## Abstract

This work experimentally investigated the flow phenomena and vortex structures in the wake of a sphere located in a water loop at Reynolds numbers of Re = 850, 1,250, and 1,700. Velocity fields in the wake region were obtained by applying the time-resolved stereoscopic particle image velocimetry (TR-SPIV) technique. From the acquired TR-SPIV velocity vector fields, the statistical values of mean and fluctuating velocities were computed. Spectral analysis, two-point velocity–velocity cross-correlation, proper orthogonal decomposition (POD) and vortex identification analyses were also performed. The velocity fields show a recirculation region that decreases in length with an increase of Reynolds numbers. The power spectra from the spectral analysis had peaks corresponding to a Strouhal number of St = 0.2, which is a value commonly found in the literature studies of flow over a sphere. The two-point cross-correlation analysis revealed elliptical structures in the wake, with estimated integral length scales ranging between 12% and 63% of the sphere diameter. The POD analysis revealed the statistically dominant flow structures that captured the most flow kinetic energy. It is seen that the flow kinetic energy captured in the smaller scale flow structures increased as Reynolds number increased. The POD modes contained smaller structure as the Reynolds number increased and as mode order increased. In addition, spectral analysis performed on the POD temporal coefficients revealed peaks corresponding to St = 0.2, similar to the spectral analysis on the fluctuating velocity. The ability of POD to produce low-order reconstructions of the flow was also utilized to facilitate vortex identification analysis, which identified average vortex sizes of 0.41D for Re1, 0.33D for Re2, and 0.32D for Re3.

## 1 Introduction

The flow over bluff bodies exhibits complex flow phenomena, even at low Reynolds numbers, and thus a variety of bluff bodies have been studied for further understanding of the flow phenomena [14]. The flow over a sphere is of particular interest and has been studied for many years due to its practical and academic importance [5,6]. Spheres can be used as idealized models for many axisymmetric bluff bodies. Flow past bluff bodies can cause many complex flow phenomena such as vortex shedding and flow separation. Although spheres are axisymmetric, the flow around spheres creates a wide variety of flow phenomena depending on the Reynolds number (Re). Flow over a sphere is laminar and axisymmetric in the Reynolds number range between 20 and 210 [7,8]. Above Re = 210, the flow becomes steady planar-symmetric up to Re = 270 [9]. The flow is still attached, with two vortex tails with equal but opposite sign up to Re = 270 [10], and transitions to an unsteady wake, shedding interconnected vortex loops until Re = 300 [11]. From Re = 400 to 1,000, vortices are shed and the vortex shedding angle oscillates causing the planar flow symmetry to be lost. In this range, the flow remains unsteady but becomes asymmetric [12]. Between Re = 650 and Re = 800, the hairpin vortices shed transition from laminar to turbulent. Starting at Re = 800, the shear layer begins to experience Kelvin–Helmholtz instabilities, smaller scales appear, and the flow becomes more chaotic [13]. Beyond Re = 1,000, the vortex loops that are shed diffuse rapidly [14]. This instability continues to develop until Re = 3,000, where the shear layer begins to transition to turbulent [15]. At Reynolds numbers higher than the critical Reynolds number of $Re=3×105$, the near-wake region shrinks and the sphere experiences a side force [16].

Experimentally investigating the wake of a sphere is challenging due to the need of fixing the sphere in the flow with minimal disturbance to the flow field. Typical supports have been azimuthally symmetric thin wires [17,18] and rigid downstream supports [19]. Upstream supports mounted to a rigid honeycomb structure have also been used to allow for particle image velocimetry (PIV) measurements to be made without the flow in the wake being disturbed [20]. The authors found that the honeycomb structure caused only 2.2% inhomogeneity in the mean flow. Many experimental and numerical studies have been conducted on flow over a sphere, and high quality experimental data are always needed for use in computational fluid dynamics (CFD) code validation.

Optical methods have become widely accepted to be accurate techniques to measure fluid velocity fields, and have been used to study flow in various geometries [2123]. Stereoscopic PIV (SPIV) has been utilized to study other fundamental fluid mechanics problems, such as flow in a pipe, in the laminar, transition, and turbulent regimes [24,25]. PIV has also been used in other bluff body studies and high shear flows, such as a truncated cylinder with a spherically blunted nose [26], and Ahmed bodies [27], and jets [28,29]. Digital PIV measurements for a sphere and a cylinder with an elliptic nose and blunt base at Re = 500, 700, and 1,000 have also been conducted [30]. The mean velocity fields were presented, and spectral analysis was used to extract the vortex shedding frequencies. The authors concluded that the loss of planar symmetry was due to the imbalance of long helical waves. Other experimental studies have been conducted at similar Reynolds numbers. One of these studies tomographic PIV measurements conducted by Eshbal et al. [31] at Reynolds numbers between 400 and 2,000. The study focused on visualization of hairpin vortices that had previously been observed by CFD simulations. Another study comparing a smooth sphere to a rough sphere similar to a pinecone was conducted by David et al. [32] at 226 < Re < 5,000. They concluded for the rough sphere that the recirculating length was minimized at Re = 900, and that the roughness enhanced mixing. Doh et al. utilized three-dimensional (3D)-particle tracking velocimetry (PTV) to quantitatively investigate the hairpin vortices shed in the wake of a sphere at Re = 1,130 [33].

Other authors have employed PIV to make measurements in the wake of a sphere at $Re=11,000$ in a water channel [17], and computed turbulent statistics with the intent of validating numerical simulations. Tomographic PIV has also been used to visualize the hairpin vortices in the wake of a sphere at Re = 465 [34]. The authors found that hairpin vortices were shed from the recirculating near-wake region, and were advected downstream, while the interaction between the recirculation region and the outer shear flow created a saddle point. They provided two-dimensional (2D) plots of the outline of the wake and ensemble-averaged statistics, and 3D surfaces of the Q criterion. Similar results can be seen from other numerical simulations [35]. Previous studies measured the wake properties of a sphere in a wind tunnel using a hot-wire probe, and reported integral length scale values around 30% of the diameter of the sphere [36]. Various simulations of flow around a sphere have also been conducted to investigate the flow phenomena in the wake of a sphere. In two studies that presented direct numerical simulation (DNS) investigations of flow over a sphere for Reynolds numbers of 3,700 and 10,000 [37,38], the authors found that the wake structure was similar between the two Reynolds numbers, exhibiting a helical-like structure due to instabilities, and that the recirculation length decreased with an increase in Reynolds number. The authors also found that the wake is dominated by three types of instability mechanism: large-scale vortex shedding, Kelvin–Helmholtz instability of the shear layer, and modulation of the recirculation region. Another DNS study of flow over a sphere for different Froude numbers (Fr) at $Re=3,700$ [39] found that the flow structure changes significantly for highly stratified wakes with $Fr≤0.125$. Another group used large eddy simulations and PIV investigations at $Re=5,000$ of flow around a sphere [40]. The authors observed that the presence of energy containing eddies increased velocity fluctuations.

