LES analysis of local isotropic turbulence depending on the spatial scale of the external forcing field

The study investigates the effect of the parameterisation of an external force scheme based on trigonometric functions on turbulence. In particular, the dependence of the generated turbulence on the Reynolds number and the influence of its spatial parameters are investigated. The turbulence used is generated by a large-eddy simulation based on a high-precision finite volume method, and the turbulence energy and isotropy of the turbulence are also investigated. The parameter of the external force field used in this study characterises the wavelength of the trigonometric function, and the influence of this parameter on the spatial scale is studied in detail. As a result, steady turbulent fields are generated and isotropic turbulence is observed at different spatial scales of the external force field with M=2, 3 and 4. The property of increasing turbulence energy with increasing Reynolds number is confirmed.


Introduction
Numerical analysis of turbulent phenomena has been carried out for many years as computer power has improved and simulations can now be carried out in cases where measurement and theoretical analysis are difficult [1].Various methods of analysis are used in numerical analysis, and Large Eddy Simulation is widely used because the computational cost is low and the analysis can be performed with sufficient accuracy for unsteady turbulent flows.The flows that occur around us are incompressible flows in a turbulent state and are widely found in flows above the earth's surface and in urban environments.Such flows are generally unsteady, randomly fluctuating and anisotropic, but in localised areas the flow field is often expected to be steady and isotropic.This has led to studies of steady isotropic turbulence.
The external force term in the governing equations is used to maintain isotropic turbulence at steady state.To generate isotropic turbulence, the external force term should be globally isotropic.External force terms for the generation of steady state turbulence have been studied previously.The forcing schemes proposed in previous studies can be broadly divided into two methods.Specifically, spectral forcing in wavenumber space and physical forcing in Eulerian coordinates.For physical forcing, linear forcing is often used [2].In linear forcing, the external force term is given based on a given velocity field.This linear forcing is often used in previous studies.
In previous studies, trigonometric external force fields have been used as a linear forcing method [2].Here, increasing the value of the parameter reduces the spatial scale of the external force field.The present study considers that the setting of this spatial scale parameter should be addressed.
The aim of this study is to determine the effect of the parameter value settings included in the trigonometric-based external force scheme on the generated turbulence.Here, as an external force scheme based on trigonometric functions, a simple scheme given by focusing on the vector potential of the external force is used in this study.The statistical properties of the disturbances obtained using this scheme are of interest.In this study, the turbulence is reproduced by a large-eddy simulation (LES) based on a high-accuracy finite volume method.The time series of the turbulent energy and the isotropy of the turbulence are also examined.

Numerical Methods
The governing equations are non-dimensionalised and subjected to a filtering operation [1].In the filtering operation, the flow field is divided into mean and fluctuating components by a filter function and the equations for the mean component are solved in LES.The averaging also introduces terms for the fluctuating quantities which are modelled and calculated.In terms of the target field, this study is concerned with local isotropic turbulence [2].The computational domain is therefore a periodic cubic domain with a side of 2π [3].
An issue in generating external force fields is that the velocity divergence of the generated external force field should also be zero due to the governing equation, the equation of continuity.The process of generating an external force field using this method is described below.First, the vector potential field (x, y, z) constructed is shown below for the x, y, z directions.
(1) It is constructed using only waves with a wavenumber of unity.The following external force field has been generated by taking a rotation of this vector potential.The process of taking the rotation necessarily sets up an external force field that satisfies the continuity equation.By assuming that the sum of the averages of the squares of the external forces in each direction is unity, the following coefficients (Cx, Cy, Cz) can be obtained.
M in the equation is the coefficient for varying the wavenumber.In order to minimise the discretisation error and to study only the effects of the external force field, a high order method was used for the spatial and temporal discretisation [4][5][6].For the analytical conditions in this study, the computational grid is a 32 3 equally spaced staggered grid.The Smagorinsky model with Cs = 0.1, a commonly used value, was used.The initial field is an isotropic velocity field.To examine the dependence on the Reynolds number by varying the Reynolds number, calculations were performed under four conditions, Re = 300, 1000, 3000 and .

