# On the Characteristics of the Super-Critical Wake behind a Circular Cylinder

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### Definition of the Problem and Boundary Conditions

## 3. Results

#### 3.1. Coherent Structures

#### 3.2. Statistical Results and Mean Wake Characteristics

#### 3.3. Time–Frequency Analysis

#### 3.4. Phase-Averaged Flow

## 4. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mesh Verification Studies

**Figure A1.**Ratio of the grid size to the Kolmogorov length scale ($h/\eta $) at different locations in the near wake.

## References

- Roshko, A. On the Development of Turbulent Wakes from Vortex Streets; Technical Report NACA TR 1191; California Institute of Technology: Pasadena, CA, USA, 1954. [Google Scholar]
- Roshko, A. Perspectives on Bluff Body Wakes. J. Wind. Eng. Ind. Aerodyn.
**1993**, 49, 79–100. [Google Scholar] [CrossRef] - Achenbach, E. Distribution of local pressure and skin friction around a circular cylinder in cross-flow up to Re = 5×10
^{6}. J. Fluid Mech.**1968**, 34, 625–639. [Google Scholar] [CrossRef] - Bearman, P.W. On vortex shedding from a circular cylinder in the critical Reynolds number regime. J. Fluid Mech.
**1969**, 37, 577–585. [Google Scholar] [CrossRef] - Shih, W.; Wang, C.; Coles, D.; Roshko, A. Experiments on flow past rough circular cylinders at large Reynolds numbers. J. Wind. Eng. Ind. Aerodyn.
**1993**, 49, 351–368. [Google Scholar] [CrossRef] - Lehmkuhl, O.; Rodríguez, I.; Borrell, R.; Chiva, J.; Oliva, A. Unsteady forces on a circular cylinder at critical Reynolds numbers. Phys. Fluids
**2014**, 26, 125110. [Google Scholar] [CrossRef] [Green Version] - Yeon, S.M.; Yang, J.; Stern, F. Large eddy simulation of the flow past a circular cylinder at sub- to super-critical Reynolds numbers. Appl. Ocean. Res.
**2016**, 59, 663–675. [Google Scholar] [CrossRef] [Green Version] - Cheng, W.; Pullin, D.I.; Samtaney, R.; Zhang, W.; Gao, W. Large-eddy simulation of flow over a cylinder with Re
_{D}from 3.9 × 10^{3}to 8.5 × 10^{5}: A skin-friction perspective. J. Fluid Mech.**2017**, 820, 121–158. [Google Scholar] [CrossRef] - Ahmadi, M.H.; Yang, Z. Large eddy simulation of the flow past a circular cylinder at super-critical reynolds numbers. Proc. Asme Turbo Expo
**2020**, 2C-2020, 663–675. [Google Scholar] [CrossRef] - Cantwell, B.; Coles, D. An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech.
**1983**, 136, 321–374. [Google Scholar] [CrossRef] [Green Version] - Unal, M.F.; Rockwell, D. On vortex formation from a cylinder. Part 1. The initial instability. J. Fluid Mech.
**1988**, 190, 491–512. [Google Scholar] [CrossRef] - Norberg, C. LDV measurements in the near wake of a circular cylinder. In Proceedings of the ASME Conference on Advances in the Understanding of Bluff Body Wakes and Vortex Induced Vibration, Washington, DC, USA, 1 June 1998. [Google Scholar]
- Ma, X.; Karamanos, G.; Karniadakis, G. Dynamics and low-dimensionality of a turbulent wake. J. Fluid Mech.