As discussed above, several investigations have reported PIV data on flow about a single sphere. These studies have provided statistical color contours, along with spectral analysis of the flow in the wake. This paper discusses the study of water flow over a single sphere at subcritical Reynolds numbers of Re = 850, 1,250, and 1,700. In this work, in addition to the high-resolution time-resolved SPIV (TR-SPIV) data on flow over a sphere, detailed analyses of the flow structures in the sphere's wake using proper orthogonal decomposition (POD) and two-point cross-correlation are presented. This type of measurement allows for investigations of the coherent structures in the near wake region, along with their temporal behavior. High fidelity experimental data with high spatial and temporal resolution is also essential for validation of numerical codes. The data acquired in this study could be used as the first step for validation of CFD codes like NEK5000. The rest of this paper is as follows: the experimental setup for TR-SPIV measurements in the wake of a sphere is discussed in Sec. 2. The statistical results are presented in Sec. 3, along with spectral analysis on the fluctuating velocity. Two-point cross-correlation and POD of the velocity fields are presented in Secs. 4 and 5. The results from the POD analysis are then used to facilitate vortex identification analysis, which is presented in Sec. 6. Finally, a summary of the work is presented in Sec. 7.

## 2 Experimental Facility and Time-Resolved-Stereoscopic Particle Image Velocimetry Experimental Setup

### 2.1 Experimental Facility.

The flow loop consisted of a 100 L storage tank, a centrifugal pump connected to a variable frequency drive to control the flow rate, a Coriolis flowmeter, and a circular acrylic test section with a 205 × 205 mm2 acrylic optical correction box attached. The total axial length of the test section was 1,400 mm, with the sphere placed at approximately 1,260 mm downstream of the test section inlet. The inner diameter of the test section was 140 mm. The stainless steel sphere, with diameter D =31.75 mm, was held in place using a 6.5 mm diameter aluminum rod mounted to a rigid aluminum honeycomb 350 mm upstream of the sphere. It is important that the support is upstream so that the flow does not reattach to the rod and distort the flow field in the wake. Based on the sphere diameter and test section diameter, approximately 5.2% of the total flow area was blocked by the sphere. Measurements were made for freestream velocities of $V0=0.024$, 0.035, and 0.047 m/s corresponding to Reynolds numbers of $Re1=850, Re2=1,250$, and $Re3=1,700$, respectively. The Reynolds number was calculated with the freestream velocity V0, the sphere diameter D, and the fluid viscosity as $Re=V0D/ν$ The obtained statistical quantities and the coordinate system are normalized to the freestream velocity, V0, and the sphere diameter, D.

### 2.2 Time-Resolved-Stereoscopic Particle Image Velocimetry Experimental Setup.

Figure 1 shows the experimental setup for the two-dimensional three-component TR-SPIV measurements. The experimental setup consisted of two Phantom Miro high-speed CMOS cameras (Vision Research, Inc., Wayne, NJ), with 12 bit image depth and $1,280×800$ pixels of resolution, and a 20 W continuous wave green laser (532 nm wavelength). The laser and cameras were mounted to 3-axis linear traverses to allow for fine control of their positions. The two cameras were both mounted on the same side of the laser sheet, viewing it at oblique angles. In order to obtain the three velocity components from these images, a calibration was performed to map the image plane to the object plane of the laser sheet. The calibration was performed using a LaVision 058-5 3D calibration plate (Ypsilanti, MI) with white markers at a spacing of 5 mm in the vertical and horizontal directions, and 1 mm in the out-of-plane direction. Using the images of the target, a mapping function was generated [41] to reconstruct the third velocity component. The laser beam was adjusted into a 2 mm thick sheet using a combined optics system. The cameras were each equipped with 100 mm Nikkor lenses. The seeding particles in the flow were hollow glass spheres with mean diameter of 10 μm and density of 1.1 g/cm3, and were injected into the test section with a syringe and mixed into the flow before the experiment. Such highly reflective seeding particles were illuminated by the laser sheet, and the reflected light from particles was captured by the camera to create an image [42]. A particle response to a constant flow acceleration [43,44] is expressed by
$τp=dp2ρp18μ$
(1)

where dp and ρp are the diameter and density of the seeding particle, respectively, and μ is the dynamic viscosity of water. Given the properties of seeding particles and the operating fluid, the particle response time was of the order of 7 × 10−6 s. The Stokes number, Stk, defined as the ratio between the particle response time τp and the flow characteristic time scale τf, was determined to vary between 9 × 10−6 and 5 × 10−6 corresponding to Re1 and Re3. According to previous studies, a value of Stokes number smaller than 10-1 will ensure the tracer particles follow the flow closely [4244].

Fig. 1
Fig. 1
Close modal

The origin of the coordinate system is set at the center of the sphere. The X-, Y-, and Z-coordinates, respectively, represent the horizontal (lateral), vertical (axial), and spanwise directions. The velocity components corresponding to the X-, Y-, and Z-directions are U, V, and W for the time-averaged velocities, and $u′, v′$, and $w′$ for the fluctuating velocities, respectively. For the experimental measurements, the cameras were operated at frame rates of 50, 75, and 100 frames per second corresponding to the Reynolds numbers of $Re1=850, Re2=1,250$, and $Re3=1,700$, respectively. For each Reynolds number, three experimental runs were performed, capturing $8,341$ images of 768 × 768 pixels on each camera per run, which yielded a total of $25,023$ images per Reynolds number used to compute the flow statistics. The images captured during the experiment were processed using advanced multipass, multigrid algorithms [45] implemented in an in-house PIV code that has been used in other studies by Nguyen et al. [4650]. The Robust Phase Correlation (RPC) algorithm [51] was used to perform the cross-correlation of the experimental images. The RPC algorithm reduces bias errors and peak locking effects when compared to the standard cross-correlation algorithm when analyzing flows with high shear or rotational motion [52]. The processing consisted of three passes with interrogation windows ranging from 64 × 64 pixels to 32 × 32 pixels. The window overlap ranged from 50% to 75%. The final vector fields had 99 × 109 vectors with 0.965 mm of spacing between each vector. A Gaussian peak fit was used to compute particle displacement to subpixel accuracy [44]. Validation was performed using a median filter [53], spurious vectors were removed and the blanks were filled via interpolation. In the measurements, the percentage of spurious vectors, computed as the number of vectors that failed to be validated in the final pass divided by the number of total vectors was less than 0.73%.

### 2.3 Convergence of the Time-Resolved-Stereoscopic Particle Image Velocimetry Measurements.

The convergence of the TR-SPIV results has been verified following the procedure laid out in Refs. [48] and [50]. The convergence function ϵN can be defined as the spatially averaged absolute difference between the statistical results computed using N instantaneous vector fields and the maximum number of vector fields, Nmax. This is expressed with the following formula, where M is the number of points in the field, and Si is a given statistical value
$ϵN=1M∑i=1N|(Si)N−(Si)Nmax|$
(2)

Bar graphs can be seen below in (Fig. 2) of ϵN computed for N = 8,000, 16,000, and 24,000 velocity snapshots. The value of N can be seen decreasing toward 0 as the number of samples approaches Nmax, and the majority of the statistical quantities are converged to within 2% by N = 8,000 samples. This shows that the data captured are enough to reach statistical convergence.

