Hadjer Soltani

Fabiola Ngomesse

Guillaume Reinhart

Mohamed Chérif

Moussa Zahzouh

Henri Nguyen-Thi

Impact of gravity on directional solidification of refined Al-20wt.%Cu alloy investigated by in situ X-radiography

Keywords: Directional solidification, equiaxed microstructure, grain characteristics, in situ Xradiography, Al-Cu alloy, gravity, convection, buoyancy

Gravity effects such as natural convection in the liquid phase and buoyancy forces acting on the solid phase have a strong influence on the grain structure and microstructure formation dynamics during the solidification of metal alloys. It is thus very useful to undertake experimental studies that will provide benchmark data for a deeper understanding of the role of such gravity effects. In this paper, we study the formation of the equiaxed grain structure during refined Al-20wt.%Cu solidification in a temperature gradient for three different configurations: horizontal, vertical upward and vertical downward solidification. The key grain characteristics, namely grain size, grain elongation and grain growth orientation, were determined for all experiments and a comparative study was performed to identify the dominant effects of gravity 2 for each case. The present study provides quantitative information on the impact of grain flotation and solute flows on the equiaxed microstructure characteristics by means of in situ laboratory X-radiography.

Introduction

Metal material properties such as ductility and hardness are closely linked to the grain structure generated during solidification processes. Ever better understanding of grain structure dynamics during its development and of related underlying mechanisms is thus of utmost importance to control the growth conditions for engineering applications and to obtain pieces with the desired and reproducible properties. During solidification in a temperature gradient, two types of grain structure can form depending on the growth conditions (alloy composition, temperature gradient and cooling rate or sample pulling velocity): the columnar grain structure, with directional properties and the equiaxed grain structure with more uniform and isotropic properties. The equiaxed grain structure is the most used in aluminium-based industrial applications and can be promoted by inoculation, which consists in adding inoculant particles to the melt. These micrometer particles act as preferential sites for heterogeneous nucleation of α-Al grains, resulting in a uniformly fine, equiaxed as-cast microstructure [START_REF] Quested | Understanding mechanisms of grain refinement of aluminium alloys by inoculation[END_REF][START_REF] Spittle | Columnar to equiaxed grain transition in as solidified alloys[END_REF]. The same principle is also successfully applied to other metallic systems, notably magnesium and zinc alloys.

On Earth, gravity has a strong impact on the formation of solidification microstructures and makes the study of alloy solidification more complex because of the combination of multiple phenomena at different scales in time and space. Firstly, natural convection takes place in the melt and is the major source of various disturbing effects, which can significantly modify or mask other physical mechanisms [START_REF] Dupouy | Influence of Convection on the Selection of Solidification Microstructures at Low Growth Rates[END_REF][START_REF] Coriell | Convective and interfacial instabilities during unidirectional solidification of a binary alloy[END_REF][START_REF] Mcfadden | Thermosolutal Convection During Directional Soliidification[END_REF][START_REF] Glicksman | Interaction of flows with the crystal-melt interface[END_REF][START_REF] Davis | Hydrodynamics Interactions in Directional Solidification[END_REF]. Indeed, flows in the liquid modify the solute transport in the melt, and generally prevent the formation of steady solidification patterns with uniform features. Thus, attempts have often been made to avoid convection in the melt in directional solidification studies to make easier the interpretation of the results [START_REF] Akamatsu | In situ observation of solidification patterns in diffusive conditions[END_REF]. A simple way of eliminating or reducing natural convection is to ensure that the density gradient in the liquid phase is vertical downward everywhere, the heavier liquid being at the bottom. This situation is more or less approached when growth occurs in a vertical upward temperature gradient, for an alloy system in which the rejected solute is denser than the solvent. For example, this is the case during upward directional solidification of hypo-eutectic Al-Cu alloys. However, even in this configuration, flow patterns adjacent to the solid-liquid interface can develop at low growth rates under the effect of a transverse horizontal temperature gradient that exists because of the difference in thermal conductivities of the solid phase, liquid phase and crucible wall. In that specific case, fluid flows can cause strong deformation (steepling and clustering phenomena) of the solid-liquid interface [START_REF] Burden | Macroscopic Stability of a planar, cellular or dendritic interface during directional freezing[END_REF][START_REF] Nguyen-Thi | Directional Solidification of Al-1.5wt% Ni alloys under diffusion transport in space and fluid flow localisation on Earth[END_REF][START_REF] Verhoeven | The Effect of Convection on the Dendrite to Eutectic Transition[END_REF] and then large segregation in the material [START_REF] Bogno | In situ analysis of the influence of convection during the initial transient of planar solidification[END_REF][START_REF] Bogno | Analysis by synchrotron X-ray radiography of convection effects on the dynamic evolution of the solidliquid interface and on solute distribution during the initial transient of solidification[END_REF]. For vertical downward solidification of hypo-eutectic Al-Cu alloys, both thermal and solutal gradients destabilize the liquid phase and strong convective motion in the melt are expected that can modulate macroscopically the solid -liquid interface, can delay or activate the morphological stability of the solid -liquid interface, and also leads to a significant longitudinal (parallel to the growth direction) macro-segregation [START_REF] Verhoeven | The Effect of Convection on the Dendrite to Eutectic Transition[END_REF][START_REF] Jamgotchian | Interaction of thermal convection with the solid-liquid interface during downward solidification of Pb-30wt\%Tl alloys[END_REF].

For vertical solidification experiments, regardless of the growth direction (upward or downward), an additional effect of gravity is buoyancy that acts on solid elements like dendrite fragments and free grains. This force provokes either grain sedimentation or flotation, depending on the relative densities of the grain and the surrounding liquid [START_REF] Nguyen-Thi | Investigation of gravity effects on solidification of binary alloys with in situ X-ray radiography on earth and in microgravity environment[END_REF][START_REF] Reinhart | In situ and real time investigation of directional solidification of Al -Ni alloys by synchrotron imaging[END_REF][START_REF] Nguyen-Thi | XRMON-GF: A novel facility for solidification of metallic alloys with in situ and timeresolved X-ray radiographic characterization in microgravity conditions[END_REF]. It is worth noticing that buoyancy force acting on secondary dendrite arms also induces dendrite bending, leading in some cases to dendrite fragmentation [START_REF] Reinhart | In Situ Investigation of Dendrite Deformation During Upward Solidification of Al-7wt.%Si[END_REF].

Diffusive mass transport without any buoyancy effect can be achieved during microgravity experiments [START_REF] Nguyen-Thi | XRMON-GF: A novel facility for solidification of metallic alloys with in situ and timeresolved X-ray radiographic characterization in microgravity conditions[END_REF][START_REF] Murphy | Meso-scale modelling of directional solidification and comparison with in situ X-ray radiographic observations made during the MASER-12 XRMON microgravity experiment[END_REF][START_REF] Mota | Initial transient behavior in directional solidification of a bulk transparent model alloy in a cylinder[END_REF][START_REF] Liu | Simulation of directional solidification of refined Al-7 wt.%Si alloys -Comparison with benchmark microgravity experiments[END_REF][START_REF] Liu | Structures in directionally solidified Al-7 wt.% Si alloys: Benchmark experiments under microgravity[END_REF][START_REF] Li | Comparative study of directional solidification of Al-7 wt% Si alloys in Space and on Earth: Effects of gravity on dendrite growth and Columnar-to-equiaxed transition[END_REF], independently of alloy composition and container shape. However, such type of experiments is rare and very expensive, because of the limited number of microgravity opportunities. In normal gravity conditions, nearly diffusive conditions can be achieved by performing horizontal solidifications of very thin samples (down to 10 µm in thickness), like in the case of transparent alloys solidification experiments [START_REF] Akamatsu | In situ observation of solidification patterns in diffusive conditions[END_REF]. In this experimental configuration, fluid flows are blocked by the strong confinement in one dimension.

For metallic samples, it is very difficult to reduce the thickness down to 10 µm in a uniform way and typical sample thickness is in the range 100-300 µm. For this value, natural convection [START_REF] Nguyen-Thi | XRMON-GF: A novel facility for solidification of metallic alloys with in situ and timeresolved X-ray radiographic characterization in microgravity conditions[END_REF] and buoyancy effects [START_REF] Murphy | XRMON-SOL: Isothermal equiaxed solidification of a grain refined Al-20 wt%Cu alloy[END_REF] during horizontal solidification are largely, but not completely, reduced.

It is now well established that in situ and time-resolved study of solidification is relevant because most of the phenomena that occur during the process are dynamic. Various X-ray imaging methods have been developed in the last two decades, in particular using synchrotron radiation [START_REF] Baruchel | Synchrotron X-ray imaging for crystal growth studies[END_REF]. However, in situ X-radiography remains the most used method for studying metal alloy solidification because of its simplicity. X-radiography enables us to follow the dynamics of the solid-liquid interface [START_REF] Mathiesen | In Situ X-Ray Video Microscopy as a Tool in Solidification Science[END_REF], the grain motion [START_REF] Abou-Khalil | Investigation of Thermo-Electro-Magnetic force on equiaxed grain motion during upward directional solidification[END_REF][START_REF] Salloum-Abou-Jaoude | Characterization of motion of dendrite fragment by X-ray radiography on Earth and under microgravity environment[END_REF][START_REF] Murphy | Equiaxed dendritic solidification and grain refiner potency characterised through in situ X-radiography[END_REF] and under certain conditions to determine quantitatively the solute concentration in the liquid phase [START_REF] Bogno | Iop, In situ and real-time analysis of the growth and interaction of equiaxed grains by synchrotron X-ray radiography[END_REF][START_REF] Becker | In-situ solute measurements with a laboratory polychromatic microfocus X-ray source during equiaxed solidification of an Al-Ge alloy[END_REF]. In the last decade, benefitting from the progress of microfocus X-ray sources and detectors, laboratory [START_REF] Shevchenko | The effect of natural and forced melt convection on dendritic solidification in Ga-In alloys[END_REF][START_REF] Becker | Near-isothermal furnace for in situ and real time X-ray radiography solidification experiments[END_REF][START_REF] Murphy | Combined in situ X-ray radiographic observations and post-solidification metallographic characterisation of eutectic transformations in Al-Cu alloy systems[END_REF][START_REF] Rakete | Compact x-ray microradiograph for in situ imaging of solidification processes: Bringing in situ x-ray micro-imaging from the synchrotron to the laboratory[END_REF] as well as microgravity [START_REF] Nguyen-Thi | XRMON-GF: A novel facility for solidification of metallic alloys with in situ and timeresolved X-ray radiographic characterization in microgravity conditions[END_REF][START_REF] Murphy | XRMON-SOL: Isothermal equiaxed solidification of a grain refined Al-20 wt%Cu alloy[END_REF][START_REF] Abou-Khalil | Influence of gravity level on Columnar-to-Equiaxed Transition during directional solidification of Al -20 wt.% Cu alloys[END_REF] apparatus have been developed to study in situ the solidification of metals. Even if microfocus X-ray sources do not generate a beam as powerful as synchrotron sources, they offer enough brilliance to conduct investigations of Albased alloy solidification.

