Experimental study of a self-excited jet precession in a sudden expansion flow

This study presents Stereoscopic particle image velocimetry measurements of a precessing motion resulting from a sudden expansion of an axisymmetric jet into a coaxial chamber. The sudden expansion has an expansion ratio of 4.9 and an aspect ratio of 2.03. The velocity profile in the inlet tube is measured upstream of the expansion using Laser doppler anemometry. The Reynolds number (Re) is varied from 3160 to 63,870. The time-averaged flow field exhibits an axisymmetric behavior, and the velocity profile near the expansion resembles that of a jet. The half-width and the opening angle of the jet are similar for all Reynolds numbers measured, with values of 3.73±0.04∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3.73\pm 0.04^{\circ }$$\end{document} and 8.73±0.19∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8.73\pm 0.19^{\circ }$$\end{document}, respectively. Despite its axisymmetry, the flow field contains unsteady coherent structures in nature, which are analyzed using both spectral proper orthogonal decomposition and phase averaging. These structures make the instantaneous flow field asymmetric and create a precession around the central axis of the geometry. The structure of this precession is similar for 8300≤Re≤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8300 \le \hbox {Re} \le$$\end{document} 63,870. For 3160≤Re≤8300\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3160 \le \hbox {Re} \le 8300$$\end{document}, the precession is very weak and hard to extract from the turbulent flow fields. The Strouhal number (St) increases linearly for 8300≤Re≤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8300 \le \hbox {Re} \le$$\end{document} 23,600, after which it reaches a value of around 0.003. Precessing structures of this kind were previously only reported for swirling sudden expansion flows, fluidic precessing jet nozzles or annular free jets. However, this study shows that they also exist in confined, non-swirling sudden expansion flows, without exiting to an ambient fluid.


