Phase-locked analysis of velocity fluctuations in a turbulent free swirling jet after vortex breakdown

Abstract

Large-scale organized vortical structures were studied experimentally in a free swirling jet of air experiencing vortex precession (PVC) at ambient conditions. Detailed measurements were performed in the region near the nozzle exit using phase-locked LDV and PIV, at a Reynolds number of Re ≈ 24,400 and a swirl parameter S ≈ 1.0. The investigation allowed reconstruction of the time-averaged flowfield, with the associated distribution of turbulent fluctuations, the phase-locked structure of the jet and the associated precessing vortex structure. An original joint analysis of power spectra and probability density functions of velocity data led to quantification of the PVC effect on turbulent fluctuations. This analysis showed that the PVC contribution can be properly separated from the background random turbulence, reproducing the results of phase-locked measurements. It is found that the background turbulence in the near field is substantially weaker if compared to the coherent fluctuations induced by vortex precession.

1 Introduction

Swirling flows are encountered in a variety of engineering applications, including flows in combustion chambers, over delta wings, in fuel injectors and cyclone separators, to name but a few. These flows are generally characterized by the presence of one or more large vortical structures, experiencing breakdown when the degree of swirl and the Reynolds number are sufficiently high. Vortex breakdown refers to an abrupt change in the flow structure of a vortex, associated with the formation of a steady or unsteady free stagnation point and a recirculation region downstream (Wang and Rusak 1997). In the case of a single large-scale vortex structure, such as in a swirling jet, the recirculation region is referred to as central recirculation zone (CRZ). Above a certain swirl intensity, the CRZ may experience an instability known as precessing vortex core (PVC): The core of the main vortical region moves almost periodically about the average axis of symmetry of the flow. Breakdown conditions are typically associated with strong unsteadiness and enhanced flow mixing (Park and Shin1993), and there is therefore a primary technological interest in understanding this phenomenon for design, optimization and control purposes.

The original definition of vortex breakdown as the critical state when an axial recirculation zone is formed is due to Harvey (1962). Vortex breakdown has been studied in detail by Sarpkaya (1971), who classified different states, introducing the concept of spiral, double-helix and axisymmetric breakdown; in the same paper, he also showed that the presence of an additional positive axial pressure gradient moves the recirculation region toward the nozzle exit. Observations by Chanaud (1965) reported on the existence of persistent oscillatory motion in swirling flows, and the same evidence was confirmed and detailed by the work of Cassidy and Falvey (1970), where also the linear dependence of the oscillation frequency against the flowrate—for large enough Reynolds numbers—was first reported. Since these seminal attempts, several works confirmed the presence of periodic or quasiperiodic motion of large-scale vortical structures in laminar and turbulent flows experiencing breakdown. Garg and Leibovich (1979), using measurements based on laser Doppler velocimetry, reported on the existence of sustained periodic oscillations in the wake of bubble-type breakdown. Chao et al. (1991) measured and discussed the effect of downstream perturbations on spectra of velocity fluctuations in swirling flows in the presence of a PVC. Wunenburger et al. (1999) analyzed some issues related to velocity measurements in a strong precessing vortex and in particular discussed the effect of precession, measurement errors, velocity gradients and inhomogeneous tracer concentration on the probability density functions (PDFs) of velocity data as measured by LDV. Grosjean et al. (1997) combined PIV and LDV to analyze the instantaneous position of the vortex core of a swirling jet during precession. Recent PIV studies by Midgley et al. (2005) and Spencer et al. (2008) reported on the effects of geometry variations on the unsteady swirling flowfields. All these works confirmed the fact that, for sufficiently high swirl intensity and Reynolds number, vortex breakdown is frequently associated with precessing motion of the vortex core. A more extensive review of the research work performed in this field is given in Escudier (1988) and, more recently, in Lucca-Negro and O’Doherty (2000) and Syred (2006), where the PVC phenomenon is analyzed in greater detail. The interested reader is referred to these reviews for additional references and information about the recent numerical attempts (mostly based on LES or DNS) to analyze these flows numerically (Gui et al. 2010; Selle et al. 2006).

In a swirling jet, the PVC is generally associated with the presence of well-defined peaks in the power spectrum of velocity fluctuations. The large-scale fluctuations associated with vortex precession induce a very high fluctuation intensity close to the vortex axis. This fact has been reported in the literature, for example, Schneider et al. (2005) reported the presence of a PVC in a free swirling jet in isothermal conditions and highlighted the substantial contribution of the PVC to the overall fluctuation levels. The inherent quasiperiodicity of the flowfield in a swirling jet in the presence of a PVC has been recently exploited by Cala et al. (2006): Leveraging the additional spatial periodicity in azimuthal direction, they used LDV measurements to reconstruct the average phase-locked three-dimensional structure of the flow and presented an interesting topology of the flowfield characterized by the presence of a main vortical structure surrounded by two secondary, counter-rotating spiraling vortices. More recently, Oberleithner et al. (2011) reconstructed the 3D time-periodic flow generated by a swirling jet from uncorrelated PIV snapshots, using proper orthogonal decomposition (POD) and a phase-averaging technique. They claim that the POD approach predicts the phase angle from spatial modes and not from one single point, such as the time-resolved LDV point measurements, yielding a more accurate prediction of the global phase. In addition, the PVC structure obtained from their data compared well with the least stable global mode computed using the measured mean flow. A successful attempt to apply a similar phase-locked analysis to a precessing flame has been performed by Stöhr et al. (2011), where simultaneous PIV and OH-PLIF data are reported.

