AIAA 2016-0339 AIAA SciTech 4-8 January 2016, San Diego, California, USA 54th AIAA Aerospace Sciences Meeting
Validation of a Window-Embedded RANS/LES Method Based on Synthetic Turbulence
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Zhao Li1, Haixin Chen2 and Yufei Zhang3 School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China
A Window-Embedded Reynolds-Average Navier-Stokes / Large Eddy Simulation (WERANS/LES) method using an explicit interface for integrated computation in different zones is presented. The scale-adaptive k-kL turbulence model is chosen in the RANS zone, whereas the implicit LES is based on the MDCD scheme and the SLAU scheme. To bridge the spatial and temporal scale gap between the RANS and the LES computations, synthetic turbulence is generated at the interface. A window boundary condition considering two-way information exchange is developed by introducing a weight function. A unified digital filter (UDF) is employed to approximate the turbulence-like spectrum with a relatively simple expression. After the UDF method is validated in the space-developed channel flow, the overall WE-RANS/LES method is applied to the simulation of the separated flows over periodic hills. The results are close to those of the full LES but the computational cost is lowered. This demonstrates the advantages of the proposed WE-RANS/LES method in simulating separated flows and encourages further practical uses.
I. Introduction
T
he detail design of modern aircrafts heavily depends upon the accurate prediction of the unsteady turbulence flows. The traditional Reynolds-Average Navier-Stokes (RANS) method fails to resolve small eddy motions, whereas the Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) methods are time-consuming and unaffordable for practical uses. The combination between LES of the high accuracy and the Unsteady RANS (URANS) of the low cost is a feasible choice and thus is increasingly attractive to industry. One of these hybrid methods is the embedded or zonal RANS/LES method (ELES), which employs LES in the key zone and the cheaper URANS or RANS elsewhere. Recently, Schlüter et al[1, 2], Quéméré and Sagaut[3], Batten[4], Menter et al[5] and Davidson and Peng[6] have developed their embedded methods. It is believed that ELES requires much lower computational cost than the full LES with flexible settings. Menter [5] pointed out that this method was especially suitable for the moderately and marginally unstable flows. Compared with other hybrid methods such as the Detached Eddy Simulation (DES)[7], the zonal interface is explicitly defined for this ELES. The two parts of the flow field could be processed simultaneously by individual technologies, and could be connected by a two-way information exchange through the interface. Particularly, turbulence fluctuations are introduced to fill in the gap of the resolved scales between RANS and LES. This may help to avoid such problems in DES according to Fröhilich and Terzi[8], Georgiadis et al[9] and Tucker[10]. Synthetic turbulence generation (STG)[11] has been widely used for this purpose ever since the
idea was proposed by Kraichnan[12]. Their increasing need of the efficient ELES method has encouraged numerous versions of the STG method to be developed. The flexibility of the ELES framework also brings about seemingly random choices of the elements such as turbulence models and interface treatment. Various combinations have been invented but the study on ELES is still preliminary and unsystematical. The synthetic turbulence added at the inlet is not always effective in improving the numerical predictions, for example, in the studies by Montorfano et al.[13] or Abboud and Smith[14]. Besides, more attention is paid on the inlet condition of LES zone than the outlet. In fact, the instantaneous flow yields two-way exchange of information at the subsonic interface. Simple time-space average at the outlet may lead to computational instability[3]. Most of the existing methods predefine the inlet and the outlet boundaries and treat them separately.
1
Ph.D,
[email protected]. Student Member AIAA. Professor,
[email protected]. Associate fellow AIAA. Corresponding Author. 3 Assistant professor,
[email protected]. Member AIAA. 1 American Institute of Aeronautics and Astronautics 2
Copyright © 2016 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
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Motivated by the above problems, a Window-Embedded RANS/LES (WE-RANS/LES) method is presented. The elements are carefully selected. The STG method employs a unified digital filter (UDF) to satisfy the two-point correlation while retaining a simple formula and low cost. A scale adaptive simulation with the k-kL turbulence model is used in the RANS zone. The adaptively unsteady computation is important especially when the LES zone does not include all the separation parts. The implicit LES (ILES) method effectively decrease the numerical dissipation for Finite Volume Method (FVM) that is widely used in aircraft industry. This combination facilitates the formation and sustain of the unsteady turbulence in the LES zone. Particularly, a weight function is introduced to handle the two-way information exchange at the zone interface, whether at the inlet or at the outlet. The rest of the paper is organized as follows: Section 2 outlines the WE-RANS/LES method including the window interface treatment and the UDF to generate the synthetic turbulence. Then this method is validated in the attached flow along the channel and the separated flow over periodic hills. Results are presented and discussed in Section 3. The final section contains concluding remarks of this study.
