# Research and Development of Criterial Correlations for the Optimal Grid Element Size Used for RANS Flow Simulation in Single and Compound Channels

^{*}

## Abstract

**:**

## 1. Introduction

- Non-stationarity;
- Irregularity, lack of strict order in time;
- Randomness;
- Three-dimensionality;
- Viscous and vortex nature of the flow;
- Coherence of large vortex structures.

- ${D}_{H}$—characteristic size, m.
- $u$—characteristic velocity, m/s;
- $\nu $—viscosity, m
^{2}/s. There are currently several main approaches to modeling turbulence:- Direct numerical simulation (DNS) [1] calculates the Navier–Stokes equations for eddies of all scales up to the Kolmogorov scale. With an increase in the size of the computational cell, the system of equations for DNS, as a rule, does not converge. At present, the DNS is used mainly for research purposes and in modeling low-Re currents.
- Large eddy modeling (LES) [2] uses the separation of eddies by scale: large eddies are directly resolved, small eddies are modeled using subgrid models. LES methods require a grid of the order of the scale of large eddies. Currently, LES is used in scientific research and technical applications as a highly accurate method.
- Modeling of Reynolds-averaged Navier-Stokes equations or unsteady Reynolds-averaged Navier-Stokes equations (RANS, URANS) [3]. Modeling is done by closing the equations through Boussinesq hypothesis and semi-empirical turbulence models over the entire energy spectrum.

_{+}< 1, as well as recommendations for choosing the size of the global mesh element, corresponding to the number of elements in the range between 500,000 and 1,500,000. The y

_{+}is the dimensionless universal flow coordinate, which is by definition:

- y—wall distance, m.
- ${u}_{\tau}$—shear velocity, m/s.
- $\nu $—viscosity, m
^{2}/s.

^{6}is provided with a number of elements of the grid model of 84 million, while the size of the element in the zones of a high-velocity gradient should not exceed 0.4 mm. It is also noted in the work that during modeling, a much greater sensitivity is manifested to a change in the size of the global element than to a change in the size of the near-wall cells.

_{+}< 5 for low Reynolds models and y

_{+}> 30 for high Reynolds ones) [11].

- The size of the grid element in correlations should be associated with characteristic hydrodynamic quantities that have a length scale and characterize the flow regime.
- The nature of the quantities used should allow one to estimate the scale of the element a priori, before conducting numerical studies.
- Correlations should take into account the results of empirical and analytical studies of turbulent flows in channels.

## 2. Research Object

^{4}–10

^{5}[14,15,16] are considered as modes.

_{0}, l

_{0}, D

_{0}are the cross-sectional area, length and diameter, respectively, of the channel until the diameter changes and F

_{1}, l

_{1}, D

_{1}are these ones after changing. The $\alpha $ is the diffuser opening angle and Re

_{0}is the inlet Reynolds number.

_{0}, D

_{0}is the length and diameter, respectively, of channel until diffuser expansion and, l

_{1}, D

_{1}are the ones after the diffuser region. The α is the diffuser opening angle and Re

_{0}is the inlet Reynolds number. The D

_{2}, l

_{2}and the D

_{3}, l

_{3}is the diameter and length of the region after sudden contraction and after sudden expansion, respectively.

## 3. Research Method

_{+}= 200, is taken as the characteristic linear dimension. Taking into account the expression for near-wall layer height [16], it is possible to convert dimensionless coordinates into dimensional ones.

- $D$—channel characteristic size, m;
- ${\Delta}_{opt}$—optimal linear size of the global element of the grid model, m;
- ${y}_{+}$—dimensionless distance from the wall corresponding to the transition to the flow core, ${y}_{+}=200$ [19];
- ${Re}^{\prime}$—the Reynolds number in the characteristic cross section of the channel, ${Re}^{\prime}=\frac{{u}^{\prime}D}{\nu}$; where ${u}^{\prime}$ is the characteristic flow velocity.

- Conducting research on grid convergence for single channels with various regime and geometric characteristics.
- Revealing transition points to grid convergence using approximation power expressions.
- Reducing the values of the size of the element corresponding to the transition to the grid convergence to the dimensionless form Ko by dividing by the thickness of the turbulent boundary layer.
- Formation of correlations Ko(Re’) for individual channels with a test of statistical significance.
- Formation of the overall correlation Ko(Re’) with a test of statistical significance.
- Verification of the obtained general correlation dependence on the compound channel.

_{0}before the inlet and 10D

_{1}after the outlet.

## 4. Results and Discussion

^{−13/14}, which indicates that the element size increases more slowly than the thickness of the boundary layers, so that higher Re corresponds to a smaller element size, which is a physical result.

