# Wavelet-Based Adaptive Eddy-Resolving Methods for Modeling and Simulation of Complex Wall-Bounded Compressible Turbulent Flows

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## Abstract

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## 1. Introduction

## 2. Wavelet-Favre Filtered/Averaged Navier–Stokes Equations

## 3. Adaptive Wavelet Collocation Method

## 4. Wall-Resolving Approach

## 5. Wall-Modeling Approaches

#### 5.1. Wavelet-Based Adaptive Unsteady Reynolds-Averaged Navier–Stokes

#### 5.2. Wavelet-Based Adaptive Delayed Detached Eddy Simulation

#### 5.3. Wavelet-Based Adaptive Wall-Modeled Large Eddy Simulation

## 6. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DNS | Direct Numerical Simulation |

WTF | Wavelet Thresholding Filter |

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

CVS | Coherent Vortex Simulation |

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

SGS | SubGrid-Scale |

RANS | Reynolds-Averaged Navier–Stokes |

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

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

CFL | Courant-Friedrichs-Lewy |

WA-WMLES | Wavelet-based Adaptive Wall Modeled Large Eddy Simulation |

AWC | Adaptive Wavelet Collocation |

A-AWC | Anisotropic Adaptive Wavelet Collocation |

AMD | Anisotropic Minimum-Dissipation |

BL | Boundary Layer |

SA | Spalart-Allmaras |

AMD | Anisotropic Minimum-Dissipation |

EL | Exchange Location |

ODE | Ordinary Differential Equation |

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**Figure 1.**WA-LES of supersonic channel flow: instantaneous adapted mesh, colored by normalized temperature field, at four different channel cross-sections.

**Figure 2.**WA-LES of supersonic channel flow: profiles of normalized mean density, temperature, and streamwise momentum, compared to reference DNS [57].

**Figure 3.**WA-LES of supersonic channel flow: profiles of turbulent shear stress and mean square temperature fluctuation, compared to reference DNS [57].

**Figure 4.**WA-URANS of NASA hump flow: contours of mean streamwise velocity, normalized by bulk flow reference velocity, around the separation bubble.

**Figure 5.**WA-URANS of NASA hump flow: adaptive mesh around the separation bubble. The colored contours represent the adaptive mesh resolution levels.

**Figure 7.**WA-URANS of Bachalo-Johnson flow: contours of mean streamwise velocity normalized by free-stream speed of sound.

**Figure 8.**WA-URANS of Bachalo-Johnson flow: adaptive mesh. The colored contours represent the adaptive mesh resolution levels.

**Figure 11.**WA-DDES of supersonic channel flow: turbulent shear stress profiles predicted without (

**left**) and with (

**right**) split adaptation, compared to reference DNS [64].

**Figure 12.**WA-DDES of periodic-hill channel: mean skin friction coefficient on the lower wall, compared to reference DNS [65].

**Figure 13.**WA-WMLES: schematic depicting the LES mesh, with exchange layer (red curve), closest LES mesh points (blue circles), line segments for RANS ODEs integration (red dashed lines), and corresponding endpoints (black and red circles).

**Figure 14.**WA-WMLES of NASA hump flow: instantaneous Q-criterion isosurfaces colored by streamwise momentum.

**Figure 15.**WA-WMLES of NASA hump flow: instantaneous skin friction coefficient on the streamwise-spanwise plane.

**Figure 16.**WA-WMLES of NASA hump flow: time and spanwise averaged skin friction (

**left**) and pressure (

**right**) coefficients over the wall.

**Table 1.**WA-LES of supersonic channel flow: mesh resolution and mean flow results compared to conventional LES and DNS.

Method | Resolution | $\mathsf{\Delta}{\mathit{x}}_{1}^{+}$ | $\mathsf{\Delta}{\mathit{x}}_{2,\mathbf{wall}}^{+}$ | $\mathsf{\Delta}{\mathit{x}}_{3}^{+}$ | ${\mathbf{Re}}_{\mathit{\tau}}$ | ${\mathbf{Ma}}_{\mathit{\tau}}$ | $-{\mathit{B}}_{\mathit{q}}$ | ${\mathit{\rho}}_{\mathit{c}}/{\mathit{\rho}}_{\mathbf{wall}}$ | ${\mathit{T}}_{\mathit{c}}/{\mathit{T}}_{\mathbf{wall}}$ | ${\mathit{u}}_{\mathit{c}}/{\mathit{U}}_{\mathbf{bulk}}$ |
---|---|---|---|---|---|---|---|---|---|---|

WA-LES | $129\times 87\times 86$ | $21.0$ | $0.12$ | $10.5$ | 216 | $0.079$ | $0.045$ | $0.718$ | $1.39$ | $1.17$ |

LES [58] | $128\times 65\times 81$ | $21.5$ | $0.11$ | $11.3$ | 219 | $0.079$ | $0.05$ | − | $1.40$ | − |

DNS [57] | $144\times 119\times 80$ | 19 | $0.1$ | 12 | 222 | $0.082$ | $0.049$ | $0.723$ | $1.38$ | $1.18$ |

**Table 2.**WA-URANS of flat plate BL: dependence study on WTF threshold for mesh size and skin friction, compared to NASA CFL3D using the SA model.