The aim of the present investigation is to examine and enlighten the major effects of gravity during the formation of an equiaxed microstructure in a fixed temperature gradient. For this purpose, directional solidifications of refined Al-20%Cu samples were carried out with different orientations of the sample with respect to the gravity direction, namely horizontal and vertical (upward and downward) solidifications. Experiments were performed in the SFINX (Solidification Furnace with IN Situ X-radiography) laboratory facility, which enabled us to follow the formation of the equiaxed microstructure using in situ X-radiography. The main grain and microstructure characteristics have been determined for all the experiments. Then, a comparative analysis between the various configurations has been conducted to emphasize the impact of natural convection as well as buoyancy force acting on the grains after nucleation during the formation of the equiaxed microstructure.

Experiment description

SFINX apparatus

Directional solidification experiments were performed in the SFINX (Solidification Furnace with IN Situ X-radiography) laboratory device, which is a duplicate of the facility used during several parabolic flight campaigns [START_REF] Abou-Khalil | Influence of gravity level on Columnar-to-Equiaxed Transition during directional solidification of Al -20 wt.% Cu alloys[END_REF] as well as during the sounding rocket experiment MASER-12 [START_REF] Nguyen-Thi | XRMON-GF: A novel facility for solidification of metallic alloys with in situ and timeresolved X-ray radiographic characterization in microgravity conditions[END_REF]. These facilities were developed within the ESA (European Space Agency) project entitled XRMON (X-ray MONitoring of advanced metallurgical processes under terrestrial and microgravity conditions) to study the microstructure formation dynamics of metal alloys onboard of microgravity platform using X-radiography [START_REF] Browne | Overview of In Situ X-Ray Studies of Light Alloy Solidification in Microgravity[END_REF]. A detailed description of those facilities has been given in our previous papers [START_REF] Nguyen-Thi | XRMON-GF: A novel facility for solidification of metallic alloys with in situ and timeresolved X-ray radiographic characterization in microgravity conditions[END_REF][START_REF] Murphy | Meso-scale modelling of directional solidification and comparison with in situ X-ray radiographic observations made during the MASER-12 XRMON microgravity experiment[END_REF]. Only the main features are summarized below.

The dimensions of the sheet-like samples of refined Al-20wt.%Cu (0.1wt.%AlTiB) were 5 mm in width, 50 mm in length and 0.25 mm in thickness. The sample thickness was chosen in order to have a good transmission of the X-ray beam and prevent the superimposing of several grains at the same location so that the interpretation of radiographs is unambiguous. The polished sample was placed into stainless-steel spacers, sandwiched between two flexible glassy carbon sheets sewn together with a silica thread. The sample-crucible assembly was then installed inside the Bridgman-type furnace (Fig. 1a). The furnace consists of two identical heaters for the "hot" and "cold" zones that are independently regulated by a PID-regulator. During the experiments, the samples were directionally solidified by applying the same cooling rate R on both heater elements, ensuring a fixed temperature gradient G during the entire experiment. The furnace enables directional solidification with temperature gradients within the range of 2.5-15 K/mm and cooling rates R within the range of 0.01-1.5 K/s. The X-radiography system features a microfocus X-ray source with a molybdenum target and a 3 mm focal spot (Fig. 1b). It provides a photon flux with two peaks of energy at 17.4 keV and 19.6 keV that ensure a good image contrast to study Al-Cu based alloys. The camera system is made of a scintillator plate that converts X-ray radiation to visible light and a digital camera with a CCD-sensor. As a result of the X-ray beam divergence, a geometric magnification of the object is observed at the detector, which is the ratio of the source-detector to source-object distances, the object in this case being the sample. In this work a magnification of ~5 was obtained for a Field-of-View (FoV) of about 5x5 mm 2 and an effective pixel size of ~4x4 µm 2 .

The acquisition rate was 2 frames/second. Grey level variations in the radiographs are related to the difference in X-ray absorption of the different parts of the sample, which depends mainly on the local density and composition.

Image quality was enhanced by applying an image processing consisting in dividing each recorded frame by a reference picture recorded just before the beginning of the solidification [START_REF] Buffet | Measurement of solute profiles by means of synchrotron X-ray radiography during directional solidification of Al -4 wt% Cu alloys[END_REF]. After image processing, radiographs showed the equiaxed grains microstructure formed during the directional solidification of the refined Al-20wt.%Cu. Liquid regions of high copper concentration showed up as dark regions in the images, while -Al dendritic grains with low copper concentration were discernible as bright regions in the FoV [START_REF] Abou-Khalil | Investigation of Thermo-Electro-Magnetic force on equiaxed grain motion during upward directional solidification[END_REF][START_REF] Bogno | Iop, In situ and real-time analysis of the growth and interaction of equiaxed grains by synchrotron X-ray radiography[END_REF].

Experimental configurations

For comparison purposes, four refined Al-20wt.%Cu specimens were solidified with similar experimental parameters but with different orientations of the gravity vector. The furnace was set in two positions, allowing us to perform directional solidification experiments in three different configurations with respect to gravity: horizontal (1 sample), vertical upward (2 samples) and vertical downward (1 sample).

-Case A: the main surface of the sample was set perpendicular to the gravity vector g (Fig. 1b) and the solidification of the sample took place horizontally. In this experimental arrangement, the effects of gravity are expected to be strongly minimized.

-Case B: the samples were set in a vertical position by rotating the whole SFINX apparatus and the temperature gradient is anti-parallel to g (bottom-up growth). During upward solidification of refined Al-20wt.%Cu alloy, both thermal and solutal gradient are stabilizing and natural convection is mainly due to the residual transverse temperature gradient [START_REF] Burden | Macroscopic Stability of a planar, cellular or dendritic interface during directional freezing[END_REF][START_REF] Nguyen-Thi | Directional Solidification of Al-1.5wt% Ni alloys under diffusion transport in space and fluid flow localisation on Earth[END_REF][START_REF] Bogno | Analysis by synchrotron X-ray radiography of convection effects on the dynamic evolution of the solidliquid interface and on solute distribution during the initial transient of solidification[END_REF]. Moreover, the impact of buoyancy on equiaxed grains will push the aluminum grains upward [START_REF] Salloum-Abou-Jaoude | Characterization of motion of dendrite fragment by X-ray radiography on Earth and under microgravity environment[END_REF].

-Case C: with the same furnace position as in Case B, downward solidification of the sample was performed with a temperature gradient parallel to the gravity vector (topdown growth). In this case, both thermal and solutal gradients are destabilizing and strong convective flows are expected.

Solidification parameters

We solidified the refined Al-20wt.%Cu specimens in different gravity conditions, as described in the previous section, and with close growth parameters. Table 1 summarizes the growth parameters of the experiments presented in this paper. The cooling rate value was identical for all the experiments, R = 0.15 K/s, but the temperatures of the heater elements had to be adjusted for each configuration to achieve nearly the same average growth velocities and temperature gradients. Indeed, because of the asymmetry of the sample holder design as visible in Fig. 1a [START_REF] Murphy | Meso-scale modelling of directional solidification and comparison with in situ X-ray radiographic observations made during the MASER-12 XRMON microgravity experiment[END_REF], the temperature interval between the two heater elements had to be increased by a factor two for the upward solidification experiments to obtain the same temperature gradient in the sample than for horizontal or downward solidifications. Furthermore, experiments with the same control parameters were carried out several times for each configuration to check the repeatability of the experiments and to improve our statistical analysis in terms of grain size, grain elongation and grain orientation (cf. section 2.4).

To determine accurate values of the growth parameters, we developed an experimental procedure based on the radiographs to measure in a first step the growth velocity [START_REF] Soltani | Impact of growth velocity on grain structure formation during directional solidification of a refined Al-20 wt.%Cu alloy[END_REF]. The position of each grain in the FoV was marked manually as soon as they became visible (diameter ≥ 10-20 µm) and the corresponding time was recorded. Then, the positions of the nucleation events were plotted as a function of time, and the average nucleation front velocity <VN> can be easily derived from this plot. Assuming that the grain nucleation always occurs at a temperature close to the liquidus temperature in the case of refined alloys [START_REF] Greer | Modelling of inoculation of metallic melts: application to grain refinement of aluminium by Al-Ti-B[END_REF], the nucleation front velocity was considered similar to the liquidus temperature velocity. The nucleation front velocity can be compared to the velocity of the eutectic front <VE> that crossed the FoV at the end of the solidification phase. A good agreement was obtained between the two types of measurement (see Table 1). Hereafter, the average nucleation front velocity <VN> will be used to compare the different experiments. In a second step, the actual temperature gradient was calculated using the relation G = R/V, which was assumed to be valid in our experiments.

Grain structure characterization

A dendritic equiaxed microstructure was obtained during the cooling of the samples due to the AlTiB refining particles added into the alloy. By using X-radiography, the solidification was observed in situ with the equiaxed grain structure propagating from the cold zone to the hot zone. For each different configuration (horizontal, vertical upward and vertical downward solidifications), significant differences were observed for the solidification process as will be presented in the next sections. To analyze the gravity effects on the final grain structure, a quantitative analysis was conducted to determine the distribution of grain size, elongation and growth orientation for all experiments. The in situ observation of the solidification process allowed us to follow the microstructure formation and thus to measure the grain characteristics without any ambiguity. The first step consisted of contouring the grains one by one with the free software ImageJ [START_REF] Abramoff | Image Processing with ImageJ[END_REF] in order to obtain the grain surface area, using a semi-automatic macro script [START_REF] Soltani | Impact of growth velocity on grain structure formation during directional solidification of a refined Al-20 wt.%Cu alloy[END_REF]. It was then possible to determine the distributions of grain size, grain elongation and grain growth orientation:

-The grain size is defined as the diameter d of a disk having the same grain surface area, named equivalent disk (white circle in Fig. 2a). The grain surface area is easily retrievable since it is given by the ImageJ software in pixels. Even if the grain size is a very important parameter, it gives no information about the grain morphology and must be completed by other measurements.

-The grain elongation factor ϕ is defined in a classical way by ϕ = L1/L2, with L1 being the length of the longest straight segment inscribed within the grain and L2 the length of the longest straight segment inscribed within the grain and orthogonal to the segment that gives L1 (Fig. 2b). Both L1 and L2 were automatically obtained from the grain contouring using a macro in ImageJ. This definition is the inverse of the one used by Biscuola and Martorano [START_REF] Biscuola | Mechanical Blocking Mechanism for the Columnar to Equiaxed Transition[END_REF] and the elongation factor varies in our paper from unity (for a perfectly equiaxed grain) to larger values (up to ten) for elongated grains.