In the presence of a PVC, the probability density functions (PDFs) of velocity fluctuations feature a bimodal character, as reported for instance by Martinelli et al. (2007). In a configuration similar to that used by Cala et al. (2006), they showed that a decomposition of the velocity field into a periodic term and a Gaussian fluctuation allows accurate reconstruction of the velocity probability density functions (PDFs) measured on the axis via a simple analytical model. Such model could be used to properly separate the contribution of the large-scale velocity fluctuation from the background turbulence.

The problem of extracting the contribution of the coherent periodic motion of the PVC from overall turbulence in swirling flows was also addressed in Grosjean et al. (1997). They explored the combined use of LDA and PIV and anticipated the possible use of POD decomposition applied to PIV data. Graftieaux et al. (2001) applied POD analysis to PIV measurements and concluded that two spatial modes are responsible for large-scale coherent fluctuations. Midgley et al. (2005) used conditional averages of the Reynolds decomposed velocity field to separate true turbulence and pseudo-turbulence generated by the large-scale rotation of the flow. Recently, Stöhr et al. (2011) performed the analysis of the velocity fields using POD and confirmed that the dominant unsteady flow structure is the PVC which is represented by a characteristic pair of POD modes.

The recent works on vortex precession in strongly swirling jets indicate phase-locked experiments as a promising tool to extract coherent flow structures. Such experiments can be performed using both LDV and PIV, thus providing complementary information. In addition, LDV measurements deliver a large amount of data that can be used to characterize statistically the velocity field in terms of PDFs. In this work, we aim at comparing results from phase-locked experiments to those obtained via an analysis of the velocity PDFs in a turbulent free swirling jet experiencing PVC. The experimental techniques encompass standard and phase-locked LDV and PIV, and details are described in Sect. 2. The general, time-averaged structure of the flow is presented in Sect. 3. Leveraging data obtained with phase-locked LDV and phase-locked PIV, we present first a spatial analysis of the flow, highlighting the features of the important spatial structures of an average flowfield at an arbitrary phase (Sect. 4). An analysis of the one-point statistics of velocity data follows (Sect. 5), where the question of quantifying the effect of the PVC on the global fluctuation intensity is addressed. In particular, a detailed analysis of the PDFs of velocity data is presented, extending the results of Martinelli et al. (2007) to generic bimodal PDFs; this approach captures the unsteadiness locally and provides a quantitative discrimination of coherent fluctuations and background turbulence in a swirling jet. We discuss the ensemble of the results in Sect. 6 and conclude in Sect. 7.

2 Experimental setup and data processing techniques

The flow is a turbulent free swirling jet of air at ambient pressure and temperature. Air flows from a nozzle placed at the exit of a swirl generator of axial-plus-tangential entry type, characterized by 4 axial and 4 tangential inlets. A schematic diagram of the swirl generator and the nozzle is given in Fig. 1. A central pipe is used for injection of seeding oil droplets. The air flows through a converging nozzle (Fig. 1b), whose exit section has a radius R = 12 mm. The total air flowrate and the swirl intensity were independently regulated by controlling the axial, tangential and axial seeding flowrates by means of thermal mass flowmeters. This experimental rig is the same described and used by Coghe et al. (2004) and Martinelli et al. (2007). A cylindrical coordinate system is introduced, whose z axis corresponds to the geometrical axis of the nozzle, and the radial and azimuthal directions are denoted by r and ϕ, respectively (see Fig. 1). Due to the geometry of the system and the angle between laser beams, we will refer to a nominal nozzle exit for LDV measurements corresponding to a plane located at z ₀ ≈ 2 mm from the geometrical exit section, as indicated in Fig. 1b.

Fig. 1

Schematic of a the swirl generator and b the nozzle, including the r − z − ϕ coordinate system employed. The origin of the coordinate system is placed at a vertical distance z ₀≈ 2 mm, the closest reachable point with LDV due to the angle between laser beams. The nozzle inlet matches the swirl generator outlet