II. Method A. Window-Embedded RANS/LES ELES is such a flexible framework that proper methods for the RANS zone, the LES zone and the interface can be chosen and combined for turbulence-resolved simulation. In the present study, a Window-Embedded RANS/LES method (WE-RANS/LES)[15] is developed by applying the ELES concept into the Window-Embedded grid technology[16] that is well incorporated in the in-house code NSAWET[17]. This allows different spatial resolutions, time resolutions, turbulence models and numerical schemes on both sides of the interface. On the LES side, a fine grid and a small time step are employed with the ILES method[18] based on the low-dissipation MDCD/SLAU scheme[19, 20]. On the other side, the grid is coarser and the time step is larger to save computational cost. The RANS model is the k-kL two-equation model[21] that exhibits the scale adaptive features in unsteady flow simulations. Both of the two turbulence simulation methods have been validated in the previous study separately[15, 18], which also benefits from the flexibility of the ELES framework.
Figure 1 Flow chart of WE-RANS/LES. VLES and VRANS denote the conservative variables for LES and RANS respectively, and KRANS denotes the modelled turbulence variables for RANS Before considering the window embedment, as shown in Fig.1, simulations in the LES region or in the RANS region work well by itself. Then data are exchanged through the interface serving as the boundary condition. The two-way information transmission is conducted by the weighted average of both the boundary cells u1mj and the internal cells u2mj . For instance, the window boundary condition for the interface reads
u mj f (U n ) u1mj (1 f (U n )) u2mj , with the weights simply defined as: 2 American Institute of Aeronautics and Astronautics
(1)
1, f (U n ) 0.5, 0,
U n / U ref 1 1 U n / U ref 1 ,
(2)
U n / U ref 1
where the reference velocity U ref can be the farfield velocity in the incompressible flow and the local speed of sound in the compressible flow. The internal cell variables employ the convective boundary condition[22]:
u2mj c1u mj 1 c2u mj 11 c3u mj 1 ,
(3)
where c1, c2 and c3 are predefined constants and the boundary cell variables can be constructed by any STG method:
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u1mj v mj .
(4)
B. A unified digital filter In the present study, a unified digital filter (UDF)[23] is used that is inspired by the Klein’s digital filter method[24] and its exponential variant developed by Xie and Castro[25]. The synthetic fluctuations are generated by filtering the white noise rn :
v(x m )
N
bn rm n ,
(5)
n N
with the three-dimensional filter as bn bi b j bk ,
bk bk /
N
(6)
bj , 2
(7)
j N
where the UDF is formulated by introducing a parameter CK to adjust the energy spectrum.
b j exp(
CK
j n
CK
).
（8）
The Klein’s method satisfying the Gaussian spectrum is obtained with CK =2 and the XC filter with CK =1. This parameter is selected as CK =0.75 here so that the spectrum agrees well with the turbulence energy spectrum[23].
III. Results and discussion A. Space-developed channel The UDF method in WE-RANS/LES is compared with other STG methods in ILES of the space-developed channel flow at Reτ=395. The outlet turbulence is decayed by coarsening the downstream mesh. Instead of the well-developed turbulence when the LES results of time-developed channel flow is used at the inlet, as shown in Figs. 2 and 3, a certain distance is required for the recovery of turbulence when STG is used. The UDF method needs a relatively shorter distance among several typical STG methods[24, 26, 27]. For the RANS region, a similar weighted average to Eq. 1 is employed, but the modelled turbulence variables of the external cell are provide by a prior steady RANS simulation. Directly filtering the modelled information from the LES
data always leads to the computational instability[3], which should be attributed to the essential distinct of the idea between LES and RANS. It is noteworthy that the one of the advantages of this WE-RANS/LES method is the flexible switch to new and advanced approaches in the future study.
