# A Method for Choosing the Spatial and Temporal Approximations for the LES Approach

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## Abstract

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## 1. Introduction

## 2. Numerical Method and Model

## 3. Computational Mesh and Initial Conditions of the DHIT Problem

- Initial turbulence kinetic energy can be estimated as ${E}_{k0}=1000\text{}{m}^{2}{s}^{2}$. Therefore, characteristic fluctuation velocity is ${u}^{\prime}=\sqrt{\frac{2}{3}{E}_{k0}}\approx 25.8\text{}\frac{m}{s}$ and the turbulent Mach number is ${M}_{t}\approx 0.08.$ Thus, the flow can be considered incompressible;
- The integral turbulence scale, ${L}_{0}=\pi L/2\approx 1.57\text{}\mathrm{m}$, is equal to a quarter of the computational domain length. This is the largest resolved scale because bigger scale eddies would be significantly deformed by the periodic conditions of the cube;
- The turbulent Reynolds number, ${\mathrm{Re}}_{t0}=\frac{\sqrt{{E}_{k0}}{L}_{0}}{\nu}\approx 3\times {10}^{6}$, is sufficiently large for turbulence to form the inertial interval. Resolved velocity scales are inviscid. Unresolved velocity scales contain a fraction of the vortices from the inertial interval and the vortices from the dissipation interval. The scales of the latter are smaller than the numerical grid size;
- Initial integral time scale, ${T}_{0}$, is estimated as ${T}_{0}={L}_{0}/\sqrt{\frac{2}{3}{E}_{k0}}\approx 0.06\text{}s$. All simulations are carried out up to physical time $2{T}_{0}\approx 0.12\text{}s$. The energy spectrum of the turbulence is assumed to reach an equilibrium shape in two large eddy turnover times, and this shape at the final moment of time is determined only by the properties of the subgrid model and the numerical method.

## 4. Results

#### 4.1. Consistent Initial Field Problem

^{3}cells. The value ${k}_{ave}$ was seen to significantly change during the first integral time scale, $t\le {T}_{0}\approx 0.06$ s. By the time equal to $2{T}_{0}\approx 0.12$ s, the ${k}_{ave}$ value had almost stopped changing (changes occurred in the fifth significant digit). A similar behavior was observed for other ${C}_{DES2}$ values on the same mesh. As a result, it was decided to carry out this procedure up to a time equal to $3{T}_{0}\approx 0.18$ s, assuming that for this time period on any of the considered meshes and for any values of the subgrid-scale parameter field, there would be enough time to adjust to the initial resolved field. At the same time, each value of the ${C}_{DES2}$ constant corresponds to its own established value, ${k}_{ave}$.

#### 4.2. Choice of Temporal Approximation

#### 4.3. Choice of Central Differences

#### 4.4. Constant Calibration

^{3}cells. It can be seen that as ${C}_{DES2}$ increased, the level of dissipation in the short-wavelength region of the spectrum became larger due to growing influence of the subgrid model. Visually, the theoretical slope corresponded to values ${C}_{DES2}=0.7$ and $0.8$; however, according to the method described above, for both cases, the error exceeded the specified threshold of $0.01$. As a result of a parabolic interpolation, the optimal value of ${C}_{DES2}$ turned out to be $0.69$. It is worth noting that in [16], the optimal value of the constant was equal to $1.2$. This difference can be explained by the fact that VTM averaging was not used in [16] (Figure 1). This led to a decrease in the value of $\Delta $ in a certain set of cells; this was compensated by an increase in ${C}_{DES2}$. It is worth noting that regardless of the local VTM averaging, the global average values over the computational domain were close to 1: $\overline{{F}_{KH}(\mathrm{VTM})}\approx 0.930$ and $\overline{{F}_{KH}(<\mathrm{VTM}>)}\approx 0.999$. Meanwhile, the optimal values of the constants differed by a factor of 1.7.

