For the closure of the subgrid-scale (SGS) stress tensor, an artificial neural network (ANN)-based SGS model that takes account of the inverse energy cascade in isotropic turbulence is developed. The data required for training this ANN-based SGS model are provided by direct numerical simulation of isotropic turbulence with an inverse energy cascade. Two input features, the root mean square of the rate-of-strain tensor and the product of the eigenvalues of the rate-of-strain tensor, are employed to characterize the inverse energy cascade. An *a priori* test reveals that the ANN-based model adequately predicts the SGS stress tensor in the backward energy transfer process, and the predictive capability of the gradient model is found to be slightly poorer than that of the ANN-based model, while that of the Smagorinsky model is not satisfactory. In comparison with the gradient model, the ANN-based model even predicts a few backward energy transfer events in the stage of excessive energy dissipation. In addition, the off-diagonal component of the SGS stress tensor, rather than the diagonal component, may be intimately associated with the inverse energy cascade. The ANN-based SGS model presented here is expected to provide inspiration for future investigations of the construction of SGS models that take account of the inverse energy cascade.

## I. INTRODUCTION

Turbulence is a common physical phenomenon that can be found in many environmental and engineering fluid flows. Turbulent dynamics are described by the Navier–Stokes equations. As is well known, these equations are highly nonlinear and therefore difficult to study analytically. Direct numerical simulation (DNS) has the advantage of resolving all scales of turbulence, but its high computational requirements mean that it is not readily applicable to high-Reynolds-number and complex flows.^{1,2} An alternative approach is large-eddy simulation (LES), which directly calculates the large-scale motions of the turbulence, leaving only the small-scale motions to be parameterized by subgrid-scale (SGS) models.^{3,4} In comparison with the Reynolds-averaged Navier–Stokes (RANS) method, which solves the ensemble-averaged variables through the introduction of turbulence models, LES is more accurate since the large-scale motions account for most energy and heat transport in turbulence. The LES technique is a balance between DNS and RANS, and it is regarded as having great potential for solving the complex flow problems that arise in many actual applications.^{5}

The direction in which energy is transferred is a very important feature of a turbulent system. In three-dimensional turbulence, as described by Kolmogorov’s theory,^{6} turbulent energy is transferred from large scales to small scales, i.e., in a forward energy cascade. As a matter of fact, real three-dimensional turbulence is much more complex, and an inverse energy cascade (from small to large scales) also exists in many cases of turbulence, such as wall-bounded turbulence,^{7,8} rotating turbulence,^{9–11} turbulent flow through an axially symmetric contraction,^{12} and internal flows in turbomachinery.^{13,14} In these cases, the backward transfer of energy is significant, but most contemporary SGS models have failed to take this phenomenon into account properly, which may lead to unreliable predictions. SGS models that incorporates backward transfer of energy are rarely mentioned in the literature. It has been suggested that the backward energy transfer process could be introduced into the SGS model by using random forcing or by extrapolation from the behavior of the resolved scales.^{15} However, such an approach has some obvious weaknesses, such as numerical instability. Owing to the complexity of the inverse energy cascade, it is challenging to perform a theoretical reconstruction of the SGS stress on the basis of he functional and structural strategies.^{16} Recently, machine learning methods have proven to be robust tools for the construction of turbulence models, including the models required for the RANS and LES methods.^{17–27} In applying data mining and machine learning to LES, Sarghini *et al.*^{20} adopted an artificial neural network (ANN) as the SGS model to perform LES, with this ANN being trained using LES data. The mathematical formulation of the SGS stress is implicitly hidden in the “black box” of the ANN, and one is unable to obtain the explicit form of ANN-based SGS models. At present, instead of modifying existing LES models, which are known to be based on unreliable assumptions, some studies have directly established mapping relationships between the known facts (DNS data) and the SGS term.^{22} By training the filtered DNS (FDNS) of turbulent channel flows, Gamahara and Hattori^{28} used an ANN to establish an SGS model. Wang *et al.*^{22} developed data-driven SGS models for decaying isotropic turbulent flows by employing an ANN as well as random forests, and they suggested that the ANN model can give better results than conventional LES models. Zhou *et al.*^{26} established an ANN-based SGS model to conduct LES of isotropic turbulence, with the filter width being considered in the training and testing procedure. Wang and co-workers^{21,29–32} have carried out a series of studies on reconstructing data-driven SGS models to perform LES of compressible turbulence.