-The growth orientation of each grain with respect to the temperature gradient direction was also measured. This grain orientation is characterized by the tilt angle θ between the temperature gradient direction and the main dendrite axis (Fig. 2c). This third grain parameter is important to measure the impact of the temperature gradient on the growth direction. A wide -distribution around a zero average value means that there is no preferential direction for the grain growth and thus that there is no influence of the temperature gradient on the grain growth direction.

It is worth noticing that similar information could be obtained by post-mortem microscopy, but only for the last solidification experiment since the samples were melted in between each solidification sequence.

Case A: horizontal solidification

Formation of the equiaxed grain structure

Fig. 3 shows a sequence of radiographs recorded during the experiments labelled A1 carried out in horizontal configuration. The solidification started with a fully melted sample and the application of the cooling rate on both heaters of the gradient furnace gave rise to the formation of grains in the cold zone of the furnace (Fig. 3a). Then, the solidification front propagated by wave-like nucleation ahead of the solidification front, in a similar way to the observations reported by many authors [START_REF] Nguyen-Thi | In-Situ and Real-Time Investigation of Columnar-to-Equiaxed Transition in Metallic Alloy[END_REF][START_REF] Prasad | A real-time synchrotron X-ray study of primary phase nucleation and formation in hypoeutectic Al-Si alloys[END_REF][START_REF] Xu | Revealing the heterogeneous nucleation behavior of equiaxed grains of inoculated Al alloys during directional solidification[END_REF][START_REF] Jia | In situ Investigation of the Heterogeneous Nucleation Sequence in Al-15 Weight Percent Cu Alloy Inoculated by Al-Ti-B[END_REF]. After their nucleation, the grains developed from almost the same initial location in the radiographs (Fig. 3a). By looking carefully at the grain dynamics, it was seen that some grains slightly rotated and seemed to move in the sample thickness. Many grains slightly moved towards the cold part of the sample due to solidification-induced shrinkage and interdendritic flow of the liquid phase as reported by Salloum et al. [START_REF] Salloum-Abou-Jaoude | Characterization of motion of dendrite fragment by X-ray radiography on Earth and under microgravity environment[END_REF] for dendrite fragments. Those effects induced only very small grain motions and did not affect the key grain structure parameters. The entire solidification process can be seen in Supplementary Video 1.

After a short stage of rapid growth, the grains interacted with each other and gradually formed a compact grain structure for which it was possible to define an effective front, at the top of the grain network (dashed line in Fig. 3c). This effective front separated the upper limit of the developing grain structure and the fully liquid phase. As the cooling of the sample continued, new layers of grains nucleated in the undercooled liquid zone ahead of the effective front (Fig. 3c). The new grains grew and eventually blocked the advancing effective front before being incorporated into the grain compact structure to form a new effective front. This mechanism was repeated over time (Fig. 3d), which lead to the formation of the equiaxed structure until the complete filling of the FoV (Fig. 3e). It is worth noticing that the grains were more elongated at the top of the FoV than in the bottom part. This was due to a higher longitudinal temperature gradient in this region close to the hot heater element of the furnace [START_REF] Soltani | Impact of growth velocity on grain structure formation during directional solidification of a refined Al-20 wt.%Cu alloy[END_REF]. The remaining intra-and inter-dendritic liquid areas solidified at a later stage, following the crossing of the eutectic front through the FoV (dotted line in Fig. 3f).

Grain structure characterization

In Case A (horizontal solidification), the final grain structure in the FoV was composed of rather homogeneous and well dispersed grains. This was confirmed by the distributions of grain diameter (Fig. 4a), grain elongation factor (Fig. 4b) and grain growth orientation (Fig. 4c) for the experiment A1, which agrees with a recent detailed analysis of the impact of growth velocity on those distributions for a similar experimental configuration [START_REF] Soltani | Impact of growth velocity on grain structure formation during directional solidification of a refined Al-20 wt.%Cu alloy[END_REF][START_REF] Soltani | Impact of gravity-related phenomena on the grain structure formation: comparative study between horizontal and vertical solidification of a refined Al-20wt.%Cu alloy[END_REF]. The distribution of grain diameter was well-fitted by a Gaussian-type curve and the microstructure was rather homogeneous in size in the FoV. For the grain size, the mean and maximal values were about dpeak  520 µm and the width at half height was  = 170 µm (similar values were obtained for experiment A2). For the elongation factor, the distribution was well-fitted by a log-normal type curve, as shown in Fig. 4b. It can be seen that the largest number of grains had an elongation factor  lower than two (the maximum peak is about 1.4, with a width at half height  =0.4 ), which means that the grain structure was of equiaxed type, according to Hunt criterion [START_REF] Hunt | Steady State Columnar and Equiaxed Growth of Dendrites and Eutectic[END_REF]. A more detailed analysis of this histogram confirmed that the rare elongated grains ( > 2) were mainly located in the top part of the FoV, in the region close to the hot heater element of the furnace where the temperature gradient was larger. Finally, the grain orientation distribution (Fig. 4c) was rather wide, with a peak at an angle peak  9°, which indicates that the grain orientation was not completely random as would have been expected for a perfectly equiaxed grain structure. The growth orientation was still influenced by the temperature gradient along the sample in these directional solidification experiments, as recently shown by H. Soltani et al. [START_REF] Soltani | Impact of growth velocity on grain structure formation during directional solidification of a refined Al-20 wt.%Cu alloy[END_REF].

Case B: Upward solidification

Impact of buoyancy on grain structure formation

A sequence of radiographs recorded during the upward solidification experiment labelled B2 is shown in Fig. 5. The solidification parameters of this experiment were close to those of the experiment A1 described in the previous section (Table 1). The major difference between Case B and Case A was the impact of buoyancy force that induced the flotation of grains after their nucleation (see Supplementary Video 2), due to the difference of densities between the solid and the liquid, the aluminum grains being much lighter than the surrounding Cu-rich liquid for equiaxed solidification of Al-20wt.% Cu alloys [START_REF] Ganesan | Densities of Aluminum-Rich Aluminum-Copper Alloys during Solidification[END_REF].

Because of buoyancy forces acting on all the nucleated grains, two different types of behavior were observed during the cooling of the sample. Firstly, some grains moved towards the hot part of the sample where they gradually melted, as described in detail in [START_REF] Salloum-Abou-Jaoude | Characterization of motion of dendrite fragment by X-ray radiography on Earth and under microgravity environment[END_REF][START_REF] Abou-Khalil | Influence of gravity level on Columnar-to-Equiaxed Transition during directional solidification of Al -20 wt.% Cu alloys[END_REF] and clearly visible in the earlier stage of the experiment when there is still no well-defined equiaxed microstructure in the FoV. The upward motion of some grains is indicated by white arrows in Fig. 5a and this phenomenon can be seen more clearly in Supplementary Video 2. The second type of grain behavior is specific to experiments carried out in thin samples (about 200 µm in thickness). Due to the strong confinement, the upward motion of some grains could be stopped when their size was of same order than the sample thickness. When this occurred, they continued to develop roughly at the same location, like the pair of grains in plain circles in Fig. 5b that remain visible in the subsequent radiographs. This "move-and-stop" behavior had two important effects on the final grain structure. Firstly, large residual liquid areas (rich of Cu) were forming in between the grains and lead to segregated regions like the one pointed out in Fig. 5b and Fig. 5d. Subsequently, a few new grains nucleated in the residual liquid areas and moved upward until they were blocked by the previously stuck grains. The possibility of grain nucleation between growing grains has also been recently reported by Jia et al. during refined Al-15wt.%Cu alloy solidification [START_REF] Jia | In situ Investigation of the Heterogeneous Nucleation Sequence in Al-15 Weight Percent Cu Alloy Inoculated by Al-Ti-B[END_REF]. In addition, another effect induced by gravity during upward solidification was the promotion of dendrite fragmentations [START_REF] Nguyen-Thi | XRMON-GF Experiments Devoted to the in Situ X-ray Radiographic Observation of Growth Process in Microgravity Conditions[END_REF], leading to the development of many small grains inside the already grown microstructure.

Grain structure characterization

Grain size distribution, grain elongation factor and growth orientation angle are depicted in Fig. 6. Despite the impact of buoyancy forces acting on solid grains, the grain size distribution remained gaussian, with a peak value at about 520 µm, which is surprisingly the same value than in the case of horizontal solidification. However, the width at half height was  = 260 µm, which is much larger than for horizontal solidification ( = 170 µm). For this experimental configuration, grains remained of equiaxed-type as shown by Fig. 6b, with peak = 1.3, also close to the value of the horizontal case. The grain orientation distribution is wide, with a peak value at  = 8°, which confirms that there is still a preferred grain orientation parallel to temperature gradient direction, despite the random motion and rotation of the equiaxed grains when floating.

Case C: Downward solidification

Solute plumes observations

For vertical downward solidification, the temperature gradient was parallel to the gravity vector, so that the temperature field was destabilizing regarding convective instabilities as the hot liquid was below the cold liquid. Moreover, the rejected solute (Cu) is heavier than the solvent (Al) yielding the solute to sink due to gravity and amplifying convective flow.

Thermosolutal convection is then expected to be the dominant gravity-driven effect in this case. Fig. 7 depicts a sequence of radiographs of the experiment C1, showing the representative behaviour of all downward solidification experiments. In this experiment, an initially small porosity defect ( 80 µm in diameter) developed into a larger hole (around 500 µm in width) visible at the bottom of the FoV as indicated in the first radiograph in Fig. 7a. The entering of the first grains in the FoV from the cold region of the sample is also visible Fig. 7a, as well as downward flows of Cu-rich liquid coming from the solidification microstructure and moving towards the bulk liquid (white arrows). These flows, also called convective plumes, were visible long before the appearance of the first grains in the FoV (readers are encouraged to see the Supplementary Video 3) and are caused by the rejection of heavy solute in the mushy zone during the solidification microstructure formation. In the radiographs, these plumes are visible because they are solute-rich and thus their X-ray absorption differs from the surrounding liquid at nominal composition.

Like in the previous experiments, the solidification microstructure progressed toward the hot part of the sample by wave-like nucleation mechanism. However, due to buoyancy force, the new solid grains floated and moved back against the existing solidification front and thus stopped rapidly the growth of existing grains (Fig. 7b to Fig. 7e). Accordingly, the equiaxed grains formed a dense and compact microstructure, whose characteristics were very different from the horizontal or upward solidification experiments due to gravity effects as discussed later in section 5.3. During the propagation of the solidification front, the plumes were drifting continuously along the existing microstructure, as clearly visible in Supplementary Video 3.

