ZHANG Bin, WANG Tong, GU Chuan-gang

Key Laboratory for Power Machinery and Engineering of Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: sjtu2009@sjtu.edu.cn

DAI Zheng-yuan

Key Laboratory for Power Machinery and Engineering of Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China

Trane’s Asia-Pacific Research Center, Shanghai 200001, China

AN ADAPTIVE CONTROL STRATEGY FOR PROPER MESH DISTRIBUTION IN LARGE EDDY SIMULATION*

ZHANG Bin, WANG Tong, GU Chuan-gang

Key Laboratory for Power Machinery and Engineering of Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: sjtu2009@sjtu.edu.cn

DAI Zheng-yuan

Key Laboratory for Power Machinery and Engineering of Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China

Trane’s Asia-Pacific Research Center, Shanghai 200001, China

(Received July 15, 2010, Revised September 2, 2010)

The Filtering Grid Scale (FGS) of sub-grid scale models does not match with the theoretical Proper FGS (PFGS) because of the improper mesh. Therefore, proper Large Eddy Simulation (LES) Mesh is very decisive for better results and more economical cost. In this work, the purpose is to provide an adaptive control strategy for proper LES mesh with turbulence theory and CFD methods. A new expression of PFGS is proposed on the basis of ?5/3 law of inertial sub-range and the proper mesh of LES can be built directly from the adjustment of RANS mesh. A benchmark of the backward facing step flow at Re=5147 is provided for application and verification. There are three kinds of mesh sizes, including the RANS mesh, LAM (LES of adaptive-control mesh), LFM (LES of fine mesh), employed here. The grid number of LAM is smaller than those of LFM evidently, and the results of LAM are in a good agreement with those of DNS and experiments. It is revealed that the results of LAM are very close to those of LFM. The conclusions provide positive evidences for the novel strategy.

Large Eddy Simulation (LES), Proper Filtering Grid Scale (PFGS), turbulence, ?5/3 law, adaptive control

1. Introduction

The Large Eddy Simulation (LES) equations with the resolved turbulent scale were established through the low-pass spatial filtering of the Navier-Stokes equations, so the Filtering Grid Scale (FGS) of the spatial filter plays a decisive role in getting LES results[1]. The Proper FGS (PFGS) should be within or close to the inertial sub-range of turbulence[1,2]. However, the FGS solutions of present SGS models are given by the local computational mesh, such as those presented by Moeng and Wyngaard[3], Metai and Ferziger[4]and others. Because of the improper mesh, the FGS of sub-grid scale models was not match with the theoretical PFGS. It may lead to lose rational results (i.e., deficient grid numbers) or cost advantage (i.e., unnecessary mesh increment). Therefore, the research on proper LES mesh distribution is helpful for better turbulent data and more economical calculation.

Recently, the studies of LES meshing have focused on more accurate LES data. Celik et al.[5], Klein[6]and Jordan[7]investigated the influence of LES mesh resolution and attempted to provide some sensitive ways to measure the quality of the LES predictions. The ways laid emphasis on the probem about whether the LES meshing was rational by analyzing the computed results, but not the proper LES meshing before computation. Degrzia et al.[8]offered the variable mesh spacing method for LES in the convective boundary layer, but their method only provided the available meshing solutions in the vertical direction yet. Leonard et al.[9]addressed theproblem of coupling an Adaptive Mesh Refinement (AMR) method with LES. A multi-grid algorithm was used in order to refine the grid in zones detected by a specific sensor based on wavelet decomposition. Naudin et al.[10]also focused on the AMR method with LES and proposed an available sensor defined by turbulent kinetic energy. However, the AMR of LES was processed on the basis of the LES results of the first mesh, but some criteria about the first mesh were not given quantitatively. It would not only increase the grid number of LES (Actually, there were twice LES calculation), but also lead to the irrational second AMR mesh because of the improper first mesh.

In this work, the efforts are made to provide an available approach or a control strategy for proper LES meshing. With the turbulent theory and CFD methods, an adaptive control strategy for proper LES mesh is proposed and the specific procedure is given. A benchmark case of the Backward Facing Step (BFS) flow is applied for discussions and validations.