Numerous studies have demonstrated that data-driven machine learning provides a new perspective and approach for the closure of the SGS stress in turbulence. This approach allows an SGS model to be built directly from reliable DNS data, without relying on potentially inappropriate assumptions. In this study, we make a preliminary attempt to use an ANN to establish a new SGS model that takes account of the inverse energy cascade in incompressible isotropic turbulence. This ANN-based SGS model is evaluated by *a priori* testing.

## II. DATA PREPARATION

*u*

_{i},

*p*,

*ρ*, and

*f*

_{i}are the velocity field, pressure, density, and large-scale forcing, respectively. For freely decaying isotropic turbulence,

*f*

_{i}= 0. The domain of the calculation is a periodic cube with edge length 4

*π*, and it is uniformly discretized using a total of 512

^{3}grids. A fourth-order Runge–Kutta scheme is used for the time integration, and the viscous term is treated by a semi-implicit method.

*u*

_{1},

*u*

_{2}, and

*u*

_{3}are the velocities in the

*x*

_{1},

*x*

_{2}, and

*x*

_{3}directions, respectively. For a full description of the numerical strategy, see Refs. 33–35. We consider a special type of isotropic turbulent flow in which the phenomenon of backward energy transfer is produced by reversing the sign of the velocity in space at a given point in time (referred to as

*t*= 0).

^{36}Note that the turbulence is fully developed before the reversal of the velocity field. Such a case of turbulent flow case has been referred to previously in the literature as “reversed–reversed” (RR).

^{33,34}At

*t*= 0, the Reynolds number based on the Taylor scale is Re

_{λ}= 132.9. In the RR case, there is initially a significant inverse energy cascade, and then the traditional forward energy cascade is recovered. The time

*t*is normalized by the eddy turnover time $T=3/2E/\epsilon $ at

*t*= 0, with

*E*being the turbulent energy and

*ɛ*the turbulent dissipation rate. The reliability of this turbulence database has been verified in our previous studies.

^{36,37}

*π*/

*k*

_{c}, with

*k*

_{c}in the inertial range. The bar over variables in this paper denotes the low-pass filtering operation. The filter width in our study is Δ = 0.3653.

*τ*

_{ij}, which is defined as $\tau ij=uiuj\u0304\u2212u\u0304iu\u0304j$ and can be calculated directly from the FDNS data. Given that the purpose of this paper is to evaluate the suitability of an ANN for modeling turbulence with an inverse energy cascade, we employ sufficient numbers of input features and hidden layers to strengthen the reliability of the ANN-based SGS model. This ANN has an input layer, an output layer, and eight hidden layers. There are over 100 × 10

^{6}neurons in the hidden layer. Figure 1 shows a sketch of the ANN. The input layers $Xj1$ are given by the filtered velocity field. Each neuron in layer

*m*receives and processes the inputs $Xjm\u22121$ from layer

*m*− 1 and then obtains the outputs $Xim$ by means of an activation function.

^{21}The transfer function is defined as

*f*is the nonlinear activation function, and $Wijm$ and $bim$ are the weight matrix and bias matrix, respectively, of layer

*m*. We use the rectified linear unit (ReLU),

*f*(

*z*) = max{0,

*z*}, as the activation function in this study. The loss function, which uses a L2 regularization term to suppress overfitting, is the mean squared error (MSE). We use the stochastic gradient descent (SGD) optimizer to train the ANN.

## III. RESULTS AND DISCUSSION

The dissipation coefficient $C\epsilon =\epsilon L/U3$ (with *ɛ* being the turbulent energy dissipation rate, $U$ the rms velocity, and *L* the integral length scale) is used to distinguish different evolutionary stages of the RR case, as suggested in Ref. 34. We divide the evolution of the RR case into four different temporal stages: (I) the stage of backward energy transfer (0 < *t* < 0.15); (II) the stage of transition (0.15 < *t* < 0.75); (III) the stage of excessive energy dissipation (0.75 < *t* < 2.8); (IV) the stage of equilibrium dissipation (*t* > 2.8). This division is supported by the relationship between *C*_{ɛ} and Re_{λ}, as shown in Fig. 2. Stages I and III scale as $C\epsilon \u221dRe\lambda \u22122$ and $C\epsilon \u221dRe\lambda \u22121$ respectively. Stage I corresponds to a turbulent state with an inverse energy cascade, and turbulent state in stage III has been termed “nonequilibrium,” which is documented by a great deal of experimental and numerical studies.^{38} In the present paper, turbulence at three typical times *t* = 0.03, 1.5, and 3.0 representing stages I, III, and IV, respectively, will be investigated. We focus mainly on the process of backward energy transfer in the RR case.