### 2.4 Estimated Uncertainties of the Time-Resolved-Stereoscopic Particle Image Velocimetry Measurements.

In this section, we present the estimated uncertainties of the TR-SPIV measurements. The uncertainty of the in-house PIV code has been evaluated by many studies [5457] to be approximately 0.1 pixels for two-dimensional two-component PIV measurements. The uncertainty quantification for three-component velocity measurements on a plane from SPIV measurements using the in-house codes was assessed in Ref. [58], in which the authors utilized the image matching [59] and correlation statistics [60] approaches. It was reported that the overall estimated uncertainty for instantaneous velocity from in-house codes based on the robust phase correlation algorithm in the TR-SPIV measurements was around 0.3 pixels, less than 5% of the mean axial velocity in the studied flow. The estimation of the total uncertainty of instantaneous velocity components is difficult due to the dependence on factors such as time interval length, tracer density, calibration, and processing algorithms [61,62].

To estimate the uncertainties of the computed SPIV statistical results, including the mean velocity, root-mean-square (RMS) fluctuating velocity, and Reynolds stress, the PIV uncertainty propagation was performed by following the approach of Sciacchitano and Wieneke [63]. The uncertainties of the mean velocities (eU, eV, and eW), RMS fluctuating velocities ($euRMS′, evRMS′$, and $ewRMS′$), and Reynolds stress ($eu′v′$) are expressed as [6365]
$eU=σuNeff,eV=σvNeff,eW=σwNeff$
(3)
$euRMS′=σU2(Neff−1),evRMS′=σv2(Neff−1),ewRMS′=σw2(Neff−1)$
(4)
$eu′v′=σuσv1+ρuv2Neff−1$
(5)

where σu, σv, and σw are the standard deviations of the measured velocity components, and ρuv is the cross-correlation coefficient between the measured velocity components. In these equations, Neff is the effective number of independent samples and can be estimated through the total time acquisition of the TR-SPIV measurements T and the integral time scale Tint as Neff = $(T/(2Tint))$. It is seen that the uncertainty of the statistical quantities depends mainly on the numbers of velocity vectors acquired from the PIV measurements [66,67].

This uncertainty propagation assumes that the errors in the measurements that contribute to the uncertainty of the statistical values are precision based rather than due to bias [67]. This allows the uncertainty of the statistical values to trend toward 0 as the number of samples, N, approaches infinity. Systematic errors that could lead to bias, such as camera focus and peak locking were accounted for during the experiment and in the PIV code, respectively, but cannot be directly quantified. Table 1 shows the uncertainty for the statistical values computed from the TR-SPIV vector fields. The quantities in Table 1 are given as a percentage of the freestream velocity, V0, and are the maximum values found for any point at each Reynolds number. The uncertainty was found to increase with an increase of Reynolds number. The uncertainties in the mean velocities were all less than 0.85%, and the uncertainties of the RMS fluctuating velocities and the Reynolds Stress were less than 0.59%.

Table 1

Estimated uncertainties, $eRe1, eRe2$, and $eRe3$, for the statistical results obtained from the TR-SPIV measurements corresponding to Re1, Re2, and Re3, respectively

$UV0(%)$$VV0(%)$$WV0(%)$$uRMS′V0(%)$$vRMS′V0(%)$$wRMS′V0(%)$$u′v′V02(%)$
$eRe1$0.16550.17730.34350.11700.12540.24290.0284
$eRe2$0.31930.32960.74430.22570.23310.52630.0986
$eRe3$0.21460.20930.84470.15170.14800.59730.0441
$UV0(%)$$VV0(%)$$WV0(%)$$uRMS′V0(%)$$vRMS′V0(%)$$wRMS′V0(%)$$u′v′V02(%)$
$eRe1$0.16550.17730.34350.11700.12540.24290.0284
$eRe2$0.31930.32960.74430.22570.23310.52630.0986
$eRe3$0.21460.20930.84470.15170.14800.59730.0441

The values are presented as a percentage of the freestream velocity V0.

## 3 Results From the Time-Resolved-Stereoscopic Particle Image Velocimetry Measurements

### 3.1 Statistical Results From the Time-Resolved-Stereoscopic Particle Image Velocimetry Measurements.

In this section, we discuss the first- and second-order statistical results from the TR-SPIV measurements for all three Reynolds numbers Re1, Re2, and Re3. The X-, Y-, and Z-coordinates are normalized by the sphere diameter D, while the mean velocity components, i.e., U, V, and W, are normalized by the freestream velocity, V0. The first- and second-order statistical results were computed from the full collection of instantaneous velocity fields.

Fig. 2
Fig. 2
Close modal

Figure 3 shows instantaneous in-plane velocity vector fields with the color contours of the out-of-plane velocity at Re3 = 1,700. The instantaneous velocity fields reveal complex flow structures, and a toroidal vortex system that, when viewed in a cross section, can be seen as a counter-rotating vortex pair in the wake of the sphere. The instantaneous fields also show the wake of the sphere oscillating as vortices are shed. The wake can be seen oscillating between $X/D=−0.75$ and $X/D=0.75$. Here, we note that in Fig. 3 and subsequent figures, the data on the bottom-right side of the sphere are absent due to the shadow of the sphere when illuminated by the laser. We also note that the wake is slightly asymmetrical, and similar asymmetry has been found by other studies [20,26,68,69]. Another clear example can be seen in Ref. [70], where the streamlines crossing the midplane of the sphere are clearly asymmetric. It is likely that the slight asymmetry of the wake seen in Fig. 4 occurs due to instability caused by a Hopf bifurcation around Re = 87.5. Past this Reynolds number, the wake may become asymmetric due to one side of the helical structure becoming dominant over the other [30,71,72].

Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal

Figure 4 shows the normalized mean out-of-plane velocity color contour, along with streamlines and a line indicating the location of the profiles shown in Fig. 5. The velocity fields reveal very similar structures with increasing Reynolds number, but with the peaks increasing in magnitude. The mean in-plane vector fields, in combination with the streamlines, show the same two counter-rotating vortices as are seen from the instantaneous fields in the cross section of the toroidal vortex system in the wake of the sphere. It can be seen from the streamlines that as Reynolds number increases, the recirculation length decreases. The recirculation length was computed from the mean fields for each case to be 1.80D, 1.65D, and 1.52D, for Re1 to Re3, respectively. This confirms the trend displayed by the streamlines. The out-of-plane color contour reveals a swirling structure that is produced as the wake moves along all of the azimuthal directions.