The experimental condition analyzed in this work is characterized by an average nozzle exit velocity U = 15.2 ± 0.2 m/s, as measured from the total flowrate; the corresponding value of the Reynolds number is Re ≈ 24,400. A nondimensional swirl parameter is defined as $S = \frac{G_\phi}{R G_z}$, where G _ϕ is the axial flux of azimuthal momentum and G _z is the axial flux of axial momentum; in this work, S is computed neglecting the contribution of pressure and velocity fluctuation, resulting in the value S ≈ 1. It has been emphasized (Panda and McLaughlin 1993; Toh et al. 2010) that different average velocity profiles at the nozzle exit may lead to the same value of the swirl parameter, as only integral quantities appear in its definition. In addition, it has been highlighted that swirl generation devices may induce nonuniformities in the average flow (Toh et al. 2010). Therefore, for the sake of completeness, in Fig. 2, we show the average flowfields as measured by PIV on cross-flow and streamwise planes, and some corresponding radial profiles extracted from PIV maps. Figure 2a shows the average velocity field at the nozzle exit; the azimuthal velocity is quite uniform in azimuthal direction. Further inspection of three azimuthal velocity profiles in Fig. 2b indicates that—apart from a slight scatter of the average vortex center location—profiles would overlap well in the solid-body rotation core r/R < 0.5. The average flowfield in a vertical plane is shown in Fig. 2c, where the axial recirculation region and the conical divergence of the jet are clearly visible. The flow shows the expected pattern, and inspection of the axial velocity profiles extracted at three different z locations indicates a slight left--right nonsymmetry. All axial velocity profiles are characterized by a minimum on the nozzle axis, and the radial gradient of axial velocity sharply increases with the radius. Since the present experimental conditions are far from critical due to the high swirl intensity and Reynolds number, we expect a minor effect of the slight mean flow nonuniformities on the dynamics of the PVC.

Fig. 2

a Horizontal PIV map showing the average flowfield at the nozzle exit. b Azimuthal velocity profiles corresponding to cuts of the PIV map a at ϕ = 0 (solid), ϕ = 2π/3 (solid-dotted) and ϕ = 4π/3 (dashed). c Vertical PIV map showing the average flowfield in a vertical symmetry plane. d Axial velocity profiles corresponding to cuts of the PIV map c at z/R = 0.5 (solid), z/R = 1.0 (solid-dotted) and z/R = 1.5 (dashed). All velocity profiles are normalized with the average axial velocity U

Instantaneous single-point velocity measurements were performed in back-scattering mode using a two-component fiber optics laser Doppler velocimeter (LDV) equipped with a 5 W Argon ion laser; a Bragg cell with 40 MHz frequency shift was used to remove directional ambiguity. The laser beam was conducted via glass fiber to a Fiber Flow probe with 310 mm focal length. The size of the probe volume is 0.08 × 0.08 × 0.71 mm³, as evaluated using the e⁻² criterion. In all measurements, the longest axis of the probe volume was kept normal to the radial direction; its radial size of 0.08 mm was estimated to be sufficiently small to avoid significant errors due to velocity gradients. Oil droplets produced by a jet atomizer were used as tracer particles (average diameter 1–2 μm); previous studies evidenced that the inertial and centrifugal effects on these particles can be considered negligible (Martinelli et al. 2007). The seeding flowrate was carefully adjusted during each experimental run in such a way to achieve mean data rates of about 5 kHz, largely sufficient to correctly resolve the typical PVC frequency f _PVC ≈ 480 Hz observed in the experimental conditions considered. LDV data were acquired together with an additional trigger signal, phase locked with the periodic PVC. The trigger signal was generated by a condenser microphone, placed on the plane of the nozzle exit at 300 mm from the nozzle axis, after a preliminary analysis confirmed that the same fundamental PVC frequency was observed in both the microphone and the LDV spectra. The microphone signal was filtered using a band-pass filter; the allowed frequency band was set to the interval [440,520] Hz, that is, f _PVC ± 40 Hz. One trigger pulse was generated by an in-house TTL generator each time the filtered signal crossed the zero level with a negative slope. All the LDV data were analyzed with in-house routines. In particular, all relevant statistical quantities including PDFs were corrected for the velocity bias using transit-time weighting. Power spectral density functions (PSDs) were computed via discrete Fourier transform of the sample-and-hold resampled time series data (Albrecht et al. 2003). Phase-locked statistics require a preliminary reordering of the data according to their phase relative to the triggering signal; such reordering is performed by first defining the pseudo-periods from the time intervals between two successive triggering instants, and by further subdivision of each pseudo-period in a fixed number of equally spaced subintervals, each of them corresponding to a phase lag. Each velocity datum is then recollected in phase, depending on the subinterval containing the corresponding arrival time. Following (Albrecht et al. 2003), the statistical error is estimated to be less than 2 % for the full data set at each measurement point, whereas the statistical error for the data subsets used in the phase analysis is estimated to be less than 4 %; these errors are given at 95 % confidence level, and the number of statistically independent samples was at least 15,000 in each case.