Introduction
Sudden expansion flows are found in a wide range of engineering applications, such as heat exchangers, chemical reactors, cyclone separators, etc. (Vanierschot 2018;Khezzar et al. 1986;Hardalupas et al. 1992). Sudden expansion flows have been extensively studied in the literature, although primarily for two-phase and/or swirling flows (Ahmed et al. 2008;Dellenback et al. 1988;Durrett et al. 1988; Vanierschot and Van den Bulck 2008; Wang et al. 2004). Most of the research focuses on the reattachment length for specific boundary conditions and far less on the temporal characteristics of the flow.
Flow field measurements and simulations can be found in the literature for sudden expansion flows, with aspect ratios either infinite or at least bigger than 10. The aspect ratio is defined as L/D where L represents the length of the confinement and D represents its diameter. Dellenback et al. (1988) conducted measurements using Laser Doppler Anemometry (LDA) in a sudden expansion with an expansion ratio of 1.94 and an aspect ratio of 10.56. The study investigated Re = 30,000 and Re = 100,000 and a range of swirl numbers from 0 to 1.2. Both the time-averaged flow fields and the temporal behavior are reported. In the low-swirl case, it was observed that a Precessing Vortex Core (PVC) rotated against the mean swirl direction. It is worth noting that no measurements were reported for S < 0.1 , and a steady flow was assumed at S = 0 , where S represents the swirl number. The assumption of steadiness at S = 0 , for an expansion ratio of 1.94, is confirmed by Unsteady Reynolds Averaged Navier-Stokes (U-RANS) simulations of Guo et al. (2001) and LDA and Large Eddy Simulations (LES) of Wang et al. (2004). Guo et al. (2001) state that a steady flow field is present up to a swirl number of 0.044 and Wang et al. (2004) confirmed this absence of low-frequency oscillations in the non-swirling case.
Low precessing instabilities can also be found in other flow topologies. Stieglmeier et al. (1989) conducted a study on axisymmetric expansions with different diffuser half-angles (i.e., 14 • , 18 • , and 90 • ) at Re = 156,000 and an expansion ratio of 2.56. They revealed unsteady and irregular behavior of the reattachment point's location without any observance of a dominant frequency. Vanierschot and Van den Bulck (2011) used phase averaging to study low precessing frequencies in the wake of a free turbulent annular jet and they found a Strouhal number St = 0.0025 of the precession with Re = 12600.
Many measurements and simulations have been done on a Fluidic Precessing Jet (FPJ). An FPJ is a particular kind of sudden expansion flow in the sense that the flow is confined for a certain length before exiting into an ambient fluid. The FPJ is patented by Nathan et al. (1999) and Luxton and Nathan (1999), mainly for application in burners for rotary cement and lime kilns. These jets show dynamic behavior for the non-swirling case as ambient air is entrained into the volume. The flow dynamics alternate between two distinct modes, namely an axial jet flow and a precessing jet mode. The probability of the precessing jet mode is dependent on the geometry of the system, i.e. the inlet condition, the outlet condition, the expansion ratio, the aspect ratio, and the Reynolds number (Mi et al. 2006;Wong et al. 2004;Madej et al. 2011;Nathan 1988). Nathan (1988) provides hot-wire anemometry data from 4 different sensors inside a sudden confined expansion of an FPJ with an aspect ratio of 2.73 and an expansion ratio of 6.43. They found a precession with St = 0.0015 at Re = 105,000. Nathan (1988) also found that the precession probability increases with increasing Re . Nathan et al. (1998) used the same geometry and measurement positions as the hot-wire sensors of Nathan (1988) for pressure sensor measurements. They found 0.0017 ≤ St ≤ 0.0014 for Re = 53,100 ≤ Re ≤ 134,000. Beside internal measurements, i.e. in the confinement of the FPJ, multiple studies focus on the flow fields exiting into the ambient. The study conducted by Mi and Nathan (2004) involved the use of an expansion ratio of 5.26 and modifications to various parameters of the setup, including changes to the aspect ratio from 2.2 to 2.54, the addition of a centrebody in some cases, and variations in inlet diameter, length, and type. The study revealed a decreasing pattern of St from 0.0014 to 0.002 for 27,000 ≤ Re ≤ 109,000, when a contraction and centrebody are combined. A similar downward trend is observed for the combination of an orifice and a centrebody. In contrast, when a contraction was used as inlet and no centrebody was present, a nearly constant value of St = 0.0021 was measured. Wong et al. (2004) studied the effect of initial conditions on an FPJ with an expansion ratio of 5.06. An LDA probe is used to measure the frequency downstream of the nozzle exit plane, where the fluid exits into an ambient fluid. The intermittency of the precessing jet mode decreases when a lip and/or a centrebody is added. This finding confirms the study of Nathan (1988) and St is around 10-20% lower with a centrebody. The data suggest that increasing Re results in a decrease of St for 50,000 ≤ Re ≤ 175,000. Mi et al. (2006) studied an FPJ with a hot-wire measurement downstream of the exit into the ambient fluid. The geometry has an expansion ratio of 5 and an aspect ratio of 2.2. The upstream flow inlet conditions are varied between an orifice, a smooth contraction, and a pipe. For 33,000 ≤ Re ≤ 71,000, St exhibits a linear decrease ranging from 0.0027 to 0.0023 for an orifice, from 0.0018 to 0.0015 for a smooth contraction, and from 0.0014 to 0.0011 for a pipe. The highest probability for a precessing jet is to be found at an aspect ratio between 2 and 2.75, in accordance with Madej et al. (2011). A probability of almost unity is achieved by adding a centrebody at the outlet (Wong et al. 2004). Guo et al. (2002) found a combination of precession and a flapping oscillation at St = 0.0026 for the no-swirl case, with an expansion ratio of 5 and an aspect ratio of 20. These results are obtained by URANS simulations with a k − turbulence model. The precession direction for low-swirl numbers is opposite to the swirl direction, which is also the case for the FPJ in experimental studies (Nathan 1988). A tomographic PIV measurement in the confinement after a sudden expansion is done by Ceglia et al. (2017), with Re = 40,800, an expansion ratio of 5 and an aspect ratio of 2.75. They state that the precession can restart in any rotational direction after going through the axial flow mode. This implies that the average flow field is expected to have no rotational component, as also stated by Nathan et al. (1998).
Even though the geometry of an FPJ is different from the geometry of this study, the FPJ is a similar case in terms of precessing instabilities in non-swirling flows. An FPJ requires a sudden expansion, after which the flow is confined for a certain length before exiting into an ambient fluid. The setup in this study uses a similar expansion with expansion and aspect ratios based on the FPJ. However, in contrast to the FPJ, the setup in this study contracts at the outlet to a round tube to close the fluidic loop. To the authors' knowledge, a detailed (temporal) analysis of confined nonswirling sudden expansion flows with a limited aspect ratio and without an exit into an ambient fluid is currently lacking. This study aims to fill this gap by performing S-PIV measurements for Re = 3160 to 63,870 and an expansion ratio of 4.9. The inlet velocity profile before the sudden expansion is measured with LDA. The mean velocity fields and profiles are analyzed, and low-frequency instabilities are identified using Spectral Proper Orthogonal Decomposition (SPOD). The intermittency of the precessing jet is quantified based on the temporal modes of the SPOD. Also, the spreading of the unstable jet is compared to a free axisymmetric jet. The dominant frequency's dependency and the spatial instabilities' shape on Re are also investigated.