2. Numerical simulation methods

3. An adaptive control strategy for proper LES mesh

3.1 PFGS solution

The famous kolmogorov ?5/3 law[2]can be given by

where α, ε, k are the Kolmogorov constant (about 1.4), turbulent dissipation rate and wavenumber respectively. Assume that the dissipation length-scale of fully developed turbulent flow is small, then the wave-number range of inertial sub-range could be set as [kc,+∞). Equation (1) can be directly integrated

It is noted that the left side of Eq.(2) is equal to the sub-grid kinetic energysq in LES, so that

Equations (2) and (3) lead to

The relation between wave-numbersk and space scalein inertial sub-range reads

Equations (4) and (5) lead to

The energy ratio coefficientsη, i.e., the ratio of sub-grid kinetic energy to total turbulent kinetic energy (k), is defined as

According to the LES theory, PFGS Δ is located in inertial sub-range. Therefore,

By using the RANS model, k and ε can be calculated well. So the PFGS is expressed as

where the subscript “RN” indicates the estimates of RANS. In Eq.(9), the reasonable range ofsη can be analyzed: (1) Since the highest turbulent kinetic energy is included in resolved scale of LES, the proper value should range from 0 to 0.5 firstly. (2) Reference [13] concluded that LES should resolve at least 80% of the total turbulent kinetic energy. The value 20% was proposed as the threshold of an acceptable quality[10]. (3) Based on the turbulence theory, the magnitude of inertial sub-range should be smaller than that that of energy containing range, so that

where, the energy containing scale l=Ck1.5/ε and C is the constant taken close to unity[14].

The energy ratio should be

On the whole, 19.6% could be recommended for the coefficient ηsin Eq.(9).

3.2 Control procedure for proper LES mesh

To match the FGS of sub-grid scale models (related to the local computational mesh) with the PFGS in LES, a novel adaptive strategy for proper LES mesh from RANS mesh adjustment is developed. The procedure can be described as follows:

(1) The LES meshing problem should be established with the detailed information such as computational domains, boundary conditions, etc..

(2) The RANS equations are solved to get the needed turbulent information, including turbulent kinetic energy, dissipation rate and energy ratio coefficient.

(3) The PFGSΔccan be calculated from the RANS results. Then, apply RANS mesh as the original LES mesh.

(4) The FGS Δ of SGS model is determined implicitly by local mesh size, expressed as

(5) The Grid ratio coefficientσΔis defined as Δ/Δc. The coefficient is selected as the mesh adjustor for LES. Then, the σΔdistribution in the flow field can be got and the threshold value for mesh adjustment should be set.

(6) This step is the test: determining if the grid ratio coefficient of the total flow field is smaller than the threshold value (Selected as 1 to match the FGS Δ with the PFGS Δc, sometimes larger according to the practical demand).

(7) If Step (6) has not been true, the mesh of those non-satisfied fields should be refined. The adjusted mesh would be the updated original LES mesh and a new process started from Step (4) until (6) becomes true.

(8) If Step (6) has been true, the adjusted mesh will be set as the proper mesh for LES and the process is over.

In LES, the Werner-Wengle wall function is adopted and the mesh size requirement for the near wall can be expressed as follows:

(1) The basic requirement for the first grid node near wall: y+＜100.

(2) Even if the mesh requirement for the near wall can be lowered due to the wall function, it is assured that y+～1 for the first grid node near wall for better LES data.

(3) When near-wall y+～1, the near-wall mesh do not need to be refined for the wall function, although the coefficient may be larger than threshold value. If not, there will be unnecessary mesh increment.

4. Application and verification

4.1 BFS flow

Many researchers[15,16]conducted studies of BFS flow. The computational domain of BFS is shown in Fig.1. The main parameters are listed in Table 1. Structured grids were used for RANS simulations. Non-uniform grid cells were adopted in the x and y directions while uniform grid cells were used in the z direction. The grid near wall was refined for the needed y+(＜100). To offer the valid data for LES meshing, the RANS mesh was obtained by the grid independence of the separation point location taken as the test quantity. The boundary conditions were expressed as: inlet boundary: velocity inflow, outlet boundary: the Sommerfeld radiation condition, upper boundary: symmetrical boundary condition, inner and outer boundary in the z direction: periodic boundary condition, the other boundaries: no-slip boundary condition.

4.2 Adaptive control for proper LES mesh

For convenience, the flow of mid-section in the z direction was selected for description. The RANS mesh, RANS-based results: PFGS distribution and the grid ratio coefficient are illustrated in Figs.2-4. And the grid ratio coefficient distribution over threshold isshown in Fig.5. The results show that the local mesh needed to be adjusted in the angular zone, recirculation zone, partial shear layer region, reattachment region and internal boundary layer. After the adaptive control form RANS mesh adjustment, the proper LES mesh (Fig.6) is got and its grid numbers are about 4.2×105. It is found that the novel adaptive LES meshing approach would lead to well-bedded and clear physics-based LES mesh distribution.