^{3}cells; they correspond to the values ${C}_{DES2}=0.6$, $0.7$ and $0.8$. A similar result is shown in Figure 7b for a mesh with 128

^{3}cells. On the coarse mesh, the optimal value turned out to be $0.70$, and on the most refined mesh, it was $0.68$. The optimal value of ${C}_{DES2}$ barely depended on mesh spacing, which is a good sign that the methods were used within their ranges of applicability in this problem; therefore, the obtained values of the coefficient can be trusted.

^{3}cells at ${C}_{DES2}=0.5$, $0.6$ and $0.7$. The optimal value of ${C}_{DES2}$ in the presence of upwind approximations turned out to be smaller than that of a pure central-difference scheme: namely, 0.56. The spectrum obtained with this value of the constant in the case of a hybrid scheme is shown in Figure 8 (dashed line). The inertial interval was as wide as it would have been in the case of a pure central-difference approximation (solid curve, Figure 8), although a slight upward convexity could be seen upon closer consideration. Despite this remark, the hybrid scheme is of the most interest for practical simulations. Indeed, in the presence of RANS regions and non-turbulent regions far from the domain of study, switching to the upwind scheme is a necessary condition for stable simulation [23].

#### 4.5. Determining the Maximum Weight of the Upwind Scheme

^{3}cells. Formally, the subgrid model was included, but taking ${C}_{DES2}$ to be equal to ${10}^{-3}$ resulted in the model working in ILES mode.

