# Hierarchical Adaptive Eddy-Capturing Approach for Modeling and Simulation of Turbulent Flows

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## Abstract

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## 1. Introduction

## 2. Wavelet-Filtered Navier–Stokes Equations

#### 2.1. Wavelet Threshold Filtering

#### 2.2. Filtered Governing Equations

#### 2.3. Closure Modeling

#### 2.4. Adaptive Wavelet Collocation Method

- (i)
- given a known solution ${\overline{u}}_{i}^{{\scriptscriptstyle >}\u03f5}$ at the current computational grid, say ${\mathcal{G}}_{>}^{t}$, the associated wavelet coefficients are computed through forward wavelet transform;
- (ii)
- the mask $\mathcal{M}$ consisting of the AWC points associated with the retained wavelets (with coefficients for which the moduli are above the prescribed threshold) is created;
- (iii)
- the extended mask ${\mathcal{M}}^{\prime}$ is generated by adding the AWC points corresponding to adjacent wavelets (for which the coefficients can potentially become significant during the next time step); and
- (iv)
- the recursive reconstruction check procedure is performed on the extended mask ${\mathcal{M}}^{\prime}$, ensuring that all the ancestry points, necessary to perform the forward wavelet transform on the updated computational grid ${\mathcal{G}}_{>}^{t+\Delta t}$, are present.

#### 2.5. Homogeneous Turbulence Simulation

## 3. Hierarchical Adaptive Eddy-Capturing Approach

#### Combined Wavelet-Collocation/Volume-Penalization Method

## 4. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

DNS | Direct Numerical Simulation |

LES | Large-Eddy Simulation |

SGS | Sub-Grid Scale |

HIT | Homogeneous Isotropic Turbulence |

WA-LES | Wavelet-based Adaptive Large-Eddy Simulation |

WTF | Wavelet Threshold Filtering |

WA-DNS | Wavelet-based Adaptive Direct Numerical Simulation |

CVS | Coherent Vortex Simulation |

LDKM | Localized Dynamic Kinetic-energy Model |

AWC | Adaptive Wavelet Collocation |

CFL | Courant–Friedrich–Lewy |

FD | Finite Difference |

GDM | Global Dynamic Model |

WA-URANS | Wavelet-based Adaptive Unsteady Reynolds-Averaged Navier–Stokes |

WA-DDES | Wavelet-based Adaptive Delayed Detached Eddy Simulation |

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**Figure 1.**Energy spectra for a wavelet-filtered instantaneous velocity field with different thresholding levels along with the unfiltered solution.

**Figure 2.**Forced homogeneous isotropic turbulence (HIT): time-averaged energy (

**left**) and enstrophy (

**right**) spectra for wavelet-based adaptive large-eddy simulation (WA-LES) with localized dynamic kinetic-energy model (LDKM) and global dynamic model (GDM), compared to pseudo-spectral direct numerical simulation (DNS), wavelet-filtered DNS (FDNS), no-model WA-LES (NOM), and nonadaptive dynamic LES.

**Figure 3.**WA-LES of square-cylinder flow: spanwise vorticity contours (

**left**) and computational mesh (

**right**) in the planes $Z/H=-1.2$, 0, and $1.2$ at a given time instant in the near wake.

**Table 1.**Fraction of active wavelets and captured energy/enstrophy for different thresholding levels.

Threshold $\mathit{\u03f5}$ | Wavelets | Energy | Enstrophy |
---|---|---|---|

0.55 | 0.15% | 95.08% | 60.06% |

0.40 | 0.46% | 98.11% | 77.08% |

0.15 | 5.07% | 99.88% | 97.53% |

0.05 | 12.50% | 99.99% | 99.98% |

**Table 2.**Square-cylinder flow: comparison of WA-DNS and WA-LES results against reference numerical and experimental data.

Study | $\mathit{b}/\mathit{H}$ | $\mathit{\beta}(\%)$ | ${\mathbf{R}\mathit{e}}_{\mathit{H}}$ | ${\overline{\mathit{C}}}_{\mathit{D}}$ | $\mathbf{St}$ |
---|---|---|---|---|---|

WA-DNS [36] | 4 | $6.25$ | 200 | 1.57 | $0.158$ |

DNS [39] | 6 | 5.56 | 200 | 1.39 | $0.157$ |

Experimental [41] | − | − | 200 | − | $0.159$ |

WA-LES [37] | 4 | $6.25$ | 2000 | $2.07$ | $0.131$ |

LES [40] | 4 | $7.69$ | 2000 | $2.6$ | $0.132$ |

Experimental [42] | $9.75$ | 7 | $21,400$ | 2.1 | $0.132$ |

$\mathit{\u03f5}$ | ${\mathit{\delta}}_{\mathbf{min}}$ | ${\overline{\mathit{C}}}_{\mathit{D}}$ | ${\mathit{C}}_{\mathit{D}}^{\prime}$ | ${\mathit{C}}_{\mathit{L}}^{\prime}$ | $\mathbf{St}$ |
---|---|---|---|---|---|

$5\times {10}^{-3}$ | ${2}^{-6}$ | 1.57 | $0.034$ | $0.366$ | $0.158$ |

$1\times {10}^{-3}$ | ${2}^{-6}$ | 1.60 | $0.030$ | $0.368$ | $0.159$ |

$5\times {10}^{-4}$ | ${2}^{-7}$ | 1.61 | $0.033$ | $0.364$ | $0.159$ |

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**MDPI and ACS Style**

De Stefano, G.; Vasilyev, O.V.
Hierarchical Adaptive Eddy-Capturing Approach for Modeling and Simulation of Turbulent Flows. *Fluids* **2021**, *6*, 83.
https://doi.org/10.3390/fluids6020083

**AMA Style**

De Stefano G, Vasilyev OV.
Hierarchical Adaptive Eddy-Capturing Approach for Modeling and Simulation of Turbulent Flows. *Fluids*. 2021; 6(2):83.
https://doi.org/10.3390/fluids6020083

**Chicago/Turabian Style**

De Stefano, Giuliano, and Oleg V. Vasilyev.
2021. "Hierarchical Adaptive Eddy-Capturing Approach for Modeling and Simulation of Turbulent Flows" *Fluids* 6, no. 2: 83.
https://doi.org/10.3390/fluids6020083