Investigation of turbulent channel flow using local mesh refinement

Large eddy simulation of turbulent channel flow is carried out at Reτ values of 150 and 395. Mesh refinement effects are evaluated on conventional meshes, then local mesh refinement in streamwise direction is applied and investigated. The goal of the paper is to give proposal about the ideal position of the merging face and to show how many cells can be saved in this case without losing too much accuracy at both Reynolds numbers. The results are compared to DNS data.


Introduction
Channel flows are important reference cases of LES development, e.g.investigating different grid structures or developing new sub-grid scale models [12,13].In finite-volume simulations of turbulent channel flows it is very important to obtain the results in relatively short time and with acceptable accuracy.The computational costs of a simulation can be decreased by using lower amount of cells, but this way the accuracy is also decreased.Grid resolution in LES also determines which scales of eddies can be resolved, thus in a channel flow it is acceptable to use a lower streamwise resolution near the centerline, where larger eddies are present.
In the literature two main techniques of local mesh refinement can be found.One technique is to use embedded or nested grids where a finer resolution grid is embedded (nested) into a zone of the coarser grid (e.g.near the wall in channels).This type of mesh refinement is used in [9,2], and [11].Using this technique, the interaction of meshes must be ensured which can make these simulations more complicated.
Another technique involves a single mesh for the whole simulation domain and in one direction the resolution can be nonhomogeneous.In this case the mesh zones are in direct interaction and the transition is ensured by some prism or distorted hexahedron cells.These kinds of meshes are used in [8,7] and [1].
Various simulation approaches using local mesh refinement are in very good agreement with the corresponding DNS or experimental data for both external and internal flows.
The goal of this paper is to simulate a turbulent channel flow using large eddy simulation, to investigate the effects of decreasing streamwise resolution above a certain distance from the wall and to give proposals about the ideal region size of refinement and the reduction of resolution.The simulations are performed at Re τ = 150 and 395 to investigate the Reynolds number dependency of the results.
In the second section the methodology of the investigation is described.The governing equations of LES and the turbulent model are shown and shortly introduced along with the setup of the simulations.The third section describes the results of the investigations.This section is separated into two subsections, which are the conventional meshes and the locally refined meshes.Using a step-by-step investigation a final, best performing mesh is obtained for both Reynolds numbers, then using these meshes the effects of local refinement in streamwise direction are investigated.In the fourth section the conclusions of the investigations are described.

Methodology 2.1 Large Eddy Simulation
LES is a widely used method for simulating fluid flows with important turbulent structures.In contrast to Reynolds Averaged Navier-Stokes (RANS) methods that do not resolve turbulent eddies, LES is capable of resolving a part of the turbulent kinetic energy spectrum.The remaining part of the spectrum needs to be modelled.Figure 1 shows the difference in the resolved and modelled turbulent kinetic energy spectrum for RANS methods and LES [14].
The governing equations are the continuity equation ( 1) and the Navier-Stokes equations (2).These are filtered equations with an added SGS viscosity.Dynamic Smagorinsky model is used to model the SGS viscosity.The classical Smagorinsky model describes the sub-grid (modelled) viscosity using a constant number, the Smagorinsky constant [15].
The dynamic Smagorinsky model calculates the value of the Smagorinsky constant in every time step locally for every cell.In order to perform this calculation a test grid is applied on which the filter width is wider than on the original grid (e.g.Δ test =2Δ) and the method of least squares is used [4,10].
The simulations are carried out using LES with the described dynamic Smagorinsky model in OpenFOAM, an open source C++ library using the built-in solver for channel flows with LES named channelFoam [17,6].

Simulation setup
Cyclic boundary condition is applied at every face except the walls.No-slip condition for the velocity field and zero-gradient for the pressure field is applied on the walls.A perturbed velocity field is used for initial condition [16, pp. 163-167].
The friction and bulk Reynolds numbers and mean velocities used are shown in Table 1.In the simulations Re m has a prescribed value and Re τ is a result of the simulation.
The constant time step is set to maintain a CFL of 0.6 -0.8 for the conventional meshes.Locally refined meshes has the same step size as their base conventional mesh.The effects of the local refinement on the CFL can be investigated this way.The results can be seen in section 3.2.

Geometry
Channel flow is a fluid flow between two infinitely large flat plates.The simulation domain is 4x2x2 meters (L x xL y xL z ).The used geometry is shown in Figure 2, where grey colour represents the channel walls and the others are cyclic faces.

Meshing techniques
Two meshing techniques are used during the investigations.First technique is used with conventional meshes: the resolution is changed in only one direction and the resolution is homogeneous for every direction.Mesh structure is shown in Figure 3.
Another technique is used with locally refined meshes.The resolution in streamwise direction is not homogeneous, but it is decreased above a certain distance from the channel wall.The structure of this kind of mesh is shown in Figure 4.
Cell size in wall normal direction is changing according to a geometric series.The quotient of the series is kept at 1.1 for every mesh. ( (2) (3)

Results
Two main parameters are used to evaluate the results of the meshes.First is the relative difference of the friction Reynolds number (Eq. ( 4)) compared to the reference, the second is the normalized mean square error, which is calculated as Eq. ( 5).In our results NMSE is applied to the dimensionless streamwise mean velocity, u + =u/u τ .
The reference values of Re τ and u + are taken from DNS database [5].