Phase-locked particle image velocimetry (PIV) measurements were performed in a plane lying on the longitudinal axis of the nozzle by using a double-pulsed Nd:YAG laser operating at 532 nm with a pulse energy of 200 mJ/pulse. Double images with an interframe time of 5 μs were acquired at a rate of 4 Hz by means of a CCD camera. The camera used a 60 mm objective at an aperture of f# = 4 and with a magnification M = 0.159. The field of view is 54.69 mm × 41.67 mm, and the CCD array is 1,344 × 1,024 pixels, thus corresponding to a resolution of 24.57 pix/mm. The thickness of the laser sheet was estimated to be about 1 mm. The same trigger system implemented for the phase-locked LDV measurements was used for PIV measurements. In this case, measurements were taken at different phase lags by properly setting the time delay from the triggering instant to the first PIV laser pulse. The single exposed image pairs were analyzed using an adaptive cross-correlation digital algorithm, including peak validation, neighborhood validation and 50 % overlap; in the adaptive algorithm, the size of the interrogation area goes from 64 × 64 pixels to 32 × 32 pixels. The corresponding in-plane spatial resolution for PIV was about 1.3 mm. By optimizing the quality of images, seeding particle density, time between frames, validation criteria and interrogation window, it is estimated that in the present experiments, the largest uncertainties in the phase-averaged velocity components are about 3 % at 95 % confidence level, with a sample size of 200 image pairs per phase angle.

3 Time-averaged flowfield

Results are reported in the r − z plane or in the r − ϕ plane, and velocity data are normalized with the average nozzle exit velocity U.

Radial profiles of average velocity, as measured using LDV, are shown in Fig. 3 at the four vertical locations z/R = 0, z/R = 0.6, z/R = 1.2 and z/R = 1.8. The average axial velocity field (Fig. 3a) features a large recirculation zone, evidenced by negative velocities, corresponding to the central recirculation zone (CRZ). In the present measurements, the CRZ extends radially up to |r/R| < 0.5; the axial length of the CRZ is z/R ≈ 3.5 (not shown from the profiles in Fig. 3). The axial velocity largely increases in the radial direction, and the maxima are located approximately at |r/R| ≈ 1. As shown in Fig. 3c, the average azimuthal velocity component is antisymmetric with respect to the vertical axis of the jet, as expected from the swirling motion. The absolute value of the azimuthal velocity is maximum at the nozzle exit and decreases along the vertical direction. Average radial velocity profiles are plotted in Fig. 3e; these profiles are not crossing the origin due to the small nonuniformities of the average field already shown in Fig. 2. Radial velocity profiles also indicate the conical divergence of the average flowfield in axial direction. The average flowfield is therefore characterized by the presence of an inner shear layer due to the central recirculation and defined by sharp gradients in the axial and azimuthal components, as well as an outer shear layer interacting with the ambient air.

Fig. 3

Average a axial, c azimuthal, e radial velocity profiles (LDV). The scale on the vertical axis indicates both z/R locations and the velocity amplitude, possibly rescaled by a factor as indicated. r.m.s. axial, azimuthal and radial velocity profiles are shown in b, d and f, respectively. All profiles are reported for vertical locations z/R = 0, z/R = 0.6, z/R = 1.2 and z/R = 1.8. Normalization with the nozzle exit velocity U

The very strong radial shear of axial velocity induces a very high r.m.s. fluctuation intensity in the region 0.5 < |r/R| < 1 as evidenced in Fig. 3b. The r.m.s. azimuthal fluctuation intensity, shown in Fig. 3d, features a pair of local maxima for 0.5 < |r/R| < 1, the same location measured for the axial fluctuations; a similar feature was found by Panda and McLaughlin in their measurements on a free swirling jet (1993). The strong fluctuations in this region can be associated with the inner shear layer. On the other hand, the r.m.s. radial fluctuation intensity (Fig. 3f) features two local maxima in the region 1 < |r/R| < 1.5, z/R > 0.5; these maxima are localized in the outer shear layer, where the effect of entrainment is important. Additionally, both radial and azimuthal r.m.s. fluctuation intensities are characterized by a local maximum close to the nozzle exit, on the axis of the jet. The high fluctuation intensity in this region is associated with a local maximum of the azimuthal or radial shear, and this feature is also attributed to the effect of the large-scale precessing vortex core; note that, on the axis of symmetry, the radial and azimuthal components have, for symmetry considerations, the same statistics.

The spatial structure of the mean velocity fields reported in Fig. 3a, c and e shows the expected symmetries, and it is in qualitative agreement with previous findings by other authors (Cala et al. 2006; Oberleithner et al. 2011); average velocity profiles obtained from LDV measurements are also in quantitative good agreement with average PIV maps in Fig. 2. Further, the r.m.s. of the axial fluctuation is in qualitative agreement with the results reported by Alekseenko et al. (2008), although in the present work the intensity is higher due to larger values of Re and S.

The precession of the vortex core is almost periodic, and its signature is clearly visible in the power spectra of single-point velocity data; an example is given in Fig. 4, where the PSDs of the axial velocity component in two locations are reported. The PVC is indicated by the sharp peak in the velocity spectrum, at f _PVC ≈ 480 Hz; the frequency of this peak has been found invariant with the spatial location. Figure 4 also shows a typical spectrum measured off the axis of symmetry; it has been found that all spectra measured on the axis feature a single peak at the fundamental harmonic of the PVC, while higher harmonics appear at off-axis locations only. A further important feature of the measured data is the nontrivial shape of the probability density functions. This is best seen from the measurements of the azimuthal and axial velocity components; representative examples of PDFs off the axis are given in Fig. 5. Away from the axis, PDFs are typically nonsymmetric and, in some locations, bimodal. They would eventually recover a Gaussian shape very far from the axis, while they are symmetric and possibly bimodal on the axis as noted in (Martinelli et al. 2007).

