# Effect of Local Grid Refinement on Performance of Scale-Resolving Models for Simulation of Complex External Flows

^{*}

## Abstract

**:**

## 1. Introduction

_{1Eq}one-equation model [28], which was itself originally derived from the k-ε closure [29]. Following this methodology, Elkhoury [30,31,32] developed several versions of these models, all of which included a second derivative of velocity and thus, operate in SRS mode.

## 2. Turbulence Models

#### 2.1. Algebraic WMLES Turbulence Model

#### 2.2. SST-Improved Delayed Detached Eddy Turbulence Model

#### 2.3. The One-Equation SAS Turbulence Model

**L**, obtained by Rotta [36], and Menter and Egorov [34], when neglecting convection and diffusion terms in the $kL$ equation and eliminating the turbulence kinetic energy with the help of the transport equation for $k$. In the original version [34], $\left({\partial}^{2}S/\partial {x}_{j}^{2}/S\right)$ was referred to by $1/{L}^{2}$, but was removed here to eliminate any misinterpretations, since occurrences of negative values are allowed in the model. In almost all flows, ${E}_{p}>>{E}_{1}$ and the original formulation is retained. Only in cases where the denominator goes to zero, the ${E}_{diff/dest}$ smoothly switches to ${E}_{p}$.

_{2}follows from the logarithmic law as ${C}_{2}=\left({C}_{1}/{\kappa}^{2}+1/{\sigma}_{v}-3\text{\hspace{0.17em}}{C}_{3}-{C}_{4}\right)$. The A

_{1}constant in the original model was set to 17.5, while it is slightly tweaked to better fit the logarithmic overlap given the present SAS formulation.

## 3. Numerical Approach

## 4. Results

^{−3}of scaled residuals of all flow variables was achieved. Furthermore, all computations were performed on the IBM HPC nextScale M5 with 112 cores of Intel(R)-Xeon(R)[email protected] GHz running with double precision at 4 Teraflops.

**Flow Past an Array of Cubes:**The flow characterization in urban regions is crucial to mitigating the pollution dispersion and load due to wind on adjacent buildings. The most common test case that to mimic the flow around buildings consists of a flow around three-dimensional cubes. Meinders [47] and Meinders et al. [48] experimentally investigated the flow around a rectangular array of 250 cubes in an aligned configuration. The cubes were placed in a channel with a configuration consisting of 25 rows of 10 cubes each, separated by a distance equal to 3 times the side length of the cube. Flow measurements were carried out using laser Doppler anemometer. In the present work, an array of 25 equally-spaced cubes was tested. The computational domain and locations of comparison between numerical and experimental results are shown in Figure 1. The mean velocity was calculated based on the average mass flow rate and normal cross-sectional area through which the flow enters the domain. Based on this bulk velocity, U

_{b}= 3.86 m/s, and the cube side length of H = 15 mm, the resulting Reynolds number was 3855. A no-slip boundary condition was imposed on all cube sides and on the top and bottom sides of the fluid domain. Additionally, Meinders et al. [48] ensured that a fully-developed periodic state was achieved at the locations where their experimental data were recorded. Therefore, the periodic boundary condition was used at the walls of the domain in both spanwise and streamwise directions to mimic experimental conditions.

_{b}, in the streamwise direction. However, in order to ensure that the initial conditions did not affect the obtained solution, the first 500 T

_{b}of flow time were disregarded, and a total period of 200 T

_{b}was averaged to ensure statistical convergence of flow variables. It is worth noting that T

_{b}is the turnover time expressed as T

_{b}= H/U

_{b}, where H is the cube height, and the time step used in the calculations was set as