Fig. 5
Fig. 5
Close modal

Figure 5 shows the statistical profiles at Y/D =1.5. The U velocity profile shows antisymmetric peaks aligned with the shear layer that had a slight decreasing trend in relative magnitude with an increase in Reynolds number due to the shortening of the wake. The $uRMS′$ profile shows a large peak inside the wake of the sphere due to the higher turbulent intensity inside the flow separation. The magnitude of the peak does not show a clear trend with a change in Reynolds number. The V velocity profile shows a negative peak in the center due to relatively small velocity inside the flow separation in the wake. The negative peak decreases in depth as Reynolds number increases due to the shrinking of the recirculation length. The $vRMS′$ profile shows two peaks due to the presence of the strong shear layer at the wake interface. The profiles of W and $wRMS′$ behaved similarly to the profiles of U and $uRMS′$ due to the symmetry of the geometry. The profiles of the RMS fluctuating velocities were the smallest along the edges where the velocities are close to that of the freestream. The dynamic range for the mean velocity measurements was [−0.0076, 0.0303] for Re1, [−0.0133, 0.0455] for Re2, and [−0.0194, 0.0599] for Re3. Likewise for the fluctuating velocities, the dynamic range was [0, 0.114] for Re1, [0, 0.238] for Re2, and [0, 0.0321] for Re3.

Figure 6 illustrates the normalized instantaneous in-plane velocity field and color contour of the normalized instantaneous spanwise vorticity corresponding to Re1, Re2, and Re3. In Fig. 6, spanwise vortices with various scales can be seen in the shear layer, detached from the sphere's surface. In this flow regime, two types of instability exist. One mode is associated with small-scale instability in the shear layer along the flow separation, and the other is the large-scale vortex shedding instability. The small-scale instability is evident from the progressively smaller scales as Reynolds number increases. These small scales arise from the presence of Kelvin–Helmholtz instabilities in the detached shear layer. To study this motion, we have followed the process proposed in Refs. [73] and [74] and identified the loci of maximum turbulence intensity. The turbulence intensity, Tuvw is defined as
$Tuvw=u′2+v′2+w′23$
(6)

where $u′, v′$, and $w′$ are the fluctuating velocity components from Reynolds decomposition. Figure 7 illustrates the color contours of Tuvw and the locations its maximum intensity along the shear layer, along with a plot of the intensities of Tuvw along the left shear layer for each Reynolds number. This plot illustrates the transition process of the shear layer such as growth, peak, and decay of intensity along the flow direction. In each case, the shear grew to a maximum value before beginning to slowly decay. The growth rate and maximum value trended upward nonlinearly with Reynolds number. The maximum value occurred between $1.5D$ and $1.8D$ downstream of the sphere. The development of this flow consists of three main parts: (i) establishment of the shear layer from the leading edge of the sphere, (ii) onset of roll near the crest of the shear layers, and (iii) Kelvin–Helmholtz instabilities developing into vortex tubes downstream. The final part creates three-dimensional flapping motion of the shear layer and vortex pairs shown by the alternate signs of vorticity in Fig. 6.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

### 3.2 Spectral Analysis on the Fluctuating Velocity Components.

The vortex shedding frequency from the sphere can be determined from the spectral analysis of the fluctuating velocity components inside the wake of the sphere [18]. Figure 8 shows the power spectrum densities (PSDs) from the spectral analysis on the streamwise fluctuating velocity component, $v′$, the lateral component, $u′$, and the spanwise fluctuating velocity component, $w′$ at the points P1, and P3 located at (X/D, Y/D) = (−0.5, 1.25), and (0.5, 1.25), respectively. The spectral analysis was performed using Welch's method [75] with a window size of 1,024 samples and window overlap of 256 samples. The PSDs revealed peak frequencies around a Strouhal number of 0.2 for both points, and at all studied Reynolds numbers. The PSDs computed using $u′$ and $v′$, and the PSD computed using the spanwise component $w′$ at P1 all contain the above mentioned peak. The absence of the peak for $w′$ at P3 is likely due to the slight asymmetry of the wake.

Fig. 8
Fig. 8
Close modal
The −5/3 slope from Kolmogorov power law theory is overlaid above the PSDs, and shows a good agreement within the inertial range. Table 2 shows the slope of the PSDs and their corresponding 95% confidence intervals. The PSDs follow this slope until the point where dissipation becomes dominant [76]. The slight difference in the power-law slope for the spanwise component is likely due to the oblique angle of the cameras, which increases the practical size of the interrogation windows, resulting in a lower frequency response [77]. The Strouhal number was calculated for each case using the freestream velocity, V0, and the diameter of the sphere, D, and the peak frequency, f, as shown in the following equation:
$St=fDV0$
(7)
Table 2

Slope fit to PSDs for U, V, and W at P1 and P3 with the associated 95% confidence interval

PointRe1Re2Re3
$P1:U$$−1.6913±0.1012$$−1.6277±0.1667$$−1.6896±0.0878$
$P3:U$$−1.6602±0.0993$$−1.6250±0.0924$$−1.6853±0.0790$
$P1:V$$−1.6132±0.1216$$−1.6108±0.1578$$−1.6488±0.0962$
$P3:V$$−1.6989±0.0523$$−1.7698±0.0507$$−1.7274±0.0651$
$P1:W$$−1.6329±0.1105$$−1.5606±0.0937$$−1.5927±0.0535$
$P3:W$$−1.5602±0.0697$$−1.5184±0.0661$$−1.5259±−0.0971$
PointRe1Re2Re3
$P1:U$$−1.6913±0.1012$$−1.6277±0.1667$$−1.6896±0.0878$
$P3:U$$−1.6602±0.0993$$−1.6250±0.0924$$−1.6853±0.0790$
$P1:V$$−1.6132±0.1216$$−1.6108±0.1578$$−1.6488±0.0962$
$P3:V$$−1.6989±0.0523$$−1.7698±0.0507$$−1.7274±0.0651$
$P1:W$$−1.6329±0.1105$$−1.5606±0.0937$$−1.5927±0.0535$
$P3:W$$−1.5602±0.0697$$−1.5184±0.0661$$−1.5259±−0.0971$

A peak near Strouhal number of St = 0.2 can be found for flow over a sphere almost universally. These Strouhal number peaks represent the vortex shedding frequencies of the flow [46].

In the study of flow around square cylinders, the peak Strouhal number was found to be strongly affected with a blockage ratio of 1/4, while the effect was much less noticeable with blockage ratios of 1/6 and 1/8 [78]. The peak Strouhal number found in this study matched well with those found in the literature. Another study on the effect of blockage ratio on the wake of a square cylinder found a strong dependence of wake behavior past a blockage ratio of 0.4 [79]. The authors found that past this blockage ratio, the flow separated from the channel and caused the wake to swell laterally. Another experimental study on blockage effects in a wind tunnel found very little blockage effect for blockage ratios up to 10% [80].