Fig. 4

Power spectral density of the axial velocity component at z/R = 0.6, r/R = 0.0 (solid) and z/R = 0.6, r/R = 0.66 (dashed). The peak at the precessing frequency f _PVC ≈ 480 Hz is clearly visible; the measurement off the axis features the presence of a second harmonic, at twice the frequency f _PVC

Fig. 5

a Probability density function of the axial component at z/R = 0.6, r/R = −0.66 (solid), z/R = 0.6, r/R = −0.33 (dashed). b Probability density function of the azimuthal component at z/R = 0.6, r/R = −0.66 (solid), z/R = 0.6, r/R = −0.33 (dashed)

4 Phase-locked spatial structure of the jet

The presence of a well-defined periodic signal in the velocity data, as evidenced in Fig. 4, suggests that the velocity field could be decomposed as follows (Reynolds and Hussain 1972):

$$ {\mathbf{v}}({\mathbf{x}},t) = \overline{{\mathbf{v}}}({\mathbf{x}}) + \tilde{{\mathbf{v}}}({\mathbf{x}},t) + {\mathbf{v}}'({\mathbf{x}},t) $$

(1)

where $\overline{\mathbf{v}}(\mathbf{x})$ is the temporal average of the velocity in a location $\mathbf{x}$, the term $\tilde{\mathbf{v}}(\mathbf{x},t)$ is a time-periodic component, and $\mathbf{v}'(\mathbf{x},t)$ represents the remaining incoherent velocity fluctuations. When a periodic signal synchronized with $\tilde{\mathbf{v}}(\mathbf{x},t)$ is available, phase averaging could be used to extract the periodic component itself, that is, the following relation holds:

$$ \overline{{\mathbf{v}}}({\mathbf{x}}) + \tilde{{\mathbf{v}}}({\mathbf{x}},t) = \lim_{M\rightarrow\infty} \frac{1}{M}\sum_{m=1}^M {\mathbf{v}}({\mathbf{x}},t + m\tau) $$

where $t\in[0,\tau]$ and τ is the period of $\tilde{\mathbf{v}}(\mathbf{x},t)$. Phase averaging can be performed either acquiring a reference signal (to be used in a post-processing) in synchronization with the measurements, or by directly performing phase-locked measurements. In this section, we present phase-averaged results of LDV measurements obtained with the former approach, as well as results of PIV measurements performed with the latter technique.

Results of PIV measurements on the r − z plane are reported in Fig. 6. In particular, these figures show contours of axial velocity for the four measured phase lags ($\Updelta\phi=0, \Updelta\phi=\pi/2, \Updelta\phi=\pi$ and $\Updelta\phi=3/2\pi$ with respect to an arbitrary initial phase), normalized with the average nozzle exit velocity U. The phase-averaged velocity contours exhibit large differences from the time-averaged maps. A first interesting feature is the asymmetry of the phase-locked axial field with respect to the geometrical axis, for all phases; further—as may be expected, since the azimuthal direction is statistically homogeneous—pairs at opposite phases are antisymmetric with respect to the geometrical axis. In each of these figures, the central recirculation bubble spans the vertical direction over an extent of approximately 3.5 R, coherently with the average axial velocity data obtained with LDV. However, along the cycle, the bubble is substantially displaced and deformed in the radial direction, causing periodic bursting events at the nozzle exit; the shedding is clearly evident by focusing on the left-hand shear layer represented in the figures, going from $\Updelta\phi=0$ to $\Updelta\phi=3/2\pi$. In fact, the edge of the nozzle imposes a fixed separation point, from which a spiral vortex structure is periodically shed. Similar structures have been recently observed (Stöhr et al. 2011) in a turbulent swirling flame. The periodic variation of the axial field in the region 0.5 < |r/R| < 1 explains the localized high fluctuation intensity in Fig. 3b. Additionally, the time-periodic displacement of the recirculation bubble causes a periodic dislocation of the free stagnation point. In turn, this affects the radial (or azimuthal) velocity at the nozzle exit, on the axis; in fact, by continuity the recirculating flow diverges radially at the unsteady free stagnation point, thus inducing a very high fluctuation intensity as shown in Fig. 3d, f. A similar effect on the dislocation of the free stagnation point was found in the measurements by Liang and Maxworthy (2005), at substantially lower Reynolds number and in breakdown conditions.