Interestingly, solidification was delayed at the root of the segregated channel due to the local increase of solute composition (Fig. 7b, 7c and7d). The remelting of some small grains entering the Cu-rich channel was also observed (see Supplementary Video 3).

Interaction between solute plumes and dendrite growth

The impact of solute plumes on dendrite growth kinetic has been analyzed by means of Xradiography visualization. Using the ImageJ software, we were able to measure (i) the tip position of several dendrites and then to deduce their growth velocities as a function of time and (ii) the gray level in front of the dendrite tip. In our experiments, it was not possible to convert in a straightforward manner the gray level into solute concentration. However, the gray level values are related to the liquid concentration in front of the dendrite tip: the lower the gray level, the higher the liquid composition.

The studied dendrites were selected according to their growth direction, parallel to both gravity and temperature gradient, and their growth duration. For all the measured dendrites, it has been found that the dendrite tip velocity was strongly influenced by solute plumes passing in front of them, as illustrated in Fig. 8. For the dendrite indicated in Fig. 8a, we had the opportunity to reveal up clearly the interaction between the solute variation induced by solute plumes flows (Fig. 8b) and the dendrite tip growth velocity (Fig. 8c). In fact, the dendrite tip growth velocity oscillated above and below an average value, with a period of  20 seconds, showing marked minimum and maximum. Comparing the two plots in Fig. 8 highlighted the close relationship between the dendrite tip growth velocity and the gray level variation. The variations of the dendrite tip velocity were approximately in phase with the liquid solute composition variations, i.e. the minimum and maximum of both curves occurred roughly simultaneously (as indicated by the dashed lines connecting the two curves). A maximum of Cu concentration in front of the dendrite tip gave a minimum of dendrite tip velocity. Moreover, in addition to the remarkable synchronization of these two curves, the oscillation amplitude variations were also in close accordance, at least qualitatively.

Grain structure characterization

When visually comparing the grain structure for Case C (downward solidification) with both other cases, the microstructure was found more compact, with both smaller and more elongated grains than for the horizontal and vertical upward solidifications. Fig. 9a depicts the grain diameter distribution, which was not of gaussian type but well fitted by a log-normal function. The peak value for this experiment was about 184 µm, which is nearly three times smaller than the value of the gaussian function of Case A or the average arithmetic mean of Case B. This log-normal distribution means that the grain structure was mainly composed of small grains completed by some longer grains. The more columnar aspect of the grain structure was confirmed by (i) the distribution of the elongation factor (Fig. 9b) that showed that the number of grains with an elongation factor higher than 2 was higher than for horizontal or upward case and (ii) the distribution of tilt angle (Fig. 9c) that is of gaussian type with a prominent maximum close to 0°.

Discussion

The investigation of solidification experiments carried out with different orientations of the sample with respect to the gravity vector is an effective approach to highlight the impact of gravity-related phenomena on the solidification microstructure formation. Comparing vertical upward and downward experiment allows the impact of convective flow and buoyancy to be studied in thermo-solutally stabilizing or destabilizing configurations, as recently illustrated by Nelson et al. using in situ 4D synchrotron tomography for the solidification of an Al-15wt.%Cu alloy [START_REF] Nelson | Gravity effect on thermal-solutal convection during solidification revealed by four-dimensional synchrotron imaging with compositional mapping[END_REF]. Valuable information can be obtained when such results are complemented by reference experiments performed in diffusive conditions, such as microgravity environment, as clearly demonstrated by Dupouy et al. [START_REF] Dupouy | Natural convective effects in directional dendritic solidification of binary metallic alloys: dendritic array morphology[END_REF] who investigated post-mortem the directional solidification of bulk hypo-and hyper-eutectic Al-Cu alloy. Although purely diffusive conditions were not achieved during the present work, the horizontal solidification experiments can be considered as reference cases for the comparison with the two vertical configurations. Indeed, in horizontal configuration buoyancy effects were strongly minimized by the confinement of the grain flotation in the small thickness of the sample. Thus, the grain motion was significantly restricted as reported in [START_REF] Murphy | XRMON-SOL: Isothermal equiaxed solidification of a grain refined Al-20 wt%Cu alloy[END_REF][START_REF] Browne | Overview of In Situ X-Ray Studies of Light Alloy Solidification in Microgravity[END_REF] and numerically studied by [START_REF] Olmedilla | Quantitative 3D mesoscopic modeling of grain interactions during equiaxed dendritic solidification in a thin sample[END_REF]. In a same way, thermally and solutally induced convection was also expected to be significantly limited, as previously discussed by Nguyen-Thi et al. using an order of magnitude analysis [START_REF] Nguyen-Thi | XRMON-GF Experiments Devoted to the in Situ X-ray Radiographic Observation of Growth Process in Microgravity Conditions[END_REF].

During the upward solidification experiments, the present observations show that the grain structure formation was disturbed mainly by the grain flotation due to buoyancy. This is at the origin of the non-homogeneous grain structure and wider distribution of the grain size. On one hand, the grains that moved towards the hot part of the sample and then melted did not participate to the final grain structure, thus leaving more space for the development both upward and on the sides of grains that remained stuck in the thickness. As a consequence, both larger and longer grains than in the horizontal case were obtained. It is worth noticing that the melted grains could strongly modify the solute composition of the bulk liquid phase, which for bulk samples such as ingot casting could be at the origin of strong macrosegregation. On the other hand, the growth of grains that nucleated in the residual liquid areas was restricted by the previously stuck grains, which resulted in the occurrence of smaller grains than in the horizontal case. The same went for the grains originating from dendrite fragments.

It is worth noting that buoyancy effects became negligible with the increase of cooling rate [START_REF] Soltani | Impact of gravity-related phenomena on the grain structure formation: comparative study between horizontal and vertical solidification of a refined Al-20wt.%Cu alloy[END_REF][START_REF] Zimmermann | Fragmentation-driven grain refinement in directional solidification of AlCu10wt-% alloy at low pulling speeds[END_REF]. Indeed, for a cooling rate of about 0.9 K/s (not shown in this paper), the growth velocity ( 120 µm/s) and the grain flotation velocity ( 100 µm/s) were of same order, yielding to an attenuation of the impact of flotation. Indeed, grains didn't have enough time to move upward after nucleation due to the rapid nucleation of a new layer of grains blocking their motion, leading the development of a grain structure similar to the one obtained in the horizontal case.

A visually striking feature of the downward experiments is the downward flows of Cu-rich liquid leading the formation of solute plumes that were continuously drifting along the effective solidification front. This non-stationary behavior may be attributed to the local change of the grain structure and its permeability during the growth process, yielding to a modification of the dissipative drag force exerted on the liquid and thus on the solute plumes [START_REF] Saad | Simulation of Channel Segregation During Directional Solidification of In-75 wt pct Ga. Qualitative Comparison with In Situ Observations[END_REF]. Similar observations were also reported by Copley et al. [START_REF] Copley | The origin of freckles in unidirectionally solidified castings[END_REF] or Hellawell et al. [START_REF] Hellawell | Channel Convection in Partly Solidified Sustems[END_REF] in transparent alloys, Hachani et al. [START_REF] Hachani | Experimental analysis of the solidification of Sn-3 wt.%Pb alloy under natural convection[END_REF] for Sn-3wt.%Pb alloy and more recently by Boden et al. [START_REF] Boden | Visualization of freckle formation induced by forced melt convection in solidifying GaIn alloys[END_REF] and Shevchenko et al. [START_REF] Shevchenko | The effect of natural and forced melt convection on dendritic solidification in Ga-In alloys[END_REF] during upward vertical solidification of Ga-In samples or Nelson et al. [START_REF] Nelson | Gravity effect on thermal-solutal convection during solidification revealed by four-dimensional synchrotron imaging with compositional mapping[END_REF] during downward solidification of Al-Cu alloys.

The close link between dendrite tip velocity and solute plume motion ahead of the dendrite tip (Fig. 8) has been already mentioned by Shevchenko et al. [START_REF] Shevchenko | The effect of natural and forced melt convection on dendritic solidification in Ga-In alloys[END_REF] for the study of Ga-In, and more recently by Reinhart et al. [START_REF] Reinhart | Impact of solute flow during directional solidification of a Ni-based alloy: In-situ and real-time Xradiography[END_REF] during directional solidification of superalloys. The explanation for these cycles running concurrently is that solute plumes locally increased the Cu concentration ahead of the dendrite tip, which reduced the constitutional undercooling intensity and thus slowed the dendrite tip growth [START_REF] Yuan | A new mechanism for freckle initiation based on microstructural level simulation[END_REF].

By comparing the grain size orientation distributions of the three cases, it appears that the effect of the temperature gradient and of gravity on the grain morphology was stronger during downward experiments than for horizontal or upward experiments. During downward directional solidification of refined alloys, gravity impacts the grain structure dynamics by three distinct mechanisms. Firstly, there is the local increase of solute composition ahead of the solidification front due to the solute plumes that reduces the constitutional undercooling intensity in front of the equiaxed structure. This modifies not only the dendrite tip velocity as discussed above but also provoked an inhibition of refiners. As a result, no nucleation of new grains was observed in the close vicinity of the solute plumes and the already existing grains have become more elongated. We can also mention that solute plumes could also change the spatial distribution of refining particles. However, this effect could not be analyzed in our experiments. Secondly, a subtler effect of gravity on grain structure formation is caused by the sinking of the Cu-enriched liquid, which stretches the solute boundary. The dilation of isoconcentrations has been also put forward by Bogno et al. [START_REF] Bogno | Growth and interaction of dendritic equiaxed grains: In situ characterization by synchrotron X-ray radiography[END_REF] to explain the difference in behavior, in terms of velocity and length, of the upward and downward arms of equiaxed grains growing in isothermal conditions but in a vertical sample. Consequently, liquid undercooling ahead of the existing grain structure becomes less intense, yielding to a reduction of the grains nucleation rate and then augmenting the growth period. A third effect comes from the fact that the grains did not nucleate directly in front of the growing dendrite tips, but often appeared on their sides. Those grains moved toward the solidification front due to buoyancy effect and were trapped in the mushy zone, in between grains that were already developing. The consequence of these combined effects was that long grains of several millimeters could easily develop toward the bottom of the FoV without being blocked, while the growth of grains trapped in the mushy zone was stopped, which agrees well with the measurement of both small equiaxed grains and long columnar dendrites as shown in Fig. 9.

