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Statistics, plumes and azimuthally traveling waves in ultimate Taylor-Couette turbulent vortices

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Pith's one-line read Turbulent Taylor-Couette flow, even in the ultimate regime at shear Reynolds numbers up to 350,000, retains large-scale azimuthally traveling waves atop its Taylor vortices, and most angular momentum transport is concentrated in the…

desk verdict Solid transport localisation, plausible but under-verified wave detection; worth refereeing with requests for CPOD diagnostics. read the letter →

arxiv 1908.11796 v1 pith:OBWWUA4O submitted 2019-08-30 physics.flu-dyn

Andreas Froitzheim , Rodrigo Ezeta , Sander G. Huisman , Sebastian Merbold , Chao Sun , Detlef Lohse , Christoph Egbers This is my paper · ORCID
classification physics.flu-dyn
keywords Taylor-CouetteflowturbulentTaylorvorticesangularmomentumtransportNusseltnumberplumesazimuthaltravelingwavescomplexproperorthogonaldecompositionwavyvortex
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper uses planar velocity measurements at 23 heights in a Taylor-Couette apparatus with radius ratio 0.714 to ask how large-scale Taylor rolls shape small-scale turbulence and angular momentum transport up to shear Reynolds numbers of 350,000. It argues that when rolls are prominent, the net angular-momentum flux is not spread through the gap but concentrated in the vortex inflow and outflow, where radial and azimuthal velocity fluctuations are strongly correlated; the outflow near the inner cylinder alone can carry about 60 percent of the transport. Those bursts of flux show up as exponential tails in local Nusselt-number probability distributions and as small-scale plumes with azimuthal extent $k_\phi d \in [10,20]$. By applying a complex proper orthogonal decomposition at the vortex center, the paper also detects azimuthally traveling waves superimposed on the turbulent Taylor vortices in both the classical and ultimate regimes, with azimuthal wavenumber about 9 and phase speeds of the same order as earlier wavy Taylor-vortex measurements. These results link small-scale plume statistics to the large-scale roll structure and suggest that the wavy-vortex phenomenon survives into a regime where the boundary layers are fully turbulent.

What carries the argument

Two tools carry the argument. The first is the net convective angular-momentum flux, expressed as a quasi-Nusselt number $r^2\langle u'_r u'_\phi \rangle / J_{\rm lam}$, evaluated on cylindrical surfaces and at specific vortex positions; this localizes the transport. The second is complex proper orthogonal decomposition (CPOD), in which each velocity-fluctuation field is made complex with a Hilbert transform and then decomposed by singular value decomposition. In CPOD, a mode whose imaginary spatial pattern is azimuthally shifted from its real part represents a wave traveling around the cylinder; the time derivative of the mode's phase gives the wave frequency, and the azimuthal derivative of its spatial phase gives the wavenumber. The first CPOD mode is the one that shows a clean traveling-wave pattern, and the same mode's structure is compared with the known wavy Taylor vortex pattern. Azimuthal energy co-spectra serve as a third piece, identifying the plume scale and showing that the large-scale spectral peak vanishes when the rolls are absent.

What would settle it

A two-dimensional Fourier transform of the first-mode space-time field, with azimuth on one axis and time on the other, would settle the matter: one sharp ridge with frequency over wavenumber equal to the quoted phase speed confirms a single traveling wave, while multiple peaks or a broad smeared ridge would show that the complex-POD phase extraction averaged over several wave components.

Watch

Extended reading notes

Core claim

The central claim is that the organization of fully turbulent Taylor-Couette flow is set by large-scale Taylor rolls whose inflow and outflow bands are the main highways for angular momentum, and that these rolls are not axisymmetric: they carry azimuthally traveling waves. This is established for radius ratio 0.714 with pure inner-cylinder rotation in the classical regime and with counter-rotation at the torque maximum in the ultimate regime. The net convective Nusselt number, built from the correlated radial and azimuthal velocity fluctuations, peaks in the inflow and outflow; its local probability distribution has its most probable value at zero and exponential positive tails that grow with Reynolds number; and pre-multiplied azimuthal co-spectra place the underlying plumes at wavenumbers $k_\phi d \approx 10$ to $20$, strongest where the rolls pull fluid off the walls. The first complex-POD mode captures about 17 percent of the fluctuation energy in the classical case and about 12 percent in the ultimate case, shows real and imaginary parts azimuthally shifted by a quarter wavelength, and yields a wave frequency of 1.80 Hz in the classical regime and 3.67 Hz in the ultimate regime, an azimuthal wavenumber of about 9, and phase speeds of roughly $0.35\Delta\omega$ and $0.11\Delta\omega$. The similarity between these Nusselt-number distributions and those of heat flux in Rayleigh-Benard convection is presented as evidence that the same plume mechanism transports angular momentum in both flows.