## 4 Analysis of the Velocity–Velocity Two-Point Cross-Correlation

The two-point cross-correlation function can be useful for providing insight into the turbulent characteristics of the flow, such as the shapes, sizes, and roles of coherent structures [46,50,8183]. The two-point cross-correlation function is defined as
$Ruu(xr,η,τ)=〈u′(xr,t)×u′(xr+η,t+τ)〉〈u′2(xr,t)〉〈u′2(xr+η,t+τ)〉$
(8)
In this equation, $〈·〉$ is the ensemble-average operator, $u′$ is the fluctuating velocity component, τ is a time delay, xr is a spatial reference location, and η is the spatial separation between the reference location and the measurement location. The value of the cross-correlation function ranges from zero to one, with lower values signifying dissimilarity in flow structures, and higher values signifying similarity between the flow structures. The values of the two point cross-correlation indicate the spatial extent of correlated regions [84], and allow the integral length scale to be calculated as
$kLx=∫0∞Ruu(xr,ηk,0)dη$
(9)

In the integral length scale equation, k is the direction along which the integral is computed, and x is the corresponding velocity component. This length scale measures the maximum correlation distance between the velocities at two points in the flow field [85]. The integral length scale in a given direction is representative of the characteristic length scale of the flow energy containing eddies along that direction [50,86], and can be considered to be the average of all turbulent length scales at that location [87]. The shapes and sizes of these eddies, along with the energy they contain, are important parameters to better understand the flow phenomena near the shear layer and inside of the wake behind the sphere.

The two-point cross-correlations, $Ruu0, Rvv0$, and $Rww0$, were computed at different points located near the shear layer and inside the recirculation bubble at the points P1 = (−0.5, 1.25), P2 = (0.0, 1.25), and P3 = (0.5, 1.25). The contours of the two-point cross-correlation shown in Fig. 9 contain elliptical shapes. Elliptical structures have been noticed in high shear flows before [88,89], and ellipses have been fit to the correlation map to compute integral length scales [90]. Following the same procedure, ellipses were fit at contour levels of 0.3 to 0.8. These ellipses were also plotted over the structures in Fig. 9. These ellipses were used to determine the direction along which the flow structure is aligned. The integral length scale was then computed along the major axis of the ellipse, rather than along the X, Y, or Z axes. The contours of $Ruu0$ display structures that were rounder than those revealed by $Rvv0$, and the angle of the major axis for these cases varied from nearly horizontal to nearly vertical. The structures in $Ruu0$ near the shear layer were usually elongated in the horizontal direction, while those inside the recirculation bubble were vertically elongated. The color maps of $Rvv0$ revealed very elongated elliptical structures both near the shear layer and inside the recirculation bubble. In all cases for $Rvv0$ the major axis of the ellipse fitted to the contour was nearly vertical. The contours of $Rww0$ revealed structures that were similar in shape to those revealed by the contours of $Ruu0$, but smaller in size. These structures were also rounder than those displayed in $Rvv0$, and all had approximately the same value of the angle for the fitted ellipses. The aspect ratios for the ellipses ranged from 0.327 to 2.559 for $Ruu0$, from 1.435 to 5.114 for $Rvv0$, and from 1.189 to 3.930 for $Rww0$.

Fig. 9
Fig. 9
Close modal

Profiles of the two-point cross-correlation were taken along the major axis of the fitted ellipses. The limits for integration in Eq. (9) were taken to be the points where the cross-correlation crossed $1/e$$≈$ 0.37 [48,82,91]. Figure 10 shows the obtained profiles of the two point cross-correlations, $Ruu0, Rvv0$, and $Rww0$ at $Re3=1,700$. The structures revealed by $Ruu0$ and $Rww0$ were significantly smaller than those from $Rvv0$, which can be seen from these profiles. The peaks in $Ruu0$ and $Rww0$ are much narrower and sharper than those of $Rvv0$. Table 3 contains the integral length scales computed at points P1, P2, and P3 located at (X/D, Y/D) = (−0.5, 1.25), (0.0, 1.25), and (0.5, 1.25), respectively, using $u′, v′$, and $w′$. The integral length scales determined using the color contour ellipse fitting did not show a significant effect of the contour level, though the contour level for $v′$ showed a slightly more pronounced effect than those for $u′$, and $w′$. The length scales along the shear layer decreased with an increase in Reynolds number, likely due to the vortex stretching that occurs due to the convective instabilities in the shear layer [92]. These instabilities are governed by momentum thickness, which is a function of the Reynolds number. The length scales for $v′$ were larger here than inside the flow separation. The length scales inside the flow separation did not seem to follow any particular trend. This is due to the fact that the flow separation is governed mainly by the geometrical parameter. The length scales for $u′$ and $w′$ were larger inside the flow separation than in the shear layer. Further knowledge about the shape and size of these eddies can provide insight into flow mixing phenomena inside the wake.

Fig. 10
Fig. 10
Close modal
Table 3

Integral length scales estimated along the major axis of ellipse fits to two-point velocity–velocity cross-correlations $Ruu0, Rvv0$, and $Rww0$

Reynolds numberCoordinatesContour level
$Re1=850$$(X/D,Y/D)$0.40.50.60.7
$Ruu0$; $Rvv0$; $Rww0$$P1=(−0.5,1.25)$0.18; 0.63; 0.140.18; 0.63; 0.140.18; 0.63; 0.140.18; 0.63; 0.14
$Ruu0$; $Rvv0$; $Rww0$$P2=(0,1.25)$0.26; 0.33; 0.170.26; 0.33; 0.170.26; 0.32; 0.160.26; 0.32; 0.16
$Ruu0$; $Rvv0$; $Rww0$$P3=(0.5,1.25)$0.25; 0.44; 0.060.25; 0.45; 0.060.25; 0.44; 0.060.24; 0.43; 0.06
$Re2=1250$$(X/D,Y/D)$0.40.50.60.7
$Ruu0$; $Rvv0$; $Rww0$$P1=(−0.5,1.25)$0.14; 0.39; 0.130.14; 0.38; 0.130.14; 0.37; 0.130.14; 0.36; 0.13
$Ruu0$; $Rvv0$; $Rww0$$P2=(0,1.25)$0.22; 0.25; 0.120.22; 0.25; 0.12;0.22; 0.25; 0.120.22; 0.24; 0.12
$Ruu0$; $Rvv0$; $Rww0$$P3=(0.5,1.25)$0.15; 0.38; 0.070.15; 0.40; 0.070.15; 0.40; 0.070.15; 0.40; 0.07
$Re3=1700$$(X/D,Y/D)$0.40.50.60.7
$Ruu0$; $Rvv0$; $Rww0$$P1=(−0.5,1.25)$0.15; 0.39; 0.130.15; 0.38; 0.130.15; 0.37; 0.130.15; 0.37; 0.12
$Ruu0$; $Rvv0$; $Rww0$$P2=(0,1.25)$0.25; 0.25; 0.100.26; 0.24; 0.100.25; 0.24; 0.100.25; 0.24; 0.10
$Ruu0$; $Rvv0$; $Rww0$$P3=(0.5,1.25)$0.13; 0.36; 0.070.13; 0.34; 0.070.13; 0.34; 0.070.13; 0.34; 0.07
Reynolds numberCoordinatesContour level
$Re1=850$$(X/D,Y/D)$0.40.50.60.7
$Ruu0$; $Rvv0$; $Rww0$$P1=(−0.5,1.25)$0.18; 0.63; 0.140.18; 0.63; 0.140.18; 0.63; 0.140.18; 0.63; 0.14
$Ruu0$; $Rvv0$; $Rww0$$P2=(0,1.25)$0.26; 0.33; 0.170.26; 0.33; 0.170.26; 0.32; 0.160.26; 0.32; 0.16
$Ruu0$; $Rvv0$; $Rww0$$P3=(0.5,1.25)$0.25; 0.44; 0.060.25; 0.45; 0.060.25; 0.44; 0.060.24; 0.43; 0.06
$Re2=1250$$(X/D,Y/D)$0.40.50.60.7
$Ruu0$; $Rvv0$; $Rww0$$P1=(−0.5,1.25)$0.14; 0.39; 0.130.14; 0.38; 0.130.14; 0.37; 0.130.14; 0.36; 0.13
$Ruu0$; $Rvv0$; $Rww0$$P2=(0,1.25)$0.22; 0.25; 0.120.22; 0.25; 0.12;0.22; 0.25; 0.120.22; 0.24; 0.12
$Ruu0$; $Rvv0$; $Rww0$$P3=(0.5,1.25)$0.15; 0.38; 0.070.15; 0.40; 0.070.15; 0.40; 0.070.15; 0.40; 0.07
$Re3=1700$$(X/D,Y/D)$0.40.50.60.7
$Ruu0$; $Rvv0$; $Rww0$$P1=(−0.5,1.25)$0.15; 0.39; 0.130.15; 0.38; 0.130.15; 0.37; 0.130.15; 0.37; 0.12
$Ruu0$; $Rvv0$; $Rww0$$P2=(0,1.25)$0.25; 0.25; 0.100.26; 0.24; 0.100.25; 0.24; 0.100.25; 0.24; 0.10
$Ruu0$; $Rvv0$; $Rww0$$P3=(0.5,1.25)$0.13; 0.36; 0.070.13; 0.34; 0.070.13; 0.34; 0.070.13; 0.34; 0.07

