Han Qi , Xinliang Li , Ning Luo , , Changping Yu a,

a LHD, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China

b School of Engineering Science, University of Chinese Academy of Sciences, Beijing 10 0 049, China

c State Key Laboratory for Geomechanics and Deep Underground Engineering,China University of Mining and Technology, Xuzhou 221116, China

Keywords: Subgrid-scale kinetic energy Eddy-viscosity model Compressible flow

ABSTRACT The subgrid-scale (SGS) kinetic energy has been used to predict the SGS stress in compressible flow and it was resolved through the SGS kinetic energy transport equation in past studies. In this paper, a new SGS eddy-viscosity model is proposed using artificial neural network to obtain the SGS kinetic energy precisely, instead of using the SGS kinetic energy equation. Using the infinite series expansion and reserving the first term of the expanded term, we obtain an approximated SGS kinetic energy, which has a high correlation with the real SGS kinetic energy. Then, the coefficient of the modelled SGS kinetic energy is resolved by the artificial neural network and the modelled SGS kinetic energy is more accurate through this method compared to the SGS kinetic energy obtained from the SGS kinetic energy equation. The coefficients of the SGS stress and SGS heat flux terms are determined by the dynamic procedure. The new model is tested in the compressible turbulent channel flow. From the a posterior tests, we know that the new model can precisely predict the mean velocity, the Reynolds stress, the mean temperature and turbulence intensities, etc.

Large-eddy simulation (LES) has been widely used in predicting turbulence and gradually applied to high Reynolds number cases. In LES, the eddy-viscosity model is a very popular model due to its strong stability. The Smagorinsky model (SM) [1] is most pop- ular eddy-viscosity model and is used to many different cases, but the SM shows excessive dissipation and cannot predict the tran- sitional flow. Thus, many different types of eddy-viscosity models are proposed. The wall-adapting local eddy-viscosity model [2] was proposed by Metais, which shows correct behavior in the near- wall region. Vreman obtained a low dissipation model [3] that can predict transitional flow well. Considering the helicity effects, Yu et al. [ 4,5 ] supplied a new eddy-viscosity model and it was ap- plied to predict compressible transitional flows Qi et al. proposed an eddy-viscosity model based on the vorticity gradient tensor for rotating turbulent flows [6] . The subgrid-scale (SGS) kinetic energy equation model (k-equation model) is also a type of eddy-viscosity model, which was added the SGS kinetic energy equation to ob- tain the eddy viscosity. Thek-equation model was proposed by Schumann [ 7 ] through dimensional analysis and Yoshizawa [ 8 ] also obtained thek-equation model by the two-scale direct interaction approximation. Then thek-equation model was applied to com- pressible flows [9] . Chai et al. [ 10 ] supplied ak-equation model where the SGS terms in the equation are modelled independently and the coefficients of the SGS models are determined dynamically . Except for the eddy-viscosity model, the structural model is an important type of SGS models, such as scale-similarity model and gradient model. The scale-similarity model was obtained based on the scale-similarity hypothesis [11] . The gradient model is derived from Taylor expansions for SGS stress [ 12 , 13 ]. The structural model has a high correlation with the real SGS stress but is unstable.

In addition, some LES methods have been introduced to help improving the predicting effects in LES. Using the Germano iden- tity, Germano et al. [ 14 ] supplied the dynamic procedure which can dynamically determine the coefficient of the SGS models. Then, Lilly [ 15 ], Ghosal et al. [ 16 ] and Meneveau et al. [ 17 ] obtained a scale-dependent dynamic SGS model and generalized the dy- namic procedure to any scalar flux model. Except for the dynamic methods, some other LES methods have been proposed recently. Chen et al. [ 18 ] suggested to constrain the SGS stress model by Reynolds stress for LES of incompressible wall-bounded turbulence to improve the prediction of the averaged quantities, and then Jiang et al. [ 19 ] generalized this method to compressible cases. Do- maradzki et al. [ 20 ] proposed a new method where the total SGS energy transfer is used to constrain the SGS models and update model constants.

