# Reynolds Stress Perturbation for Epistemic Uncertainty Quantification of RANS Models Implemented in OpenFOAM

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Epistemic Uncertainty

#### 2.2. Decomposition of the Reynolds Stress Tensor

- ${\Lambda}_{nl}$ is a diagonal tensor containing the eigenvalues of the anisotropy tensor in an order such that ${\lambda}_{1}>{\lambda}_{2}>{\lambda}_{3}$,
- ${v}_{ij}$ is a tensor containing the eigenvectors of the anisotropy tensor in the same order as the eigenvalues.

#### 2.3. Perturbation of the Reynolds Stress Tensor

- ${v}^{\ast}=v+\Delta v$ is the perturbation on the orientation of the Reynolds stresses,
- ${\Lambda}^{\ast}=\Lambda +\Delta \Lambda $ is the perturbation on the anisotropy of the Reynolds stresses,
- and ${k}^{\ast}=[k/{n}_{k},{n}_{k}k]$, is the amplitude of the perturbation of the turbulent kinetic energy. It is established as a range with a minimum and a maximum ${k}^{\ast}$.

## 3. Results and Analysis

#### 3.1. Pressure Coefficient

#### 3.2. Friction Coefficient

#### 3.3. Mean Velocity in the x-Direction

#### 3.4. Reynolds Shear Stress

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Implementation of the Reynolds Stress Perturbation in OpenFOAM

**Figure A2.**Mean velocity field in the x-direction, normalized with respect to the inlet velocity ($\overline{u}/{u}_{0}$) obtained from the baseline case. Three different locations have been chosen in the flow domain to study their positions in the barycentric map. The points 1, 2 and 3 are respectively located at (h, $0.7h$), ($10h$, $0.7h$) and ($8h$, $5h$).

**Figure A3.**Locations of the chosen points shown in Figure A2 in the barycentric map. The points were perturbed towards the three limiting states and the amount of perturbation is ${\delta}_{B}=0.25$.

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**Figure 1.**Barycentric map. The limiting states are represented in the vertices as one-component ${X}_{1c}$, two-component ${X}_{2c}$ and three-component (isotropic) ${X}_{3c}$ turbulence. The anisotropy is zero at ${X}_{3c}$ and maximum at ${X}_{1c}$. The dashed line denotes plane shear flow, where at least one of the eigenvalues ${\lambda}_{l}$ is zero. The arrows show perturbations toward limiting states of turbulence [6].

**Figure 2.**Dimensions of the flow domain as a function of the step height h. The domain is divided in seven blocks in order to design and optimize the mesh. It also shows the location of the boundaries.

**Figure 4.**Mesh topology in the flow domain zoomed in where the step is located and expansion starts, $x/h=0$.

**Figure 5.**(

**a**) Convergence of the pressure coefficients calculated at the bottom of the flow domain after and before the expansion, computed as $(\overline{p}-{p}_{0})/(0.5\rho {u}_{0}^{2})$. (

**b**) Convergence of the friction coefficients calculated at the bottom of the flow domain after and before the expansion, computed as ${\tau}_{w}/(0.5\rho {u}_{0}^{2})$. (

**c**) Convergence of the mean streamwise velocity profiles calculated at $x/h=4$, $x/h=6$, $x/h=10$ and normalized with respect to the inlet mean velocity ($\overline{u}/{u}_{0}$). (

**d**) Convergence of the Reynolds shear stress component calculated at $x/h=4$, $x/h=6$, $x/h=10$ and normalized with respect to the inlet mean velocity ($-\overline{{u}^{\prime}{v}^{\prime}}/{u}_{0}^{2}$).

**Figure 6.**Pressure coefficients at $y/h=0$, computed as: $(\overline{p}-{p}_{0})/(0.5\rho {u}_{0}^{2})$. (

**a**) Turbulent kinetic energy perturbation ${n}_{k}=1.5$, (

**b**) turbulent kinetic energy perturbation ${n}_{k}=2$, (

**c**) eigenvalue perturbation with ${\delta}_{B}=0.1$ and (

**d**) eigenvalue perturbation with ${\delta}_{B}=0.25$. ${X}_{1c}$, ${X}_{2c}$ and ${X}_{3c}$ represent perturbations toward the three corners of the barycentric map. direct numerical simulation (DNS) data [15] (— ·); experiment data [9] (•); Reynolds-averaged Navier-Stokes (RANS) k-$\omega $ SST model (−). The uncertainty bounds are shown with gray areas.

**Figure 7.**Friction coefficients at $y/h=0$. Computed as: ${\tau}_{w}/(0.5\rho {u}_{0}^{2})$. (

**a**) Turbulent kinetic energy perturbation ${n}_{k}=1.5$, (

**b**) turbulent kinetic energy perturbation ${n}_{k}=2$, (

**c**) eigenvalue perturbation with ${\delta}_{B}=0.1$ and (

**d**) eigenvalue perturbation with ${\delta}_{B}=0.25$. ${X}_{1c}$, ${X}_{2c}$ and ${X}_{3c}$ represent perturbations toward the three corners of the barycentric map. DNS data [15] (— ·); experiment data [9] (•); RANS k-$\omega $ SST model (−). The uncertainty bounds are shown with gray areas.