## 5. Conclusions

- Firstly, for simulations using the current implementation of DDES in zFlare, a hybrid scheme based on a central-difference scheme of the second order of accuracy and the explicit three-step Heun method (which has a weaker time-step constraint than the midpoint method) is recommended to maximize computational efficiency, at least if the computational mesh is close to uniform;
- Secondly, with the recommended hybrid numerical method, the optimal value of ${C}_{DES2}$ was found to be 0.56. This value was almost independent of the mesh spacing, at least if its cutoff scale fell within the inertial interval. At the same time, the optimal value of ${C}_{DES2}$ for a pure central-difference scheme of the second order of accuracy equal to 0.69 was found;
- Thirdly, the influence of the subgrid model very quickly decreased with an increase in the weight of the upwind part of the numerical scheme. It became insignificant at values as low as $\sigma =0.07$, which indicates a possibility of using these schemes with the ILES method in eddy-resolving regions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Spalart, P.R. Strategies for turbulence modelling and simulations. Int. J. Heat Fluid Flow
**2000**, 21, 252–263. [Google Scholar] [CrossRef] - Reynolds, O. On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion. Phil. Trans. R. Soc. Lond.
**1895**, 186, 123–164. [Google Scholar] - Smagorinsky, J. General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weather. Rev.
**1963**, 91, 99–164. [Google Scholar] [CrossRef] - Lilly, D.K. The Representation of Small-Scale Turbulence in Numerical Simulation Experiments; IBM Form: Yorktown Heights, NY, USA, 1967; pp. 195–210. [Google Scholar]
- Deardorff, J.W. A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech.
**1970**, 41, 453–480. [Google Scholar] [CrossRef] - Kolmogorov, A.N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. CR Acad. Sci. USSR
**1941**, 30, 301–305. [Google Scholar] - Chaouat, B. The state of the art of hybrid RANS/LES modeling for the simulation of turbulent flows. Flow Turbul. Combust.
**2017**, 99, 279–327. [Google Scholar] [CrossRef] [Green Version] - Shur, M.; Spalart, P.R.; Strelets, M.; Travin, A. Detached-eddy simulation of an airfoil at high angle of attack. In Engineering Turbulence Modelling and Experiments 4; Elsevier Science Ltd.: Amsterdam, The Netherlands, 1999; pp. 669–678. [Google Scholar] [CrossRef]
- Spalart, P.R.; Deck, S.; Shur, M.L.; Squires, K.D.; Strelets, M.K.; Travin, A. A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theor. Comput. Fluid Dyn.
**2006**, 20, 181–195. [Google Scholar] [CrossRef] - Travin, A.; Shur, M.; Strelets, M.; Spalart, P.R. Physical and Numerical Upgrades in the Detached-Eddy Simulation of Complex Turbulent Flows. Fluid Mech. Appl.
**2002**, 65, 239–254. [Google Scholar] [CrossRef] - Gritskevich, M.S.; Garbaruk, A.V.; Schütze, J.; Menter, F.R. Development of DDES and IDDES Formulations for the k-ω Shear Stress Transport Model. Flow Turbul. Combust.
**2012**, 88, 431–449. [Google Scholar] [CrossRef] - Yu, H.; Grimaji, S.S.; Luo, L.S. DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann. J. Comp. Phys.
**2005**, 209, 599–616. [Google Scholar] [CrossRef] - Hansen, A.; Sørensen, N.N.; Johansen, J.; Michelsen, J.A. Detached-eddy simulation of decaying homogeneous isotropic turbulence. In Proceedings of the 43rd AIAA Aerospace Sciences Meeting and Exhibit 2005, Reno, NV, USA, 10–13 January 2005; p. 885. [Google Scholar] [CrossRef]
- Chumakov, S.G.; Rutland, C.J. Dynamic structure subgrid-scale models for large eddy simulation. Int. J. Numer. Methods Fluids
**2005**, 47, 911–923. [Google Scholar] [CrossRef] - Zhou, Z.; He, G.; Wang, S.; Jin, G. Subgrid-scale model for large-eddy simulation of isotropic turbulent flows using an artificial neural network. Comput. Fluids
**2019**, 195, 104319. [Google Scholar] [CrossRef] [Green Version] - Bakhne, S. Comparison of convective terms’ approximations in DES family methods. Math. Model. Comput. Sim.
**2022**, 14, 99–109. [Google Scholar] [CrossRef] - Bakhne, S.; Bosniakov, S.M.; Mikhailov, S.V.; Troshin, A.I. Comparison of gradient approximation methods in schemes designed for scale-resolving simulations. Math. Model. Comput. Sim.
**2020**, 12, 357–367. [Google Scholar] [CrossRef] - Troshin, A.; Bakhne, S.; Sabelnikov, V. Numerical and physical aspects of large-eddy simulation of turbulent mixing in a helium-air supersonic co-flowing jet. In Progress in Turbulence IX; Örlü, R., Talamelli, A., Peinke, J., Oberlack, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2021; pp. 297–302. [Google Scholar] [CrossRef]
- Bosnyakov, S.; Kursakov, I.; Lysenkov, A.; Matyash, S.; Mikhailov, S.; Vlasenko, V.; Quest, J. Computational tools for supporting the testing of civil aircraft configurations in wind tunnels. Prog. Aerosp. Sci.
**2008**, 44, 67–120. [Google Scholar] [CrossRef] - Zhang, R.; Zhang, M.; Shu, C.W. On the order of accuracy and numerical performance of two classes of finite volume WENO schemes. Commun. Comput. Phys.
**2011**, 9, 807–827. [Google Scholar] [CrossRef] [Green Version] - Suresh, A.; Huynh, H. Accurate Monotonicity-Preserving Schemes with Runge–Kutta Time Stepping. J. Comput. Phys.
**1997**, 136, 83–99. [Google Scholar] [CrossRef] [Green Version] - Godunov, S.K.; Zabrodin, A.V.; Ivanov, M.I.; Kraiko, A.N.; Prokopov, G.P. Numerical Solution of Multidimensional Problems of Gas Dynamics; Nauka: Moscow, Russia, 1976; p. 400. [Google Scholar]
- Guseva, E.K.; Garbaruk, A.V.; Strelets, M.K. An automatic hybrid numerical scheme for global RANS-LES approaches. J. Phys. Conf. Ser.
**2017**, 929, 83–99. [Google Scholar] [CrossRef] [Green Version] - Fornberg, B. Generation of Finite Difference Formulas on Arbitrarily Spaced Grids. Math. Comput.
**1988**, 51, 699–706. [Google Scholar] [CrossRef] - van Leer, B.; Lee, W.T.; Roe, P.L.; Powell, K.G.; Tai, C.H. Design of Optimally Smoothing Multistage Schemes for the Euler Equations. Commun. Appl. Numer. Methods
**1992**, 8, 761–769. [Google Scholar] [CrossRef] [Green Version] - Shur, M.L.; Spalart, P.R.; Strelets, M.K.; Travin, A.K. An Enhanced Version of DES with Rapid Transition from RANS to LES in Separated Flows. Flow Turbul. Combust.
**2015**, 95, 709–737. [Google Scholar] [CrossRef] - Batchelor, G.K. The Theory of Homogeneous Turbulence; Cambridge University Press: Cambridge, UK, 1953; 197p. [Google Scholar]
- Shur, M.L.; Spalart, P.R.; Strelets, M.K.; Travin, A.K. Synthetic Turbulence Generators for RANS-LES Interfaces in Zonal Simulations of Aerodynamic and Aeroacoustic Problems. Flow Turbul. Combust.
**2014**, 93, 63–92. [Google Scholar] [CrossRef] - Etkin, B. Dynamics of Atmospheric Flight; John Wiley & Sons, Inc.: New York, NY, USA, 1972; 340p. [Google Scholar]
- Comte-Bellot, G.; Corrsin, S. The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech.
**1966**, 25, 657–682. [Google Scholar] [CrossRef] - Comte-Bellot, G.; Corrsin, S. Simple Eulerian time correlation of full- and narrow-band velocity signals in grid-generated, ‘isotropic’ turbulence. J. Fluid Mech.
**1971**, 48, 273–337. [Google Scholar] [CrossRef] - Sagaut, P. Large Eddy Simulation for Incompressible Flows: An Introduction, 3rd ed.; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Godunov, S.K.; Ryabenkii, V.S. Difference Schemes an Introduction to the Underlying Theory; Elsevier Science Publ.: Amsterdam, The Netherlands, 1987; 490p. [Google Scholar]