Experimental setup
A view of the experimental setup from the PIV laser perspective is shown in Fig. 1. The geometry is a sudden 107 Page 4 of 15 expansion with a round nozzle that contracts at the outlet in a 45 • angle, seen in the left of Fig. 2. The geometry is based on a reactor design for liquid-solid mixing, as discussed by Yang et al. (2022). Before the test section, the flow is split into six channels and fed into a movable block swirl generator, as shown in Fig. 3. However, the guide vanes have been rotated to produce no swirl, as can be seen in Fig. 4. This swirl generator originates from an International Flame Research Foundation burner of 30kW (Dugué and Weber 1992). It has also been used in previous studies, where the outlet flow is symmetrical within measurement accuracy (Vanierschot et al. 2014;Vanierschot and Ogus 2019). The diameter of the inlet tube is d i = 22.5 mm, and the one of the confinement D = 110 mm, giving an expansion ratio D∕d i = 4.9. The nozzle consists of multiple parts. First, a small step of 0.15d i , then a small conical part at an angle of 20 • and last, a larger conical part at an angle of 70 • (diffuser half-angle), where both angles are measured from the central axis. The length (L) of the confinement is 223.5 mm, from the end of the nozzle up to the start of the outlet contraction, which results in an aspect ratio (L/D) of 2.03, ideally for a high precession probability (Nathan 1988;Wong et al. 2004). The outlet contracts with an angle of 45 • to an outlet diameter of 30 mm, which gives a contraction ratio of 3.67. A frequency-controlled pump (Packo NP/68-50/152) is used with water as a circulating medium. The flow rate is adjusted by changing the input frequency of the pump. Reynolds numbers from 3160 to 63,870 are obtained, as shown in Table 1. The bulk inlet velocity ( u inlet ) is calculated from the flow rate. This has a maximum measurement error of 6.3% , 1.3% , and 1.4% for the data sets S-PIV 1, S-PIV 2, and S-PIV 3. The water temperature is measured at the start of each measurement to obtain the correct density and viscosity for the calculation of Re . The cylindrical and octagonal confinement are made of transparent Polymethyl Methacrylate (PMMA). The volume between the confinement and the cylinder is filled with water to minimize refraction issues.

Methodology for the stereoscopic particle image velocimetry (S-PIV) measurements
The velocity field is measured in three different S-PIV measurement campaigns: S-PIV 1, S-PIV 2, and S-PIV 3. Each measurement is done on the same setup, but the measurement parameters are slightly changed, as shown by Table 1. Between the measurements, the setups have been disassembled and reassembled. These parameter changes are done to check the repeatability and the influence of different measurement settings. A schematic representation of the S-PIV configuration is shown in the right of Fig. 2. The laser sheet contains the central axis of the geometry, measuring the velocity field in a median plane (XY-plane). The two cameras are positioned in a forward scattering position, as shown in Fig. 2, and the angles for are given in Table 1 Table 1. The overlap is 50% for all measurements. The spatial resolution between adjacent vectors for every measurement is shown in Table 1. Two criteria are used to delete possible erroneous vectors: any vector with a correlation value lower than 0.5 is removed, and a 4-pass regional median filter checks the correlation map for the second and third highest peak, and if these are less than 4 standard deviations different to the neighboring vectors average, they are used. These two post-processing steps are applied to the final vector field, and the remaining missing vectors are interpolated from their neighbors.