**∆**t = T

_{b}/32. To prove that the solution is time-independent, the time step was further reduced to half of its original value for the simulation on the regular mesh using WMLES. The average velocities remained unchanged with the smaller time step, proving that the selection of the original time step was adequate. Figure 3 depicts the variation the streamwise velocity at vertical lines located in the XY plane (Z = 0) at various x-locations (Figure 1B) for all three tested turbulence models compared to the experimental data my Meinders et al. The plot was done on the three meshes discussed earlier; the unclustrered low- and high-density meshes are shown in Figure 3a, and a comparison between the clustered and unclustered meshes is shown in Figure 3b. All three turbulence models accurately predict the velocity variation when used on the unclustered high-density mesh. However, the One-Equation SAS model had the most accurate predictions of the coarse, low-density mesh. The advantage of using the One-Equation SAS model can be clearly inferred from Figure 3b; it is the only model that succeeds in reproducing the experimental velocity profiles on the clustered mesh with a 0.4% deviation from experimental data at x/H = 2.8, followed by the WMLES having a total error of 3.2%. Consequently, its superiority in generating accurate results on both uniform and non-uniform meshes is demonstrated. In Figure 4, the streamwise velocity profiles on horizontal lines (parallel to the z-axis) for the same meshes and models as in Figure 3, but on a plane parallel to the XZ plane at Y = 0.5 H, were plotted. The One-Equation SAS model predicts the most accurate results on the low-density mesh with a total deviation from experimental data of 1.62% at x/H = 2.8, followed by the IDDES, having an error of 3.6%, while the WMLES deviates by 6.53%. On the other hand, the WMLES model generates better predictions on the clustered mesh, with the highest deviations from experimental data at x/H = 3.8. On the other hand, the IDDES model performs poorly near the wall (0.0 < z/H < 0.6) on the clustered mesh and in the region where 1.0 < z/H < 2.0 on the unclustered uniform mesh. Figure 5 depicts the variation of the spanwise velocity on same plane and lines and for similar models and conditions as Figure 4. Of all the considered meshes, the IDDES and WMLES models produce the least accurate results. Additionally, predictions of the IDDES model on the clustered mesh show large deviations from the experimental data. On the other hand, the One-Equation SAS model shows the lowest deviation from experimental data on all three grids tested. Additionally, it is worth noting that the One-Equation SAS model succeeded in predicting the streamwise velocity with better accuracy (Figure 3 and Figure 4).

**Flow Past a Circular Cylinder**: This test case involves massive flow separation behind a circular cylinder. The objective here is to assess the extent to which the considered models are able to reproduce the flow features downstream the cylinder when subject to mesh refinements applied only in the wake region. A coarse mesh, consisting of 1.4762 × 10

^{6}nodes, was used to assess the ability of the subgrid modeling. Two levels of grid adaptation were then applied adjacent to the cylinder in the wake region, resulting in a final mesh density of 4.8764 × 10

^{6}nodes. Figure 11 depicts the original and the refined mesh. Grid adaptation was used to investigate the response of the models to local refinements, i.e., when resolving flow structures is of interest in local regions, which could arise in urban city layout. In the present case, refinements that encompass the cylinder, i.e., the separated shear layers, yielded similar results to those of the literature [52], and thus, are not included herein as they do not fall within the previously-set objectives.

^{−4}m

^{2}/s were set at the inlet. A zero-gauge pressure was set at the outlet boundary condition, while the turbulent viscosity value was extrapolated from the interior domain. A CFL of 0.7 was used with default relaxation and under-relaxation factors for all flow variables. For all models, the bounded central differencing scheme was selected as it involves acceptable numerical dissipation that is small enough not to affect the evolution of small-scale turbulent structures. A time step of 5 × 10

^{−4}s was set for all models, which was several fold less than that of the experimental shedding period computed from the Strouhal number St of 0.179, and smaller by an order of magnitude when compared to simulations found in the literature using scale resolving models [52,54].