**2000**, 410, 29–65. [Google Scholar] [CrossRef] [Green Version] - Dong, S.; Karniadakis, G.E.; Ekmekci, A.; Rockwell, D. A combined direct numerical simulation-particle image velocimetry study of the turbulent near wake. J. Fluid Mech.
**2006**, 569, 185. [Google Scholar] [CrossRef] [Green Version] - Lehmkuhl, O.; Rodríguez, I.; Borrell, R.; Oliva, A. Low-frequency unsteadiness in the vortex formation region of a circular cylinder. Phys. Fluids
**2013**, 25, 085109. [Google Scholar] [CrossRef] [Green Version] - Rodríguez, I.; Lehmkuhl, O.; Chiva, J.; Borrell, R.; Oliva, A. On the flow past a circular cylinder from critical to super-critical Reynolds numbers: Wake topology and vortex shedding. Int. J. Heat Fluid Flow
**2015**. [Google Scholar] [CrossRef] [Green Version] - Schewe, G. On the force fluctuations acting on a circular cylinder in crossflow from subcritical up to transcritical Reynolds numbers. J. Fluid Mech.
**1983**, 133, 265–285. [Google Scholar] [CrossRef] - Roshko, A. Experiments on the flow past a circular cylinder at very high Reynolds number. J. Fluid Mech.
**1961**, 10, 345–356. [Google Scholar] [CrossRef] [Green Version] - Zdravkovich, M.M. Conceptual Overview of Laminar and Turbulent Flows Past Smooth and Rough Circular Cylinders. J. Wind. Eng. Ind. Aerodyn.
**1990**, 33, 53–62. [Google Scholar] [CrossRef] - Vreman, A.W. An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications. Phys. Fluids
**2004**, 16, 3670–3681. [Google Scholar] [CrossRef] - Vázquez, M.; Houzeaux, G.; Koric, S.; Artigues, A.; Aguado-Sierra, J.; Arís, R.; Mira, D.; Calmet, H.; Cucchietti, F.; Owen, H.; et al. Alya: Multiphysics engineering simulation toward exascale. J. Comput. Sci.
**2016**, 14, 15–27. [Google Scholar] [CrossRef] [Green Version] - Lehmkuhl, O.; Houzeaux, G.; Owen, H.; Chrysokentis, G.; Rodriguez, I. A low-dissipation finite element scheme for scale resolving simulations of turbulent flows. J. Comput. Phys.
**2019**, 390, 51–65. [Google Scholar] [CrossRef] - Capuano, F.; Coppola, G.; Rández, L.; de Luca, L. Explicit Runge-Kutta schemes for incompressible flow with improved energy-conservation properties. J. Comput. Phys.
**2017**, 328, 86–94. [Google Scholar] [CrossRef] - Trias, F.X.; Lehmkuhl, O. A self-adaptive strategy for the time integration of Navier-Stokes equations. Numer. Heat Transf. Part B
**2011**, 60, 116–134. [Google Scholar] [CrossRef] - Pastrana, D.; Cajas, J.C.; Lehmkuhl, O.; Rodríguez, I.; Houzeaux, G. Large-eddy simulations of the vortex-induced vibration of a low mass ratio two-degree-of-freedom circular cylinder at subcritical Reynolds numbers. Comput. Fluids
**2018**, 173, 118–132. [Google Scholar] [CrossRef] [Green Version] - Rodriguez, I.; Lehmkuhl, O.; Soria, M.; Gomez, S.; Domınguez-Pumar, M.; Kowalski, L. Fluid dynamics and heat transfer in the wake of a sphere. Int. J. Heat Fluid Flow
**2019**, 76, 141–153. [Google Scholar] [CrossRef] - Rodriguez, I.; Lehmkuhl, O.; Borrell, R. Effects of the actuation on the boundary layer of an airfoil at Reynolds number Re = 60000. Flow Turbul. Combust.
**2020**, 390, 51–65. [Google Scholar] [CrossRef] - Rodriguez, I.; Lehmkuhl, O.; Soria, M. On the effects of the free-stream turbulence on the heat transfer from a sphere. Int. J. Heat Mass Transf.
**2021**, 164, 120579. [Google Scholar] [CrossRef] - Schewe, G. Sensitivity of transition phenomena to small perturbations in flow round a circular cylinder. J. Fluid Mech.
**1986**, 172, 33–46. [Google Scholar] [CrossRef] - Jeong, J.; Hussain, F. On the identification of a vortex. J. Fluids Mech.
**1995**, 285. [Google Scholar] [CrossRef] - Achenbach, E.; Heinecke, E. On vortex shedding from smooth and rough cylinders in the range of Reynolds numbers 6e3 to 5e6. J. Fluid Mech.
**1981**, 109, 239–251. [Google Scholar] [CrossRef] - Williamson, C.H.K. Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech.
**1996**, 28, 477–539. [Google Scholar] [CrossRef] - Spitzer, R. Measurements of Unsteady Pressures and Wake Fluctuations for Flow over a Cylinder at Supercritical Reynolds Number. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, USA, 1965. [Google Scholar]
- Aljure, D.E.; Rodríguez, I.; Lehmkuhl, O.; Pérez-Segarra, C.D.; Oliva, A. Influence of rotation on the flow over a cylinder at Re = 5000. Int. J. Heat Fluid Flow