Methodology for the laser doppler anemometry (LDA) measurements
LDA measurements are performed to provide a better understanding of the inlet flow conditions of the specific jet. Despite the addition of a confinement to minimize refraction errors, the LDA measurements are only reliable in the jet's center Fig. 3 30kW movable block swirl generator as designed by Dugué and Weber (1992) and used in several previous studies (Vanierschot et al. 2014;Vanierschot and Ogus 2019). The swirl generator is set up to produce no swirl Fig. 4 Guide vanes of the swirl generator, rotated to produce a nonswirling inlet  [-] 9-70 47-58 13-29 Number of data points per half period [-] 21-157 8-11 34-85 region ( 0.86d i ), as the boundary layer cannot be measured due to reflections near the walls. The measurement locations inside the tube are corrected for refraction inside the pipe based on the formulas found in Gardavsky et al. (1989). The LDA system is a FiberFlow LDA system from Dantec Dynamics with a single measurement volume. The used optics have a focal length of 310 mm, an expansion factor of 1.98, and a wavelength of 532 nm. This results in a fringe spacing of 2.21 μ m and a measurement volume of 652μ m by 79μ m (Durst et al. 1976). Considering refraction, these values change to a fringe spacing of 2.95 μ m and a measurement volume of 871 μ m by 79 μ m inside the inlet tube. The LDA measurements are performed 6.8 mm below the first step of the nozzle ( x∕d i = −0.3 ), seen as an olive dotted line in Fig. 2. The velocity is corrected using a transit time weighting to correct for any statistical bias towards higher velocities, as described by Buchhave et al. (1979). A power law velocity profile (Eq. 1) was fitted to the inlet profile (Çengel 2022) with the following function, where u max is the maximum axial velocity, r is the radial position and n is generally equal to 7, which approximates many engineering flows in practice. n is optimized for this case by minimizing the following objective function.
where R 2 is the coefficient of determination (Eq. 3), ṁ power law is the mass flow rate obtained by the power law velocity profile and ṁ is the measured flow rate during the experiment.
where K is the total number of LDA measurement samples. The turbulent intensity is calculated as where N is the total number of snapshots and n is a specific snapshot.

Spectral proper orthogonal decomposition (SPOD) to extract the precessing structure
To capture the relevant flow dynamics of the jet flow, snapshot SPOD is applied as a reduced order modeling technique (Sieber et al. 2016). Snapshot SPOD has already shown its capabilities for both swirling and non-swirling jet flows (Rovira et al. 2021;Vanierschot et al. 2020Vanierschot et al. , 2021Zhang and Vanierschot 2021). The velocity field is decomposed into a mean and fluctuating part: In Eq. (5), the fluctuating part is written as a set of temporal coefficients a i (t) and spatial modes i (x) . V is a velocity vector composed of components u, v and w and the overbar denotes time averaged quantities. By solving the eigenvalue problem, the temporal coefficients where the elements of the correlation matrix R are given by The fluctuating velocity fields are projected onto the temporal coefficients to obtain the spatial modes.
The matrix R has a diagonal wave-like structure (diagonal similarity) if periodic coherent structures exist in the flow field (Sieber et al. 2016). A simple low-pass filter to filter the correlation matrix R augments this similarity. This introduces the correlation matrix S , defined as where g is a vector of length 2N f + 1 with filter coefficients and N f is the filter width. The correlation matrix S substitutes R in Eq. (6) and the velocity field is decomposed using S instead of R . The optimal filter width was shown to be one or two times the period of the coherent structure of interest if it is pronounceable present (Sieber et al. 2016) or 4 to 8 times if intermittently present (Vanierschot et al. 2020). In this study, the filter width is either a factor 1.5, 4, or 8 times the frequency of the coherent structure. The factor is chosen by quantitative identification of the precessing structure in the spatial modes and by checking for a distinct peak in the power spectral density of the temporal mode.