## 5. Conclusions

- There are regularities that relate the size of the grid model element, which ensures convergence along the grid, with the regime and geometric parameters of the flow in the channel;
- As a dimensionless similarity criterion, one can introduce the coefficient Ko, the ratio of the size of the grid model element that ensures grid convergence to the thickness of the turbulent boundary layer;
- There are statistically significant correlations Ko(Re’) for channels with sudden expansion, sudden contraction and diffusers, and there is also an overall statistically significant correlation Ko(Re’);
- This correlation makes it possible to a priori estimate the required size of the grid model element, including for compound channels, the simulation results using the obtained grid settings are within acceptable limits compared to the literature data.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Sketches of the considered geometries: (

**a**)—channel with a sudden expansion; (

**b**)—channel with a sudden contraction; (

**c**)—diffuser channel; (

**d**)—compound channel.

**Figure 5.**Plots δ(1/Δ) for channel with sudden expansion: (

**a**) F

_{0}/F

_{1}= 0.1; (

**b**) F

_{0}/F

_{1}= 0.3; (

**c**) F

_{0}/F

_{1}= 0.5.

**Figure 6.**Plots δ(1/Δ) for channel with sudden contraction: (

**a**) F

_{0}/F

_{1}= 0.1; (

**b**) F

_{0}/F

_{1}= 0.3; (

**c**) F

_{0}/F

_{1}= 0.5.

**Figure 7.**Plots δ(1/Δ) for diffuser: (

**a**) F

_{0}/F

_{1}= 0.1; (

**b**) F

_{0}/F

_{1}= 0.3; (

**c**) F

_{0}/F

_{1}= 0.5.

**Figure 9.**Graph of the coefficient of hydraulic resistance of the compound channel from the Reynolds number.

Channel with a Sudden Expansion | |||
---|---|---|---|

$\frac{{F}_{0}}{{F}_{1}}$ | 0.1 | 0.3 | 0.5 |

${D}_{0}$, mm | 84 | 84 | 84 |

${D}_{1}$, mm | 265.6 | 153.4 | 118.8 |

${l}_{0}$, mm | 50 | 50 | 50 |

${l}_{1}$, mm | 1400 | 700 | 700 |

$R{e}_{0}$ | 20,000; 60,000; 100,000 | 20,000; 60,000; 100,000 | 20,000; 60,000; 100,000 |

Channel with a sudden contraction | |||

$\frac{{F}_{1}}{{F}_{0}}$ | 0.1 | 0.3 | 0.5 |

${D}_{0}$, mm | 84 | 84 | 84 |

${D}_{1}$, mm | 48.1 | 83.3 | 107.5 |

${l}_{0}$, mm | 50 | 50 | 50 |

${l}_{1}$, mm | 700 | 700 | 700 |

$R{e}_{0}$ | 20,000 60,000 100,000 | 20,000 60,000 100,000 | 20,000 60,000 100,000 |

Diffuser channel | |||

$\alpha ,\xb0$ | 10 | 15 | 20 |

${D}_{0}$, mm | 84 | 84 | 84 |

${D}_{1}$, mm | 220.4 | 289.32 | 359 |

${l}_{0}$, mm | 168 | 168 | 168 |

${l}_{1}$, mm | 700 | 700 | 1078 |

$R{e}_{0}$ | 20,000; 60,000; 100,000 | 20,000; 60,000; 100,000 | 20,000; 60,000; 100,000 |

${\mathit{D}}_{0},\mathbf{mm}$ | ${\mathit{l}}_{0},\mathbf{mm}$ | ${\mathit{D}}_{1},\mathbf{mm}$ | ${\mathit{l}}_{1},\mathbf{mm}$ | ${\mathit{D}}_{2},\mathit{m}\mathit{m}$ | ${\mathit{l}}_{2},\mathbf{mm}$ | ${\mathit{D}}_{3},\mathbf{mm}$ | ${\mathit{l}}_{3},\mathbf{mm}$ |
---|---|---|---|---|---|---|---|

84 | 100 | 150 | 100 | 120 | 189 | 150 | 250 |

$\alpha ,\xb0$ | 14 | ||||||

$R{e}_{0}$ | 20,000 60,000 100,000 |

General | Steady State RANS, 2D Axisymmetric | Turbulence Model | k-ω SST |
---|---|---|---|

Velocity inlet, m/s | 3.478 10.434 17.39 | Gauge pressure outlet, Pa | 0 |

Fluid | $\mathrm{Air}\mathrm{at}25\mathbb{C}$ | First near-wall prismatic layer y_{+} | 1 |

