# Accuracy of the Gamma Re-Theta Transition Model for Simulating the DU-91-W2-250 Airfoil at High Reynolds Numbers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{6}to 6 × 10

^{6}. The primary goal of the present paper is to validate the unsteady Reynolds averaged Navier-Stokes (URANS) approach together with the four-equation transition SST turbulence model with experimental data from a wind tunnel. The main computational fluid dynamics (CFD) code used in this work was ANSYS Fluent. For comparison, two more CFD codes with the Transition SST model were used: FLOWer and STAR-CCM +. The obtained airfoil characteristics were also compared with the results of fully turbulent models published in other works. The XFOIL approach was also used in this work for comparison. The aerodynamic force coefficients obtained with the Transition SST model implemented in different CFD codes do not differ significantly from each other despite the different mesh distributions used. The drag coefficients obtained with fully turbulent models are too high. With the lowest Reynolds numbers analyzed in this work, the error in estimating the location of the transition was significant. This error decreases as the Reynolds number increases. The applicability of the uncalibrated transition SST approach for a two-dimensional thick airfoil is up to the critical angle of attack.

## 1. Introduction

^{N}method [36,37,38] implemented in the XFOIL code [39]. However, they are not compatible with common general-purpose CFD codes [40].

## 2. The DU 91-W2-250 Airfoil and the Flow Regime

## 3. Numerical Procedure

#### 3.1. Computational Domain, Mesh and Boundary Conditions

^{+}parameter along the airfoil surfaces is equal to 0.1 for all simulations presented in this paper. A literature review shows that this wall mesh spacing should be considered enough to capture the boundary layer separation [75]. A growth rate of quadrilateral boundary layer elements was 1.1 in the wall-normal direction. Figure 2a shows the distribution of the mesh elements throughout the computing domain whereas Figure 2b presents the mesh around the airfoil.

#### 3.2. CFD Procedure and the Turbulence Model

#### 3.3. Verification to Grid Sensitivity

#### 3.4. Verification of the Length of the Simulation Time

#### 3.5. Verification of the Length of Time Step

#### 3.6. Other CFD Procedures

#### 3.7. XFOIL Procedure

^{N}method to detect the location of transition. In this paper, the N value is set to 9. This value corresponds to a smooth wing surface in a flow with a low free-stream turbulence level [81,82].

## 4. Results

#### 4.1. Lift and Drag Airfoil Characteristics

#### 4.2. Static Pressure Coefficients

#### 4.3. Skin Friction Coefficient

^{N}transition method based on linear stability and implemented in the XFOIL code are dependent on the N-factor, which should be determined by wind tunnel or flight test calibration [35].

#### 4.4. Variation of Transition Location with the Angle of Attack

## 5. Conclusions

- The aerodynamic characteristics obtained by means of classical turbulence models prove the important role of transition phenomena in the boundary layer.
- For the studied range of Reynolds numbers, the static pressure distributions do not significantly depend on the Reynolds number.
- The angle of attack has a much more significant influence on the pressure around the airfoil.
- Contrary to static pressure distributions, the skin friction coefficient distributions depend on both the angle of attack and the Reynolds number. However, the Reynolds number effect is mainly seen on the suction side of the airfoil. An increase in the Reynolds number causes an increase in the value of this coefficient on the suction edge and its shift towards the leading edge.
- For all the angles of attack analyzed in this study, on the pressure side of the profile, the decrease in the maximum value of the skin friction coefficient with the increase in the angle of attack is almost linear.
- On the suction side of the profile, the increase in the maximum value of the skin friction coefficient with the increase in the angle of attack is an exponential function.
- As with static pressure, the angle of attack has a larger effect on the distribution of the skin friction coefficient than the Reynolds number, but mainly for the suction side of the airfoil.
- With the increase of the angle of attack, the maximum value of the skin friction coefficient increases on the suction side and decreases on the pressure side.
- For the angle of attack range investigated, the maximum values of the skin friction coefficient are larger on the suction side of the airfoil compared to the pressure side. As the Reynolds number increases, the difference is larger.
- The deviations of the instantaneous pressure values from the average value are minimal.
- The maximum values of the standard deviation of the static pressure coefficients are concentrated around the areas of the laminar-turbulent transition.
- The deviations of the instantaneous values of the skin friction coefficients from the averaged values are almost constant in time and close to zero.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | |

c | chord length |

$\alpha $ | angle of attack |

$L$ | lift force |

$D$ | drag force |

${\overline{C}}_{L}$ | mean lift coefficient |

${\overline{C}}_{D}$ | mean drag coefficient |

$\Delta t$ | time step size |

${V}_{\infty}$ | undisturbed flow velocity |

K | lift-to-drag ratio |

$P$ | static pressure |

${P}_{ref}$ | reference static pressure |

${q}_{ref}$ | reference dynamic pressure |

${\rho}_{\infty}$ | free stream density |

${V}_{\infty}$ | free stream velocity |

$STD$ | standard deviation |

${C}_{f}$ | skin friction coefficient |

${C}_{fmax}$ | maximum skin friction coefficient |

${C}_{P}$ | static pressure coefficient |

${X}_{tr}$ | location of the laminar-turbulent transition |

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**Figure 1.**Numerical model: (

**a**) DU-91-W2-250 airfoil; (

**b**) computational domain and boundary conditions.

**Figure 2.**Mesh: (

**a**) The mesh in the entire computing domain; (

**b**) The distribution of the mesh around the airfoil.

**Figure 3.**The averaged aerodynamic force coefficients and their standard deviation at the angle of attack of 4°: (

**a**) lift coefficient; (

**b**) drag coefficient.

**Figure 5.**DU-91-W2-250 airfoil performance for two Reynolds numbers of $3\times {10}^{6}$ and $6\times {10}^{6}$.

**Figure 6.**The drag and lift coefficients as a function of the Reynolds number for six angles of attack: (

**a**,

**b**) drag coefficients; (

**c**,

**d**) lift coefficients; (

**e**,

**f**) torque coefficients. The results are given only for URANS with the Transition SST turbulence model.

**Figure 7.**Static pressure distributions calculated for different numerical approaches. The results are presented for one angle of attack equal to 4 degrees.

**Figure 8.**Static pressure distributions calculated for different angles of attack and for different Reynolds numbers.

**Figure 9.**Effect of the Reynolds number on static pressure coefficient characteristics: (

**a**) airfoil shape; (

**b**) distribution of the static pressure coefficients on the pressure side of the airfoil; (

**c**,

**d**) details of the static pressure distributions on the pressure side of the airfoil—the details are marked in these figures with the appropriate colors.

**Figure 10.**Static pressure coefficient for Reynolds number of $6\times {10}^{6}$. The influence of the angle of attack.

**Figure 11.**Standard deviation of the static pressure coefficient for the suction side of the airfoil and for the angle of attack of 4 degrees: (

**a**) as a function of the angle of attack and Reynolds number; (

**b**) as a function of x/c and Reynolds number.

**Figure 12.**Skin friction coefficient at the angle of attack of 4° for two Reynolds number of $3\times {10}^{6}$ (

**a**) and $6\times {10}^{6}$ (

**b**). The results are given for three numerical approaches: ANSYS Fluent code together with the URANS method; the k-ε RNG turbulence model; the XFOIL code.

