. This observation is consistent with the lower fluctuation levels at the location of the first shock in Fig. 6(b).

*r*/

*D*

_{eq}= 0.45). The reference point for the cross-correlation is fixed at

*z*/

*D*

_{eq}= 1.95. The corresponding line in the circular shear layer is line

*L*

_{2}shown in Fig. 8. The cross-correlation is performed between the reference point and the other 80 points spread along the line in jet shear layer between

*z*/

*D*

_{eq}= 1 and

*z*/

*D*

_{eq}= 12. The results are shown in Fig. 11(b). It shows the time-lag in seconds as a function of the axial distance along the flow direction. Negative slopes indicate downstream propagating hydrodynamic pressure corresponding to the convection of fluid flow structures in the jet shear layer. The maximum cross-correlation level is obtained for the reference point and corresponds to auto-correlation. Periodic patterns with positive slopes indicate upstream propagating acoustic pressure. These patterns are only apparent at distances

*z*/

*D*

_{eq}< 8 where a shock-cell structure is still present (see Fig. 6(a)). The observed time-lag for a single periodic pattern is around Δ

*t*= 4.16 × 10

^{−4}s which corresponds to a Strouhal number St = 0.51. Using the vortex sheet model for a circular jet and the approximation for the first harmonic the screech tone of a circular jet [29,31], the screech frequency assuming a circular jet downstream of the aerospike is the following:

*u*

_{c},

*M*

_{c}=

*u*

_{c}/

*c*

_{0}are the convection velocity and the convective Mach number, respectively.

*L*

_{circ.}is the average length of the first five shock-cells in the circular jet taken from Table 3. The convection velocity

*u*

_{c}is determined using cross-correlation for the axial velocity along lines in the annular and circular shear layers. We obtain

*u*

_{c}= 0.72

*u*

_{j}, which is in good agreement with other cold jet simulations [13].

A similar procedure is applied for the annular jet between *z*/*D*_{eq} = 0 and *z*/*D*_{eq} = 2 (see Fig. 8). Two-point cross-correlation for the pressure is performed along line *L*_{1} consisting of 20 monitoring points starting at (*r*/*D*_{eq}, *z*/*D*_{eq}) = (0.85, 0.47) to (*r*/*D*_{eq}, *z*/*D*_{eq}) = (0.64, 1.42) located in the annular shear layers. The corresponding line in the annular shear layer is shown in Fig. 8. Several band- and high-pass filters have been tested to evaluate the different frequency contents due to the various flow patterns. Figure 11(a) shows the two-point cross-correlation for the pressure on line *L*_{1} using a high-pass filter with a cut-on Strouhal number of St = 0.75 to filter out the upstream propagating waves associated with the region of the jet located downstream of the aerospike tip (*z*/*D*_{eq} > 2). Stronger cross-correlation levels are observed for this cross-correlation compared to Fig. 11(b). Less broadband structures are propagating in the vicinity of the aerospike. However, no positive slope patterns are notable for this cross-correlation. This suggests that there is no feedback mechanism between the annular shock-cell structure and the nozzle lip and hence no screech phenomenon associated with the annular jet region (0 < *z*/*D*_{eq} < 2). On the other hand, the shedding of coherent flow structures corresponding to the negative slopes is periodic. The time-lag associated with this shedding is Δ*t* = 1.75 × 10^{−4}s which corresponds to St = 1.21. This matches with the oscillation frequency of the shock-cell structure in radial direction (see also Fig. 10).