Recently, artificial neural networks (ANNs) have been increas- ingly applied to develop turbulence models [ 21 ]. Ling et al. [ 22 ] supplied a new Reynolds stress anisotropic tensor through a new multiplicative-layer neural network with an invariant tensor firstly, which can obtain obvious improvement in simulation result. Us- ing machine learning and through optimal evaluation theory anal- ysis, Vollant et al. [ 23 ] obtained a new SGS scalar flux model which can predict results much closer to the DNS results. With the ANN method, Xie et al. [ 24 ] supplied the coefficients of the mixed model which combined the Smagorinsky model and the gradient model for compressible isotropic turbulence and the new method have better behaviours than the traditional LES models. Through the relationship between the resolved-scale velocity gradient ten- sor and the SGS stress tensor, Zhou et al. [ 25 ] proposed a new SGS model using the ANN method for isotropic turbulence. Park and Choi [ 26 ] obtained an SGS model for LES of turbulent channel flows using a fully connected neural network . The new model can have good performance and is not affected by the grid resolution Yuan et al. [27] .

In this paper, a new dynamic eddy-viscosity model (NDKM) is proposed for LES of compressible flow. In this new model, the eddy-viscosity is supplied by the SGS kinetic energy, which is ob- tained by the infinite series expansion. And the coefficient of the modelled SGS kinetic energy is determined by the artificial neural network.

Filter the Navier-Stokes (N-S) equations and the filtered N-S equations for compressible in LES can be written as

And(ˉ·)represents spatial filtering with a low-pass filter at scaleΔandrepresents density-weighted (Favre) filteringIn the filtered N-S equations,andare the filtered den- sity, velocity, pressure and total energy, respectively. The filtered pressure is determined by, whereRis the specific gas constant. In the equations,Pris the molecular Prandtl number and the molecular viscosityμtakes the formac- cording to Sutherland’s law, in whichTsis 110.3 K, the Reynolds numberRetakes the formRe=ρ∞U∞L/μ∞.

Based on Boussinesq type hypothesis, the eddy-viscosity model can be written as

For the SGS heat flux model, the commonly used SGS diffusion model is as

andPrsgsis the SGS Prandtl number.

In the compressible dk-equation model, the modelled SGS stress and SGS heat flux can be written as

In compressible dk-equation model,ksgsis solved by the SGS ki- netic energy equation. In the following, we will introduce another method to obtainksgs.

First, we introduce the infinite series expansion [28] as

where

In Eq. (17) ,G(x,y) is the kernel of the filter and designated as the box filter inaprioriand the grid filter inaposterior.

When applying the infinite series expansion to SGS kinetic en- ergy, one can obtain

where Ckis the coefficient andΔkis the grid width in thexkdi- rection.

For avoiding the complexity of additional boundary conditions and due to the other higher-order terms are small enough com- pared to the first term, we only reserve the first term and it can be expressed as

For obtaining the coefficientCk, we will use artificial neural net- work in the next part.

The coefficientCsandPrsgsare determined dynamically by the Germano identity. For any terma=, we assume thatA=holds on the test filter level, wheredenotes test filter- ing. The Germano identity is then defined byL=A?Assume that the model foraisa=Cm, wheremis a function of the resolved (grid filter level) quantities; then at the test filter level,A=CM, whereMtakes similar form tombut is a function of the test-filtered quantities. Substituting the models forAanda, the Germano identity becomes

The model coefficientCcan be solved dynamically as

The coefficientCvaries with time and space. To avoid compu- tational instability,Cis regularized using a combination of least- square method and volume averaging. For the coefficient of SGS stressCsis

where

For the coefficient of SGS heat flux model, it can be determined dynamically as

where

In our study, we use an ANN to construct the coefficientCkin compressible turbulent channel flow. The data selected for train- ing and testing in this study are obtained from the direct numeri- cal simulation (DNS) data of a temporally compressible isothermal- wall turbulent channel flow [29] . In this case, the Mach num- berMa= 1.5 , the Reynolds numberRe= 30 0 0 , and the friction Reynolds numberReτ=uτδ/ν= 220 (uτandδare the friction ve- locity and the half width of the channel). The computation domain for the DNS of channel flow is a box with a size of 4 π×2 ×4/3 π, and the grids for DNS are 900 ×201 ×300 andΔz+= 3 ×0.32 ×3 , whereandare the mesh spacings of wall units in streamwise, wall-normal, and spanwise directions. During the course of training and test- ing, the DNS data are filtered in streamwise and spanwise direc- tions with a box filter. The input features of the ANN are critical to the performance of predicting the coefficientCk. A set of in- put variables are dimensionless quantities, where several variables may be selected in compressible wall-bounded turbulence, such asΔ+,y+,ReΔ, and. In this paper,is the nor- malized filter width,is the dimensionless normal distance,ReΔ=is the mesh Reynolds number, and. The filtered wall friction ve- locity isis the wall shear stress and 〈·〉 is denoted as the spatial average along the homogeneous directions. Figure 1 shows schematic diagram of the artificial neu- ral network for predicting the model coefficientsCk. Table 1 shows a set of inputs and outputs for different ANN models. In this pa- per, a total of four layers (an input layer, two hidden layers and an output layer) with neurons in the ratioM: 100: 100: 1 andMis the number of input variables listed in Table 1 . The activa- tion functions of the hidden layers and output layer are the hy- perbolic tangent function (σh(x)=(ex?e?x)/(ex+ e?x)) and lin- ear function (σo(x)=x). The mean-squared error (MSE) function is chosen as the loss function of the ANNwhereanddenote the true and predicted values of the ANN.)