**Figure 8.**Mean streamwise velocity profiles calculated at $x/h=4$, $x/h=6$ and $x/h=10$ and normalized by the inlet mean velocity ($\overline{u}/{u}_{0}$). (

**a**) k perturbation with ${n}_{k}=1.5$ at $x/h=4$, (

**b**) k perturbation with ${n}_{k}=1.5$ at $x/h=6$, (

**c**) k perturbation with ${n}_{k}=1.5$ at $x/h=10$, (

**d**) k perturbation with ${n}_{k}=2$ at $x/h=4$, (

**e**) k perturbation with ${n}_{k}=2$ at $x/h=6$, (

**f**) k perturbation with ${n}_{k}=2$ at $x/h=10$, (

**g**) eigenvalue perturbation with ${\delta}_{B}=0.1$ at $x/h=4$, (

**h**) eigenvalue perturbation with ${\delta}_{B}=0.1$ at $x/h=6$, (

**i**) eigenvalue perturbation with ${\delta}_{B}=0.1$ at $x/h=10$, (

**j**) eigenvalue perturbation with ${\delta}_{B}=0.25$ at $x/h=4$, (

**k**) eigenvalue perturbation with ${\delta}_{B}=0.25$ at $x/h=6$ and (

**l**) eigenvalue perturbation with ${\delta}_{B}=0.25$ at $x/h=10$. ${X}_{1c}$, ${X}_{2c}$ and ${X}_{3c}$ represent perturbations toward the three corners of the barycentric map. DNS data [15] (— ·); experimental data [9] (•); RANS k-$\omega $ SST model (−). The uncertainty bounds are shown with gray areas.

**Figure 9.**Reynolds shear stress calculated at $x/h=4$, $x/h=6$ and $x/h=10$ and normalized by the inlet mean velocity ($-\overline{{u}^{\prime}{v}^{\prime}}/{u}_{0}^{2}$). (

**a**) k perturbation with ${n}_{k}=1.5$ at $x/h=4$, (

**b**) k perturbation with ${n}_{k}=1.5$ at $x/h=6$, (

**c**) k perturbation with ${n}_{k}=1.5$ at $x/h=10$, (

**d**) k perturbation with ${n}_{k}=2$ at $x/h=4$, (

**e**) k perturbation with ${n}_{k}=2$ at $x/h=6$, (

**f**) k perturbation with ${n}_{k}=2$ at $x/h=10$, (

**g**) eigenvalue perturbation with ${\delta}_{B}=0.1$ at $x/h=4$, (

**h**) eigenvalue perturbation with ${\delta}_{B}=0.1$ at $x/h=6$, (

**i**) eigenvalue perturbation with ${\delta}_{B}=0.1$ at $x/h=10$, (

**j**) eigenvalue perturbation with ${\delta}_{B}=0.25$ at $x/h=4$, (

**k**) eigenvalue perturbation with ${\delta}_{B}=0.25$ at $x/h=6$ and (

**l**) eigenvalue perturbation with ${\delta}_{B}=0.25$ at $x/h=10$. ${X}_{1c}$, ${X}_{2c}$ and ${X}_{3c}$ represent perturbations toward the three corners of the barycentric map. DNS data [15] (— ·); experimental data [9] (•); RANS k-$\omega $ SST model (−). The uncertainty bounds are shown with gray areas.

Boundary | Velocity $(\overline{\mathit{u}})$ | Pressure $(\overline{\mathit{p}})$ |
---|---|---|

Upper Wall | No-stress wall | Zero gradient |

Lower Wall | No-slip condition | Zero gradient |

Inlet | Non-uniform inlet | Zero gradient |

Outlet | Zero gradient | Uniform, $\overline{p}=0$ |

**Table 2.**Mesh refinement and topology for each of the three types of refinements applied. The Cells columns represent the amount of cells in each block in the x- and y-direction, respectively. The Grading columns represent the refinement in each block in the x- and y-direction, respectively.

Coarse Mesh | Intermediate Mesh | Fine Mesh | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Cells | Grading | Cells | Grading | Cells | Grading | |||||||

Block | x | y | x | y | x | y | x | y | x | y | x | y |

A | 40 | 11 | 50 | 11.29 | 80 | 21 | 50 | 10.61 | 160 | 42 | 50 | 10.91 |

B | 21 | 11 | 1 | 11.29 | 41 | 21 | 1 | 10.61 | 81 | 42 | 1 | 10.91 |

C | 40 | 11 | 50 | 0.09 | 80 | 21 | 50 | 0.09 | 160 | 42 | 50 | 0.09 |

D | 21 | 11 | 1 | 0.09 | 41 | 21 | 1 | 0.09 | 81 | 42 | 1 | 0.09 |

E | 21 | 20 | 1 | 100 | 41 | 40 | 1 | 100 | 81 | 80 | 1 | 100 |

F | 40 | 20 | 50 | 100 | 80 | 40 | 50 | 100 | 160 | 80 | 50 | 100 |

G | 40 | 20 | 0.02 | 100 | 80 | 40 | 0.02 | 100 | 160 | 80 | 0.02 | 100 |

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**MDPI and ACS Style**

Cremades Rey, L.F.; Hinz, D.F.; Abkar, M.
Reynolds Stress Perturbation for Epistemic Uncertainty Quantification of RANS Models Implemented in OpenFOAM. *Fluids* **2019**, *4*, 113.
https://doi.org/10.3390/fluids4020113

**AMA Style**

Cremades Rey LF, Hinz DF, Abkar M.
Reynolds Stress Perturbation for Epistemic Uncertainty Quantification of RANS Models Implemented in OpenFOAM. *Fluids*. 2019; 4(2):113.
https://doi.org/10.3390/fluids4020113

**Chicago/Turabian Style**

Cremades Rey, Luis F., Denis F. Hinz, and Mahdi Abkar.
2019. "Reynolds Stress Perturbation for Epistemic Uncertainty Quantification of RANS Models Implemented in OpenFOAM" *Fluids* 4, no. 2: 113.
https://doi.org/10.3390/fluids4020113