Triple velocity decomposition to extract the coherent motion, based on the SPOD modes
A Reynolds decomposition of the flow field into an average and fluctuating component is unsuitable for flow fields with a coherent motion. The Root Mean Square (RMS) values of the coherent motion and turbulence are indistinguishable, resulting in overestimating the Reynolds stresses. Note that these RMS values are calculated on the fluctuating field with the average field already subtracted. A more appropriate decomposition is the triple velocity decomposition (Hussain and Reynolds 1970). The flow field is decomposed into a time average ( u(x) ), turbulent ( u �� (x, t) ) and coherent motion ( u � (x, t) ), as seen in Eq. (10). The coherent motion is extracted from the SPOD decomposition by finding the relevant modepairs and adding them together, according to Eq. (11).

3D flow field reconstruction
A 3D flow field reconstruction can be done based on the S-PIV data and their corresponding temporal analysis. The S-PIV coordinate system is fixed, and the precessing jet rotates with a given frequency. Concerning the coherent motion, the measurement plane seems to rotate with the same frequency. A 3D reconstruction can be done by rotating the 2D flow fields from the S-PIV measurements with an angle Θ.
where f SPOD is the frequency of the specific mode pair, obtained by determining the distance between each minima and maxima of the temporal coefficient, f PIV is the sampling frequency, and i is the number of the specific snapshot. Note that the value of f SPOD changes due to the intermittent behavior of precessing jets and frequency jitter (Nathan et al. 1998). More information on this procedure is found in the work of Vanierschot and Ogus (2019). The most pronounced period in the temporal evolution is used for the 3D reconstructed flow fields based on SPOD. A 3D phase-averaged solution is obtained for the phase-averaged flow fields. The data sets are interpolated on a polar grid with a 1 • resolution. It is important to note that the data is rotated around the central axis, with data on both sides. For the data sets which have a clear precession ( Re ≥ 8, 300 ), the number of measured periods of the precessing jet and the number of data points per half period in each data set is shown in Table 1. After interpolating all half periods on the interpolation grid, the average of all half periods is taken, and this is visualized in Figs. 12 and 13. Analysing coherent 3D structures based on 2D S-PIV measurements with SPOD introduces a maximum frequency deviation of 10% due to the loss of spatial information ).

Accuracy of S-PIV measurements
The errors involved in the S-PIV measurements can be divided into systematic errors and random errors (Sciacchitano 2019). The primary systematic errors are misalignment of the PIV setup and tracing accuracy of the seeding particles. The former is kept as small as possible by carefully aligning the laser sheet and the calibration plate. The calibration of the two cameras gives an RMS fit error, shown in Table 1. To ensure accurate flow tracing, the Stokes number ( Stk ) of the seeding is 0.021 for the highest flow rate measured, according to Eq. (13). A good flow tracing accuracy is achieved when Stk < 0.1 (Raffel et al. 2018).
where f is the characteristic time scale of the flow, which is the ratio of the nozzle diameter ( d i ) and bulk inlet velocity of the jet ( u inlet ). P is given by: where p is the density of the particle, d p the diameter of the particle and is the dynamic viscosity of the fluid. The uncertainties on both the mean and the standard deviation can be calculated as where u is the uncertainty on the time-averaged velocity component u and u is the uncertainty of the standard deviation. v true is determined by the following formula, where u is the standard deviation calculated based on the data set, which is the sum of v true (true standard deviation) and 2 u (mean square of measurement uncertainty) (Sciacchitano and Wieneke 2016). Z ∕2 is 1.96 for a 95% confidence interval. The effective number of samples N eff is determined for each location by Eq. (17).
where (nΔt) is the auto-correlation of these samples, of which a summation is taken. The summation is stopped once .
, the correlation values reach zero for the first time (Sciacchitano and Wieneke 2016). Representative profiles of u and u with their corresponding uncertainties, u and u , are plotted at multiple sections in Figs. 5, 6 and 7, for the axial and tangential velocities respectively.

