# Effect of Wall Boundary Conditions on a Wall-Modeled Large-Eddy Simulation in a Finite-Difference Framework

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## Abstract

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## 1. Introduction

## 2. Numerical Simulations

- Neumann boundary condition with zero eddy viscosity at the wall (N-ZEV):$${\left.\frac{\partial \overline{u}}{\partial y}\right|}_{w}=\frac{{\tau}_{w}}{\rho \nu},\phantom{\rule{1.em}{0ex}}\overline{v}{{|}_{w}=0,\phantom{\rule{1.em}{0ex}}{\left.\frac{\partial \overline{w}}{\partial y}\right|}_{w}=0,\phantom{\rule{1.em}{0ex}}{\nu}_{t}|}_{w}=0,$$
- A Neumann boundary condition with nonzero eddy viscosity computed from the SGS model at the wall (N-EV):$${\left.\frac{\partial \overline{u}}{\partial y}\right|}_{w}=\frac{{\tau}_{w}}{\rho (\nu +{\nu}_{t}){|}_{w}},\phantom{\rule{1.em}{0ex}}\overline{v}{|}_{w}=0,\phantom{\rule{1.em}{0ex}}{\left.\frac{\partial \overline{w}}{\partial y}\right|}_{w}=0,$$
- A Dirichlet boundary condition with eddy viscosity augmentation (D-EV):$$\overline{u}{|}_{w}=0,\phantom{\rule{1.em}{0ex}}\overline{v}{{|}_{w}=0,\phantom{\rule{1.em}{0ex}}\overline{w}|}_{w}=0,\phantom{\rule{1.em}{0ex}}{\left.{\nu}_{t}\right|}_{w}={\left.\left(\frac{\partial \overline{u}}{\partial y}\right)\right|}_{w}^{-1}\frac{{\tau}_{w}}{\rho}-\nu .$$

## 3. Comparison of the Boundary Conditions

#### 3.1. Effect of Reynolds Number

#### 3.2. Effect of Grid Resolution

## 4. Application to Wall Models

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Mean velocity profile in (

**a**) semi-log and (

**b**) defect forms for N-ZEV (circles), N-EV (crosses),and D-EV (triangles) for the $R{e}_{\tau}=5200$ case with ${\Delta}_{x}/\delta \simeq {\Delta}_{y}/\delta \simeq {\Delta}_{z}/\delta \simeq 0.1$. The dashed line indicates the direct numerical simulation (DNS) profile by Lee and Moser [35].

**Figure 2.**Eddy viscosity as a function of wall-normal distance for N-ZEV (circles), N-EV (crosses), and D-EV (triangles) for the $R{e}_{\tau}=5200$ case with ${\Delta}_{x}/\delta \simeq {\Delta}_{y}/\delta \simeq {\Delta}_{z}/\delta \simeq 0.1$.

**Figure 3.**Root-mean-square (

**a**) streamwise, (

**b**) wall-normal, and (

**c**) spanwise velocity fluctuations and the (

**d**) resolved tangential Reynolds stress for N-ZEV (circles), N-EV (crosses), D-EV (triangles) for the $R{e}_{\tau}=5200$ case with ${\Delta}_{x}/\delta \simeq {\Delta}_{y}/\delta \simeq {\Delta}_{z}/\delta \simeq 0.1$. The dashed line indicates the DNS profile by Lee and Moser [35].

**Figure 4.**Deviatoric parts of the Reynolds stress tensor components: (

**a**) ${R}_{11}^{D}$, (

**b**) ${R}_{22}^{D}$, (

**c**) ${R}_{33}^{D}$, and (

**d**) ${R}_{12}$ for N-ZEV (circles), N-EV (crosses), and D-EV (triangles) for the $R{e}_{\tau}=5200$ case with ${\Delta}_{x}/\delta \simeq {\Delta}_{y}/\delta \simeq {\Delta}_{z}/\delta \simeq 0.1$. The dashed line indicates the deviatoric part of the Reynolds stress tensor computed from DNS by Lee and Moser [35].