Finally, it can be mentioned that the formation of segregated channels in the mushy zone was not observed in our experiments as (i) either the secondary arms of dendrites on both sides of the plumes grew more rapidly than the average transport of solute or (ii) free equiaxed grains moved upward and plugged the root of the plumes (see Supplementary Video 3). Only a residual liquid area developed above the porosity (Fig. 7d) and lead the formation of a segregated area.

For a sake of completeness, we can mention that stable liquid channels on one side of the sample were observed during downward solidification of refined Al-20wt.%Cu alloys with the application of permanent magnetic field [START_REF] Fautrelle | Thermo-Electric-Magnetic Hydrodynamics in Solidification: In Situ Observations and Theory[END_REF]. In those experiments, it has been demonstrated that an additional thermo-electro-magnetic force pushes the Cu-enriched liquid in the transverse direction of the sample. As a result, a liquid channel was created on one side of the sample, depending on the magnetic field direction.

Conclusion

Gravity effects on equiaxed solidification were investigated during horizontal, vertical upward and vertical downward directional solidifications of refined Al-20wt.%Cu (0.1wt.% AlTiB) samples. The solidification has been provoked by lowering the temperature of both heater elements and the observation of the grain structure has been done in situ and in real-time by X-radiography using the SFINX laboratory facility. The comparison between the three experimental configurations enlightened the marked effects of gravity on the grain structure formation and then on the final grain structure:

-For upward solidification, the dominant effect was buoyancy that provoked the grain flotation in the case of Al-20wt.%Cu alloy and caused the formation of liquid pockets at low growth rates. The grain motion following the nucleation phase yielded a non-homogeneous grain structure.

-For downward solidification, the most striking effect was the formation of solute flow, also called solute plumes. These fluid flows strongly affected the dendrite growth kinetic, which oscillated in phase with solute variations in the liquid ahead of the dendrite tips. Moreover, the local increase in solute due to the plumes prevented grain nucleation in their close vicinity.

Another influence of gravity is related to buoyancy that kept the grains very close to the downward growing microstructure and then significantly shortened their growth duration.

As a result, the average grain size was three times smaller for this type of experiment than for horizontal or upward solidifications.

All these gravity-driven effects have been observed only at low growth rates. For higher growth rates, neither grain flotation nor solute plume movement affected the microstructure.

To continue the study and overcome the gravity effects, a microgravity solidification experiment was conducted on board of the sounding rocket MASER-14 in June 2019 to obtain benchmark experimental data. The columnar-to-equiaxed transition was investigated during this campaign, for a grain-refined Al-20wt.%Cu sample, which is also the alloy composition used in this paper. A comparison will be made between the experiments carried out in microgravity and on ground, as the one presented in this paper, to investigate further the impact

Fig. 1 :Fig. 2 :Fig. 3 :

 123 Fig. 1: Schematic layout of (a) the SFINX furnace (courtesy of A. G. Murphy, UCD), and (b) the SFINX experimental arrangement (furnace and X-radiography system) for horizontal solidification experiment.

Fig. 4 :FrequencyFig. 5 :

 45 Fig. 4: Distributions of (a) grain size, (b) grain elongation and (c) growth orientation for the experiment A1 (horizontal solidification of a refined Al-20wt.%Cu, <VN> = 25.4 µm/s and <GN> = 5.9 K/mm).

Acknowledgements

This work is supported by the XRMON project (AO-2004-046) of the MAP program of the European Space Agency (ESA), by the French National Space Agency (CNES) and the French-Algerian doctoral fellowship program PROFAS B+. The authors would also like to thank the Swedish Space Corporation (SSC) for the development of the SFINX facility and the technical support.

Appendix A. Supplementary data

Supplementary data related to this article can be found at (link to supplementary material)

of gravity and allow a reliable comparison with numerical simulations.


Marcus H Wong

Rhiannon Kirby

Peter Jordan

Daniel Edgington-Mitchell

Azimuthal decomposition of the radiated noise from supersonic shock-containing jets

teaching and research institutions in France or abroad, or from public or private research centers.

I. INTRODUCTION

Broadband shock-associated noise (BBSAN) is a high-frequency noise component emanating from shockcontaining supersonic jets. Observed in both military [START_REF] Vaughn | Broadband shock-associated noise from a high-performance military aircraft[END_REF] and civilian aircraft [START_REF] Huber | Understanding and reduction of cruise jet noise at aircraft level[END_REF], it is more prominent in the upstream and sideline directions. BBSAN is produced by the interaction of downstream-travelling turbulent structures with the train of shock-cells in the jet plume [START_REF] Tam | Supersonic jet noise[END_REF]. Unlike the discrete component known as screech [START_REF] Edgington-Mitchell | Aeroacoustic resonance and selfexcitation in screeching and impinging supersonic jets-a review[END_REF][START_REF] Raman | Supersonic jet screech: half-century from powell to the present[END_REF], where noise is generated by an easily-disrupted feedback loop, suppression of BB-SAN is not straightforward. The lack of simple noise mitigation strategies means BBSAN remains an important component of jet noise.

The work by [START_REF] Harper-Bourne | The noise from shock waves in supersonic jets[END_REF] identified the main features of BBSAN; the sound field is characterised by a broad spectral lobe which varies in frequency as a function of observer position and the ideallyexpanded Mach number (M j ) of the jet. The distinctive directivity arises from the constructive and destructive interference within the axially-extended acoustic source. Using the interpretation of [START_REF] Tam | Shock associated noise of supersonic jets from convergent-divergent nozzles[END_REF], the shock-cell structure can be represented as a series of waveguide modes with wavenumbers k n and shock-cell length approximated by L 1 = 2π/k 1 . Tam and Tanna a marcus.wong@monash.edu (1982) also showed that the primary BBSAN peak frequency, f p , varies with observer angle θ, measured from the downstream jet axis, according to

f p = u c L n (1 -M c cos(θ)) , n = 1, (1)
where u c is the convection velocity of the large-scale coherent structures, and L 1 is the shock-cell spacing of the first shock-cell mode n = 1. Eq. 1 can also be used to predict peaks generated by higher-order shock-cell modes (n ≥ 2). The width of the BBSAN peak narrows as observer angle increases. The amplitude of BBSAN scales to the fourth power of the off-design parameter, defined as β = (M 2 j -M 2 d ) 1/2 where M d is the design Mach number of the nozzle.

When studying noise generated from a circular nozzle, it is useful to exploit the symmetry inherent in the flow. By casting the problem in cylindrical coordinates, both the sound field and its underlying source [START_REF] Lighthill | On sound generated aerodynamically. i. general theory[END_REF] can be decomposed in the azimuthal direction and represented as a Fourier series due to circumferential homogeneity. The advantage of using such a decomposition is exemplified in subsonic jets; the sound field is dominated by the lowest-order azimuthal modes at low polar angles [START_REF] Cavalieri | Axisymmetric superdirectivity in subsonic jets[END_REF][START_REF] Michalke | On turbulence and noise of an axisymmetric shear flow[END_REF]. The acoustic efficiency of higher-order azimuthal modes is reduced due to azimuthal source interference. The interference is explicitly expressed by the Bessel function term in line-source models [START_REF] Michalke | A wave model for sound generation in circular jets[END_REF]. [START_REF] Cavalieri | Axisymmetric superdirectivity in subsonic jets[END_REF] also showed there exists a direct correspon-dence between the acoustic source and sound field of the same azimuthal mode. From a reduced-order modelling perspective, this is appealing, as the desired azimuthal component of the far-field sound can be obtained by computing only the corresponding sound-producing wavepacket [START_REF] Cavalieri | Wavepacket models for jet dynamics and sound radiation[END_REF][START_REF] Jordan | Wave packets and turbulent jet noise[END_REF].

While our understanding of the role of wavepackets in subsonic and ideally-expanded jets [START_REF] Sinha | Wavepacket models for supersonic jet noise[END_REF] is now relatively mature, their application to model BB-SAN is less common. By artificially separating the shock and turbulence components, [START_REF] Tam | Shock associated noise of supersonic jets from convergent-divergent nozzles[END_REF] first proposed the use of spatially-extended instability waves to model the downstream-travelling coherent structures. A BBSAN model [START_REF] Tam | Stochastic model theory of broadband shock associated noise from supersonic jets[END_REF] was developed based on the interaction of these proposed instability waves with stationary shock-cells; experimental sound spectra were reproduced with a high-degree of accuracy over a wide range of operating conditions. The model predictions, however, were obtained semi-empirically by adopting approximations and scaling arguments. One such simplification was the removal of the dependence on azimuthal wavenumber, treating the source as effectively axisymmetric. Hence, comparisons were only made with the total sound field [START_REF] Norum | Measurements of mean static pressure and far field acoustics of shock containing supersonic jets[END_REF]. In their model, [START_REF] Ray | Sound generated by instability wave/shock-cell interaction in supersonic jets[END_REF] also made a comparable simplification, whereby far-field pressure predictions for only the first three azimuthal modes were combined and scaled to match the total acoustic signal. Recently, wavepacket models for BBSAN have been proposed [START_REF] Lele | Phased array models of shock-cell noise sources[END_REF]Wong et al., 2019a,b). Similar to [START_REF] Tam | Stochastic model theory of broadband shock associated noise from supersonic jets[END_REF], they have shown fair agreement with experimental spectral shape, but relied on empirical adjustments to match measured amplitudes since no azimuthally-decomposed far-field data was available. The validation of BBSAN models, in particular those predicated on instability theory, is hindered by the lack of azimuthally-decomposed sound-field data.

While azimuthal decomposition has previously been performed on shock-containing jets, the focus has, until now, largely been on the identification of different screech oscillation modes (Edgington-Mitchell, 2019, see references therein). Only a handful of studies have examined the azimuthal modes of BBSAN in detail. [START_REF] Arroyo | Azimuthal mode analysis of broadband shock-associated noise in an under-expanded axisymmetric jet[END_REF] investigated azimuthal modes using an LES of a non-screeching underexpanded jet at one exit condition. Successive azimuthal modes were found to peak at higher frequencies. Suzuki (2016) also arrived at similar conclusions by observing the azimuthal modal distributions in an LES of an underexpanded jet. An upward shift in emission angle and frequency led to higher-order modes featuring more prominently. Experimentally, this behaviour was also measured by [START_REF] Vold | Axisymmetry and azimuthal modes in jet noise[END_REF] for an overexpanded jet, although the dependence on observer position was not investigated. Of the above studies, only [START_REF] Suzuki | Wave-packet representation of shock-cell noise for a single round jet[END_REF] made mode-bymode comparisons between the decomposed far-field data and BBSAN model predictions, but poor agreement was found. Furthermore, the mentioned studies considered only weakly off-design conditions (β < 0.8); whether the same observations apply to stronger levels of under-or over-expansion is not known.

