# Evaluation of Turbulence Models in Unsteady Separation

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Literature Review

#### 1.3. Objectives

## 2. Methodology

#### 2.1. Governing Equations

#### 2.2. Problem Formulation

## 3. Results

#### 3.1. Effect of Reduced Frequency

#### 3.2. Effect of Pressure Distribution

## 4. Summary and Conclusions

- The turbulence model’s accuracy is comparable for the three models considered. They uniformly predict early separation, and the length of the recirculation region is generally overestimated.
- At the intermediate frequency, all turbulence models predict the downstream advection of the recirculation region that was observed in the resolved LES.
- The mean velocity profiles are reasonably accurate in the outer layer, the errors being concentrated in the near-wall region, especially near the separation and reattachment points.
- The Reynolds shear stresses are over-predicted during the acceleration phases, as observed previously [32].
- The drag is predicted with good accuracy, the integrated error (IE) remains always below 10% for all models.
- The results of Park et al. [32] resemble qualitatively our results, but the error there is larger. We conjecture that the error, in this case, is due to a combination of factors. In particular, the more dissipative character of the numerical method used for the RANS calculations, coupled with the coarser grid used, could result in additional diffusion. In our case, numerical errors were the same for the RANS calculations and the LES.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Detachment of a turbulent boundary layer. $\gamma $ is the percentage of backflow (fluid moving upstream): (

**Top**) mean streamwise velocity contours; (

**bottom**) instantaneous contours of the skin-friction coefficient (red: positive; blue: negative. Data from [8].

**Figure 3.**Freestream velocity distribution. (

**a**–

**c**) ${U}_{\infty}/{U}_{o}$; (

**d**–

**f**) ${V}_{\infty}/{U}_{o}$. (

**a**,

**d**) Case A; (

**b**,

**e**) Case B; (

**c**,

**f**) Case C. The gray area is the envelope of the velocity profile through the cycle, while the lines show the four extreme phases.

**Figure 4.**Contours of the phase-averaged streamwise velocity at $\Phi ={90}^{\circ}$ (

**a**–

**d**) and $\Phi ={270}^{\circ}$ (

**e**–

**h**), $k=10$.

**Figure 5.**Contours of the phase-averaged streamwise velocity at $\Phi ={90}^{\circ}$ (

**a**–

**d**) and $\Phi ={270}^{\circ}$ (

**e**–

**h**), $k=1.0$. The Supplementary Material contains an animation showing the entire cycle (Supplemental File S1).

**Figure 6.**Contours of the phase-averaged streamwise velocity at $\Phi ={90}^{\circ}$ (

**a**–

**d**) and $\Phi ={270}^{\circ}$ (

**e**–

**h**), $k=0.2$.

**Figure 7.**Skin-friction coefficient at four phases in the cycle. (

**a**–

**d**) $k=10$; (

**e**–

**h**) $k=1.0$; (

**i**–

**l**) $k=0.2$; (

**m**–

**p**) steady.

**Figure 8.**Time evolution of (

**a**–

**c**) ${C}_{f}$ and (

**d**–

**f**) ${U}_{\infty}/{U}_{o}$ at the center of the recirculation region ($x/{\delta}_{o}^{*}=300$). (

**a**,

**d**) $k=10$; (

**b**,

**e**) $k=1.0$; (

**c**,

**f**) $k=0.2$.

**Figure 10.**Reynolds shear stress profiles ($\times {10}^{3}$) at three streamwise locations ($x/{\delta}_{o}^{*}=200$, 300, and 400) and four phases, $\Phi ={0}^{\circ},\phantom{\rule{3.33333pt}{0ex}}{90}^{\circ},\phantom{\rule{3.33333pt}{0ex}}{180}^{\circ}$, and ${270}^{\circ}$. (

**a**–

**d**) $k=10$; (

**e**–

**h**) $k=1.0$; (

**i**–

**l**) $k=0.2$.

**Figure 11.**(

**a**–

**c**) Normalized drag and (

**d**–

**f**) Integrated error. (

**a**,

**d**) $k=10$; (

**b**,

**e**) $k=1.0$; (

**c**,

**f**) $k=0.2$.

**Figure 12.**Case B. Contours of the phase-averaged streamwise velocity at (

**a**–

**d**) $\varphi ={90}^{\circ}$; (

**e**–

**h**) $\varphi ={270}^{\circ}$. (

**a**,

**e**) $\mathcal{K}$–$\omega $ model; (

**b**,

**f**) $\mathcal{K}$–$\epsilon $ model; (

**c**,

**g**) SA model; (

**d**,

**h**) LES. The Supplementary Material contains an animation showing the entire cycle (Supplementary File S2).

**Figure 13.**Case C. Contours of the phase-averaged streamwise velocity at (

**a**–

**d**) $\varphi ={90}^{\circ}$; (

**e**–

**h**) $\varphi ={270}^{\circ}$. (

**a**,

**e**) $\mathcal{K}$–$\omega $ model; (

**b**,

**f**) $\mathcal{K}$–$\epsilon $ model; (

**c**,

**g**) SA model; (

**d**,

**h**) LES. The Supplementary Material contains an animation showing the entire cycle (Supplementary File S3).

**Figure 15.**Skin-friction coefficient at four phases in the cycle. (

**a**–

**d**) Case A; (

**e**–

**h**) Case B; (

**i**–

**l**) Case C.

**Figure 16.**Reynolds shear stress profiles at three streamwise locations ($x/{\delta}_{o}^{*}=200$, 300, and 400) and four phases in the cycle. (

**a**–

**d**) Case A; (

**e**–

**h**) Case B; (

**i**–

**l**) Case C.

**Figure 17.**Comparison of the skin-friction coefficient with the results of Park et al. [32]. (

**a**) DNS, present LES and $\mathcal{K}$–$\omega $ model; (

**b**) DNS, present LES and SA model.

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**MDPI and ACS Style**

MacDougall, C.Y.; Piomelli, U.; Ambrogi, F.
Evaluation of Turbulence Models in Unsteady Separation. *Fluids* **2023**, *8*, 273.
https://doi.org/10.3390/fluids8100273

**AMA Style**

MacDougall CY, Piomelli U, Ambrogi F.
Evaluation of Turbulence Models in Unsteady Separation. *Fluids*. 2023; 8(10):273.
https://doi.org/10.3390/fluids8100273

**Chicago/Turabian Style**

MacDougall, Claire Yeo, Ugo Piomelli, and Francesco Ambrogi.
2023. "Evaluation of Turbulence Models in Unsteady Separation" *Fluids* 8, no. 10: 273.
https://doi.org/10.3390/fluids8100273