Fig. 6

Contours of phase-averaged axial velocity at different phases, as measured by phase-locked PIV. Velocity vector field superimposed to the contour plot. Normalization with the average axial velocity U

Phase-locked LDV measurements can be used to gain additional insight into the spatial structure of the flowfield; as suggested by Cala et al. (2006), the key idea is to exploit the natural periodicity of the flow in the azimuthal direction, in order to relate a spatial phase angle to each temporal phase determined via phase averaging. In this way, the phase-averaged flowfield is assumed to rigidly rotate about the nozzle axis at the PVC frequency. Results are reported in Fig. 7, showing an horizontal cross-section of the precessing vortex. As a vortex marker, we use the λ₂ criterion by Jeong and Hussain (1995). Upon indicating with $\mathbf{S}$ and $\mathbf{\Upomega}$ the strain rate and vorticity tensors, respectively, a vortex is identified as a spatial region where the intermediate eigenvalue of the tensor $\mathbf{S}^2 + \mathbf{\Upomega}^2$ is negative. In incompressible flows, this criterion relates vortical motion to the existence of an instantaneous local pressure minimum in a two-dimensional plane. In Fig. 7, contours of negative values of the vortex marker λ₂—normalized with the precessing frequency squared—are shown together with the planar velocity field in the crossflow plane, as well as the axial recirculation zone, at four distances from the nozzle exit. The phase-averaged field in Fig. 7a, b presents a single, kidney-shaped nonsymmetric vortical structure, whose maximum intensity is located in a position off the axis; similar structures have been observed by other authors (Cala et al. 2006; Oberleithner et al. 2011; Stöhr et al. 2011), who indicated the dominant structure as an helical mode with azimuthal wavenumber m = 1 winding around the recirculation bubble. The tail of the structure has a lower intensity, and it embraces the axial recirculation region, indicated by the black thick line. Moving in the axial direction (i.e., at increasing distance from the nozzle exit), results show that this vortex structure spirals within the inner shear layer, in opposite direction with respect to the swirl direction, and that its intensity progressively decreases far away from the nozzle exit. Furthermore, a second weaker vortical structure is present within the outer shear layer; this structure spirals around the main vortex. Note that this structure is not evident in Fig. 7a, b because no measured data were collected in the relevant region. These results compare well to those reported by Cala et al. (2006) in a similar flow configuration. However, similarly to the results reported in Stöhr et al. (2011), from our data, it is possible to evince two large-scale vortical structures only. Additional analysis (not shown) of our results indicates that the minimum of the λ₂ marker in Fig. 7 corresponds to a maximum in both the axial and azimuthal vorticity; it is therefore difficult to clearly establish whether the main contribution to the vortical structure is due to a locally high radial shear of axial or azimuthal velocity. Note that the periodic shedding of a secondary vortex structure from the nozzle edge, evidenced in Fig. 7, is consistent with the results shown in Fig. 6. The coherent structures have a decreasing strength as larger distances from the nozzle exit are considered.

Fig. 7

Contours of the vortex marker λ₂/f ²_PVC . Also shown is the planar phase-averaged vector field. The black thick line indicates the outer boundary of the axial recirculation zone

The analysis of phase-locked LDV and PIV measurements allowed us to characterize the average flowfield as well as its large-scale unsteady structures. The flow topology and the structure of the fluctuation fields obtained from our results compare well with those reported by other authors in similar configurations (Cala et al. 2006; Liang and Maxworthy 2005; Oberleithner et al. 2011; Stöhr et al. 2011). Our results have been obtained with data analysis methods that have been used successfully in these recent works. In the following section, we move one step forward by analyzing the flow from another perspective, focusing on the PDFs of velocity fluctuations. This allows us to confirm the interpretation of phase-locked results, to explain the shape of PDFs of velocity fluctuations coherently and to quantify the background turbulence intensity after filtering out the effect of the PVC.

5 One-point temporal statistics of the flowfield

As shown in Sect. 3, the presence of a time-periodic component in the velocity signal is evidenced by the presence of well-defined peaks in the spectra of velocity fluctuations. Its effects are also evidenced by the bimodal structure of some of the measured PDFs, as well as the high level of r.m.s. fluctuation intensity in regions close to the nozzle exit. The particular shape of bimodal PDFs measured on the axis of the jet was investigated in Martinelli et al. (2007), where an analytical model was proposed and used to separate the effect of the periodic vortex precession from the background turbulence. However, that model was inherently symmetric, and it failed to explain the (possibly bi-modal) asymmetry of PDFs measured in locations away from the axis. In this section, we propose an analytical model for the velocity PDFs that accounts for nonsymmetric effects and that can be used to fit experimental data.

