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

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## Abstract

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## 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.

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**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) |

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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