**2015**, 55, 76–90. [Google Scholar] [CrossRef] [Green Version] - Tani, I. Low-speed flows involving bubble separations. Prog. Aerosp. Sci.
**1964**, 5, 70–103. [Google Scholar] [CrossRef] - Palkin, E.; Mullyadzhanov, R.; Hadžiabdić, M.; Hanjalić, K. Scrutinizing URANS in Shedding Flows: The Case of Cylinder in Cross-Flow in the Subcritical Regime. Flow Turbul. Combust.
**2016**, 97, 1017–1046. [Google Scholar] [CrossRef] - Rodriguez, I.; Lehmkuhl, O.; Piomelli, U.; Chiva, J.; Borrell, R.; Oliva, A. LES-based Study of the Roughness Effects on the Wake of a Circular Cylinder from Subcritical to Transcritical Reynolds Numbers. Flow Turbul. Combust.
**2017**, 99, 729–763. [Google Scholar] [CrossRef] - Olhede, S.C.; Walden, A.T. Generalized Morse wavelets. IEEE Trans. Signal Process.
**2002**, 50, 2661–2670. [Google Scholar] [CrossRef] [Green Version] - Farge, M. Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech.
**1992**, 24, 395–457. [Google Scholar] [CrossRef] - Bloor, M. The transition to turbulence in the wake of a circular cylinder. J. Fluid Mech.
**1964**, 19, 290–304. [Google Scholar] [CrossRef] - Klebanoff, P.S.; Tidstrom, K.D.; Sargent, L.M. The three-dimensional nature of boundary-layer instability. J. Fluid Mech.
**1962**, 12, 1–34. [Google Scholar] [CrossRef] [Green Version] - Hall, P. On the stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech.
**1984**, 146, 347–367. [Google Scholar] [CrossRef] [Green Version] - Saric, W.S. Gortler vortices. Annu. Rev. Fluid Mech.
**1994**, 26, 379–409. [Google Scholar] [CrossRef] - Karp, M.; Hack, P. Transition to turbulence over convex surfaces. J. Fluid Mech.
**2018**, 855, 1208–1237. [Google Scholar] [CrossRef] - Pastrana, D.; Rodriguez, I.; Cajas, J.; Lehmkuhl, O.; Houzeaux, G. On the formation of Taylor-Görtler structures in the vortex induced vibration phenomenon. Int. J. Heat Fluid Flow
**2020**, 83, 108573. [Google Scholar] [CrossRef] - Reynolds, W.C.; Hussain, A.K.M.F. The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech.
**1972**, 54, 263–288. [Google Scholar] [CrossRef] - Braza, M.; Perrin, R.; Hoarau, Y. Turbulence properties in the cylinder wake at high Reynolds numbers. J. Fluids Struct.
**2006**, 22, 757–771. [Google Scholar] [CrossRef] [Green Version] - Perrin, R.; Cid, E.; Cazin, S.; Sevrain, A.; Braza, M.; Moradei, F.; Harran, G. Phase-averaged measurements of the turbulence properties in the near wake of a circular cylinder at high Reynolds number by 2C-PIV and 3C-PIV. Exp. Fluids
**2007**, 42, 93–109. [Google Scholar] [CrossRef] - Rai, M.M. Flow physics in the turbulent near wake of a flat plate. J. Fluid Mech.
**2013**, 724, 704–733. [Google Scholar] [CrossRef] - Piomelli, U.; Chasnov, J.R. Transition and Turbulence Modelling; Chapter Large-eddy Simulations and Applications; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996; pp. 269–331. [Google Scholar]
- Pope, S. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]