The aim of this study is to provide azimuthallydecomposed sound-field measurements of shockcontaining jets for comparison with models. Specifically, we examine the azimuthal modal composition for BB-SAN and its variation with Mach number and observer position. The findings should better inform future BBSAN models and allow a basis for comparison between model and experiment. The paper is organised as follows. The experimental facility and setup is discussed in section II, which includes the microphone apparatus, acquisition details and post-processing techniques.

In section III, we present both the far-field acoustic results and azimuthally-decomposed mid-field spectra. A summary of the findings and future perspectives is provided in section IV.

II. METHODOLOGY A. Facility Overview

The acoustic measurements were acquired in the Monash University LTRAC Supersonic Jet Anechoic Facility (SJAF). The facility is designed for the study of acoustics and flow-fields of cold small-scale supersonic jets in a controlled environment. The key components of the facility are presented below.

Anechoic Chamber

The jet exhausts into an anechoic chamber designed according to the best practices suggested by [START_REF] Ahuja | Designing clean jet-noise facilities and making accurate jet-noise measurements[END_REF]. The interior chamber dimensions (wedge-tip to wedge-tip) are 1.5m × 1.2m × 1.4m. The anechoic chamber walls are acoustically treated using foam wedges with a depth of 400mm, corresponding to a nominal cut-on frequency of 500Hz. The floor of the anechoic chamber is also lined with wedge panels. Entrained and exhausted air is replaced via two open sections along the upstream wall. The rectangular openings are acoustically treated with an over-hanging wall to ensure no external noise propagates into the chamber. The acoustic performance of the chamber is provided in Appendix A.

Air Supply

Compressed air is passed through a heatlessdesiccant dryer which is connected to a 5m 3 air reservoir. The air supply allows for a continuous operation up to a nozzle pressure ratio (NPR) of five. The jet operating condition is determined by the NPR, defined as N P R = P t /P ∞ , where P t and P ∞ are the total and ambient pressures respectively. Since the facility shares the same air supply as a supersonic wind tunnel [START_REF] Knast | Design of a supersonic wind tunnel facility[END_REF], a manual butterfly valve is used to direct air into a large mixing chamber. The air flow is controlled by an eccentric rotary plug control valve. In order to manually select the desired NPR, a Fairchild 100 High-Flow Pressure Regulator is used [START_REF] Edgington-Mitchell | Coherent structure and sound production in the helical mode of a screeching axisymmetric jet[END_REF]. Pressurised air from the mixing chamber is then fed into the jet plenum chamber via a flexible wire-braided high pressure PTFE hose. The stagnation pressure in the plenum, as used to determine NPR, was measured using an RS-3100 pressure transducer, with a range of 0-10 bar. The stated manufacturer accuracy of the transducer is ±0.25%, but the measured range in the plenum was approximately ±1%. The size of the plenum chamber was designed to minimise fluid velocity and to provide a large contraction ratio at the nozzle exit plane. No flow conditioning was used upstream of the nozzle exit. A schematic of the facility is shown in Fig. 1(a).

Nozzle Geometry and Design

The jet issues from a converging round nozzle which has an exit diameter of D = 8mm and a lip thickness of 0.5mm. The mild-steel plenum chambers and nozzles are assembled in a modular fashion via flange connections (Fig. 1(a)). The modular nature of the jet rig allows for future modifications and upgrades. The jet rig is mounted onto an adjustable steel stand which traverses linear rails to allow constrained axial motion.

The plenum is connected to the nozzle via a bent and straight circular section with a total length of 800mm and a diameter of 25.4mm. The nozzle contour follows a fifth-order polynomial [START_REF] Morel | Comprehensive design of axisymmetric wind tunnel contractions[END_REF] ensuring that a smooth connection with the straight section is achieved. The resulting contraction ratio is 10.

The small nozzle exit diameter is dictated by the restrictions on air supply and the internal dimensions of the anechoic chamber. Using a small nozzle ensures that the jet is able to operate continuously, even at high NPR. This allows for jet conditions to stabilise, and for lengthy data acquisition. Importantly, the acoustic frequency range of interest is significantly higher than the cut-on frequency of the anechoic chamber.

B. Coordinate Systems

To facilitate the following discussion, two coordinate systems are used in a similar fashion to [START_REF] Cavalieri | Axisymmetric superdirectivity in subsonic jets[END_REF] as shown in Fig. 1(b). The cylindrical coordinate system of the jet is represented by (x, r, φ), where x is the downstream jet axis, r is the radial direction and φ is in azimuth. This is overlaid with a spherical coordinate system (R, θ, Φ) representing the acoustic field, where R is the distance, θ is the polar angle and Φ the azimuth.

C. Operating Conditions

Acoustic measurements were performed on cold supersonic underexpanded jets. The jet operating conditions are characterised by the NPR and the off-design parameter β [START_REF] Norum | Measurements of mean static pressure and far field acoustics of shock containing supersonic jets[END_REF]. For a converging nozzle, the design Mach number M d is unity, and so β = (M 2 j -1) 1/2 . The Reynolds number at the nozzle exit is of order 10 5 based on the assumption of isentropic flow. We note that the Reynolds numbers of all three conditions correspond to the turbulent transition region of a supersonic jet (Re ≈ 10 5 ). Hence, the measured acoustic field may differ from jets at higher Reynolds numbers which have a fully-turbulent shear layer at the nozzle exit [START_REF] Bogey | Effects of moderate reynolds numbers on subsonic round jets with highly disturbed nozzle-exit boundary layers[END_REF][START_REF] Ozawa | Effect of the reynolds number on the aeroacoustic fields of a transitional supersonic jet[END_REF]. We define D j and U j as the ideally-expanded diameter [START_REF] Tam | Shock associated noise of supersonic jets from convergent-divergent nozzles[END_REF] and jet velocity respectively. The operating conditions for the mid-field measurements are summarised in Table I.

D. Data acquisition

Acoustic measurements were made using G.R.A.S. Type 46BE 1/4" pre-amplified microphones with a frequency range of 4Hz-100kHz and a nominal sensitivity of 3.6mV/Pa. The microphone signals were passed through an amplifier and acquired using a National Instruments DAQ. Each recording consisted of 400,000 samples and was acquired at a rate of 200kHz, corresponding to a Nyquist frequency of 100kHz. A low-pass filter was used in the post-processing stage to remove aliasing effects. The microphones were calibrated using a G.R.A.S. pistonphone with a standard calibration level of 114dB at 1kHz. The reference pressure used for all acoustic measurements is P ref = 20µPa. The methodology to obtain acoustic spectra from measured time signals is presented in Appendix B. To facilitate comparison with acoustic measurements of shock-containing jets in literature, nondimensionalised frequency St is used. We define the Strouhal number as St = f D j /U j where f is frequency in Hz. For all acoustic measurements, solid surfaces including the ring are covered with acoustic foam (Fig. 1(c)).

E. Microphone Array & Azimuthal Decomposition

We obtain sound pressure level readings at two different distances. At a distance of R = 50D from the nozzle exit, which we refer to as 'far-field', noise measurements were obtained at a range of polar angles from θ = 60 • -120 • . The aim of performing these simple single-microphone far-field measurements is to validate the new facility against trends in the existing literature.

We then present acoustic measurements using an azimuthal ring at a constant radial distance of r = 11D from the jet axis. These measurements are performed to interrogate the modal composition of BBSAN. The azimuthal ring set-up is shown in Fig. 1(c). The location of the microphone is assumed to be well outside the hydrodynamic field for a shock-containing supersonic jet [START_REF] Seiner | Acoustic near-field properties associated with broadband shock noise[END_REF][START_REF] Suzuki | Wave-packet representation of shock-cell noise for a single round jet[END_REF][START_REF] Tam | Stochastic model theory of broadband shock associated noise from supersonic jets[END_REF], ensuring that only acoustic fluctuations are detected. These measurements, which we refer to as 'mid-field', are used to decompose the acoustic field into azimuthal modes using a methodology similar to [START_REF] Sinha | Active control of noise from hot supersonic jets[END_REF].

The methodology involves two microphones, one fixed and the other moved in 10 • azimuthal increments. By tracing the cross-correlation between the two microphones as a function of azimuthal separation, the crosscorrelation matrix R pp is constructed and used to ob-tain the azimuthal decomposition of the radiated acoustic field. The processing steps are detailed in Appendix C. While the process is more time-consuming than for a fixed microphone array [START_REF] Suzuki | Instability waves in a subsonic round jet detected using a near-field phased microphone array[END_REF], we effectively reproduce the same results with a much smaller microphone count. It was found that sweeping over an azimuthal range of 180 • was sufficient for the frequency range of interest. The azimuthal array is traversed in the axial direction in order to obtain measurements at different polar angles. Since the azimuthal ring is at a fixed distance to the jet axis, the radial distance R between the nozzle and microphone differs for each polar angle.

Before azimuthally decomposing the mid-acoustic field, it is instructive to first look at the far-field behaviour of the jet. Narrow-band far-field noise spectra are shown in Fig. 2 at θ = 60 • and 90 • for a range of NPR. Typical BBSAN directivity and amplitude traits are clearly observed; the peak BBSAN frequency decreases as NPR and θ increase. Moreover, it can be seen that the intensity of BBSAN, represented by overall sound pressure level (OASPL), follows the well-known β 4 proportionality (Harper-Bourne and Fisher, 1973) for β 4 > 0.5. Strong screech peaks are also evident in the acoustic spectra. The presence of screech is known to affect BBSAN, lowering the BBSAN peak frequency for the same operating condition [START_REF] André | Broadband shockassociated noise in screeching and non-screeching underexpanded supersonic jets[END_REF] and also modulating the lengthscales of velocity fluctuations in the jet plume [START_REF] Tan | Novel method for investigating broadband velocity fluctuations in axisymmetric screeching jets[END_REF]. Since the nozzles were not designed with screech suppression features, comparison with other BBSAN studies in literature where screech was suppressed [START_REF] André | Broadband shockassociated noise in screeching and non-screeching underexpanded supersonic jets[END_REF][START_REF] Norum | Measurements of mean static pressure and far field acoustics of shock containing supersonic jets[END_REF][START_REF] Savarese | Experimental study of shock-cell noise in underexpanded supersonic jets[END_REF][START_REF] Viswanathan | Quality of jet noise data: Issues, implications and needs[END_REF], was not undertaken.