Load-bearing premise

The traveling-wave result rests on the assumption that the strongest decomposition mode is a single coherent disturbance circling the cylinder; if that mode instead mixes several disturbances or contains a standing part, the quoted frequency, wavenumber, and speed are not well defined.

Editorial extensions

If this is right

  • At torque-maximum rotation, global torque measurements alone understate the spatial concentration of transport: up to about 60 percent of the net convective flux occurs in the outflow band near the inner cylinder, so models of ultimate Taylor-Couette transport should treat the inflow and outflow bands as the controlling regions.
  • Local Nusselt-number probability distributions peak at zero and develop exponential tails, implying that the time-averaged flux is carried by rare, strong plume events; statistical descriptions of the transport must reproduce this intermittency.
  • Azimuthal traveling waves with wavenumber about 9 appear whenever Taylor rolls are present, in both the classical and ultimate regimes, and not when the rolls are absent, so the large-scale roll state is intrinsically three-dimensional rather than axisymmetric.
  • Because the detected wave speeds are of the same order as the classical wavy Taylor vortex speeds, the wavy-vortex instability is a plausible continuation into the turbulent regime rather than an unrelated large-scale mode.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If confirmed, the detected wave provides a candidate mechanism for setting the spanwise scale of large-scale superstructures in Taylor-Couette and related turbulent flows; checking whether the same $k_\phi \approx 9$ signature appears in direct numerical simulations at comparable parameters would test that idea.
  • The wave was characterized only at the vortex center; a natural extension is to track the phase across several heights to test whether the wave is a rigidly traveling azimuthal modulation or a more complex three-dimensional pattern.
  • Because the ultimate-regime phase speed is slower relative to the imposed shear, the wave may be a passive remnant of the roll structure rather than an active participant in the momentum transport; varying the rotation ratio at fixed shear Reynolds number while monitoring wave amplitude would separate roll strength from Reynolds-number effects.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. This experimental paper reports planar PIV measurements in horizontal planes of a Taylor-Couette facility with radius ratio η=0.714 and aspect ratio Γ=18.3, covering shear Reynolds numbers up to ReS=3.5×10^5 and rotation ratios µ=0 and µ=µmax. The authors validate their velocity statistics and torque results against literature data and DNS, then use the data to characterize the local angular momentum transport, the statistics of the net convective Nusselt number, azimuthal energy co-spectra, and large-scale flow organization. The two headline claims are that the local angular momentum transport is concentrated in the vortex inflow and outflow regions, and that a complex proper orthogonal decomposition reveals azimuthally traveling waves superimposed on turbulent Taylor vortices in both the classical and ultimate regimes, with wavenumber approximately 9 and phase speeds 0.35Δω and 0.11Δω, respectively. The paper also draws an analogy to Rayleigh-Bénard convection based on plume-like small-scale structures and exponential tails in the transport PDFs.

Significance. The transport-location claim is well supported: the meridional maps of the net convective Nusselt number, the joint PDFs of the radial and azimuthal velocity fluctuations, and the inflow/outflow contribution profiles form a coherent picture, and the global Nusselt numbers agree with independent torque measurements and DNS. The comparison of the PDFs to the product-of-Gaussians prediction is a reference normalization rather than a fitted model, so the statistics section is not circular. If the traveling-wave claim holds, the paper would provide experimental evidence of wavy-Taylor-vortex-like azimuthal waves in highly turbulent Taylor-Couette flow, a result of considerable interest to the rotating-turbulence and convection communities, and it would extend the TC-RB analogy. The main weakness is that the traveling-wave conclusion currently rests on the interpretation of a single CPOD mode that captures only 12–17% of the fluctuation energy, and the paper does not provide the mode diagnostics needed to exclude alternative interpretations.