^{22}In addition, we choose the rms of the rate-of-strain tensor $(SijSij)$ and the product of eigenvalues of the rate-of-strain tensor (

*abc*) as two particular ANN input features, since the backward energy transfer can be described with

*abc*and

*S*

_{ij}

*S*

_{ij}.

^{37}Here, $Sij=12(\u2202ui/\u2202xj+\u2202uj/\u2202xi)$, with eigenvalues

*a*,

*b*, and

*c*. In homogeneous and isotropic flows, the ratio of

*abc*and

*S*

_{ij}

*S*

_{ij}can be written as

^{37}

*k*is the wavenumber, with

*T*(

*k*) and

*E*(

*k*) being the transfer spectrum and energy spectrum, respectively. This means that −⟨

*abc*⟩/⟨

*S*

_{ij}

*S*

_{ij}⟩ represents the relationship between energy transfer and energy dissipation.

In homogeneous isotropic turbulence, the diagonal components *τ*_{11}, *τ*_{22}, and *τ*_{33} are almost equivalent, and so are the relations for the off-diagonal components *τ*_{12}, *τ*_{13}, and *τ*_{23},^{28} and thus we use one diagonal component *τ*_{11} and one off-diagonal component *τ*_{12} of *τ*_{ij} as the ANN output labels to train the SGS models. *τ*_{11} and *τ*_{12} are calculated directly from the FDNS data. Four different ANN-based SGS models are trained on the basis of the above four different instantaneous flow fields. For the training and testing datasets, we randomly select 15% of the data from an instantaneous turbulence field, with the ratio of training and testing being 7:3. In addition, cross-validation is used to avoid the overfitting problem, and the untrained data are used to estimate the performance of the ANN. In this study, the training of the ANN takes place over a long period (1000 training steps) until the learning rate reaches a minimum, i.e., when the training and validation losses exhibit a similar variation, which indicates successful training of the ANN model.

*a priori*test. The SGS stress tensor in the Smagorinsky model is expressed as

*C*

_{S}= 0.14.

^{26}The gradient model is defined by

^{39}

Figure 3 shows the contours of the SGS stress tensor components *τ*_{11} and *τ*_{12} at *t* = 0.03, calculated by the Smagorinsky model, the gradient model, the ANN-based model, and the FDNS data. It should be kept in mind, as we mentioned before, that the turbulence is undergoing a backward energy transfer at *t* = 0.03. It is well known that the SGS stress tensor calculated by the FDNS data is considered to be the true SGS stress. As can be seen, the contours of *τ*_{11} and *τ*_{12} calculated by the ANN-based model are very similar to those from the FDNS data, which means that the ANN model can precisely predict the distribution of the true SGS stress. The predictions of the gradient model are slightly worse than those of the FDNS data, especially for *τ*_{12}, which exhibits some larger values compared with those from the FDNS data. The trend of the true *τ*_{11} is roughly captured by the Smagorinsky model, whereas it is difficult to detect any correlation between the *τ*_{12} calculated by the Smagorinsky model and the true *τ*_{12}. This indicates that the Smagorinsky model is incapable of capturing the backward energy transfer phenomenon, which is in line with the results of Hartel *et al.*^{40} The relative error $E(\tau ijM)$ of the SGS model with respect to the DNS data can be taken as a measure of the accuracy of the model, where $E(\tau ijM)=\u27e8(\tau ijM\u2212\tau ijD)2\u27e9/\u27e8(\tau ijD)2\u27e9$, with $\tau ijM$ and $\tau ijD$ being the SGS stresses calculated by the model and from the DNS data, respectively. The errors $E(\tau ijM)$ of the gradient model and Smagorinsky model are quite different from that of the ANN-based model; for example, the relative errors of *τ*_{11} calculated by the Smagorinsky, gradient, and ANN-based models at *t* = 0.03 are 46.4%, 25.3%, and 1.2%, respectively. In addition, compared with the diagonal component *τ*_{11}, the off-diagonal component *τ*_{12} may be more closely associated with the inverse energy cascade. The differences between the ANN-based and gradient models will be evaluated and examined in the following paragraphs.