Cross-correlation is also performed in the azimuthal direction to detect the presence of rotating patterns coupled with the radial oscillations of the separation bubble observed in Fig. 10(a). Two-point cross-correlation for the pressure is performed along line *L*_{3} consisting of 17 monitoring points in an angular sector of $90deg$ in the *xy*-plane at an axial distance *z*/*D*_{eq} = 0.60 and radial distance *r*/*D*_{eq} = 0.67 located at the limit between the shock-cell structure and the separation bubble (at the same location as point *P*_{2}). The corresponding line in the annular shear layer is shown in Fig. 9. The reference point for the cross-correlation is set at an angle of $45deg$ in the middle of the angular sector. Several low- and high-pass filters have been applied in order to make the frequency patterns easier to visualize. Figure 11(c) shows the two-point cross-correlation for the pressure on line *L*_{3} using a low-pass filter with a cut off Strouhal number St = 0.85 to filter out the higher order azimuthal patterns. Strong cross-correlation levels are observed with a periodic behavior. The patterns are almost horizontal since the convection direction of the vortices is the *z*-direction. A slight negative slope is noticeable. The time-lag between the highest cross-correlation levels is on average Δ*t* = 3.96 × 10^{−4}s which corresponds to a Strouhal number St = 0.54. This frequency is also observed in the power spectral density for the radial velocity in the separation bubble (see Fig. 10(a)). This suggests that there is a coupling mechanism between the oscillations in radial and azimuthal directions and the screech phenomenon.

Figure 11(d) depicts the cross-correlation on line *L*_{3} after application of a high-pass filter at cut-on Strouhal number St = 0.85. Lower cross-correlation levels are observed compared to Fig. 11(c). Both positive and negative slopes are noticeable suggesting that pressure patterns are rotating in both positive and negative azimuthal directions. Furthermore, the observed patterns indicate that several components of higher frequency are superposed. Band-pass filters were applied to further identify these higher frequency components. However, it requires to narrow the filter bandwidth which in turn affected the time-lag. This dependency on the quality of the filter does not allow to properly compute the time-lag and to draw conclusions about the higher frequency components. Further analyses are required to better frame the coupling between the various azimuthal and radial oscillation modes with the screech frequency.

### Far-Field Acoustic Spectra.

With use of the Ffowcs Williams–Hawkings equation, the far-field aeroacoustic spectrum is computed. Flow data are exported at 400,000 nodes located on the FWH surface (shown as a thin line in Fig. 2) every 3 × 10^{−6}s. The original formulation is found in Ref. [34]. A detailed formulation applied to jet noise is given in Ref. [35]. The observation points are considered on an arc at a radial distance of 60*D*_{eq} from the aerospike body tip.

The far-field acoustic spectrum is presented in Fig. 12 as a function of the Strouhal number St and the directivity angle *θ* in degrees. $20deg$ correspond to the upstream direction, toward the aerospike, and $160deg$ correspond to the downstream side angle along the flow direction. Several acoustic components can be observed. At downstream angles $\theta >140deg$, mixing noise produced by large turbulent flow structures propagate at low frequencies ranging in the neighborhood of St ∼ 0.2 in agreement with previous observations [36,37]. A first tonal component is observed at St = 0.51 at upstream angles $\theta <60deg$. This corresponds to the screech in the circular region of the jet (*z*/*D*_{eq} > 2). The Strouhal number of this component matches the Strouhal number computed from the two-point cross-correlation in the circular jet shear layer shown in Fig. 11(b). Another tonal component at upstream angles is observed at St = 0.54 at a lower frequency. It corresponds to the peak observed in the PSD for the separation bubble (see in Fig. 10(a)) which was identified as an azimuthal motion by the two-point cross-correlation in Fig. 11(c). A third strong tonal noise is noticeable at upstream angles for St = 0.68. This frequency corresponds to one of the radial oscillation modes of the separation bubble in the annular jet highlighted in Fig. 10(a). The third strong noise component is located at St = 1.21. It has a slight broadband character and is spreading over the frequency range St ∼ 1.15 to St ∼ 1.35. This noise component propagates more strongly in the upstream direction but is also present in the downstream direction. It is associated with the radial oscillation of the annular shock-cells and of the separation bubble shown in Figs. 10(a), 10(c), and 10(d). The radial oscillation mode observed in the latter figures also exhibits a broadband character. These different sound components propagating at upstream angles are superposed.

