# A Hybrid Large Eddy Simulation Algorithm Based on the Implicit Domain Decomposition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Near-Wall Model

#### 2.1. Slip Boundary Conditions

#### 2.2. Turbulent Viscosity Estimation

## 3. Implicit Near-Wall Domain Decomposition

#### 3.1. Algorithm

- Initialize the coarse grid for LES.
- Initialize the interface boundary ${y}^{*}$ and a sub-grid for RANS.
- Initialize the flow fields for both the unresolved LES grid and RANS sub-grid.
- Compute the turbulent viscosity.
- Compute ${f}_{1}$ and ${\tilde{f}}_{2}$ (Equation (3)) based on the chosen turbulent viscosity model of Section 2.2.
- Compute the coefficients ${f}_{w1}$ and ${f}_{w2}$ (Equation (5)) to impose the slip boundary condition at the wall.
- Solve the LES governing equations on the coarse grid with the slip boundary condition (Equation (4)).
- Transfer the LES streamwise velocity and turbulent kinetic energy at ${y}^{*}$ (i.e., ${u}^{*}$ and ${k}_{tot}^{*}$) to the embedded RANS model.
- Compute the wall shear stress in the inner region using ${u}^{*}$ value (Equation (6)).
- Compute the RANS velocity solution in the inner region (Equation (7)) with the updated ${\tau}_{w}$.
- Compute the turbulent kinetic energy (Equation (14)), if needed.
- Repeat the procedure from step 4.

#### 3.2. Discussion

## 4. Test Cases

#### 4.1. Setup

#### 4.2. Simulation Results

#### 4.3. Effect of Eddy Viscosity Model

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Roman symbols | |

t | time (s) |

${u}_{\tau}$ | friction velocity (m/s) |

k | turbulent kinetic energy (m^{2}/s^{2}) |

$Re$ | Reynolds number (-) |

u | flow velocity (m/s) |

${C}_{\mu}$ | model constant = 0.09 (-) |

x | streamwise direction (-) |

y | wall-normal direction (-) |

z | spanwise direction (-) |

D | damping function (-) |

P | turbulence production term |

Greek symbols | |

${\tau}_{w}$ | wall shear stress (N/m^{2}) |

$\rho $ | density (kg/m^{3}) |

$\mu $ | dynamic viscosity (Pa·s) |

$\nu $ | kinematic viscosity (Pa·s) |

$\u03f5$ | turbulent kinetic energy dissipation (m^{2}/s^{3}) |

$\omega $ | specific dissipation rate (1/s) |

$\alpha $ | model constant (-) |

$\beta $ | model constant (-) |

$\sigma $ | model constant (-) |

Subscripts and superscripts | |

res | resolved |

sgs | subgrid scale |

u | velocity |

l | laminar |

T | turbulent |

int | interface |

* | interface location |

w | wall |

$\tau $ | friction-related parameter |

Abbreviations | |

LES | large eddy simulation |

WMLES | wall-modeled LES |

DNS | direct numerical simulation |

TBLE | thin boundary layer equation |

RANS | Reynolds-averaged Navier–Stokes |

WSM | wall-stress model |

LLM | log–layer mismatch |

DES | detached eddy simulation |

NDD | near-wall domain decomposition |

BC | boundary condition |

IBC | interface boundary condition |

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**Figure 2.**Mean streamwise velocity for C950 compared to DNS [48].

**Figure 3.**Reynolds normal stress intensities for C950. lines: DNS [48]; circles: INDD.

**Figure 4.**Reynolds shear stress for C950. Red line: DNS [48]; blue circles: INDD.

**Figure 8.**Reynolds shear stress for C2000. Red line: DNS [48]; blue circles: INDD.

**Figure 9.**Mean streamwise velocity for C2000. Comparison of INDD (blue triangles) and conventional hybrid methods without a blending function (green squares).

**Figure 12.**Reynolds shear stress intensities for C4200. Red line: DNS [50]; blue circles: INDD.

Case | ${\mathbf{Re}}_{\mathit{\tau}}$ | ${\mathit{L}}_{\mathit{x}}\times {\mathit{L}}_{\mathit{y}}\times {\mathit{L}}_{\mathit{z}}$ | Resolution | $\Delta {\mathit{x}}^{+}$ | $\Delta {\mathit{y}}^{+}$ | $\Delta {\mathit{z}}^{+}$ | $\Delta {\mathit{y}}_{\mathit{j}+1}/\Delta {\mathit{y}}_{\mathit{j}}$ | ${\mathit{y}}^{*}/\mathit{h}$ | ${{\mathit{y}}^{*}}^{+}$ |
---|---|---|---|---|---|---|---|---|---|

C950 | 950 | $2\pi h\times 2h\times \pi h$ | $40\times 64\times 36$ | 148 | $1.57\to 126$ | 83 | 1.15 | 0.063 | 60 |

C2000 | 2000 | $2\pi h\times 2h\times \pi h$ | $40\times 72\times 36$ | 312 | $1.78\to 267$ | 174 | 1.15 | 0.065 | 129 |

C4200 | 4200 | $3\pi h\times 2h\times \pi h$ | $60\times 84\times 42$ | 659 | $1.82\to 547$ | 314 | 1.15 | 0.0476 | 200 |

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**MDPI and ACS Style**

E. Fard, A.; Utyuzhnikov, S.
A Hybrid Large Eddy Simulation Algorithm Based on the Implicit Domain Decomposition. *Mathematics* **2023**, *11*, 4340.
https://doi.org/10.3390/math11204340

**AMA Style**

E. Fard A, Utyuzhnikov S.
A Hybrid Large Eddy Simulation Algorithm Based on the Implicit Domain Decomposition. *Mathematics*. 2023; 11(20):4340.
https://doi.org/10.3390/math11204340

**Chicago/Turabian Style**

E. Fard, Amir, and Sergey Utyuzhnikov.
2023. "A Hybrid Large Eddy Simulation Algorithm Based on the Implicit Domain Decomposition" *Mathematics* 11, no. 20: 4340.
https://doi.org/10.3390/math11204340