The key fact to notice is that, as shown in the example in Fig. 4, spectra of velocity fluctuations on the axis feature a single peak, whereas spectra measured off the axis of symmetry feature multiple harmonics. Therefore, the decomposition in Eq. 1 can be further refined to consider explicitly the presence of N harmonics in the periodic term; for instance, considering a single velocity component, the Fourier expansion reads:

$$ v_k(t) = \underbrace{\sum_{n=1}^N A_n \sin(2\pi {f_{\rm PVC}} n t + \psi_{n} + \theta_k)}_{x} + \underbrace{v_k'(t) + \mu}_{y}. $$

(2)

In this equation, the subscript k denotes the k-th term in an ensemble of experiments, A _n and ψ _n denote the amplitude and phase of the nth harmonic, θ _k is the random initial phase, $v_{k}'$ denotes a fluctuation, and μ is the time-averaged mean value. The time-periodic velocity fluctuation associated with the PVC would be represented by the term x; the steady average velocity in each measurement point is represented by the mean value μ, and the background turbulence intensity is quantified by the variance of the fluctuation $v_{k}'$. Following (Martinelli et al. 2007), we include the average and turbulent components in the erm y and assume a Gaussian PDF for it, that is,

$$ p_y(y) = \frac{1}{\sqrt{2\pi}\sigma_y} e^{-\frac{(y-\mu)^2}{2\sigma_y^2}}; $$

this PDF is uniquely defined by the mean μ and the standard deviation σ _y . Further, we assume that θ _k is uniformly distributed in [0, 2π], and under this assumption, an analytical expression for the PDF of x is available when N = 1. In the general case of N > 1, similar analytical expressions are not available, but nevertheless, the PDF p _x (x) can be approximated numerically by evaluating its histogram on a large number of samples. Assuming that x and y are statistically independent, the PDF of the sum v = x + y is then directly computed from the convolution integral (Bendat and Piersol 1986)

$$ p_v(v) = \int\limits_{-\infty}^{+\infty} p_x(x) p_y(v-x) \rm dx, $$

(3)

once p _x (x) and p _y (y) are known for some values of their parameters. This convolution can be easily computed numerically and used to fit the PDFs measured in the experiments. Fitting is performed by iterative minimization of the square error, leading to the determination of the amplitudes and phase angles in Eq. (2), to the value of μ and, more importantly, to the value of σ _y . This latter parameter can be interpreted as the “true” small-scale background turbulence intensity, properly separated from the large-scale fluctuation due to the PVC; in turn, the PVC fluctuation is represented by the parameters A _n and ψ _n .

The procedure previously described is applied to the single-point velocity data measured by LDV. To ensure a faster convergence of the procedure, we use data obtained from the phase-averaging described in Sect. 4 as an initial guess for the optimization. The optimization performed with N = 3 (i.e., approximating the periodic signal using the fundamental PVC frequency and the corresponding two higher harmonics) resulted in PDFs that approximate very well the experimental ones; the relative energy content of higher harmonics was found negligible in our measurements. Note that the phase-averaging technique presented in Sect. 4 implicitly provides information about the phase-locked coherent fluctuation and its harmonics (Panda and MacLaughlin1993); Fourier analysis of these phase-averaged velocity data confirms that the first three harmonics are sufficient to provide an accurate reconstruction of the phase-locked flowfield. The value of the correlation coefficient between the experimental PDFs and the model PDFs obtained after fitting was almost always larger than 0.98; typical fitting results are reported in Fig. 8, for the same PDFs in off-axis locations shown in Fig. 5.

Fig. 8

a Fitting of the PDF of the axial component at z/R = 0.6, r/R = −0.66 (solid), z/R = 0.6, r/R = −0.33 (dashed). b Fitting of the PDF of the azimuthal component at z/R = 0.6, r/R = −0.66 (solid), z/R = 0.6, r/R = −0.33 (dashed). Experimental data, corresponding to those shown in Fig. 5, are indicated using symbols

The fitting procedure allows separation of the PVC contribution from the random, background turbulence. Results are shown in Fig. 9, for the azimuthal component. In particular, the PVC intensity [i.e., the r.m.s. of the coherent fluctuation $\sqrt{A_1^2 + A_2^2 + A_3^2}/\sqrt{2}$, computed from the Fourier coefficients appearing in (2)], is shown in Fig. 9a, whereas the intensity of the random background turbulence (i.e., the term σ _y ) is shown in Fig. 9c. Radial profiles are plotted for the four vertical locations z/R = 0, z/R = 0.6, z/R = 1.2 and z/R = 1.8, using the same notation as in Fig. 3. It is instructive to compare these figures with Fig. 3d. Comparison shows that the PVC is indeed responsible for the high fluctuation intensity close to the nozzle exit, both on the axis and close to the nozzle edges, while the corresponding σ _y in these locations is small. The background turbulence increases further away from the nozzle exit, and the PVC intensity decreases, indicating that the vortex structures evidenced in Fig. 7 become weaker and lose their coherence. Additionally, it is noteworthy that the background turbulence is quite uniform in radial direction if compared to the PVC intensity.