Table 1 A set of inputs and outputs for different ANN models.

In this study, we select 2 ×104samples from 20 snapshots of the filtered DNS data with a ratio of the filter width Δ/ ΔDNSrang- ing from 2 to 20 (Δ/ ΔDNS∈ {2, 4, …, 20}). The cross-validation strategy is used and the dataset is divided into a training set and testing set to suppress parameter overfitting of the ANN. We randomly extract seventy percent of the samples from the total dataset and are used as the training set, while the others are used for testing. The weights of the ANN are initialized by the Glorot- uniform algorithm and optimized by the Adam algorithm [30] for 1 ×104iterations, with a batch size and learning rate of 10 0 0 and 0.01, respectively. To determine the optimal hyperparameters, such as the numbers of layers and neurons and the types of activation functions, we choose the grid search method as the hyperparame- ter pruning method of the ANN.

For seeing the rationality of the selected hyperparameters, in Table 2 , we supply the correlation coefficientC(Ck), the relative er- rorEr(Ck), and the ratio of the root-mean-square valueR(Ck), which are defined, respectively, as

Table 2 Correlation coefficient(C), relative error(Er), and ratio of root-mean-square value

From the results in Table 2 , we know that the selected hyper- parameters are reasonable and the ANN models are well trained.

(R) of the coefficientCkin different datasets for different ANN models.

Figure 2 shows comparisons of the coefficientCkreconstructed by different ANN models along the normal direction with different filter widths. From the figures, we know that the modelled SGS kinetic energy modified by the ANNs models can have similar re- sults and have good agreement with the DNS results. We choose the ANN3 model in the following testing because the ANN3 model has the best behaviours in the three ANN models.

In this section, the new model is tested in compressible turbu- lent channel flow. The case setting of the LES in this part is same as that of the DNS in previous part. The governing equations are solved by a high-precision non-dimensional finite difference solver in Cartesian coordinates: the third-orderR- scheme is chosen as the time integrating method, and a sixth-order central difference scheme is used for the discretization of both the convective and viscous terms. The grid filter width isΔ=(ΔxΔyΔz)1/3withΔx,ΔyandΔzrepresenting the local grid width, and the test-filter width is set as 2Δ. Table 3 shows the grid setting and the main pa- rameters for the simulations in the compressible turbulent channel flow.

Table 3 The grid setting and the main parameters for the simulations in the compressible turbulent channel flow (Ma = 1 . 5 and Re = 30 0 0).

Figure 3 shows the profiles of Van Driest transformed mean velocityand the mean temperature=(Tw?〈T〉)/Tτobtained from DNS, the NDKM and the dk-equation model.Tτ=BqTwis the friction temperature,is the nondimensional heat flux, andqwis the wall-normal heat flux. From the figures, we know that the NDKM can have perfect agreement with the DNS results, but the dk-equation model shows deviations aty+>30.

The profiles of the total Reynolds stress and the total turbu- lent heat flux from DNS and different SGS models are shown in Fig. 4 . In Fig. 4 a, the NDKM can have perfect behavior at almost regions but the dk-equation model shows worse results. In Fig. 4 b, the NDKM has good performance but the results from the NDKM are a little higher than results from DNS at 15

Fig. 1. Schematic diagram of the artificial neural network for predicting the model coefficient C k .

Fig. 2. Comparisons of the coefficient C k reconstructed by different ANN models along the normal direction with different filter widths: (a) Δ/ ΔDNS = 4 ; (b) Δ/ ΔDNS = 8 ; (c) Δ/ ΔDNS = 12 ; (d) Δ/ ΔDNS = 16 (the Model in the pictures means that C k is 1/12 and.

Fig. 3. The profiles of Van Driest transformed mean velocity and the mean temperature from different SGS models and results from DNS is as comparison (‘Coleman et al.’ is data from Coleman et al. [29]).

Fig. 4. The profiles of the total Reynolds stress normalized by ρw and u τ, and the total turbulent heat flux normalized by ρw , u τand T w from DNS and different SGS models.