**Figure 1.**Vortical structures identified by means of Q iso-contours colored by the velocity magnitude.

**Figure 3.**Instantaneous vorticity magnitude superimposed with pressure coefficient iso-contours (red) in the vortex formation zone.

**Figure 6.**(

**a**) Stream-wise velocity and (

**b**) its fluctuations in the wake centerline. Comparison with sub-critical and super-critical results. The red solid line present the results; (black dash-dot line) $Re=7.2\times {10}^{5}$ [16]; (blue line) $Re=3900$ [15]; (purple circles) $Re=5000$ [34]; (green squares) $Re=1.4\times {10}^{5}$ [10].

**Figure 7.**Near wake mean field. (

**a**) Streamlines; non-dimensional iso-contours of (

**b**) pressure coefficient ($-2.2\le \langle Cp\rangle \le 1:0.36$); (

**c**) stream-wise velocity ($-0.25\le \langle {u}_{1}/{U}_{ref}\rangle \le 1.8:\phantom{\rule{0.222222em}{0ex}}0.205$); (

**d**) cross-stream velocity ($-0.9\le \langle {u}_{2}/{U}_{ref}\rangle \le 0.9:\phantom{\rule{0.222222em}{0ex}}0.2$).

**Figure 8.**Second order turbulent statistics. (

**a**) Stream-wise normal stresses $\langle \overline{{u}_{1}^{\prime}{u}_{1}^{\prime}}\rangle /{U}_{ref}^{2}$, (

**b**) cross-stream wise normal $\langle \overline{{u}_{2}^{\prime}{u}_{2}^{\prime}}\rangle /{U}_{ref}^{2}$, (

**c**) shear stresses $\langle \overline{{u}_{1}^{\prime}{u}_{2}^{\prime}}\rangle /{U}_{ref}^{2}$, (

**d**) turbulent kinetic energy $\langle k\rangle /{U}_{ref}^{2}$.

**Figure 9.**(

**a**) Mean turbulent kinetic energy production ${P}_{k}$. (

**b**–

**d**) Production for the normal and shear stresses $\langle {u}_{1}^{\prime}{u}_{1}^{\prime}\rangle /{U}_{ref}^{2}$, $\langle {u}_{2}^{\prime}{u}_{2}^{\prime}\rangle /{U}_{ref}^{2}$ and $\langle {u}_{1}^{\prime}{u}_{2}^{\prime}\rangle /{U}_{ref}^{2}$. (

**b**) ${P}_{11}$, (

**c**) ${P}_{22}$, (

**d**) ${P}_{12}$.

**Figure 10.**Production of turbulent kinetic energy in the wake centerline and the contribution of the different terms in Equations in Section 3.1.

**Figure 12.**Energy spectra at different locations. (

**a**) for stream-wise velocity fluctuations, (

**b**) for cross-stream wise velocity fluctuations.

**Figure 13.**{Time–frequency} analysis using the continuous wavelet transform of the streamwise velocity component at different locations. (

**a**) Cylinder boundary layer at ${70}^{\xb0}$ from the front stagnation point; (

**b**) cylinder boundary layer at ${90}^{\xb0}$ from the front stagnation point; (

**c**) in the separated shear layer at ($x/D,y/D)\sim (0.6,0.33)$; (

**d**) at the wake centerline at ($x/D,y/D)\sim (1.3,0)$.

**Figure 14.**(

**a**) Reference signal used as periodic oscillator. (

**b**) Power spectrum of the reference oscillator signal.

**Figure 15.**Phase-averaged streamlines. (

**a**) $\varphi =0$, (

**b**) $\varphi =\pi /4$, (

**c**) $\varphi =\pi /2$, (

**d**) $\varphi =3\pi /4$. The location of the saddle point is also marked.

**Figure 16.**Phase averaged spanwise vorticity superimposed with pressure contours in the wake at constant phase. (

**a**) $\varphi =0$, (

**b**) $\varphi =\pi /4$, (

**c**) $\varphi =\pi /2$, (

**d**) $\varphi =3\pi /4$.

**Figure 17.**Contours of the random component of the normal and shear stresses at constant phase, $\varphi =0$, plotted over pressure coefficient contours (in black) (

