Large Eddy Simulation with wall functions of Ahmed body

Large Eddy Simulation with Wall Function (WFLES) is known to be a cheap alternative to classical LES methods for simulation of flow where large and complex computational meshes are typically required. This makes it attractive for engineering applications. However experience of applying such methods to complex turbulent flows with flow separation and reattachment is still little-known in literature. In this work WFLES of flow around simplified car body with slant angle equal to 25 degrees and ReL = 2.8 · 106 is carried out on Octree mesh to demonstrate the capabilities and limitations of the method in such type of the flow. The results on a series of meshes show that even though the general flow topology is well captured, the critical part of the flow on the slant is hardly predicted even on 100 mln mesh. It is concluded that the prediction of separation above the slant requires significant mesh refinement even in the frame of WFLES.


Introduction
Nowadays capabilities of RANS models are known to be not sufficient for accurate prediction of turbulent flows with separation and reattachment. For improvement of the prediction accuracy of such flows the Scale-Resolving Simulation (SRS) methods are required, where at least a part of the flow is resolved, and another part is modeled. The most popular approaches are hybrid RANS-LES methods [1][2][3], which contrary to classical LES methods allow to significantly reduce the computational requirements due to near wall treatment using RANS models, however the methods still suffer from the limitations of the RANS model.
Contrary to hybrid methods classical LES approach combined with such models as Smagorinsky [4], WALE [5], Dynamical [6] and other suffer from near quadratic increase of the required number of grid points within inner part of the attached boundary layer in respect to Reynolds number [7]. This makes the classical LES approach improper for use in high Re number flows which appear in most of the industrial applications.
This fact stimulates attempts to apply and develop less expensive alternatives, such as LES modifications where the near wall regions are specially treated. One of the promising approaches for engineering applications is LES combined with near wall modeling (extensive review of such methods see in [7]). The approach has two important benefits: 1) it allows to use automative mesh generation tools (to build meshes such as octree or mosaic); 2) due to near wall treatment 2 the meshes within the wall can be substantially coarser than in classical LES which results in significant decreasing of grid size.
The family of near wall models in LES is large and a number of simple and sophisticated methods exists [8][9][10][11][12][13][14][15][16][17][18], however they are typically calibrated and tested using canonical flows with attached boundary layer flows (flat plate, channel flow etc). In the current study we focus on a LES with classical wall modeling approach [8,11,19] here called as LES with Wall Function (WFLES). As shown in [20], when applied to problems with attached boundary layers good solution accuracy is obtained, however application to industrial type of the flows (with complex flow topology with physical effects such as flow separation and reattachment) is still rather questionable. Knowing the advantages of LES with near wall modeling listed above, defining its potential and limits in realistic flow configurations is important for engineering applications.
In this work we demonstrate capabilities of WFLES for turbulent flow around a simplified car body (known as Ahmed car) [21,22]. The configuration with 25 o slant angle is considered, which provides relatively complex flow topology with boundary layer separation and reattachment above the slant. It is known that simulation the flow with LES or hybrid RANS-LES approaches requires substantially large computational resources. Therefore in literature simulations at lower Re number are considered (see e.g. [23,24]), or computational results are sensitive to model, mesh or employed numerical method, see e.g. [25]. The goal of the current study is to justify the capabilities of WFLES to capture such a complex flow phenomena. The obtained results are compared with experiment data of [21] and [22].

Numerical setup
Simulation of flow around Ahmed body is carried out using incompressible branch of general purpose CFD solver ANSYS Fluent R21.1 [26] at Re = 2.8 · 10 6 based on the body length,   Table 1. WFLES is used with WALE turbulence model [5] with coefficient 0.5. The near wall treatment is based on classical wall function similar to [19]. The governing equations for the incompressible fluid are solved in the transient formulation by means of the finite volume method on collocated grids. Modification of the BCD scheme [27], namely the Weakly Bounded Central Difference (WBCD), is employed. According to the author experience the scheme is robust and accurate for LES simulation of both wall bounded and free shear problems. The pressure gradient is discretised using a second order pressure scheme; gradient terms are approximated using the Least Square Cell Based (LSCB) scheme. SIMPLEC [28] approach with 5 subiterations per time step is used to ensure residual drop up to 1 order for pressure and 4 orders for velocity components. Pressure velocity coupling is ensured using a modified version of Rhie-Chow interpolation [29]. The time step ∆ t is set such that the solution is time step independent. All simulations were performed in two stages. During a first transient period a statistically mature solution was obtained. After that averaging of the solution was started. Each of the periods were set to 5 flow passes over the body length.

Results
As seen on Figure (    Computed and experimental lengths of the recirculation zone as well as shape and position of the vortices are in a good agreement. Frequency of the vortex shedding behind the body is mesh independent as seen from the velocity spectra at a control point behind the body, see Figure   As was mentioned in other WFLES studies, see e.g. [15], the reason of inaccurate flow field prediction with WFLES can be twofold: 1) lack of mesh resolution in the near wall areas with high gradients or 2) modeling errors of wall function approach employed. Note that Mesh 3 satisfies the minimum requirements for WFLES [20], namely it has cubic cells and the cell size is less than 10% of boundary layer thickness. However, near the slant beginning flow acceleration is observed which leads to boundary layer thinning. In this area even Mesh 3 becomes insufficient (see Figure (7)) and, as a result, velocity profile is predicted with significant error. So, for the prediction of the separation above the slant further mesh refinement is required. However, one may note that this would result in mesh size of hundreds of million cells, which are comparable with meshes for LES approach and are inappropriate for engineering applications.  mln of cells. It is shown that this mesh is insufficient for resolution of thin boundary layer before separation, so prediction of separation above the slant requires significant mesh refinement even in the frame of WFLES.