^{+}< 0.092 was attained with all three models. It is clear that with this mesh coarseness, none of the models was able to predict subgrid scales. Similarly, Travin et al. [55] numerically assessed the impact of the grid size on the predictions of the roll-up shear vortices that occur right after boundary layer transition around a cylinder. Unfortunately, their grid sizes were not fine enough, leading to an under-resolved flow transition. The IDDES and the One-Equation SAS models slightly underpredict the extent of the recirculation bubble, while the WMLES model predicts an acceptable length of the separation bubble, albeit with a fast recovery that overpredicts the centerline mean velocity ratio. Refinement of the mesh in the wake region shows improvement in velocity profiles, as predicted by the IDDES and the One-Equation SAS models. The WMLES model shows no improvements; on the contrary, the model predicts a faster recovery. In both mesh scenarios, the WMLES model performs poorly in comparison to the other two models. This could be related to the simple nature of the model, which is based on a Smogarinsky, rendering it incapable of accounting for the near-wall turbulence at this mesh density. It is worth noting that the flow structure downstream of the cylinder is very much affected by its boundary layer development. Therefore, irrespective of the refinement in the wake region, the WMLES model fails to predict the wake velocity profile. On the contrary, the other two models performed better, as they rely on transport equation(s) to model turbulent mechanism.

^{2}colored by eddy viscosity ratio is shown in Figure 15. The effect of low mesh resolution on the resolved scales and viscosity ratio is clearly visible in the upper part of Figure 15. By comparing the turbulent flow structures of the considered models, it becomes evident that with a decrease in mesh resolution, small scales disappear and large scales emerge, accompanied by higher levels of eddy turbulent viscosity ratios. Mesh-refined flow scales are depicted in the bottom part of Figure 15. Despite the mesh refinement, the WMLES does not seem to be able to resolve as many scales as the IDDES and the One-Equation SAS models; this could be due to the near-wall eddy viscosity modeling that was mentioned earlier. As expected, all models dissipate lower levels of eddy viscosity behind the cylinder with smaller scales on the refined mesh. It is important to note that while the eddy viscosity is proportional to the grid scale in both of the IDDES and WMLES models, the One-Equation SAS model adjusts the eddy viscosity levels based on the Von Karman length scale, and subsequently, the flow structures in the wake region.

## 5. Conclusions

_{τ}. The most accurate predictions of the velocity and normalized and Reynolds stress profiles on coarse or non-uniform clustered grids were achieved by the One-Equation SAS model, followed by the WMLES model. It is worth noting that the IDDES model required the lowest number of iterations per time-step to reach convergence. This could be attributed to the elevated levels of dissipation compared to the other two models, a point that merits further investigation in future work.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**A**) Schematic of the 3D computational domain for the flow past an array of cubes, (

**B**) 2-D view of an XY plane with locations at which comparisons between numerical and experimental results were performed.

**Figure 2.**2D view of the three different meshes used in the simulations: (

**a**) uniform coarse-mesh, (

**b**) uniform fine mesh and (

**c**) non-uniform clustered mesh.

**Figure 3.**Effect of (

**a**) mesh density, and (

**b**) mesh clustering on the average streamwise velocity in the XY plane cutting through the center of the cube at z/H = 0. Each successive x/H profile starting from the one at x/H =1.2 is offset by one unit from the previous profile.

**Figure 4.**Effect of (

**a**) mesh density, and (

**b**) mesh clustering on the average streamwise velocity in the XZ plane cutting through the center of the cube at y/H = 0.5. Each successive x/H profile starting from the one at x/H = 1.2 is offset by one unit from the previous profile.

**Figure 5.**Effect of (

**a**) mesh density, and (

**b**) mesh clustering on the average sapnwise velocity in the XZ plane cutting through the center of the cube at y/H = 0.5. Each successive x/H profile starting from the one at x/H = 1.2 is offset by one unit from the previous profile.

**Figure 6.**Effect of (

**a**) mesh density, and (

**b**) mesh clustering on the streamwise normalized Reynolds stress in the XY plane cutting through the center of the cube at z/H = 0.

**Figure 7.**Effect of (

**a**) mesh density, and (

**b**) mesh clustering on the streamwise normalized Reynolds stress in the XZ plane cutting through the center of the cube at y/H = 0.5.

**Figure 8.**Effect of (

**a**) mesh density, and (

**b**) mesh clustering on the spanwise normalized Reynolds stress in the XY plane cutting through the center of the cube at z/H = 0.