The observed sound components can also interact. Tonal noise at lower amplitudes is visible at St = 0.59 and St = 0.86 at angles $\theta \u223c50deg$ and at upstream angles $\theta <45deg$, respectively. These two observed frequencies correspond to the interaction between the screech noise with characteristic frequency St = 0.51 and the two strong noise components with characteristic frequencies St = 0.68 and St = 1.21 respectively. At downstream angles $\theta >140deg$, a tonal component at lower amplitude is observed at St ∼ 2.15. This Strouhal number matches the observed Strouhal number of the double peak at lower amplitude for the radial oscillation of the separation bubble in Fig. 10(a) at St ∼ 2.20.

Stronger SPL are observed at downstream angles around $\theta \u223c120deg$ at St = 2.60. This sound component corresponds to the main oscillation mode of the first shock-cell observed in Fig. 10(b) and of the second oscillation mode of the second shock-cell shown in Fig. 10(d).

*L*

_{j}is the shock-cell length and

*θ*is the observation angle. The three black dashed lines correspond to the central frequency of the BBSAN component plotted using the average shock-cell length of the non-attached (BBSAN N.-A.) and attached (BBSAN A.) annular shock-cell structures and the circular shock-cell structure (BBSAN Circ.). The lengths

*L*

_{j}for each component are taken from Tables 2 and 3. The high SPL in Fig. 12 are located around these frequency branches with asymptotes at $St0deg=0.51$ (corresponding to screech) for the circular jet, $St0deg=1.06$ for the attached annular jet and $St0deg=1.42$ for the non-attached annular jet. At upstream angles and slightly above the right angle ($\theta \u2264110deg$), the higher SPL are observed close to the BBSAN lines corresponding to the attached and non-attached annular shock-cells. In particular, the aeroacoustic signature in the domain bounded by the angles $\theta \u223c70deg$ and $\theta \u223c120deg$ for Strouhal numbers between St ∼ 1.3 and St ∼ 2.7 displays the highest SPL. In the far-field, the different waves generated in the jet are interfering. This leads to secondary sound components whose frequencies are close to the frequencies of the original sound component. This amplifies the broadband character of the aeroacoustic signature.

At larger observation angles corresponding to the downstream (above $\theta >120deg$), the high SPL are located close to the BBSAN line for the circular shock-cell. The BBSAN lines for the annular shock-cell are located in a silent zone (blue region at the higher frequencies St > 3.2) and are no longer useful to predict the aeroacoustic far-field signature. Due to the inherent broadband character of the BBSAN, linked to convection of vortices of different sizes through the shock-cell structures, high SPL are generally spread close to the BBSAN lines.

## Conclusions

Large eddy simulations of a cold over-expanded aerospike nozzle are performed. The grid sensitivity study shows that the 170 million nodes grid is sufficient to capture the flow features relevant for noise generation. The non-ideally expanded jet at the outlet of the annular nozzle displays two distinct shock-cell structures. The annular shock-cell structure and the associated separation bubble oscillate at Strouhal numbers St = 0.68 and St = 1.21. This leads to strong tonal components in the far-field aeroacoustics spectra in upstream direction at these Strouhal numbers. Screech noise at St = 0.51 produced by the circular shock-cell structure is detected using two-point cross-correlation in the jet shear layers. These main modes interact and lead to sub-harmonic components that propagate in the far-field. Starting from the model developed by Tam [29,31] for a circular jet, a vortex sheet model is developed to predict the annular shock-cell length and a good agreement is obtained between the developed model and the simulations. This vortex sheet model is used to predict the central frequency of the BBSAN component due to the annular structure. High SPL are observed between the lines corresponding to the central frequencies of BBSAN for the annular and circular jets. Mixing noise is observed radiating in the low frequency range at the downstream angles.

## Acknowledgment

This project has received funding from the “INSPIRE” EU Project H2020-MSCA-ITN-2020, Marie Skłodowska-Curie Innovative Training Networks, Project No. 956803. The computations were performed on resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS, Project NAISS 2023/1-19) at PDC Centre for High-Performance Computing (PDC-HPC) and at National Supercomputer Center (NSC). The authors would like to thank Dr. Stefan Wallin, Dr. Peter Eliasson, and Dr. Myeong-Hwan Ahn for the assistance of the code implementations and running the solver.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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