Jet inlet profile
Following the suggestion of Hussain and Reynolds (1970) to always document the initial conditions for studies of free shear flows, the jet inlet profile in the inlet tube is measured using LDA. The measured profile is shown in Fig. 5, including error bars which are calculated similar to Eq. (15). The turbulent intensity of each measurement point is calculated using Eq. (4). The LDA measurements are taken at Re = 22,350, a representative flow measurement for the Reynolds number range considered in this study. Based on the centerline velocity measured with S-PIV (discussed in Sect. 3.2) of the different Reynolds numbers, the value of n would be proportional to Re . The objective function (Eq. 2) is minimized, resulting in n = 9.2 for Re = 22,350. This results in an analytical flow field, shown in Fig. 5 by the solid line. This analytical flow field has a R 2 value of 0.98, and the resulting volume flow rate is within 2.5% accurate of the measured volume flow rate. The profile is situated between a pipe ( n = 7 ) and a contraction flow (top-hat profile, n >> 7 ). Example measurements for both profiles are shown by Wong et al. (2004), , and , which all measured the profiles in a free jet flow. Besides the axial flow profiles, the tangential velocity profile is also measured to check if the movable block swirl generator obtains a zero-swirl inlet, which was confirmed by these measurements.

Time-averaged flow fields
The time-averaged and Root Mean Square (RMS) of the fluctuations at Re = 38,400 are shown for the axial and tangential velocity respectively in Figs. 6 and 7. The other Reynolds numbers measured show a similar shape and distribution, hence Figs. 6 and 7 are representative for all measurements. Close to the expansion, the axial velocity profile resembles a jet. However, the shear layer with the surroundings is influenced by the recirculation zone induced by the specific nozzle geometry (Fig. 6). Further downstream, the jet spreads similarly to a free jet due to the large expansion ratio with significant growth in shear layer thickness. Both the half-width (also called spreading rate) and the opening angle are shown in, respectively, a red and a blue line on Fig. 6. Both angles are similar throughout all the Reynolds numbers and are 3.73 ± 0.04 • and 8.73 ± 0.19 • , respectively. The angles are shown in Fig. 8 for all measurements. Due to the confinement, the angles are lower than those for free axial jets, where the half-width and the opening angle are respectively 5.8 • 11.8 • (Cushman-Roisin 2006), with some deviation due to the inlet condition used. An angle of 5.5 • for half-width of the scalar temperature field is noted by . In free jets, the centerline velocity only starts decaying linearly after x∕d i ≥ 5 (Mi et al. 2007;Karimipanah and Sandberg 1994;Papadopoulos and Pitts 1999;Crow and Champagne 1971), while in our measurements, it linearly decays in the entire measurement area, 1 ≤ x∕d i ≤ 5 . A similar behavior is seen for an FPJ, where the centerline velocity linearly decays from x � ∕d i ≤ 1.2 (Wong et al. 2008), where x ′ is the axial distance after the expansion to ambient air. The outlet condition is much less critical for these differences between the free and confined jet (Husain and Hussain 1979). The RMS values shown in the right of Fig. 6 also show a typical jet profile with high values in the shear layer between the jet and recirculation zone. The RMS values in the center linearly increase, as is also the case in free jets (Karimipanah and Sandberg 1994). As the jet spreads, the RMS values in the recirculation zone decrease further downstream, and the growth of the shear layer increases the width of the region with high RMS values. Further downstream, the values increase due to The geometry of the vessel is added in full black lines. Only 1 of every 2 data points is plotted to improve the readability of the figure   Fig. 8 Jet half-width angle and opening angle, shown respectively in red and blue for all the measured Reynolds numbers and different data sets the large-scale precession of the jet. Section 3.3 gives more details about this precession.
The velocity fields in Figs. 6 and 7, are very similar to the one measured using Tomographic-PIV by Cafiero et al. (2014), which also shows a net non-zero swirl component. This arises to balance the overall angular momentum, conform Dellenback et al. (1988). Another similarity can be noted with a free annular jet (Vanierschot and Van den Bulck 2011), where also a net non-zero swirl component is noted.
The net swirling component decreases if the Reynolds numbers decrease for the low Reynolds number measurements. This decrease is due to the increased intermittency of the precessing jet.