Results
Figure 1 shows time series results for the turbulence energy of the turbulence field generated by the vector potential external force field at M = 2.The Reynolds number varies in this plot with the blue line showing results for Re = 300, the green line for Re = 1000, the red line for Re = 3000 and the black line for infinite Re.The graph also shows the results after t = 200.This is because the turbulence field generated in the initial stage from 0 to 200 is not sufficiently developed and does not become a steady turbulence field.The results will be discussed below.The graph shows that the turbulence energy is maintained at a constant value, which means that a macroscopically stationary turbulence field is generated by the vector potential external force field, independent of the Reynolds number.It is also maintained up to a time t of 3000, indicating that the calculation is stable over a sufficiently long time period to provide sufficient statistics.Comparison of the results at each Reynolds number shows that there is little difference in wavelength and amplitude, but the distribution of the waveforms varies with Reynolds number.
Figure 2 shows time series of velocity fluctuation intensity results for M = 2.Only the results for Re = 1000, which is the reference value in this study, out of the four Reynolds number conditions are shown here.The black lines in the diagram show the instantaneous values, while the red lines show the time averaged values.The averages are obtained by integrating over the period t = 200 -3000.A comparison of the time averaged values shown in the red line between the three directions confirms the agreement of the results.This shows that an isotropic turbulence field is generated by the vector potential external force field at M = 2.The amplitude and wavelength also show similar results.It can therefore be expected that the instantaneous turbulence field is also isotropic.
Then, Figure 2   for t = 200 -3000.From the results it can be seen that the amplitude is smaller compared to the results for M = 2 and the time averages are also smaller for the results for M = 3, regardless of the direction; for the results for M = 4 the amplitude and the time averages are also smaller compared to the results for M = 2.However, compared to the results for M = 3, the mean values are smaller, but there are small differences between the waveforms.Time series of turbulence energy results for M = 3 and M = 4 are shown in Figure 1.The Reynolds number also changes in this plot, with the blue line at Re = 300, the green line at Re = 1000, the red line at Re=3000 and the black line at Re=infinity.In terms of the results, a comparison with the results for M = 2 shows that the M = 3 results have smaller amplitude waveforms with shorter wavelengths, indicating that the scale of the turbulence generated is smaller.Furthermore, the results for M = 4 show that the scale of the turbulence is smaller than for M = 2, but there is no significant difference in the waveforms compared to the results for M = 3.However, it can be seen that a steady turbulence field is also generated by the vector potential external force field for M = 3 and M = 4.
Figure 3 shows the dependence of the mean turbulence energy on the Reynolds number.The abscissa is set to Reynolds number fractions.For the plot points, the black dots represent M = 1, the blue triangles M = 2, the green squares M = 3 and the red diamonds M = 4.In general, as the Reynolds number increases, the turbulent energy is also expected to increase due to the greater influence of inertia in the flow field.Taking this into account, the results for M = 1 show that the turbulence energy also increases with increasing Reynolds number.This indicates that the vector potential external force field at M = 1 generates a turbulence field with the expected Reynolds number dependence.The results for M = 2, M = 3 and M = 4 are presented next.Firstly, for the M = 2 results it can be seen that the mean value of the mean turbulence energy has decreased, but the Reynolds number dependence is maintained without relaxation.For the M = 3 and M = 4 results it can also be seen that the mean values decrease from M = 2 to M = 3 and M = 4, but the Reynolds number dependence is maintained.These results show that the Reynolds number dependence is maintained irrespective of the spatial scale.

Discussion
The value of the turbulence energy can be determined from its viscous dissipation.The results suggest that the wavelength of the external force field does not qualitatively change the nature of the dissipation rate of the turbulent energy.Previous studies have been set up to include multiple wavelengths of the external force term within the computational domain.This setting is used to fully account for the influence of scales longer than the wavelength of the external force terms that make up the large-scale turbulence.The results of the present study suggest that the influence of scale variations longer than the wavelength of the external force term in the large-scale turbulence is small in the main external force term.Further analysis is considered necessary to clarify whether this insensitivity is limited to the present external force term.
In this study, a forcing scheme in physical space is used.Here, in the present study, the external force term is set up by focusing on the vector potential of the external force.In the present study, however, the turbulence energy increases with increasing Reynolds number.This result suggests that the configuration of the external force field can significantly change the nature of the turbulence energy of turbulence.Studies to clarify this point should be carried out in future research.

Conclusion
A method for setting up an external force field that necessarily satisfies the continuity equation and enables the generation of steady isotropic turbulence was investigated by focusing on the vector potential.As an additional validation, numerical analyses were carried out to verify whether a steady isotropic turbulence field can be generated.As a result of the investigation, the turbulence energy remains constant up to t = 3000 regardless of the Reynolds number, confirming that a steady turbulence field is generated.The time series results of the velocity fluctuation intensity at M=2 show that the timeaveraged values are consistent in the u, v.w.direction, confirming that isotropic turbulence is generated.The time series results of the velocity fluctuation intensity show that the time-averaged values for the M = 3 and M = 4 results are also consistent between the three directions, indicating that isotropic turbulence has been generated.The turbulence energy time series results also confirm that steady turbulence is generated over a long period of time, although the magnitude of the turbulence generated becomes smaller and converges to a constant magnitude.

Figure 1 .
Figure 1.Time series of the turbulence energy K as an external force field parameter M.
also shows the time series of velocity fluctuation intensities for M = 3 and M = 4 are shown.Again the results are shown for Re = 1000, which is the reference Reynolds number in this study.The black line shows the instantaneous values and the red line shows the time averaged values

Figure 2 .
Figure 2. Time series of three spatially averaged velocity fluctuation intensity components, u 2 , v 2 and w 2 , at each time.Here cases with the Reynolds number of 1000 are shown.

Figure 3 .
Figure 3.The dependence of the space-time averaged turbulence energy K on the inverse of the Reynolds number is shown as a function of the external force term parameter M.