Conventional meshes
Conventional meshes refer to homogeneous resolution in the main directions.Figure 3 shows this mesh structure.
According to [3] the suggested minimal resolution for LES is Δx + =100, y + =1 and Δz + =30.Starting from this resolution at Re τ =150 a step-by-step investigation is carried out in order to reach mesh independent results.In every step the resolution is changed only in one direction.One mesh is considered better from another if the error of Re τ and the NMSE also decrease.Table 2 and Table 3 show the investigation steps and results at Re τ =150.
The last row of    The results of the best performing mesh in Table 3 are in good agreement with the reference considering Re τ error, which is 3%.The 10 -4 order of magnitude of the NMSE is a reasonably good result too, especially when taking the total number of cells into consideration, which is 57000 cells.In contrast the reference DNS grid has more than 1.5 million cells at the same Re τ [5].
Table 4 shows the investigation steps and results for Re τ =395 and the last row represents the best performing mesh.
Results of the best performing mesh in Table 4 show that the error of Re τ is 1% and the order of magnitude of the NMSE is 10 -4 .The total number of cells is 195920 (the DNS reference has almost 9.5 million cells).The resolved part of the turbulent kinetic energy is above 90% for both Reynolds numbers.
In Figure 7 -Figure 10 "base mesh" legend represents the velocity and RMS profiles for these meshes.
Summarizing the results the minimal required resolution in our implementation can be described as Eq. ( 6)-(8).

Locally refined meshes
Locally refined meshes refer to meshes, where the streamwise resolution is decreased above a certain distance from the channel walls.Figure 4 shows a locally refined mesh.
The two varied parameters during the local mesh refinement study are the wall-normal size of refined region (y M F + ) and the streamwise resolution near the centerline (Δx c + ).For Re τ =150 the value of y M F + is varied between 10…75 and Δx c + is varied between 35…50.For Re τ =395 the following ranges are studied: y M F + =55…95 and Δx c + =53…79.In this interval the relative error of the friction Reynolds number remains between ±5%.
Table 5 and Table 6 show the worst and the best results along with the parameters of the base meshes.
Table 5 and Table 6 show that 13-17% of the cells can be saved without significantly reducing accuracy (compared to the base mesh).At Re τ =150 the maximum CFL is roughly 6% lower than on the base mesh, thus the time step can be somewhat higher.At Re τ =395 the maximum CFL does not change significantly on the locally refined meshes compared to the base mesh.
Figure 5 and Figure 6 show tendencies of NMSE results on the investigated interval of the varied parameters.In the figures the two lines represent two typical values for both cases, i.e. one close to the wall and one far from the wall.The tendencies between these parameters are similar.
The farther the position of the merging face is from the channel walls, the better the results become for every investigated Δx c + at both Reynolds numbers.According to Figure 5, decreasing the streamwise resolution near the centerline has a significant effect on the results at Re τ =150 if y M F + is low, i.e. the merging face is close to the wall, but has almost no effect if y M F + is high enough.In case of Re τ =395 decreasing Δx c + also affects the NMSE results even if y M F + is higher.In the case of Re τ =150 the best result is achieved by placing the merging face around y M F + =75.In wall normal distance this is equal to y=0.5•δ.In this case the streamwise resolution can be decreased by 47%.In the case of Re τ =395 the best y + ≤ 1…1.5      position of the merging face is around y M F + =95, which means y=0.24•δ and the resolution in the near centerline region can be decreased by around 25%.
Figure 7-10 show profiles of the streamwise mean velocity and RMS.For the best meshes the profiles correspond fairly well with the base mesh as the streamwise resolution is decreased near the centerline.
Locally refined meshing technique is proposed in order to maintain computational resources and accuracy.The investigation of locally refined meshes shows that acceptable results can be achieved in terms of NMSE and the error of Re τ by decreasing the streamwise resolution above a certain value of wall distance.The resolution can be decreased by 25-47% without losing too much accuracy.
As the log-law region of the turbulent boundary layer is closer to the wall for higher Re τ , more cells can be saved without loosing accuracy by moving the merging face closer to the wall.The results obtained for both Re τ values are in good agreement with this theory.The ideal position of the merging face gets closer to the wall at a higher Re τ as it was described in section 3.2.
In future, the proposed non-dimensional parameters of locally refined meshes should be tested for higher Reynolds numbers.

Fig. 2 .
Fig. 2. Geometry of the channel: rectangular fluid domain with two parallel solid walls and cyclic faces in the homogeneous directions (x and z directions)

Fig. 5 .
Fig. 5. NMSE results normalized by the NMSE of the base mesh for two typical y M F + values.Re τ =150.

Fig. 6 .
Fig. 6.NMSE results normalized by the NMSE of the base mesh for two typical y M F + values.Re τ =395.
Table 2 represents an intermediate state.Around this resolution other investigations are made, these steps are shown in Table 3, where the last row represents the final, best performing mesh.
Tab. 2. Results of conventional mesh investigation steps, first study, Re τ =150 3ab.3.Results of conventional mesh investigation steps, second study, Re τ =150 5ab.5.Comparison of the base, best and worst cases, Re τ =150 base mesh best case worst case Comparison of the base, best and worst cases, Re τ =395 base mesh best case worst case ] molecular viscosity v SGS [m/s 2 ] sub-grid scale viscosity