major comments (3)
  1. [§7.2–7.3, Eqs. (7.5)–(7.6)] The traveling-wave claim, which is a headline result, is not yet fully established because the phase-based extraction of fw and k assumes that CPOD1 is a single quasi-monochromatic traveling wave. The paper reports only the averaged values fw=1.80 Hz, fw=3.67 Hz, and k≈9, but does not show that the temporal phase φ1(t) advances linearly in time, that the azimuthal phase profile Φ1(r=0.5,ϕ) is linear, or that the mode's azimuthal Fourier spectrum contains a single dominant wavenumber. This matters because CPOD1 captures only 17% (classical) and 12% (ultimate) of the fluctuation energy, and because §6.1 states that the limited azimuthal field of view prevents resolving the large-scale peak. I request that the authors add (i) the time trace of φ1(t) and its time derivative, (ii) the azimuthal profile of Φ1 at r≈0.5 and its derivative, (iii) the azimuthal wavenumber spectrum of CPOD1, and (iv) a statement of the angular extent of the PIV field of view. Without these diagnostics, the global wavenumber and phase speed are not well defined, and the result should be presented as a local traveling wave within the measured sector.
  2. [§4, Fig. 11] The quantitative statement that the vortex outflow contributes up to approximately 60% of the net convective transport in the inner gap region depends on the operational definition of the in- and outflow regions. The manuscript describes locating the in- and outflow by the extrema of the mean radial velocity profile and trimming the axial interval to one vortex pair by excluding the gray data points in Fig. 3e, but it does not specify how the axial bands assigned to the in- and outflow are chosen or how sensitive the resulting fractions are to that choice and to the number of heights included. Since this 60% figure is presented as a striking result, I ask the authors to provide a sensitivity test with respect to the band width and the axial averaging interval, or to soften the quantitative claim accordingly.
  3. [§7.3] The claim that azimuthally traveling waves are detected in the ultimate regime is currently based on a single flow state, case C4 at ReS=6.68×10^4 and µ=−0.36. The higher-Reynolds-number cases at µmax, namely C5 and C7, are not analyzed with the CPOD procedure, even though these are the cases for which the inflow/outflow transport asymmetry is strongest. To support the general statement 'in the ultimate regime,' the authors should either extend the CPOD analysis to the highest-Re µmax case or explicitly limit the claim to the specific Reynolds number studied in §7.3 and note this limitation in the abstract.
minor comments (8)
  1. [§7.1, Eq. (7.3)] The sentence 'the columns are assigned to the spatial grid points (1:M) and rows to the time signals (1:N)' contradicts the displayed matrix, where rows correspond to spatial points; please correct the dimension description.
  2. [References and §1] There are typographical errors in the references: 'Donelly' should be 'Donnelly' and 'Suanto et al.' appears to be a misspelling of 'Sutanto et al.'; also, the LaTeX control word 'greaterorequalslant' appears in the introduction and should be fixed.
  3. [§5, Fig. 12] The Gaussian-product prediction of Eq. (5.2) is evaluated using ρP from case C8 (ReS=3.51×10^5, µ=0); please state whether the shape comparison is robust to using another pure-inner-rotation case, since the predicted tail slope depends on ρP.
  4. [§6.1] The phrase 'the limited range of the azimuthal coordinate' is never quantified; please state the angular extent of the PIV field of view, because this directly affects the interpretation of the large-scale spectral peak and the CPOD results.
  5. [Figs. 18 and 21] The CPOD mode plots are shown without colorbars or quantitative scales; adding them would help the reader judge the mode amplitude and the relative strength of the radial and azimuthal components.
  6. [§7.3] The text states that the temporal power spectrum for the ultimate-regime case is 'not shown'; please include the spectrum or at least report the peak frequency and its width quantitatively in the text.
  7. [Table 2] The classical-regime phase speed is given as 0.35ω1 in the text and 0.35Δω in Table 2; the two expressions coincide for µ=0, but the notation should be unified to avoid confusion.
  8. [§8] The generalization from Taylor-Couette flow to pipe and channel flow is a speculation; please mark it as a hypothesis rather than a conclusion, or remove it from the summary.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are based on direct PIV measurements, standard complex-POD/SVD processing, and comparisons with external data, not on fitted parameters or self-referential definitions.