In Fig. 3 we show sound pressure level (SPL) as a function of polar angle (60 • < θ < 120 • ) and frequency (0.1 < St < 1.5) at three different NPRs. The BBSAN peak frequency predictions from Eq. 1 are also plotted for comparison. Both N P R = 3.10 and 3.67, in Fig. 3(b) and 3(c) respectively, display agreement with Eq. 1, where we have defined the convection velocity as u c = 0.7u j and the shock-cell spacing L 1 using the vortex sheet approximation from the Pack and Prandtl (P&P) model [START_REF] Pack | A note on prandtl's formula for the wave-length of a supersonic gas jet[END_REF]. The presence of a higherorder shock-cell mode (n = 2) is evident as the second BBSAN lobe. There is a slight underprediction of BBSAN frequency for N P R = 2.25 (Fig. 3(a)), possibly due to the mismatch in shock-cell spacing between the P&P model and experiment. Discrete screech peaks are also observed. Mixing noise is not prominent as this noise component has only small contribution at sideline and upstream positions [START_REF] Sinha | Wavepacket models for supersonic jet noise[END_REF]. The results from far-field measurements are in agreement with wellvalidated models from literature and give us confidence in the mid-field measurements presented below.

B. Azimuthal Modal Analysis of the Mid-field

Following the method shown in Appendix C, we obtain the azimuthal decomposition of the radiated noise. Due to the microphone array being only 11D from the jet axis, the origin of the polar angle θ is now taken at a downstream location of x 0 = 5D, in similar fashion to [START_REF] Gojon | Numerical study of the flow and the near acoustic fields of an underexpanded round free jet generating two screech tones[END_REF]. The origin is redefined in an effort to more faithfully capture the origin of the acoustic source, and is an indirect demonstration that BB-SAN is generated at the downstream shock-cells [START_REF] Tam | A multiple-scales model of the shock-cell structure of imperfectly expanded supersonic jets[END_REF]. Sound pressure level measurements are only shown up to St = 1.5 due to the upper frequency limit of the microphones. Hereafter, we shall focus only on the BBSAN component in our discussion.

Azimuthally Decomposed Spectra

The acoustic pressure spectra at θ = 80 • , 95 • and 115 • , for the range of NPRs in Table I, is shown in Fig. 4. The first four azimuthal modes (m = 0, ±1, ±2, ±3) are shown along with the full signal. For m > 0, we combine both the negative and positive components (Eq. C8) and hence drop the '±' for compactness. The BBSAN lobe is present for all azimuthal modes and follows the known trends as Mach number and observer location vary. For both N P R = 3.10 and 3.67, the lobe due to the second shock-cell mode, viewed at the sideline and upstream angles, is clearly present.

We first focus on the relative importance of each azimuthal mode across the frequency range of interest. At low frequencies (St 0.6 -0.7), the single helical mode is dominant. There is a clear distinction between each successive mode (m > 1), with decreasing amplitude as mode number increases. The separation between modes becomes less pronounced as frequency increases. Fig. 4 also clearly shows that m = 0 is not the dominant azimuthal mode for the angles of interest. The sound field of the axisymmetric mode is on average 3-4dB/St less than the dominant mode(s). This observation is also consistent with subsonic jet noise and source models; higher azimuthal modes (m = 1 and 2) become increasingly important at higher frequencies and for positions further upstream, while downstream noise is dominated by mode m = 0 [START_REF] Jordan | Wave packets and turbulent jet noise[END_REF].

At St ≈ 0.6 -0.7, the distinction between non-zero azimuthal modes is absent and we observe an approximately equal contribution from modes m = 1, 2 and 3. This trend is observed at all three NPRs shown and continues as mode number increases (not shown for brevity). The increasing contribution from successive modes as frequency increases is consistent with the findings of [START_REF] Kuo | Experimental investigation of near-field pressure fluctuations generated by supersonic jets[END_REF] and [START_REF] Vold | Axisymmetry and azimuthal modes in jet noise[END_REF], who measured the same behaviour in both ideally-expanded and overexpanded jets.

By comparing these results against well-known BB-SAN trends, we arrive at some observations of interest. Depending on the frequencies of the BBSAN lobe, the relative contribution of each azimuthal mode varies. For instance, looking at the N P R = 2.25 jet at θ = 80 • , the m = 1, 2 and 3 azimuthal modes are of similar strength for the BBSAN lobe, while those same modes are distinct (with m = 1 being the strongest) at the lobe for the N P R = 3.67 case at θ = 115 • . A similar trend is also observed when comparing the azimuthal content of higher frequency secondary lobes (n = 2) to the primary one (n = 1).

The trends observed in Fig. 4 are shown in a more comprehensive manner in the St -θ contour diagrams in Fig. 5. Decomposed acoustic spectra up to m = 4 are shown between 50 • < θ < 125 • . For both N P R = 3.10 and 3.67 jets, the BBSAN directivity of the primary (n = 1) and secondary (n = 2) peaks agrees favourably with Eq. 1 (dashed lines). For N P R = 2.25, the experimental peak frequency is higher than that suggested by Eq. 1, as expected from the far-field data in Fig. 3(a). The same inputs as section III A were used in Eq. 1 for all azimuthal modes. This suggests that the instability waves associated with each azimuthal mode convect at a similar velocity. From Eq. 1, the BBSAN lobe shifts upward as Mach number decreases and observer position moves downstream. Hence, as θ and NPR decrease, the azimuthal content of BBSAN becomes increasingly multi-modal. Fig. 5 illustrates this trend. At low Mach numbers (N P R = 2.25), since the BBSAN lobe is above St ≈ 0.7, there is largely equal contribution from the first four nonzero modes. At higher Mach numbers, the BBSAN lobe shifts down and we see a difference in modal behaviour. At N P R = 3.67, BBSAN that peaks at sideline and upstream angles (θ > 90 • ) is dominated by m = 0, 1 and 2. However, at lower emission angles (θ < 90 • ), the BB-SAN lobe shifts to higher frequencies and the contribution from modes m = 3 and 4 is non-negligible. Similar behaviour is also seen for N P R = 3.1. This trend is in agreement with [START_REF] Arroyo | Azimuthal mode analysis of broadband shock-associated noise in an under-expanded axisymmetric jet[END_REF] where LES of a non-screeching underexpanded jet with an ideallyexpanded Mach number of M j = 1.15 was investigated. Although the experimental measurements of this study are conducted in the presence of screech, this does not seem to affect the frequency dependence of the azimuthal modes that contribute to BBSAN.

In regions where the difference between azimuthal modes is pronounced (low frequency, upstream angles), the acoustic field exhibits low-rank behaviour. Here, 'low-rank' is referred to as a rank-N truncation of the azimuthal Fourier basis which provides an approximate description of the sound field to within 1dB/St precision of the full-rank data. The connection between BBSAN directivity and variation in azimuthal modal quality is further explored in the following section.

Acoustic Field Reconstruction

To investigate the relevance of subsequent lowerorder azimuthal modes to the full BBSAN signal as a function of θ and NPR, we perform reconstructions of the acoustic field using an increasing number of azimuthal modes (Eq. C6). The upper row of Fig. 6 contains rank maps showing the number of modes required to attain 1dB/St precision for the total signal. The modes are added successively. The solid red contour provides a visual aid for the position of the BBSAN lobe. As expected from Fig. 4, a single-mode (m = 0) representa-tion of the acoustic field is insufficient. At low frequencies (St 0.6 -0.7), apart from some inconsistencies for the N P R = 3.10 case, low-rank behaviour is observed, with the combination of 2-3 modes sufficient to recover the full signal. Comparing this trend with the position of the BBSAN lobes, we hence observe low-rank behaviour for N P R = 3.10 and 3.67; 2-3 modes again provide a sufficient approximation for BBSAN. As alluded to in section III B 1, however, at lower emission angles the number of required modes increases. For example, at N P R = 3.10, five modes are required at θ = 70 • , while only three modes are used at θ = 100 • to obtain the same precision. The latter observation, however, does not apply at N P R = 2.25, where the higher-frequency lobe consistently requires 4-5 modes to capture the total BBSAN amplitude.

To further illustrate this behaviour, reconstructed acoustic spectra at θ = 95 • are shown in the lower row of Fig. 6. The corresponding position on the rank maps is indicated by the vertical dashed line. While convergence is rapid at low frequencies, the low-rank approximation deteriorates as frequency increases. Consequently, we observe that the secondary lobe is of increased modal complexity relative to the primary lobe (Fig. 6b andc). To summarise, BBSAN can be approximated by an azimuthally low-rank approximation. As the peak frequency varies, the exact number of modes required for reconstruction of the total sound field will depend on both the operating condition of the jet and the observer position, with an increasing number of modes required at higher frequency.

These observations suggest that previous instability wave BBSAN models neglected some relevant features. For example, the removal of azimuthal dependence by [START_REF] Tam | Stochastic model theory of broadband shock associated noise from supersonic jets[END_REF] does not adequately address the multi-modal behaviour at high frequencies. Similarly, we might infer that the truncation of modes to m ≤ 2 by [START_REF] Ray | Sound generated by instability wave/shock-cell interaction in supersonic jets[END_REF] likely contributed to the observed 'missing sound' at high frequencies. These approximations may explain why such models either had poor amplitude predictions at high frequencies, or resorted to empiricism to match the total measured sound field. A more insightful means of validating such models may be to undertake azimuthal mode-by-mode comparisons, similar to the process of [START_REF] Cavalieri | Axisymmetric superdirectivity in subsonic jets[END_REF] for subsonic jets.

IV. SUMMARY & OUTLOOK

Acoustic measurements were obtained from an underexpanded jet operating at three conditions. Far-field single microphone measurements were acquired to val- idate measurements from the new facility. Using two microphones mounted on a traversable array, the sound field was decomposed for the purposes of exploring the modal composition of BBSAN.

The single microphone measurements were in agreement with previous observations. Strong screech peaks were observed and the well-known BBSAN directivity shape was recovered, in agreement with [START_REF] Tam | Shock associated noise of supersonic jets from convergent-divergent nozzles[END_REF]. The BBSAN amplitude was also found to follow the β 4 scaling.

Using a similar methodology to [START_REF] Sinha | Active control of noise from hot supersonic jets[END_REF], azimuthally-decomposed acoustic spectra were obtained. It was found that, unlike in subsonic and ideallyexpanded jets where the dominant sound radiation is in the downstream direction, the axisymmetric mode is not the dominant contribution to BBSAN in the acoustic field. Rather, the m = 1 mode is the most important for St 0.7. Further, we show that modes m > 1 are increasingly important for higher frequencies. Successive reconstructions of the acoustic field demonstrate that approximately 3-5 azimuthal modes are required to recover the total BBSAN signal; the minimum number of requisite modes varies depending on the peak frequency. When correlated with the known directivity of BBSAN, we arrive at an interesting corollary; the total BBSAN sound field becomes progressively low-rank in azimuth as polar angle and Mach number increase. For the secondary lobe, at higher frequencies than the primary one, we clearly see higher-rank behaviour.