Case | $\mathit{\u03f5}=4.0\times {10}^{-3}$ | $\mathit{\u03f5}=2.0\times {10}^{-3}$ | $\mathit{\u03f5}=1.0\times {10}^{-3}$ | $\mathit{\u03f5}=5.0\times {10}^{-4}$ | CFL3D |
---|---|---|---|---|---|

Mesh size | 6.6 K | 7.5 K | 9.3 K | 11.0 K | 209.8 K |

${C}_{f}$ | 0.0026879 | 0.0026900 | 0.0026972 | 0.0026984 | 0.0027056 |

Error | 0.654% | 0.576% | 0.310% | 0.266% | – |

**Table 3.**Grid resolution for the NASA hump flow simulations, in both wall and chord units. The wall unit is normalised by the kinematic viscosity over the friction velocity. All data are evaluated at the inflow flat-plate turbulent boundary layer, while the subscript 1 denotes the first wall-normal mesh spacing.

Grid | WA-WMLES | WMLES [67] | WMLES [68] | WMLES [69] | WRLES [70] |
---|---|---|---|---|---|

$\Delta {x}^{+}$ | 90 | 600 | 300 | 360 | 25 |

$\Delta x/c$ | (3.8 ∼ 38) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | (12 ∼ 180) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | (12 ∼ 200) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | (15 ∼ 100) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 7.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

$\Delta {z}^{+}$ | 40 | 100 | 120 | 180 | 12.5 |

$\Delta z/c$ | 1.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $3.1\times {10}^{-3}$ | $3.8\times {10}^{-3}$ | $5.0\times {10}^{-3}$ | 3.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

$\Delta {y}_{1}^{+}$ | 13 | 20 | 50 | 36 | 0.8 |

$\Delta {y}_{1}/c$ | $3.6\times {10}^{-4}$ | $5.5\times {10}^{-4}$ | $13\times {10}^{-4}$ | (2.0 ∼ 33) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | $2.2\times {10}^{-5}$ |

Span size/c | 0.3 | 0.4 | 0.6 | 0.3 | 0.4 |

Total size/million | 7.5 | 9.4 | 12.9 | 11 | 420 |

**Table 4.**Comparison of separation and reattachment locations for NASA hump flow. Note that all WMLES cases use the equilibrium wall model.

Case | Separation ($\mathit{x}/\mathit{c}$) | Reattachment ($\mathit{x}/\mathit{c}$) | Bubble Length ($\mathsf{\Delta}\mathit{x}/\mathit{c}$) | Error in Bubble |
---|---|---|---|---|

WA-WMLES | 0.677 | 1.138 | 0.461 | 6.0% |

WMLES [68] | 0.680 | 1.084 | 0.404 | −7.1% |

WMLES [69] | 0.655 | 1.105 | 0.450 | 3.4% |

WRLES [70] | 0.641 | 1.09 | 0.449 | 3.2% |

Experiment [71] | 0.665 ($\pm 0.005$) | 1.10 ($\pm 0.005$) | 0.435 | – |

**Table 5.**Summary of wavevet-based adaptive eddy-resolving methods for wall-bounded compressible turbulent flows.

Method | Model Form Adaptation | Wall Effect | Mesh Size | Computational Cost | Fidelity | |
---|---|---|---|---|---|---|

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**MDPI and ACS Style**

Ge, X.; De Stefano, G.; Hussaini, M.Y.; Vasilyev, O.V.
Wavelet-Based Adaptive Eddy-Resolving Methods for Modeling and Simulation of Complex Wall-Bounded Compressible Turbulent Flows. *Fluids* **2021**, *6*, 331.
https://doi.org/10.3390/fluids6090331

**AMA Style**

Ge X, De Stefano G, Hussaini MY, Vasilyev OV.
Wavelet-Based Adaptive Eddy-Resolving Methods for Modeling and Simulation of Complex Wall-Bounded Compressible Turbulent Flows. *Fluids*. 2021; 6(9):331.
https://doi.org/10.3390/fluids6090331

**Chicago/Turabian Style**

Ge, Xuan, Giuliano De Stefano, M. Yousuff Hussaini, and Oleg V. Vasilyev.
2021. "Wavelet-Based Adaptive Eddy-Resolving Methods for Modeling and Simulation of Complex Wall-Bounded Compressible Turbulent Flows" *Fluids* 6, no. 9: 331.
https://doi.org/10.3390/fluids6090331