full rationale

The paper's two central claims are (i) local angular momentum transport is concentrated in Taylor-vortex in- and outflow regions where radial and azimuthal velocity fluctuations are strongly correlated, and (ii) azimuthally traveling waves are superimposed on turbulent Taylor vortices in both the classical and ultimate regimes. Neither claim reduces to an input by construction. The net convective Nusselt number is defined in Eq. (4.2) as Nu_c,net = r^2/J_lam <u'_φ u'_r>, so the statement that transport is largest where u'_r and u'_φ are highly correlated is an empirical result obtained from measured velocity fields, not a tautology. The in- and outflow contributions are computed from independently identified axial locations of the vortex, and the totals are benchmarked against torque measurements of Lewis & Swinney (1999) and DNS of Ostilla-Mónico et al. (2014c). The PDF analysis uses the Gaussian-product distribution of Eq. (5.2) as a reference shape for normalization; the paper deliberately uses one measured case (C8) as the reference rather than fitting a parameter to each dataset, and the comparison to the measured µ_max PDFs is a diagnostic, not a fit renamed as a prediction. The CPOD analysis in Sec. 7 follows the standard Hilbert-transform-plus-SVD procedure of Eqs. (7.1)-(7.6). The wave frequency f_w = ∂_t φ_1/(2π) and wavenumber k_φ,w = ∂_φ Φ_1 are extracted from the phase of the dominant mode after the decomposition; they are not imposed by the method, and the space-time diagrams and power spectra provide independent evidence of a propagating pattern. The comparison with wavy Taylor vortex flow uses external data from King et al. (1984) and Wang et al. (2005). Self-citations to Froitzheim et al. (2017), van der Veen et al. (2016a), and Harlander et al. (2011) concern the facility, the PIV configuration, and the CPOD algorithm provenance; they are not load-bearing for the physical conclusions. The skeptical concern that mode 1 may not be quasi-monochromatic is a robustness/interpretation issue, not a circularity: the reported wave parameters could be poorly defined if the mode is broadband, but the paper does not derive the wave's existence from the definition of the phase functions alone. No equation in the paper is equivalent to its own input by construction, and no fitted parameter is relabeled as a prediction. Score 0 is therefore appropriate.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central claims rest on standard statistical and spectral methods and on the validity of PIV measurements in the Twente facility. The main non-standard choices are the axial evaluation window and the PDF normalization reference, both of which are disclosed but could affect quantitative comparisons.

free parameters (1)
  • PDF normalization reference case = C8 (ReS=3.51e5, mu=0)
    All Nusselt PDFs are normalized using the theoretical product-of-Gaussians prediction for case C8, which is a data-dependent choice that affects the comparison of tail shapes.
assumptions (4)
  • standard math The discrete Fourier transform, Hilbert transform, and SVD used in the spectral and CPOD analyses are correctly applied to the Cartesian-to-polar interpolated velocity fields.
    These are standard tools; the paper follows cited methodology (Marple 1999, Harlander et al. 2011).
  • domain assumption The flow is statistically stationary and homogeneous in the azimuthal direction, so temporal and azimuthal averages are representative.
    Used throughout for Reynolds averaging and spectral estimation; could be violated if the Taylor rolls drift or oscillate on time scales longer than the measurement window.
  • domain assumption The 23 measured heights with 4 mm spacing resolve the axial structure of the Taylor rolls and allow identification of vortex inflow, center, and outflow.
    The vortex position detection in Section 3 relies on the axial profile of the mean radial velocity; if the axial spacing is too coarse, the in/outflow locations could be misidentified.
  • ad hoc to paper The axial average is restricted to one vortex pair by excluding gray data points so that the interval starts and ends at a vortex center.
    This selection is introduced in Section 3 and is necessary to enforce periodicity, but it assumes the chosen vortex pair is representative and that end effects are negligible.