The profiles of the resolved turbulence intensities from DNS, the NDKM and dk-equation model are shown in Fig. 5 a, 5b and 5c. From the figures, we know that the NDKM shows better predic- tions than the dk-equation at almost regions. Figure 5 d shows the turbulent kinetic energy from DNS and different SGS models. In the figure, we know that the NDKM can well predict the total tur- bulent kinetic energy and the SGS part. We also can infer that the NDKM can obtain better SGS kinetic energy than the dk-equation model.

Fig. 5. The profiles of the resolved turbulence intensities normalized by the friction velocity u τand the turbulent kinetic energy from DNS and different SGS models: (a) Streamwise turbulence intensity; (b) Wall-normal turbulence intensity; (c) Spanwise turbulence intensity; (d) The turbulent kinetic energy.

Fig. 6. The profiles of the resolved RMS density fluctuations normalized by averaged density ρa v , and the resolved RMS temperature fluctuations normalized by averaged temperature T a v from DNS and different SGS models: (a) density fluctuations; (b) temperature fluctuations.

Fig. 7. The profiles of the van Driest transformed mean velocity U vd and mean temperature T+ av obtained from different SGS models and DNS. The DNS results are from Coleman et al. [29] .

Fig. 8. The profiles of the total Reynolds stress and the turbulent heat flux normalized by ρw , u τand T w from different SGS models and DNS. The DNS results are from Coleman et al. [29] .

Figure 6 shows that the profiles of the resolved RMS density fluctuations and the resolved RMS temperature fluctuations from DNS and different SGS models. From the figures, we know that the NDKM can well predict the resolved RMS density fluctuations and the resolved RMS temperature fluctuations. The dk-equation model can also show good behaviours but still behaves a little worse than the NDKM.

Next, we will test the new model in the case ofMa= 3.0 andRe= 4880 . The size of the computational domain, the Prandtl numberPr, the boundary conditions, the ratio of specifific heats and the setting of LES solver are the same with the case ofMa= 1.5 andRe= 30 0 0 . The grid setting and the main parameters for this case are shown in Table 4 .

Table 4 The grid setting and the main parameters for the simulations in the compressible turbu- lent channel flow (Ma = 3 . 0 and Re = 4880).

Figure 7 shows the profiles of the Van Driest transformed mean velocity U vd and mean temperatureT+av obtained from different SGS models and DNS. Figure 8 shows the profiles of the total Reynolds stress and the turbulent heat flux from different SGS models and DNS. From the figures, we can see that the NDKM can also obtain better results than dk-equation model in the case of higher Mach number.

In this paper, we propose a new eddy-viscosity model for large- eddy simulation of compressible flow. In this new model, the eddy viscosity is obtained by the subgrid-scale kinetic energy. The SGS kinetic energy is an unclosed term and for resolving this quan- tity, we apply the infinity series expansion to it. Since the other higher-order terms are small enough compared to the first term, and for avoiding the complexity of additional boundary conditions, the first term of the expanded quantity is reserved as the modelled SGS kinetic energy. And the coefficient of the modelled SGS kinetic energy is determined by the artificial neural network. The coef- ficients of the eddy-viscosity model are determined dynamically by the Germano identity. The new model is tested in compress- ible turbulent channel flow and it shows that the new model can show better behavior than the dynamic SGS kinetic energy equa- tion model, including the mean velocity profile, the mean temper- ature profile, the RMS quantities, the total Reynolds stress and the turbulent heat flux, etc. Compared to the SGS kinetic energy equa- tion model, the NDKM can obtain precise SGS kinetic energy and has higher computational efficiency.

In summary, a new SGS eddy-viscosity model is proposed using the artificial neural network to predict the SGS kinetic energy for LES of compressible flow, and it can present good results. In future researches, the new model will be applied to high Mach number flows in complex geometries.

Declaration of Competing Interest

The authors declare that they have no competing interests and all author(s) read and approved the final manuscript.

Acknowledgements

This work was supported by the National Key Research and Development Program of China (Grant Nos. 2020YFA0711800 , 2019YFA0405302) and NSFC Projects (Grant Nos. 12072349, 91852203), National Numerical Windtunnel Project, Science Chal- lenge Project (Grant No. TZ2016001), and Strategic Priority Re- search Program of Chinese Academy of Sciences (Grant No. XDC010 0 0 0 0 0). The authors thank the National Supercomputer Center in Tianjin (NSCC-TJ) and the National Supercomputer Center in GuangZhou (NSCC-GZ) for providing computer time .