$\rho $, kg/m^{3} | 1.225 | Number of prismatic layers | 10 |

$\nu $, M^{2}/c | 1.46 ·10^{−5} | Growth coefficient | 1.1 |

Meshing method | Unstructured, triangles | Global element size, mm | 0.2–40 |

Sudden Expansion | |||||||||
---|---|---|---|---|---|---|---|---|---|

Re | 20,000 | 60,000 | 100,000 | 20,000 | 60,000 | 100,000 | 20,000 | 60,000 | 100,000 |

$\frac{{F}_{0}}{{F}_{1}}$ | 0.1 | 0.1 | 0.1 | 0.3 | 0.1 | 0.1 | 0.5 | 0.5 | 0.5 |

Re′ | 6324 | 18,973 | 31,622 | 10,959 | 32,879 | 54,799 | 14,142 | 42,426 | 70,710 |

${\Delta}_{opt}$, mm | 14.48 | 5.4 | 3.1 | 6.8 | 2.67 | 1.28 | 3.37 | 2.02 | 1.09 |

Ko | 0.107 | 0.111 | 0.103 | 0.146 | 0.158 | 0.122 | 0.117 | 0.196 | 0.169 |

Sudden contraction | |||||||||

Re | 20,000 | 60,000 | 100,000 | 20,000 | 60,000 | 100,000 | 20,000 | 60,000 | 100,000 |

$\frac{{F}_{1}}{{F}_{0}}$ | 0.1 | 0.1 | 0.1 | 0.3 | 0.1 | 0.1 | 0.5 | 0.5 | 0.5 |

Re′ | 61,632 | 184,704 | 308,160 | 106,735 | 319,872 | 533,675 | 137,743 | 412,800 | 688,716 |

${\Delta}_{opt}$, MM | 0.63 | 0.40 | 0.85 | 0.74 | 1.01 | 0.88 | 1.16 | 0.75 | 0.89 |

Ko | 0.213 | 0.591 | 1.294 | 0.087 | 0.329 | 0.462 | 0.083 | 0.149 | 0.414 |

Diffuser | |||||||||

Re | 20,000 | 60,000 | 100,000 | 20,000 | 60,000 | 100,000 | 20,000 | 60,000 | 100,000 |

10 | 10 | 10 | 15 | 15 | 15 | 20 | 20 | 20 | |

Re′ | 15,245 | 45,735 | 76,225 | 11,613 | 34,840 | 58,067 | 9,359 | 28,077 | 46,796 |

${\Delta}_{opt}$, mm | 0.4 | 0.3 | 0.25 | 1 | 0.5 | 0.3 | 2.5 | 1 | 0.8 |

Ko | 0.016 | 0.033 | 0.045 | 0.023 | 0.033 | 0.031 | 0.039 | 0.043 | 0.056 |

Geometry | Correlation | c_{p} | $\left|{\mathit{r}}_{\mathit{p}}\right|$ | ${\mathit{r}}_{\mathit{k}\mathit{r}\mathit{i}\mathit{t}}$ |
---|---|---|---|---|

Sudden expansion | $Ko=0.031{{Re}^{\prime}}^{0.1424}$ | 0.497 | 2.29 | 1.74 |

Sudden contraction | $Ko=8\xb7{10}^{-7}{{Re}^{\prime}}^{1.108}$ | 0.97 | 16.06 | 1.74 |

Diffuser | $Ko=0.0027{{Re}^{\prime}}^{0.2456}$ | 0.492 | 2.26 | 1.74 |

General | $Ko=8\xb7{10}^{-5}{{Re}^{\prime}}^{0.7062}$ | 0.652 | 3.44 | 1.70 |

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**MDPI and ACS Style**

Bryzgunov, P.; Osipov, S.; Komarov, I.; Rogalev, A.; Rogalev, N. Research and Development of Criterial Correlations for the Optimal Grid Element Size Used for RANS Flow Simulation in Single and Compound Channels. *Inventions* **2023**, *8*, 4.
https://doi.org/10.3390/inventions8010004

**AMA Style**

Bryzgunov P, Osipov S, Komarov I, Rogalev A, Rogalev N. Research and Development of Criterial Correlations for the Optimal Grid Element Size Used for RANS Flow Simulation in Single and Compound Channels. *Inventions*. 2023; 8(1):4.
https://doi.org/10.3390/inventions8010004

**Chicago/Turabian Style**

Bryzgunov, Pavel, Sergey Osipov, Ivan Komarov, Andrey Rogalev, and Nikolay Rogalev. 2023. "Research and Development of Criterial Correlations for the Optimal Grid Element Size Used for RANS Flow Simulation in Single and Compound Channels" *Inventions* 8, no. 1: 4.
https://doi.org/10.3390/inventions8010004