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Pith. "Pith review of Statistics, plumes and azimuthally traveling waves in ultimate Taylor-Couette turbulent vortices." pith.science (2026). https://pith.science/paper/OBWWUA4O

@misc{pith2026190811796,
  author       = {Pith},
  title        = {Pith review of: Statistics, plumes and azimuthally traveling waves in ultimate Taylor-Couette turbulent vortices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBWWUA4O}},
  note         = {Machine review of arXiv:1908.11796}
}
abstract

In this paper, we experimentally study the influence of large-scale Taylor rolls on the small-scale statistics and the flow organization in fully turbulent Taylor-Couette flow {for Reynolds numbers up to $\text{Re}_S=3\times 10^5$}. The velocity field in the gap confined by coaxial and independently rotating cylinders at a radius ratio of $\eta=0.714$ is measured using planar {particle image velocimetry} in horizontal planes at different cylinder heights. Flow regions with and without prominent Taylor vortices are compared. We show that the local angular momentum transport (expressed in terms of a Nusselt number) mainly takes place in the regions of the vortex in- and outflow, where the radial and azimuthal velocity components are highly correlated. The efficient momentum transfer is reflected in intermittent bursts, which becomes visible in the exponential tails of the probability density functions of the local Nusselt number. In addition, by calculating azimuthal energy co-spectra, small-scale plumes are revealed to be the underlying structure of these bursts. These flow features are very similar to the one observed in Rayleigh-B\'{e}nard convection, which emphasizes the analogies of these both systems. By performing a {complex proper orthogonal decomposition}, we remarkably detect azimuthally traveling waves superimposed on the turbulent Taylor vortices, not only in the classical but also in the ultimate regime. This very large-scale flow pattern{,} which is most pronounced at the axial location of the vortex center, is similar to the well-known wavy Taylor vortex flow{,} which has comparable wave speeds, but much larger azimuthal wave numbers.

Figures

Figures reproduced from arXiv: 1908.11796 by the authors.