**Figure 5.**(

**a**) Mean streamwise velocity profile and (

**b**) mean eddy viscosity for N-ZEV (circles), N-EV (crosses), and D-EV (triangles) for $R{e}_{\tau}=5200$ (red), ${10}^{4}$ (green), and ${10}^{5}$ (blue) with ${\Delta}_{x}/\delta \simeq {\Delta}_{y}/\delta \simeq {\Delta}_{z}/\delta \simeq 0.1$. Mean velocity profiles are offset by 5 for each Reynolds number. The dashed line indicates logarithmic law $u/{u}_{\tau}=1/\kappa log(y{u}_{\tau}/\nu )+B$, with $\kappa =0.41$ and $B=5.2$.

**Figure 6.**(

**a**) Mean streamwise velocity profile and (

**b**) mean eddy viscosity for N-ZEV (circles), N-EV (crosses), and D-EV (triangles) for $R{e}_{\tau}=5200$ with ${\Delta}_{x}/\delta \simeq {\Delta}_{y}/\delta \simeq {\Delta}_{z}/\delta \simeq 0.05$ (blue), $0.1$ (red), and $0.2$ (green). Mean velocity profiles are offset by 5 for each grid resolution. The dashed line indicates the DNS profile by Lee and Moser [35].

**Figure 7.**Mean streamwise velocity profile with EQWM (${y}_{m}={\Delta}_{y}/2$) with boundary conditions N-ZEV (circles) and D-EV (triangles) for $R{e}_{\tau}=5200$ with ${\Delta}_{x}/\delta \simeq {\Delta}_{y}/\delta \simeq {\Delta}_{z}/\delta \simeq 0.1$. The dashed line indicates the DNS profile by Lee and Moser [35].

**Table 1.**Mean discrete wall normal derivative of the streamwise velocity at the wall, the mean streamwise velocity at the wall, and the eddy viscosity at the first off-wall grid point for the three different boundary conditions at $R{e}_{\tau}=5200$ with grid resolution ${\Delta}_{x}/\delta \simeq {\Delta}_{y}/\delta \simeq {\Delta}_{z}/\delta \simeq 0.1$. DNS results computed with ${\Delta}_{y}/\delta =0.1$.

Case | $\frac{\left[\langle \overline{\mathit{u}}\rangle (\mathit{y}={\mathbf{\Delta}}_{\mathit{y}}/2)-\langle \overline{\mathit{u}}\rangle (\mathit{y}=0)\right]}{{\mathbf{\Delta}}_{\mathit{y}}/2}\frac{\mathit{\delta}}{{\mathit{u}}_{\mathit{\tau}}}$ | $\frac{\langle \overline{\mathit{u}}(\mathit{y}=0)\rangle}{{\mathit{u}}_{\mathit{\tau}}}$ | $\frac{\langle {\mathit{\nu}}_{\mathit{t}}\rangle (\mathit{y}={\mathbf{\Delta}}_{\mathit{y}})}{\mathit{\nu}}$ |
---|---|---|---|

DNS | 375.8 | 0 | - |

N-ZEV | 5186 | −241.1 | 51.3 |

N-EV | 2130 | −88.3 | 48.8 |

D-EV | 383.0 | 0 | 33.2 |

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Bae, H.J.; Lozano-Durán, A.
Effect of Wall Boundary Conditions on a Wall-Modeled Large-Eddy Simulation in a Finite-Difference Framework. *Fluids* **2021**, *6*, 112.
https://doi.org/10.3390/fluids6030112

**AMA Style**

Bae HJ, Lozano-Durán A.
Effect of Wall Boundary Conditions on a Wall-Modeled Large-Eddy Simulation in a Finite-Difference Framework. *Fluids*. 2021; 6(3):112.
https://doi.org/10.3390/fluids6030112

**Chicago/Turabian Style**

Bae, H. Jane, and Adrián Lozano-Durán.
2021. "Effect of Wall Boundary Conditions on a Wall-Modeled Large-Eddy Simulation in a Finite-Difference Framework" *Fluids* 6, no. 3: 112.
https://doi.org/10.3390/fluids6030112