Unsteady flow field analysis using SPOD
Two instantaneous flow fields are shown in Fig. 9, with a time difference of one-half period of the precessing structure between the two velocity fields. Despite the axisymmetric nature of the time-averaged flow field, one can see that the instantaneous flow fields are asymmetric where the jet in the left figure is deflected towards the left. In contrast, the jet in the right figure is deflected to the right. The spatial scale of this asymmetry is much larger than the ones associated with turbulence, and hence it is caused by a large-scale instability in the flow field. To study the dynamics of this instability, the flow is decomposed using SPOD (Sieber et al. 2016).
The energy content of each mode pair and its corresponding Strouhal number is shown in Fig. 10 for a Re = 38,400, from S-PIV 1. The Strouhal number is defined as, St = where f is the frequency and u inlet is the bulk inlet velocity. The most pronounced mode pair, mode pair 1, has an energy content of around 11% of the total turbulent kinetic energy of the flow. St ≈ 0.003 for this mode pair. Since the data acquisition frequency is maximally 12.95 Hz, only lowfrequency dynamics of the flow field, i.e. the precession, are captured. The harmonic correlation of a mode pair is shown by the 'yellowness' of the dot. The high harmonic correlation of mode pair 1 can be visually represented by plotting the temporal coefficients, which are shown in Fig. 11. The rectangle identifies the maximum of the temporal mode and one period surrounding this maximum, which is used for the 3D SPOD-reconstruction (described in Sect. 2.6). Unlike an  FPJ, no axial jet structure is notable in this SPOD analysis. Different methods of quantifying the intermittency of a precessing jet have been discussed by Madej et al. (2011). An intermittency parameter is proposed based on the temporal mode of the SPOD. First, all signal peaks (negative and positive) are identified. A precessing jet is present if its respective temporal mode peak is in the intermittency region defined by the 50% of the average of the peaks. These limiting values are indicated in Fig. 11, using dashed lines. The intermittency is then calculated as the percentage of peaks inside the intermittency region.

3D reconstruction of the precession
From the SPOD analysis, a 3D flow field reconstruction can be done based on the temporal mode of any mode pair using Eq. (12). The instantaneous flow fields are rotated in the physical space, and for every rotation of , the data are interpolated on a uniform cylindrical grid with an increased resolution of . The data are phase averaged with the [0 − ] interval added to the [ − 2 ] with a phase shift of , similar to the method used in Madej et al. (2011) andVanierschot andVan den Bulck (2011). Only the precessing jet mode, which is the most harmonically correlated and highest energy mode pair, is considered in the reconstruction. The highest temporal mode peak is identified, and one period around this temporal mode peak is considered. A specific measurement's mode pair is shown in Fig. 11, and the maxima and the corresponding period are enlarged, indicated by the rectangle.
The reconstructed 3D flow fields of the precessing jet mode are plotted in Figs. 12 and 13, for the axial and tangential velocity flow field respectively. There are three planes shown in these figures: a XZ-plane at x∕d i = 3.5 , a XY-plane, and a YZ-plane. A thin black line indicates all the planes on the other figures. In Fig. 12, a similar shape in the phase-averaged and SPOD reconstruction is seen. The SPOD reconstruction is 'smoother' since it omits all turbulence, and the phase averaged field is statistically not fully converged. The SPOD reconstruction also shows higher values since only the most prominent period of the temporal mode is plotted, while in reality there is intermittency which attenuates the structure. The shapes of both reconstructions are very similar. In the YZ-plane, a jet is off-center towards the top right of the figure. This off-center jet is also visible on the XY-plane, where the jet deviates towards the right. In the XZ-plane, the jet is almost centered in the middle. In Fig. 13, a similar shape in the phase-averaged and SPOD reconstruction is visible. The axial and tangential velocity reconstruction show a similar flow field for both the SPOD and the phase-averaged flow field, confirming that the SPOD-mode pair is a physical coherent motion. A clockwise tangential velocity is considered a negative velocity. The jet deviates towards the top right, as shown in Fig. 12, which is also where the tangential velocity is pointing towards. This tangential velocity comes from a recirculation, which results in the precessing jet.