Figure 1
Figure 1. Sketch of the ejecting regions bounded by the inner (IC) and outer (OC) cylinders within a Taylor roll: (a) vortex inflow, (b) vortex center, (c) vortex outflow. measurements performed at different heights, which further confirms the connection between small-scale plumes and large-scale vortices. Within the context of both TC and RB flow, many studies in the literature focus on establishing a distinct connection bet… view at source ↗
Figure 2
Figure 2. (a) Sketch of the experimental apparatus (BTTC). Shown is a vertical section. A mirror set at 45 degrees is used so the camera captures the velocity field in the r − ϕ plane. The picture shows an exaggeration of the 23 positions of the laser sheet along ∆ℓ. (b) Temporally averaged flow field in a horizontal plane at the axial height of the vortex center for ReS = 3.51 × 105 and µ = 0. Colors represent the azimuthal … view at source ↗
Figure 3
Figure 3. Representation of the flow states in terms of temporal and azimuthally averaged velocities. The contour plots depict the azimuthal velocity component for (a) ReS = 9.30 × 103 and µ = 0 (C1), (b) 9.30 × 103 and µ = −0.15 (C2), (c) ReS = 2.15 × 105 and µ = 0 (C6) and (d) ReS = 2.14 × 105 and µ = −0.36 (C7). (e) Axial profile of the radial velocity component for ReS = 2.14 × 105 and µ = −0.36 (C7) at ˜r = 0.5 with mark… view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: Radial profiles of the (a) azimuthal and (b) radial velocity component, averaged over space (cylindrical surfaces) and time. Tilde symbols denote normalized quantities. The radial coordinate is normalized as ˜r = (r − r1)/(r2 − r1), where 0 means the location of the in…
Figure 5
Figure 5. Figure 5: Radial profiles of the standard deviation of the (a) azimuthal and (b) radial velocity component, calculated over space (cylindrical surfaces) and time. The profiles are normalized by the shear velocity uS. Legend abbreviations represent C#| µ ReS . figure 3e. This rep…
Figure 6
Figure 6. Figure 6: Probability density functions of the (a,b) azimuthal and (c,d) radial velocity component calculated over space (cylindrical surfaces) and time at ˜r = 0.5 for ReS = 2.1 × 105 . Red dashed lines correspond to µ = 0 (C6) and red solid lines to µ = −0.36 (C7). Dark gray, …
Figure 7
Figure 7. Figure 7: Probability density functions of the azimuthal velocity component calculated over space (cylindrical surfaces) and time for the radial locations (a) ˜r = 0.1, (b) ˜r = 0.3 and (c) r˜ = 0.5. Legend abbreviations represent C#| µ ReS . (a) (b) [PITH_FULL_IMAGE:figures/fu…
Figure 8
Figure 8. Figure 8: Radial profiles of the (a) skewness and (b) kurtosis of the azimuthal velocity component, calculated over space (cylindrical surfaces) and time. Legend abbreviations represent C#| µ ReS . we calculate the PDFs of the azimuthal velocity component for different radial lo…
Figure 9
Figure 9. Figure 9: (a) Azimuthal and time averaged profiles of the total Nusselt number indicated by lines and of the net convective Nusselt number indicated by diamonds (µmax) and circles (µ = 0). The evaluation is restricted to the interval 0.1 6 r˜ 6 0.9 to exclude the boundary layers…
Figure 10
Figure 10. Figure 10: Contour plot of the dimensionless (a) convective and (b) net convective local angular momentum transport in a meridonal plane for ReS = 2.14×105 and µ = −0.36 (C7). Black lines indicate the zero line of the depicted quantities. The color codes are given in the legends…
Figure 11
Figure 11. Figure 11: Radial profiles of the contribution of the net convective momentum transport of the (a) inflow and (b) outflow to the total transport. nz represents the number of heights included into the average over cylindrical surfaces. Legend abbreviations represent C#| µ ReS . t…
Figure 12
Figure 12. Figure 12: Probability density functions of the local net convective angular momentum transport calculated over space (cylindrical surfaces) and time for ˜r = 0.5. (a) PDFs of all investigated flow states are depicted. (b) PDF for ReS = 2.15 × 105 and µ = 0 (C6) with the corresp…
Figure 13
Figure 13. Figure 13: (a) Probability density functions of the local net convective angular momentum transport calculated over space (cylindrical surfaces) and time for ReS = 2.14×105 at µ = −0.36 at different radial positions (C7). (b,c) Same PDFs at ˜r = 0.3 and ˜r = 0.7, respectively wi…
Figure 14
Figure 14. Figure 14: (a) Temporally and azimuthally averaged azimuthal kinetic energy co-spectra evaluated at ˜r = 0.5. Spectra are normalized to cover an area of 1 under their respective curves and by the gap width d. (b) Local scaling exponent γ of the co-spectra for Er,ϕ ∼ k γ ϕ calcul…
Figure 15
Figure 15. Figure 15: Temporally averaged pre-multiplied azimuthal kinetic energy co-spectra in the classical regime for ReS = 9.30×103 and µ = −0.15 (C2) at different radial positions at the axial height of (a) vortex inflow, (b) vortex center and (c) vortex outflow. The region of kϕd ∈ […
Figure 16
Figure 16. Figure 16: Upper row: Temporally averaged pre-multiplied azimuthal kinetic energy co-spectra for ReS = 2.15×105 and µ = 0 (C6) at different radial positions at the axial height of (a) vortex inflow, (b) vortex center and (c) vortex outflow. Region of kϕd ∈ [10, 20] is marked in …
Figure 17
Figure 17. Figure 17: Upper row: Azimuthal two-point autocorrelation function of the fluctuation radial velocity component at the axial height of vortex in- and outflow at (a) ˜r = 0.1 and (b) ˜r = 0.9. Lower row: Azimuthal two-point autocorrelation function of the fluctuation (c) azimutha…
Figure 18
Figure 18. Figure 18: CPOD1 for ReS = 9.32 × 103 and µ = 0 (C1) at the axial height of the vortex center for both velocity components. (a) Real and (c) imaginary part of CPOD1 for the radial velocity component. (b) Real and (d) imaginary part of CPOD1 for the azimuthal velocity component. …
Figure 19
Figure 19. Figure 19: (a) Turbulent energy fraction captured by the CPOD modes for the case ReS = 9.32 × 103 and µ = 0 (C1) at the axial height of the vortex center. Only the first 20 of the total 1500 modes are plotted. (b) Temporal power spectrum of the real part of the temporal coeffici…
Figure 20
Figure 20. Figure 20: Space-time diagram of (a) the reconstructed radial velocity component based on the first CPOD mode and (b) the full field radial velocity component for ReS = 9.32 × 103 and µ = 0 (C1) at the axial height of the vortex center as function of ϕ and t. The radial coordina…
Figure 21
Figure 21. Figure 21: CPOD1 for ReS = 6.68× 104 and µ = −0.36 (C4) at the axial height of the vortex center for both velocity components. (a) Real and (c) imaginary part of CPOD1 for the radial velocity component. (b) Real and (d) imaginary part of CPOD1 for the azimuthal velocity componen…

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Reviewed August 14, 2026 · model on record in the stance chip above.