**Figure 9.**Effect of (

**a**) mesh density, and (

**b**) mesh clustering on the spanwise normalized Reynolds stress in the XZ plane cutting through the center of the cube at y/H = 0.5.

**Figure 10.**Contours of normalized Reynolds stress $\overline{\mathrm{w}\text{}{}^{\prime}\text{}\mathrm{w}\text{}{}^{\prime}}{/\mathrm{U}}_{\mathrm{b}}^{2}$ in the XY plane cutting through the center of the cube at z/H = 0 for all three tested turbulence models on the uniform coarse and regular meshes and the non-uniform clustered mesh.

**Figure 11.**Mesh in the x-y plane that passes through the center of the cylinder, (

**a**) coarse mesh, and (

**b**) clustered mesh.

**Figure 12.**Centerline mean streamwise velocity distribution showing the effect of coarse and clustered mesh on the predicted profile for the One-Equation SAS, WMLES, and IDDES models.

**Figure 13.**Normalized Reynolds stress distribution $\overline{\mathrm{u}\text{}{}^{\prime}\text{}\mathrm{u}\text{}{}^{\prime}}{/\mathrm{U}}_{\mathrm{b}}^{2}$ on the centerline behind the cylinder in the wake region showing effect of mesh density for all three models.

**Figure 14.**Normalized Reynolds stress distribution $\overline{\mathrm{v}\text{}{}^{\prime}\text{}\mathrm{v}\text{}{}^{\prime}}{/\mathrm{U}}_{\mathrm{b}}^{2}$ on the centerline behind the cylinder in the wake region showing effect of mesh density for all three models.

**Figure 15.**Effect of low mesh resolution on turbulent flow structures. Iso-surface of Q = S

^{2}− Ω

^{2}= 1, colored by eddy viscosity ratio. (

**A**) One-Equation SAS model, (

**B**) IDDES model, (

**C**) WMLES model.

Contributor | Model | Grid, Nx, Ny, Nz | Number of Points on Cube Side |
---|---|---|---|

Hsieh et al. (2010) [50] | LES-Standard Smogarinsky, PANS, k-ε RANS | 49 × 49 × 49 | -- |

Yang et al. (2015) [51] | Integral wall-modeled LES | 32 × 32 × 32 | 8 |

Goodfriend et al. (2016) | LES-Standard and mixed model | 84 × 96 × 84 | 28 |

Present | Wmles | 55 × 45 × 55 | 15 |

SST-IDDES | 150 × 180 × 150 | 70 | |

One-Equation SAS | 150 × 180 × 150 Clustered | 70 |

**Table 2.**Values of y+, Δ+ and growth rate for all models and grids used for the flow past an array of cubes.

Model | Uniform Coarse-Mesh | Uniform Fine Mesh | Non-Uniform Clustered Mesh | |
---|---|---|---|---|

WMLES | y^{+} | 8.66 | 5.27 | 0.699 |

SST-IDDES | 8.14 | 5.5 | 0.648 | |

One-Equation SAS | 8.5 | 5.05 | 0.568 | |

Growth Rate | 1.088 | 1 | 1.1727 | |

∆^{+}(∆/H) | 0.012 | 0.0085(vertical direction) 0.01(horizontal direction) | 0.0002 |

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**MDPI and ACS Style**

ElCheikh, A.; ElKhoury, M. Effect of Local Grid Refinement on Performance of Scale-Resolving Models for Simulation of Complex External Flows. *Aerospace* **2019**, *6*, 86.
https://doi.org/10.3390/aerospace6080086

**AMA Style**

ElCheikh A, ElKhoury M. Effect of Local Grid Refinement on Performance of Scale-Resolving Models for Simulation of Complex External Flows. *Aerospace*. 2019; 6(8):86.
https://doi.org/10.3390/aerospace6080086

**Chicago/Turabian Style**

ElCheikh, Amne, and Michel ElKhoury. 2019. "Effect of Local Grid Refinement on Performance of Scale-Resolving Models for Simulation of Complex External Flows" *Aerospace* 6, no. 8: 86.
https://doi.org/10.3390/aerospace6080086