Fig. 9

a R.m.s. intensity of PVC fluctuations ($\sqrt{A_1^2 + A_2^2 + A_3^2}/\sqrt{2}$), as obtained from the fitting of PDFs. b R.m.s. intensity of PVC fluctuations from phase-locked measurements. c R.m.s. intensity of the background turbulence σ _y , as obtained from the fitting of PDFs. d R.m.s. intensity of the background turbulence from phase-locked measurements. All profiles obtained from LDV data and normalized using the nozzle exit velocity U

It is worth emphasizing that this fitting technique for the PDFs is alternative to phase averaging; even if the basic modeling assumptions are the same, analysis of the PDFs does not require phase-locked measurements. Fig. 9b, d show again the PVC intensity and background turbulence, respectively, as directly computed from the phase-locked LDV data; in particular, results in Fig. 9d are obtained by averaging the r.m.s. fluctuation intensity over the phase. Good agreement is found between the results on the PVC intensity, obtained with the two different techniques. The agreement on the random background turbulence intensity is relatively good, although larger r.m.s. intensity is obtained by direct use of phase-locked measurements. This effect can be attributed to the residual variation of the r.m.s. fluctuation intensity with respect to the phase; uncertainty in the phase locking can be attributed to residual frequency jitter and amplitude modulation.

6 Discussion

The phase-locked analysis of the PVC reported in the previous sections highlights the fact that the high fluctuation intensity in flow regions close to the nozzle exit is due to large-scale quasideterministic motions of the coherent vortical structures. High values of r.m.s. intensity are apparent, and therefore, it is of interest to quantify the effects of deterministic motions and random background. In particular, precession of the recirculation bubble about the geometrical axis of symmetry induces a very high intensity of turbulent fluctuations on the radial and azimuthal components at the nozzle exit, on the nozzle axis, as shown in Fig. 3d, f. As a consequence of the oscillations of the recirculation bubble, the free stagnation point moves periodically; by continuity, the axial recirculating flow expands radially from the stagnation point, thus inducing a quasiperiodic oscillation of the radial and azimuthal velocity components measured with LDV on the nozzle axis.

The second effect related to the displacement of the recirculation bubble is the release of a secondary vortex structure which spirals around the main vortex core, and which is initiated by the instantaneous asymmetry of the flowfield and maintained by the presence of a fixed separation point corresponding to the nozzle edge. This flow structure is coherent with that obtained in recent studies leveraging POD analysis (Oberleithner et al. 2011; Stöhr et al. 2011). The periodicity of the motion of this secondary structure is the same of that of the main vortex core, and the dynamics of the two structures is interdependent. In addition, successful separation of the periodic coherent motion from the background turbulence in Fig. 9a, c confirms that this structure is shed at the same precessing frequency of the PVC.

In terms of spatial structure, our results indicate the presence of two main vortical structures, one in the inner shear layer and one in the outer one. These structures are rotating with the flow, but winding in opposite direction, and they lose coherence for z/R > 1 (Panda and MacLaughlin 1993). This justifies the increase in the intensity of the background turbulence reported in Fig. 9c. It is also worth noting that the present analysis supports the fact that the bimodality of the measured PDFs is due to a single, large-scale flow phenomenon, perturbed with a random noise. This is in contrast with the common interpretation of bimodal PDFs as a result of intermittency between two independent, competing dynamics within the shear layers.

It is interesting to notice that the spatial scale of these two vortical structures is large, of the same order of the nozzle radius; the timescale of the corresponding dynamics is relatively slow and well separated from that of the small-scale turbulence, and this is strongly supported by the analysis and the results reported in Sect. 5. In particular, these results emphasize that vortex precession leaves relatively unaffected the small-scale mixing, but increases the large-scale mixing, and promotes entrainment (Park and Shin 1993). This can be interesting under the modeling point of view. Our results suggest that an unsteady phase-averaged analysis as proposed by Reynolds and Hussain (1972), together with a simple closure scheme for the phase-averaged stresses, could be sufficient to simulate the dynamics of the large-scale motion satisfactorily.

The PDFs analysis method proposed in this paper provides a viable and effective alternative to a pure phase-locked analysis, as it does not require triggering signals. In addition, it provides a straightforward way to evaluate the intensity of the background turbulence in this type of flows. Previous authors (Cala et al. 2006; Liang and Maxworthy 2005; Oberleithner et al. 2011; Stöhr et al. 2011) have mostly focused on the large-scale dynamics of coherent structures rather than quantifying the small-scale turbulence intensity. This information, together with its relative contribution to the total turbulence intensity, is reported in this paper for a swirling jet at high Re and S, an experimental condition that has not received significant attention.

7 Conclusions

Large-scale organized structures were studied experimentally in a free swirling jet of air at ambient pressure and temperature by using phase-locked LDV and PIV. Detailed measurements were focused on the near-field region of the jet, in a parameter regime characterized by strong vortex precession. The phase-averaged analysis indicated that the high turbulence intensity observed in localized regions of the flow in time-averaged data is mostly due to quasideterministic motions of the coherent vortical structures characterizing the PVC instability. These structures were identified using the λ₂ indicator. Joint analysis of the power spectra and PDFs of velocity data led also to quantify the PVC effect on turbulent fluctuations and showed that the PVC contribution to the total turbulent fluctuation can be properly separated from the background random turbulence. This approach has the advantage of being applicable directly to measured data without the need of phase-locked measurements, and results obtained are in good agreement with phase-locked results.