## 5 Proper Orthogonal Decomposition Analysis on the Velocity Fields

Proper orthogonal decomposition is a tool introduced by Lumley [93] to identify large-scale energy containing features or coherent structures in turbulent flow. POD decomposes a set of spatiotemporal data into a set of spatial basis functions and corresponding temporal coefficients. The POD spatial modes display large-scale orthogonal flow structures, and the order of the POD spatial modes is ranked by the kinetic energy they contain.

A given velocity field, $u(x,t)$, over a finite time interval $(0 can be decomposed into a set of spatial modes and temporal coefficients as follows:
$u(x,t)=∑k=1Nζk(t)ψ(x)$
(10)
where N is the number of velocity field samples, $ζk(t)$ are the POD temporal coefficients, and $ψ(x)$ are the POD spatial basis functions. The POD basis functions, $ψ(x)$ are the eigenvectors of the two-point correlation matrix. The two-point correlation is denoted as Cij in the discrete case, and is computed as
$Cij=1N∫u(x,ti)·u(x,tj)dx$
(11)
We can then define αki as
$αki=vikN∑m=1N∑r=1NvmkvrkCmr$
(12)
where $vik$ is the ith element of the eigenvector vk corresponding to the eigenvalue λk of the two-point correlation matrix, Cij. These values can then be used to compute the basis functions and temporal coefficients as follows:
$ψ(x)=∑k=1Nαkiu(x,ti)$
(13)
$ζk=∫u(x,t)·ψ(x)dx=N∑k=1NαkiCij$
(14)

Results from the POD analysis applied to the velocity fields for each Reynolds number are shown in Figs. 1117. Figure 11 shows the kinetic energy spectra and cumulative kinetic energy contained within the modes. The first POD mode followed a trend of decreasing with an increase in Reynolds number, and contained 95.68%, 93.15%, and 90.11%, of the total kinetic energy for Re1, Re2, and Re3, respectively. This trend indicates that the flow became more turbulent and more of the kinetic energy was distributed over structures with a smaller scale than the mean flow as the Reynolds number increased. The maximum amount of energy contained in the modes after the first was about 0.5% of total kinetic energy, which resulted in a quick convergence of the cumulative kinetic energy. The cumulative energy contained within the first 50 modes was 99.10%, 97.31%, and 95.42%, for the three Reynolds numbers considered in this study. Table 4 shows the kinetic energy contained in the first five modes for each Reynolds number.

Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal
Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal
Fig. 17
Fig. 17
Close modal
Table 4

Percentage of kinetic energy contained in the first five POD modes at each Reynolds number

ψ1ψ2ψ3ψ4ψ5
Re195.680.340.240.190.19
Re293.150.410.280.270.21
Re390.110.500.380.340.28
ψ1ψ2ψ3ψ4ψ5
Re195.680.340.240.190.19
Re293.150.410.280.270.21
Re390.110.500.380.340.28

Figure 12 shows a comparison of the mean velocity field with the first POD mode. It can be seen that the first POD mode for each case is approximately equivalent to the mean velocity field. Figure 13 illustrates the POD modes 2, 3, 4, and 8 for Reynolds numbers of Re1 and Re3. It can be seen in Fig. 13 that dominant flow structures revealed in the POD modes 2, 3, and 4 had large sizes, while those depicted in the higher POD mode 8 had relatively smaller sizes. Such smaller scale flow structures contained in the higher POD modes represent the unsteady chaotic nature of the flow. The rest of the POD modes, however, showed significantly different structures when comparing across the different Reynolds numbers. The size of the structures contained in the POD modes tended to reduce as the mode order increased and as Reynolds number increased.

Figure 14 shows the POD temporal coefficients 1–4 for all three Reynolds numbers. The POD temporal coefficients were rescaled by applying a z-score normalization, which modified the signal to have a mean of 0 and standard deviation of 1. The POD temporal coefficients did not reveal any obvious patterns, which suggests that the flow is a complex dynamical process. It can also be seen from Fig. 14 that the temporal coefficients fluctuate more heavily with an increase in Reynolds number. This occurs due to the strong increase in turbulence in the wake, and the transition between a laminar shear layer to a turbulent shear layer.

The PSDs computed from the POD temporal coefficients are shown in Fig. 15, and were computed with Welch's method similar to those computed for the fluctuating velocity. The −5/3 slope from Kolmogorov power law theory is overlaid above the PSDs from the POD temporal coefficients similar to those from the fluctuating velocity. The PSDs computed using the POD temporal coefficients also show very good agreement with the −5/3 slope. The PSDs of the POD temporal coefficients covered a large frequency range, which could suggest that there are multiple scales of turbulent structures contained within the flow. The frequency peaks in the PSDs computed using the POD temporal coefficients were very close to those revealed by the PSDs from the spectral analysis of the vertical fluctuating velocity, which corresponded to a St = 0.2.