**Figure 1.**The field of the weight, ${F}_{KH}(\mathrm{VTM})$, on mesh with 64

^{3}cells. (

**a**) Without VTM averaging; (

**b**) with VTM averaging.

**Figure 3.**The turbulent-energy spectrum for three different time schemes on the mesh with 64

^{3}cells; ${C}_{DES2}=1$.

**Figure 4.**The energy spectrum for various central-difference approximations on a mesh with 64

^{3}cells and heun3; ${C}_{DES2}=1$.

**Figure 5.**The energy spectrum for various central-difference approximations on a mesh with 64

^{3}cells and euler2; ${C}_{DES2}=1$.

**Figure 7.**The energy spectrum with different ${C}_{DES2}$ values: (

**a**) on a mesh with 32

^{3}cells and (

**b**) on a mesh with 128

^{3}cells.

**Figure 8.**The energy spectra at optimal values of ${C}_{DES2}$ (0.69 for CD-2 and 0.56 for the hybrid scheme) on a mesh with 64

^{3}cells.

**Figure 9.**The energy spectra on a mesh with 64

^{3}cells at different values (0.5, 0.6, 0.7) using a hybrid scheme.

**Figure 10.**The energy spectra in a series of ILES at various constant values of $\sigma $ on a mesh with 64

^{3}cells.

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Bakhne, S.; Sabelnikov, V.
A Method for Choosing the Spatial and Temporal Approximations for the LES Approach. *Fluids* **2022**, *7*, 376.
https://doi.org/10.3390/fluids7120376

**AMA Style**

Bakhne S, Sabelnikov V.
A Method for Choosing the Spatial and Temporal Approximations for the LES Approach. *Fluids*. 2022; 7(12):376.
https://doi.org/10.3390/fluids7120376

**Chicago/Turabian Style**

Bakhne, Sergei, and Vladimir Sabelnikov.
2022. "A Method for Choosing the Spatial and Temporal Approximations for the LES Approach" *Fluids* 7, no. 12: 376.
https://doi.org/10.3390/fluids7120376