4.3 Validation and comparison

In order to validate the above approach for proper LES mesh, three kinds of mesh sizes (see Table 2), including the LAM (LES of adaptive-control mesh), LFM (LES of fine mesh (coarsening DNS mesh by a factor of 2 in each direction)) and DNS (hyperfine), are employed for comparison and validation. The LES results would be compared withDNS data[15]and experimental data[16]. It is shown that the number of LAM is smaller than those of LFM and DNS evidently. In LES, the time step is 0.05 (non-dimensional time). Time averaging is obtained for 500 non-dimensional times and spatial averaging was accomplished at the last time step. The mean velocity profiles, turbulent intensity and Reynolds shear stress are normalized with the inflow free stream velocity. Fine, but not the same accordance could be seen between the experimental and DNS data. However, DNS and experimental data could be combined to verify the LES data better. Based on LES results, DNS and experimental data, the detailed analysis could be expressed below:

(1) Velocity profiles

Figure 7 shows the comparison among the computed LES data (LAM and LFM), DNS and experimental data for the non-dimensional mean stream velocity profiles. The comparisons are made at four locations in the recirculation zone (x/H=4), reattachment region (x/H=6) and recovery regions (x/H=10and x/H=19) respectively.

The discrepancy between the mean velocity profiles of the DNS and the experiments is observed, especially for the region y / H ＜ 1. But, the difference between them does not affect the following validation. At x/H=4, the results of LES are over-predicted in the region y / H ＜ 0.5 and under-predicted in the region 0.5 ＜ y / H ＜ 0.5 relatively. The computed LES results of LAM and LFM accord well with the DNS results at x / H = 6. The stream velocity of LES data is over-predicated slightly atx/H=10, especially y / H ＜ 0.5. It is observed that the stream velocities of LES data are over-predicated at x/H=19, y / H ＜ 1.5. On the whole, the results of LAM are in a relatively good agreement with those of DNS and experimental data. It is also found that the results of LAM with relatively smaller cells are.

(2) Turbulent fluctuation characteristics

Figures 8-10 give the streamwise turbulent intensity, normal turbulent intensity and Reynolds shear stress, as compared to DNS and experimental data, where u′ and v′ were the velocity fluctuations in the streamwise and normal directions respectively. The comparison is made at the same four locations. When x/H=4 and 6, the streamwise turbulent intensity of LES data (LAM and LFM) accords well with DNS especially for the region y / H ＜1.25 and is under-predicted for the other regions. In the recovery region (x / H=10 and 19), it is close at y / H ＜1 and slightly over-predicted at y / H ＞ 1. At x/H=4, the normal turbulent intensity of LES is under-predicted but relatively close to those of experiments. At x/H=6, 10 and 19, the results of LES are compared well with those of DNS data at y / H ＞ 1 and under-predicted in the other regions. As for the Reynolds shear stress, those of LAM are a little superior to the results of LFM actually. When the grid number of LAM is smaller than those of LFM evidently, the results of LAM are very close to those of LFM and in a good agreement with those of DNS and experiments.

5. Conclusions

(1) Based on turbulence theory and CFD methods, a novel solution of PFGS has been put forward on the basis of ?5/3 law of inertial sub-range. It is found that the PFGS is related to turbulent kinetic energy, turbulent dissipation rate and energy ratio coefficient.

(2) To match FGS of sub-grid scale models with PFGS, an adaptive control strategy for proper LES mesh has been developed. The grid ratio coefficient is selected as the parameter for the mesh adjustment. The proper mesh of LES could be generated from the adjustment of RANS mesh.

(3) A benchmark of backward facing step flow at Re = 5 174 has been tested. Different kinds of mesh size are employed for the comparison and analysis. It is shown that the grid number of LAM is smaller than those of LFM obviously for getting the same results and the results of LAM are very close to those of LFM. The conclusions provide positive evidences.

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10.1016/S1001-6058(09)60127-X

* Project supported by the National Natural Science Foundation of china (Grant No. 50776056) the National High Technology Research and Development of China (863 Program, Grant No. 2009AA05Z201).

Biography: ZHANG Bin (1983- ), Male, Ph. D. Candidate

WANG Tong,

E-mail: twang@sjtu.edu.cn