Figure 16 depicts the trajectories of the POD temporal coefficients in the phase space, along with the cross-correlation of POD coefficients ζ2 and ζ3 at each studied Reynolds number. The coefficients ζ2 and ζ3 form a nearly circular trajectory in the phase space, which suggests that they dynamic pair of modes that coexist within the flow domain [94]. This dynamic mode pair is likely the result of a traveling structure within the flow, which is represented as a pair of modes that appear similar but shifted along the main advection direction [95]. The maximum magnitude of $Rζ2ζ3$ were 0.73 for Re1, 0.78 for Re2, and 0.73 for Re3, and occurred at 1.27 for Re1, 0.97 for Re2, and 1.23 for Re3, respectively. These high values of the cross-correlation further confirm the pairing of the POD modes, and can be used to estimate the time delay between them.

Figure 17 shows a low‐order reconstruction of an instantaneous velocity field at Re3 using the low‐order POD spatial modes and POD temporal coefficients. The original velocity field contains small scale vortex structures. The velocity field was reconstructed using 10, 25, and 50 modes, which captured 92.61%, 94.12%, and 95.42% of the total flow kinetic energy. When reconstructing the velocity fields, the flow reconstruction using ten modes depicted only the dominant large-scale structures contained in the flow. As the number of modes used to reconstruct the velocity fields increased to 25 or 50, the smaller scale flow structures seen in the original velocity field began to reveal themselves. Overall, as Reynolds number increased, the flow contained more small-scale flow structures, and thus higher numbers of modes were needed to accurately reconstruct the flow. These reconstructed velocity fields can also be used to facilitate the vortex identification analysis following the procedure laid out in Ref. [96].

Performing POD on the velocity fields has allowed us to extract the coherent structures contained within the flow in the wake of the sphere. POD analysis is beneficial to characterize the flow structures and the energy contained within them. A low‐order model of the flow can also be extracted from the POD analysis. The POD analysis on the velocity fields can provide information on the energy contained in the structures, but more direct characterization the size and shape of the coherent structures in the wake are also needed.

## 6 Identification and Distribution of Vortices in the Central Time-Resolved-Particle Image Velocimetry Measurement Plane

In this section, we utilize the POD analysis and its capability of flow reconstruction using low-order POD modes to investigate the characteristics of z-vortices within the central TR-SPIV measurement plane for Reynolds numbers Re1, Re2, and Re3. The statistical characteristics of the vortices, such as population densities, spatial distributions, and circulations, are described here. Many methods have been proposed to identify vortices within a flow field. She et al. [97] used DNS results from an isotropic turbulence field to identify vortices based on the magnitude of vorticity. This method was unable to distinguish the flow regions induced by vortex cores from shear motion. More robust methods have been proposed, including the Q criterion [98], Δ criterion [99], λ2 criterion [100], and λci criterion [101], which better separate vortex cores against shear motion [102]. In PIV measurements, the projection of a 3D vortex on 2D velocity fields must be extracted. Adrian et al. [103] introduced the 2D form of the λci criterion, which utilizes a 2D form of the 3D velocity gradient tensor. Carlier and Stanislas [104] utilized pattern recognition to identify vortices based on a reference eddy structure with Gaussian damping. Another method was proposed based on triple decomposition of the velocity fields [105], which was further enhanced by Maciel et al. [106]. This method utilizes the rigid-body rotation vorticity, which can be considered a local measure of vortex-related vorticity. This allows the swirling of vortices to be separated from pure shear motion. Further review of vortex identification methods can be found in the works of Kolář [105] and Chen et al. [102]. In this study, we combined POD analysis and the vortex identification algorithms previously discussed in Ref. [96] to identify the in-plane z-vortices from instantaneous velocity vector fields, i.e., in-plane u- and v-velocity components are considered. First, POD analysis was applied to the snapshots of instantaneous velocity vector fields from the TR-SPIV measurements; the POD spatial modes and POD temporal coefficients were acquired. Low-order POD modes and associated POD coefficients were then used to reconstruct the collection of instantaneous velocity fields to facilitate the separation of large-scale flow structures from the small-scale flow structures [42,96]. In the studies of Refs. [42,48], and [50], the authors demonstrated the capabilities of flow reconstruction using low-order POD modes such that the low-order reconstructed velocity and vorticity fields clearly depicted the large-scale coherent flow structures, i.e., energy-containing eddies, compared to the flow structures that appeared in the original instantaneous velocity snapshots. Using the reconstructed instantaneous velocity fields, centers and cores of the in-plane z-vortices are identified. For a given fixed point P in the measurement domain, the nondimensional scalar function $Γ1$ at P is defined as follows:
$Γ1(P)=1N∑S(PM∧UM)·z||PM||·||UM||=1N∑S sin (θM)$
(15)
where S is the 2D area surrounding P, point M is within S, z is the normal vector of the measurement region $(||z||=1)$, N is the number of points M inside S, and θM represents the angle between the velocity vector UM and the radius vector PM. $|Γ1|$ has a maximum value of 1 and reaches values ranging from 0.9 to 1 near the vortex center, which allows us to identify the center of vortical structures based on a threshold calculation [96]. The vortex boundary identification can be derived as follows:
$Γ2(P)=1N∑S[PM∧(UM−ŨP)]·z||PM||·||UM−ŨP||$
(16)

where the local convection velocity $ŨP$ around point P is defined as $ŨP=1S∫SUdS$. Graftieaux et al. [96] discussed that $Γ2$ is a local function depending only on Ω, i.e., the rate of rotation of the tensor corresponding to the antisymmetrical part of the velocity gradient $▽$u, and μ, i.e., the eigenvalue of the symmetrical part of $▽$u. In the region where the flow is locally dominated by rotation, i.e., $|Ωμ>1|$ and $|Γ2>2Π|$, can be chosen as a threshold [96] to define the vortex core region and to estimate the area and circulation of identified vortices.