**a**) $\langle {u}_{1}^{\prime}{u}_{1}^{\prime}\rangle /{U}_{ref}^{2}$, (

**b**) $\langle {u}_{2}^{\prime}{u}_{2}^{\prime}\rangle /{U}_{ref}^{2}$, (

**c**) $\langle {u}_{3}^{\prime}{u}_{3}^{\prime}\rangle /{U}_{ref}^{2}$, (

**d**) $\langle {u}_{1}^{\prime}{u}_{2}^{\prime}\rangle /{U}_{ref}^{2}$.

**Table 1.**Statistics in the near-wake: maximum normal and shear Reynolds stresses $\langle {u}_{1}^{\prime}{u}_{1}^{\prime}\rangle /{U}_{ref}^{2}$, $\langle {u}_{2}^{\prime}{u}_{2}^{\prime}\rangle /{U}_{ref}^{2}$,$\langle {u}_{1}^{\prime}{u}_{2}^{\prime}\rangle /{U}_{ref}^{2}$, maximum turbulent kinetic energy $\langle k\rangle /{U}_{ref}^{2}$ and minimum stream-wise velocity in the wake centerline. The positions of these extrema are also given. † [10], ‡ [34], § [15].

$\mathit{Re}$ | ||||
---|---|---|---|---|

$\mathbf{7}.\mathbf{2}\times {\mathbf{10}}^{\mathbf{5}}$ | $\mathbf{1.4}\times {\mathbf{10}}^{\mathbf{5}}$${}^{\u2020}$ | 5000 ${}^{\u2021}$ | 3900 ^{§} | |

${\langle {u}_{1}^{\prime}{u}_{1}^{\prime}\rangle}_{max}/{U}_{ref}^{2}$ | 0.102 | 0.22 | 0.239 | 0.237 |

$({x}_{1}/D,{x}_{2}/D)$ | (1.033, ±0.149) | (1.587, ±0.297) | (1.576, ±0.310) | |

${\langle {u}_{2}^{\prime}{u}_{2}^{\prime}\rangle}_{max}/{U}_{ref}^{2}$ | 0.201 | 0.43 | 0.467 | 0.468 |

$({x}_{1}/D,{x}_{2}/D)$ | (1.216, 0) | (1.992, 0) | (2.000, 0) | |

${\langle {u}_{1}^{\prime}{u}_{2}^{\prime}\rangle}_{min}/{U}_{ref}^{2}$ | ±0.06 | ±0.19 | ±0.128 | ±0.125 |

$({x}_{1}/D,{x}_{2}/D)$ | (1.161, ±0.138) | (1.901, ±0.422) | (1.941, ±0.391) | |

${\langle k\rangle}_{max}/{U}_{ref}^{2}$ | 0.138 | - | 0.331 | 0.335 |

$({x}_{1}/D,{x}_{2}/D)$ | (1.106, ±0.109) | (1.764, ±0.202) | (1.775, ±0.216) | |

${\langle {u}_{1}\rangle}_{min}/{U}_{ref}$ | −0.255 | - | −0.284 | −0.261 |

$({x}_{1}/D,{x}_{2}/D)$ | (0.845, 0) | (1.399, 0) | (1.396, 0) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rodriguez, I.; Lehmkuhl, O.
On the Characteristics of the Super-Critical Wake behind a Circular Cylinder. *Fluids* **2021**, *6*, 396.
https://doi.org/10.3390/fluids6110396

**AMA Style**

Rodriguez I, Lehmkuhl O.
On the Characteristics of the Super-Critical Wake behind a Circular Cylinder. *Fluids*. 2021; 6(11):396.
https://doi.org/10.3390/fluids6110396

**Chicago/Turabian Style**

Rodriguez, Ivette, and Oriol Lehmkuhl.
2021. "On the Characteristics of the Super-Critical Wake behind a Circular Cylinder" *Fluids* 6, no. 11: 396.
https://doi.org/10.3390/fluids6110396