**Figure 13.**Skin friction coefficient at the angle of attack of 4°—Reynolds number effect for (

**a**) the pressure side of the airfoil and (

**b**) the suction side of the airfoil. These results are given for URANS with the Transition SST model.

**Figure 14.**Skin friction coefficient—the effect of the angle of attack for (

**a**,

**b**) $Re=3\times {10}^{6}$ and (

**c**,

**d**) $Re=6\times {10}^{6}$. Moreover, ${C}_{f}$ for the pressure side of the airfoil are given in (

**a**,

**c**), whereas for the suction side of the airfoil are given in (

**b**,

**d**). These results are given for URANS with the Transition SST model.

**Figure 15.**Standard deviation of the skin friction coefficient for the suction side of the airfoil—the effect of the angle of attack and the Reynolds number (

**a**); the effect of the Reynolds number and x/c for: $\alpha =2\xb0$ (

**b**); $\alpha =4\xb0$ (

**c**); and $\alpha =6\xb0$ (

**d**).

**Figure 16.**Variation of transition location with the angle of attack for two Reynolds numbers. For the suction side of the airfoil: validation of the Transition SST turbulence model and the XFOIL code using the experimental data; for the pressure side: comparison of the results obtained from CFD and XFOIL simulations.

**Table 1.**Mesh convergence study. The drag coefficient for a different number of nodes on the airfoil surface for the Reynolds number of $4\times {10}^{6}$.

Test Case (No. of Nodes on Airfoil Surface) | $\mathbf{Drag}\mathbf{Coefficient},{\mathit{C}}_{\mathit{D}}$ | Percentage Difference of Drag Coefficient to Case 1 |
---|---|---|

Case 2 (N/4 = 155) | 0.009173 | 19.77% |

Case 3 (N/2 = 310) | 0.008326 | 8.71% |

Case 1 (N = 620) | 0.007659 | 0.00% |

Case 4 (2N = 1240) | 0.007459 | −2.60% |

**Table 2.**The derivative of the lift-to-drag ratio $dK/d\alpha $ for the range of angles of attack from 0° to 4° and for different aerodynamic methods.

Aerodynamic Method | $\mathit{d}\mathit{K}/\mathit{d}\mathit{\alpha}$$(\mathit{R}\mathit{e}=3\times {10}^{3})$ | $\mathbf{Relative}\mathbf{Error}(\mathit{R}\mathit{e}=3\times {10}^{3})$ | $\mathit{d}\mathit{K}/\mathit{d}\mathit{\alpha}$$(\mathit{R}\mathit{e}=6\times {10}^{3})$ | $\mathbf{Relative}\mathbf{Error}(\mathit{R}\mathit{e}=6\times {10}^{3})$ |
---|---|---|---|---|

Experiment | 819.74 | 0.00 | 978.81 | 0.00 |

URANS with the Transition SST | 688.39 | 16.02 | 748.22 | 23.56 |

RANS with the Transition SST | 822.90 | 0.39 | 789.68 | 19.32 |

k-ω SST | 439.08 | 46.44 | 470.90 | 51.89 |

k-ε | 414.39 | 49.45 | 471.11 | 51.87 |

RNG k-ε | 420.98 | 48.64 | 451.06 | 53.92 |

STAR-CCM + | 660.36 | 19.44 | 735.75 | 29.65 |

FLOWer | 665.11 | 18.86 | 771.97 | 25.23 |

XFOIL | 917.16 | 11.88 | 1088.33 | 11.19 |

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## Share and Cite

**MDPI and ACS Style**

Michna, J.; Rogowski, K.; Bangga, G.; Hansen, M.O.L.
Accuracy of the Gamma Re-Theta Transition Model for Simulating the DU-91-W2-250 Airfoil at High Reynolds Numbers. *Energies* **2021**, *14*, 8224.
https://doi.org/10.3390/en14248224

**AMA Style**

Michna J, Rogowski K, Bangga G, Hansen MOL.
Accuracy of the Gamma Re-Theta Transition Model for Simulating the DU-91-W2-250 Airfoil at High Reynolds Numbers. *Energies*. 2021; 14(24):8224.
https://doi.org/10.3390/en14248224

**Chicago/Turabian Style**

Michna, Jan, Krzysztof Rogowski, Galih Bangga, and Martin O. L. Hansen.
2021. "Accuracy of the Gamma Re-Theta Transition Model for Simulating the DU-91-W2-250 Airfoil at High Reynolds Numbers" *Energies* 14, no. 24: 8224.
https://doi.org/10.3390/en14248224