The statistical information of the z-vortices was analyzed from a total of 25,023 instantaneous velocity vector fields corresponding to each Reynolds number of Re1, Re2, and Re3. Based on the suggestions of Nguyen et al. [50,86], the flow reconstructions were performed using the numbers of low-order POD modes that contained 99% of the total flow kinetic energy. Figure 18 showed the results obtained from the vortex identification in the wake region of the sphere. Figure 18(a) depicts the spatial distributions and sizes of identified z-vortices overplotted over the mean velocity vector fields for Re1, Re2, and Re3. In this figure, the color scale shows the vortex strength. In Fig. 18(b), histograms illustrate fluid areas covered by the z-vortices identified for Re1, Re2, and Re3. Table 5 summarized the statistical results of the z-vortices, such as the total number of detected vortices, $NΩ$, the vortex population, $NΩ/Af$ (Af is the flow area of wake region downstream of the sphere), the normalized mean and standard deviation of the z-vortex area, i.e., $μA/D2$ and $σA/D2$, respectively, and the normalized mean and standard deviation of the z-vortex strength, i.e., $μΓ/DV0$ and $σΓ/DV0$, respectively. It is noted that the $μΓ$ and $σΓ$ of z-vortex strength were calculated considering the absolute strength values. Results from the vortex identification have shown that an increase of the Reynolds number increased the number of detected vortices, i.e., the greatest vortex population density was found at Re3, along with a reduction of average vortex area. In addition, the spatial extent of the vortex distribution was shortened with an increase of the Reynolds number. For instance, vortices were detected further downstream at approximately 2D for Re1, while they were identified at approximately $1.67D$ for Re3. This is because when the recirculation length is reduced, vortices generated from the sphere's trailing edge and shear layers are dynamically evolved within the wake region of the sphere. These observations are in accordance with the reduced recirculation lengths in the wake region when Reynolds numbers increased from Re1 to Re3. For the studied Reynolds numbers Re1, Re2, and Re3, the detected vortices varied with the ranges of minimum and maximum covered areas, i.e., $0.01D2$ to $1.53D2$, respectively. If the vortex area is considered as a circle with a radius of rvor, the identified z-vortices in the wake region of the sphere had mean radii of $rvor/D$ = 0.41, 0.33, and 0.32 for Re1, Re2, and Re3, respectively. The normalized mean and standard deviation of the vortex strength were found to decrease when Reynolds numbers increased from Re1 to Re3, although the mean vortex strengths increased with an increase of Reynolds number.

Fig. 18
Fig. 18
Close modal
Table 5

Statistical results of z-vortices identified from the instantaneous velocity vector fields in the wake region of the sphere for Re1, Re2, and Re3

Re1Re2Re3
$NΩ$23,13126,07527,710
$NΩAF$4.595.175.49
$μAD2$0.5210.3380.318
$σAD2$0.3120.2720.193
$μΓDV0$1.6191.3141.305
$σΓDV0$0.8590.7770.657
Re1Re2Re3
$NΩ$23,13126,07527,710
$NΩAF$4.595.175.49
$μAD2$0.5210.3380.318
$σAD2$0.3120.2720.193
$μΓDV0$1.6191.3141.305
$σΓDV0$0.8590.7770.657

Results are shown here for the total counts of identified z-vortices, $NΩ$, the vortex population density, $NΩ/AF$, the normalized mean and standard deviation of the vortex area, i.e., $μA/D2$ and $σA/D2$, respectively, and the normalized mean and standard deviation of the vortex strength, i.e., $μΓ/DV0$ and $σΓ/DV0$, respectively.

## 7 Summary

The purpose of this study is to investigate the flow characteristics in the wake of a sphere at subcritical Reynolds numbers. TR-SPIV measurements were performed in the wake of a sphere at three Reynolds numbers between 850 and 1,700. The instantaneous velocity fields revealed small vortex structures and an oscillating recirculation region. The mean velocity fields showed the cross section of the toroidal vortex system in the wake, which presents itself as two counter-rotating vortices in the wake of the sphere. The length of the recirculation region was found to decrease with an increase in Reynolds number. The profiles of the mean velocities showed the same structures with peaks and valleys increasing in magnitude with increase in Reynolds number. The PSD plots from the spectral analysis of both the fluctuating velocity and the POD temporal coefficients contained peaks around St = 0.2, corresponding to the vortex shedding frequency that is commonly found in literature for flow over a sphere. The two-point cross-correlation analysis revealed elliptical structures within the flow. The turbulent length scales computed along the major axis of the elliptical structures revealed by the two-point cross-correlation analysis ranged between $0.12D$ and $0.63D$. The computed integral length scales generally tended to decrease with an increase in Reynolds number, which shows a similar trend of smaller coherent structures being dominant in the wake. The POD analysis and two-point cross-correlation analysis applied to the TR-SPIV snapshots extracted the coherent flow structures that capture important characteristics related to the vortices in the wake the sphere. The POD analysis revealed the dominant coherent structures that occur in the wake of a sphere at these Reynolds numbers, and allowed the energy levels of these structures to be studied. The energy contained in the first POD mode for each Reynolds number considered in this study was 95.68%, 93.15%, and 90.11%, for Reynolds numbers 1–3, respectively. The first POD mode is considered approximately equal to the mean velocity field, so these values can be considered representative of the amount of kinetic energy contained within the mean velocity field. Modes 2 and 3 were found to be a traveling structure contained within a mode pair that were in phase. The decreasing trend of energy captured by higher order POD modes shows that the smaller scales within the wake contained more energy as Reynolds number increases. As Reynolds number increases, more energy content is pushed into smaller structures, and thus the higher order POD modes contain more energy. This leads to stronger momentum transfer in the shear layer, which causes the recirculation length to shrink. The ability of POD to produce low-order reconstructions of the velocity fields was also utilized to facilitate vortex identification analysis, which also revealed smaller, stronger structures in the wake with increasing Reynolds number, a shrinking wake, and an average vortex size of 0.41D for Re1, 0.33D for Re2, and 0.32D for Re3.

## Acknowledgment

This research is financially supported by the U.S. Department of Energy, NEAMS project and under a Contract No. DE-NE0008983. The authors also thank the support from the U.S. Argonne National Laboratory.

## Funding Data

• Nuclear Energy University Program (No. DE-NE0008983; Funder ID: 10.13039/100000015).

## Nomenclature

• Af =

flow area

•
• Cij =

two-point correlation matrix

•
• D =

sphere diameter

•
• dp =

seeding particle diameter

•
• eRe =

uncertainty for the statistics at a given Reynolds number

•
• f =

frequency

•
• $kLx$ =

integral length scale

•
• N =

number of samples

•
• $NΩ$ =

number of detected vortices

•
• rvor =

radius of equivalent circular vortex

•
• $Ruu0, Rvv0, Rww0$ =

velocity–velocity cross-correlation coefficients

•
• Re =

Reynolds number

•
• St =

Strouhal number

•
• Stk =

Stokes number

•
• t =

time

•
• T =

total measurement time

•
• Tint =

integral time scale

•
• TUVW =

time

•
• U, V, W =

mean velocities in the X, Y, and Z directions

•
• $u′, v′, w′$ =

fluctuating velocities in the X, Y, and Z directions

•
• $uRMS′, vRMS′, wRMS′$ =

root-mean-square fluctuating velocities in the X, Y, and Z directions

•
• V0 =

freestream velocity

•
• X, Y, Z =

horizontal, vertical and spanwise directions

•
• $Γi$ =

vortex identification scalar functions

•
• ϵ =

convergence function

•
• $ζk(t)$ =

POD temporal coefficient

•
• η =

separation distance

•
• μA =

mean vortex area

•
• $μΓ$ =

mean vortex strength

•
• ν =

viscosity

•
• ρp =

seeding particle density

•
• σA =

vortex area standard deviation

•
• $σΓ$ =

vortex strength standard deviation

•
• τ =

time delay

•
• τf =

characteristic time scale

•
• τp =

particle response time

•
• $ψk(x)$ =

POD spatial basis function

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