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(b)
(c) (d) Figure 2 Instantaneous vortical structures calculated by the different inlet boundary conditions. (a) LES data; (b) RFG[27]; (c) DFSEM[26]; (d) Klein’s digital filter[24]. Shown by the iso-surfaces of Q/U2=2. Data_LES RFG DFSEM DF_K UDF
0.008
0.006
Cf
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(a)
0.004
0.002
0
5
10
x
15
20
25
Figure 3 Skin friction coefficients along the streamwise direction. B. Periodic hills The WE-RANS/LES method is then applied to the flow over periodic hills as illustrated in Fig. 4. Flöhilch and Terzi[8] has employed a three-section simulation with the outlet boundary not exactly at the section interface, which makes the simulation more challenging. Different from their case, this study focuses on the two-way exchange without differentiating inlet and outlet. Therefore, two periodic sections are used with one for RANS and the other for LES. At the two interfaces between the sections, the window condition is implemented with the velocity fluctuations synthesized from UDF. The SST model and the k-kL model are compared in the RANS region. 4 American Institute of Aeronautics and Astronautics
The Reynolds number is 10600 based on the bulk velocity above the crest and the hill height, and the bulk Mach number is set to be 0.2 to satisfy the incompressibility. The cell numbers of the RANS and LES zones are 72×144 ×40 and 130×144×120 respectively. In contrast, the cell numbers of the separated RANS and LES simulations for comparison are 96×112×64 and 192×224×160. It is noteworthy that there is no significant difference between the 2D steady simulation and the 3D unsteady simulation for the SST model[15]. Inlet
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3D LES Periodic Boundaries
Convective Boundary 3D LES
2D RANS
(a) Interface (Periodic Boundary)
Interface (Periodic Boundary)
Interface
RANS Region
LES Region
(b)
(c) Figure 4 Mean streamwise velocity and stream lines. (a) Flöhilch and Terzi[8]; (b) WE-RANS/LES using the SST model in the RANS region; (c) WE-RANS/LES using the k-kL model in the RANS region. Shown are 14 levels from u/U=-0.25 to u/U=1.05.
(a) (b) Figure 5 Instantaneous vortical structures shown by the iso-surfaces of Q/U2=2.5. (a) WE-RANS/LES using SST model in the RANS region; (b) WE-RANS/LES using k-kL model in the RANS region. 5 American Institute of Aeronautics and Astronautics
For the WE-RANS/LES computation, Fig. 4 shows that the two RANS models perform similarly with a slightly overpredicted recirculation bubble, but the results of the LES part with the k-kL model are much closer to the LES data of Flöhilch and Terzi[8] than those with the SST model. The scale adaptability of the k-kL model is beneficial for the transport of the key flow structure downstream as the LES inlet condition as shown in Fig.5. Figure 6 compares the mean velocity distributions of the LES part of WE-RANS/LES with those of the independent LES calculation using single periodic section[18], whereas Fig. 7 compares the results of the RANS part[15]. It can also be seen that the results with the k-kL model agree better with the experimental measurement and the full LES data than those with the SST model. Surprisingly, as shown in Fig. 7, the window embedded simulation improves the mean streamwise velocity predicted by the SST model. Its effect on the k-kL simulation is minor. This indicates that the LES zone provides reasonable boundary conditions for the RANS simulation through the window condition. 3.0
2.5 Exp. LES WE+k-kL WE+SST
2.0
y/h
y/h
2.5
2.0
1.5
1.0 1.5
0.5
1.0
0.0
0.5 /U
1.0
0.0 /U
0.0 -0.5
0.1
0.0
0.5 /U
1.0
(a)
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(b)
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y/h
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3.0
1.5
1.5 1.0
1.0 0.5
0.0
0.5 0.0
0.5 /U
1.0
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1.0
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0.1 0.2 /U
0.3
(c) (d) Figure 6 Mean velocity of the periodic hill flow predicted by LES at different streamwise locations. (a) x/h=0.05; (b) x/h=2.0; (c) x/h=5.0; (d) x/h=8.0. The experimental results are obtained from online database[28].
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3.0
3.0
2.5 Exp. k-kL SST WE+k-kL WE+SST
2.0
y/h
y/h
2.5
2.0
1.5
1.0 1.5
1.0
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0.5
1.5
1.5 1.0
1.0 0.5
0.0
0.5 0.0
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1.0
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0.0 /U
0.0
0.5 /U
1.0
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0.3
(c) (d) Figure 7 Mean velocity of the periodic hill flow predicted by RANS at different streamwise locations. (a) x/h=0.05; (b) x/h=2.0; (c) x/h=5.0; (d) x/h=8.0. The experimental results are obtained from online database [23].
IV. Conclusions In summary, the WE-RANS/LES method is introduced by combining the k-kL turbulence model, the ILES method and the window condition with the UDF method. Numerical simulations verify its cost-effective feature in predicting complex separated flows with lower cost than the full LES. The window condition also reflects the good balance between the effectiveness and the code feasibility. This indicates the promising applicability of the proposed WE-RANS/LES method in simulating similar separated flows.
Acknowledgments This work was supported by National Key Basic Research Program of China (2014CB744801) and National Natural Science Foundation of China (11102098 and 11372160).
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References
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