From a modelling perspective, the findings are consistent with the predictions made by wavepacket BBSAN models. In such models, the shock-cells are axisymmetric and assumed to be stationary, while the frequency and azimuthal characteristics are solely defined by the wavepacket. It is well-known that the source location shifts towards the nozzle as frequency increases [START_REF] Suzuki | Wave-packet representation of shock-cell noise for a single round jet[END_REF]. Previous parabolised stability equation (PSE) calculations of hydrodynamic wavepackets in supersonic jets [START_REF] Ray | Sound generated by instability wave/shock-cell interaction in supersonic jets[END_REF][START_REF] Rodríguez | A study of linear wavepacket models for subsonic turbulent jets using local eigenmode decomposition of piv data[END_REF] have shown that the spatial extent of wavepackets contracts as azimuthal mode number increases, shifting the peak of the envelope closer to the nozzle exit. Hence, only for the high frequency BBSAN components can wavepackets of higher-order modes be acoustically efficient. [START_REF] Ray | Sound generated by instability wave/shock-cell interaction in supersonic jets[END_REF] also found that the energy is more evenly distributed across azimuthal modes of instability waves as frequency increases, in accordance with our findings here.

The nontrivial contribution of higher-order azimuthal modes to the full spectrum necessitates that wavepacket models for BBSAN should be able to accurately predict their evolution, particularly for higher frequencies, as confidently as for the lower-order modes. [START_REF] Sasaki | High-frequency wavepackets in turbulent jets[END_REF] have demonstrated the validity of wavepacket models up to a Strouhal number of St = 4.0 and azimuthal wavenumber m = 4 for a M = 0.9 jet. This suggests wavepackets are well-placed to model the large-scale structures over the BBSAN spectral range. While outside the scope of the present study, there is clear motivation for future work on the construction and refinement of these wavepacket models for shockcontaining jets.

The results presented not only directly address the issue of the lack of azimuthally-decomposed acoustic data, but also shed light on the relative contribution of each azimuthal mode to the full BBSAN signature. The importance of higher-order modes, including their relevance in shaping overall BBSAN directivity, should be addressed in future models. The method of [START_REF] Ahuja | Designing clean jet-noise facilities and making accurate jet-noise measurements[END_REF] is used to calibrate the anechoic chamber and determine its acoustic 'cleanliness'. For a point source, the measured sound amplitude should obey the inverse-square law as distance increases in any direction. Apart from the removal of the nozzle and azimuthal ring, no further modifications were made inside the chamber. A 45mm diameter speaker, with a range of 140-20,000Hz, is placed along the jet centreline at the same position as the nozzle exit. The same G.R.A.S. Type 46BE 1/4" microphones were used and mounted on a traverse line at different directions from the source. To determine the deviation from the inversesquare law, the method prescribed by [START_REF] Cunefare | Anechoic chamber qualification: Traverse method, inverse square law analysis method, and nature of test signal[END_REF] was used. The theoretical free-field SPL (SP L p ) at distance r i is given by where a and r 0 are the apparent acoustic source strength and the offset distance between the physical location of the source and its acoustic centre respectively. For N number of measurements in the radial direction, the values of a and r 0 are given as

SP L p (r i ) = 20 log 10 a r i -r 0 , (
a = ( N i=1 r i ) 2 -N N i=1 r 2 i N i=1 r i N i=1 q i -N N i=1 r i q i , (A2) r 0 = N i=1 r i N i=1 r i q i - N i=1 r 2 i N i=1 q i N i=1 r i N i=1 q i -N N i=1 r i q i , (A3)
where q i = 10 -0.05 (SP Lpi) . The deviation in dB between the measured SP L pi and theoretical value at distance r i is hence given by

∆SP L p (r i ) = SP L pi -SP L p (r i ). (A4)
The curves in Fig. 7 show that the frequencies of interest are within the required tolerance as specified in ISO 3745:2012 and confirms that the inverse-square law holds for far-field measurements. Some slight deviations are seen closer to the nozzle at upstream positions (θ = 140 • ) due to the presence of upstream piping.

noise spikes, and a low-pass filter is applied at 80kHz. The mean of the signal is removed and then de-trended using a linear regression line P (t) The power spectral density is computed by

P SD(f ) = 1 N B K k=1 p k (f )p * k (f ), ( B3
)
where the signal has been averaged over N B = 974 blocks and * denotes the complex conjugate. We convert the PSD into SPL using the classic definition and a reference pressure of P ref = 20µPa giving SP L(f ) = 10 log 10 P SD(f )

P 2 ref . ( B4
)
The presented SPL is in units of dB/St since the PSD is nondimensionalised by the factor U j /D j , and frequency is thus defined as St = f D j /U j .

APPENDIX C:

We present the methodology used to compute the two-point cross correlation matrix R pp (φ 1 , φ 2 ). The Fourier-transformed signals from the two microphones (Eq. B2), are used to compute the cross spectrum,

R pp (φ 1 , φ 2 ; f ) = 1 N B K k=1 p k (φ 1 ; f )p * k (φ 2 ; f ), ( C1
)
where ψ = φ 2 -φ 1 and we drop the x and r coordinates for compactness as both microphones are on the same circumferential plane. The azimuthal homogeneity of the round jet allows R pp to only depend on ψ.

The increment in φ is fixed at 10 • , but a finer discretisation is desired for our analysis. We employ interpolation to reconstruct R pp for each desired St. We note that R pp , from Eq. C1, is a complex quantity and can be separated into the synchronous (real) and asynchronous (imaginary) component, R pp (ψ; f ) = a(ψ; f ) + ib(ψ; f ).

(C2)

The real component, a(ψ; f ) is a symmetric function and hence a sinc function was found to be an appropriate fit where A, B, C, D are fitting parameters. An example of the raw data R pp with an interpolated fit R pp is shown in figure 8. Where a sinc function was found to be inappropriate, the values at the grid points are connected with a linear fit. We note here that the goal is to have a curve that is faithful to the measured data rather than to infer physical meaning from the fit itself.

Using the interpolated fits R pp , the m th azimuthal Fourier mode is obtained by performing a Fourier transform in azimuth, giving

R pp (f, m) = 1 2π π -π
R pp (φ 1 , φ 2 ; f )e -imψ dψ, (C5) and its reconstruction is given by

R pp (ψ; f ) = ∞ m=-∞ R pp (f, m). ( C6
)
where for the axisymmetric mode (m = 0), Using the circumferential isotropy and homogeneity properties [START_REF] Michalke | On turbulence and noise of an axisymmetric shear flow[END_REF], the Fourier coefficients of R pp are equivalent to the PSD of pressure fluctuations when ψ = 0, which gives P SD(f ) = R pp (ψ = 0; f ), (C9)

and hence the amplitude | R pp,m (ψ = 0; f )| is substituted into Eq. B4 to compute the power spectrum for each azimuthal mode.

  FIG. 1. Schematics of facility and experimental setup; (a) sectioned plan and side view schematic of SJAF showing key components of the facility (dimensions in mm), (b) schematic of adopted coordinate system, (c) azimuthal ring set-up (colour Online) .

  FIG. 2. (Colour Online) Effect of changing NPR at (a) θ = 60 • , (b) θ = 90 • and the dependence on β 4 as a function of Mj.

  FIG. 4. (Colour Online) Decomposed acoustic spectra in the mid-field (R = 11D) for different NPRs; N P R = 2.25 (upper row), 3.10 (middle row) and 3.67 (lower row). Polar angles taken with respect to downstream jet axis; θ = 80 • (left column), θ = 95 • (middle column) and θ = 115 • (right column).

FIG. 5

 5 FIG. 5. (Colour Online) Full and decomposed spectra in the mid-field (R = 11D) for different conditions; N P R = 2.25 (upper row), N P R = 3.10 (middle row) and N P R = 3.67 (lower row). Polar angles taken with respect to downstream jet axis and contours are in dB/St. Dashed lines indicate peak frequency predictions from Eq. 1.

  FIG. 6. (Colour Online) Reconstruction of the acoustic field with modes added successively (m = 0, ±1, ±2...). Top row; St -θ maps displaying the number of modes required to reconstruct the total acoustic signal to within 1dB/St for N P R = a) 2.25, b) 3.10 and c) 3.67. The 130dB/St contour (red line) identifies the primary BBSAN lobe seen in Fig. 5. The vertical dashed lines locate θ = 95 • . Bottom row; Reconstruction of acoustic spectrum at θ = 95 • with increasing number of modes.

ACKNOWLEDGMENTS

  This research is supported by an Australian Government Research Training Program (RTP) Scholarship and the Endeavour Research Leadership Award -an Australian Government initiative. The authors also received funding from the Australian Research Council through the Discovery Projects scheme. M.H.W would also like to acknowledge Mr Danny Lynch, Mr Phil Nicholson and the support of the Monash Mechanical and Aerospace Technical Services Group in the construction of SJAF. APPENDIX A:

FIG. 7 .

 7 FIG. 7. Deviation from the inverse-square law at three different angles for a range of frequencies and polar angles. Red dashed line indicates maximum permissible deviation as specified by ISO 3745.

  FIG. 8. Example of an interpolated fit to the Rpp two-point data as a function of azimuthal separation for N P R = 3.67 at θ = 95 • for a) St = 0.2 and b) St = 0.7.

R

  pp,0 (ψ; f ) = R pp (f, m = 0), (C7) and for m ≥ 1 R pp,m (ψ; f ) = R pp (f, m) + R pp (f, -m).(C8)

TABLE I .

 I Summary of jet operating conditions.

	NPR	Mj	Dj/D	β	Re
	2.25	1.14	1.01	0.55	2.8 × 10 5
	3.10	1.38	1.05	0.95	3.8 × 10 5
	3.67	1.50	1.09	1.12	4.5 × 10 5

  Before carrying out the Fast Fourier transforms (FFTs) of p(t), to improve convergence, we separate the time signal into N B blocks of data each containing N f f t = 2048 points. To suppress side-lobe leakage, a Hanning window H 0 (t) is used. The FFT of each block p k (t) is then given by p k (f ) = 8/3 N f f t (FFT(H 0 (t)p k (t))) .

	p(t) = P (t) -	1 N t	Nt n=1	P (t) -P (t).	(B1)
					(B2)

J. Acoust. Soc. Am. / 26 November 2020 Azimuthal decomposition of shock-containing jets

J. Acoust. Soc. Am. / 26 November 2020Azimuthal decomposition of shock-containing jets

APPENDIX B:

The following outlines the steps required to process microphone data to obtain the ensemble-averaged power spectral density (PSD) spectra shown in section III. All acoustic data were acquired at a sample rate of 200kHz for two seconds, giving a total of N t = 4 × 10 5 sample points P (t). The signal is checked for spurious trends and