Influence of the Reynolds number
Analysis of the flow fields at different Reynolds numbers reveals very similar SPOD modes of the precessing structure and shows that the large-scale instability is present in the entire Reynolds number regime measured. The SPOD analyses have a similar structure as the one shown in Fig. 10. The frequency and associated St are shown in Fig. 14. The filter width, for each data set, is determined according to Sect. 2.4. The multiplication factor, which is the number of periods of the coherent motion, is indicated in color in Fig. 14. The frequency of the precession increases linearly with increasing Reynolds number. The error bars plotted are calculated by combining the errors on flow rate, frequency, d i , viscosity, and density. The error bars on the frequency are for most data points 10% due to the window length of 10 periods for Welch's power spectral density. The unexpected low frequencies at low Reynolds-number flow regimes have motivated new measurements (S-PIV 2 and S-PIV 3) with lower recording rates tailored to the expected frequency. A precesssing jet in the studied geometry exists or Re ≤ 8300 , which is similar to an FPJ, which only has a probability of existing at Re ≥ 10,000 (Madej et al. 2011). There are measurements of 2360 ≤ Re ≤ 5360 performed, for which Re = 2360 showed no resemblance towards a precession in its SPOD analysis. The measurements for 3160 ≤ Re ≤ 5360 showed some resemblance in their SPOD analysis towards a precession, but the structures are dominated by noise as the precession is very weak. At these low Reynolds numbers, the flow starts to become turbulent, which is most likely the reason for the absence/decrease of the precession. The turbulent kinetic energy attributed to the precession also decreases with decreasing Reynolds number.
St first increases for Re < 23,600 and then reaches a value of about 0.003 for 23,600 ≤ Re ≤ 63,870. The measured St is very close to the ones observed for annular free jets, where values of 0.0025 have been reported by Vanierschot and Van den Bulck (2011), which seems to suggest that the physical phenomena driving these instabilities are very similar for both flow topologies. The literature for FPJs shows 0.0014 ≤ St ≤ 0.0020 . The differences between an FPJ and the studied setup here may originate from one of the following reasons: • A different flow topology; an FPJ uses a confined sudden expansion that exits into an ambient fluid. This study studies a fully confined setup that doesn't experience any influence of ambient fluid and has a 45 • contraction at its outlet to close a fluidic loop. • A non-developed inlet flow and no specific flow conditioning. Before the vessel, the flow is split into 6 and fed into a movable block swirl generator, where the guide vanes have been rotated to generate no swirl. After this, there is a distance of around 1.5d i before the first expansion. Ceglia et al. (2017), showed that after adding a turbulence promoter in the inlet, like a grid, a precessing motion is still present and very similar to the one present without this grid.
The average intermittency for the whole data set is 20.7% with a standard deviation of 5.6%, determined as discussed in Sect. 3.3,. It shows a slightly decreasing pattern with an increasing Reynolds number, which is similar to the works of Nathan (1988) and Madej et al. (2011) for an FPJ. In the studied flow case, the precessing structure mainly changes in amplitude with the addition of some irregular phase shifts. In contrast to the FPJ, there is no second flow structure, like the axial jet, present in the SPOD analysis.

Conclusions
The precessing motion in a sudden expansion of an axisymmetric jet issuing in a coaxial chamber was studied for Re ranging from 3, 160 to 63, 870. The inlet profile is measured using LDA measurements in the inlet tube, slightly uptream of the expansion. The inlet flow exhibits a profile between that of a pipe flow and a contraction flow due to the limited development length ( 1.56d i ) of the inlet tube. The flow field in the 1 ≤ x∕d i ≤ 5.5 region after the expansion was measured using S-PIV. The time-averaged flow fields resemble a jet flow. The half-width and the opening angle are similar for all Reynolds numbers measured with values of 3.73 ± 0.04 • and 8.73 ± 0.19 • , respectively. The decay of the centerline velocity shows a resemblance to the behavior noted in an FPJ and is faster compared to a free axial jet. Despite the axisymmetric nature of the time-averaged flow field, the instantaneous flow is asymmetric due to the presence of a large-scale precession of the jet around the central axis, which has similarities with the precessing jets in an FPJ. SPOD extracts the dynamic behavior of the jet, where the filter width is dependent on the energy of the coherent motion. The precession exists for 8300 ≤ Re ≤ 63,870, while it is very weak for 3160 ≤ Re ≤ 5360 and no longer present at Re = 2360 . The Strouhal number increases linearly for 8300 ≤ Re ≤ 23,600, after which it stagnates at St = 0.003 until at least Re = 63,870. The average flow field shows a net nonzero swirl component to balance the angular momentum created by the precession, conforming the literature. The average intermittency of the precessing structure for all the data sets is 20.7%, with a standard deviation of 5.6%. In contrast to an FPJ, there is no axial jet structure notable in the SPOD analysis.