*C*

_{ij}is given by

^{28}

*y*and

*z*directions, and we observe the variation of

*C*

_{ij}along the

*x*direction in the present paper. The correlation coefficients between the SGS stress tensors calculated by the Smagorinsky, gradient, and ANN-based models and that from the FDNS data are depicted in Fig. 4. The correlation coefficients determined by the ANN-based model are close to 1 (or larger than 0.95), which indicates that the ANN-based model estimates the SGS stress tensor accurately regardless of the turbulent state of the RR case. The correlation coefficients according to the gradient model are ∼0.9, slightly smaller than those of the ANN-based model. For the Smagorinsky model, the correlation coefficients are very small, or even have negative values for

*τ*

_{12}[Fig. 4(b)], which means that the SGS stress from the Smagorinsky model correlates poorly with that from the DNS data. The observation of negative values of

*C*

_{ij}at

*t*= 0.03 also explains the inability of the Smagorinsky model to predict the inverse energy cascade, especially for the off-diagonal component

*τ*

_{12}in the context of LES. This agrees with the results in Fig. 3(c).

The SGS dissipation, defined as $\epsilon SGS=\u2212\tau ijS\u0304ij$, is used to characterize the energy transfer between the resolved scales and the subgrid scales.^{41,42} Turbulent energy is transferred from the resolved scales to the subgrid scales in the traditional forward energy cascade and *ɛ*_{SGS} has a positive value. However, in the inverse energy cascade, the opposite occurs, with the turbulent energy being transferred from the subgrid scales to the resolved scales, and *ɛ*_{SGS} has a negative value. We also use the mean ⟨*ɛ*_{SGS}⟩ to evaluate the effectiveness of the SGS model, with the averaging being performed in the *y* and *z* directions. Figure 5 shows the *x*-direction evolution of ⟨*ɛ*_{SGS}⟩ calculated by the Smagorinsky, gradient, and ANN-based models and from the FDNS data. As expected, the energy transfer predicted by the ANN-based model shows good consistency with that calculated by DNS and is more precise compared with the result from the gradient model. Of the three models, the Smagorinsky model performs the worst. The value of ⟨*ɛ*_{SGS}⟩ from the Smagorinsky model is the largest, which agrees with the purely dissipative nature of the Smagorinsky model. It is worth mentioning here that the Smagorinsky model gives a qualitatively wrong result when the turbulence is in the state of backward energy transfer [⟨*ɛ*_{SGS}⟩ > 0 in Fig. 5(a)]. Figure 6 shows the probability density function (PDF) of *ɛ*_{SGS} for the three models and the FDNS data at *t* = 0.03, 1.5, and 3.0. In the stage of backward energy transfer, the PDF of *ɛ*_{SGS} is in theory, negatively skewed. However, this is not reflected by the Smagorinsky model, which gives an approximately symmetrically shaped PDF, as shown in Fig. 5(a). In the stages of excessive energy dissipation and equilibrium dissipation, the PDF of *ɛ*_{SGS} is positively skewed, as can be seen in Figs. 5(b) and 5(c). Overall, the ANN-based model accurately evaluates the transfer of energy between the resolved scales and the subgrid scales, and the performance of the gradient model is also acceptable, while that of the Smagorinsky model is not satisfactory.

On the basis of the above observations, the performance of the gradient model seems to be comparable to that of the ANN-based model in the construction of SGS stress tensor in turbulence. For a more in-depth examination of the performance of these models in evaluating the backward energy transfer process, the SGS backscatter *ɛ*_{SGS-back} = (⟨*ɛ*_{SGS}⟩ − |⟨*ɛ*_{SGS}⟩|)/2 is adopted to investigate the presence of the inverse energy cascade. Obviously, *ɛ*_{SGS-back} is negative in the inverse energy cascade, while *ɛ*_{SGS-back} is zero in the forward energy cascade. Figure 7 plots the *x*-direction evolution of *ɛ*_{SGS-back} for three models and the FDNS data at *t* = 0.03, 1.5, and 3.0. It is clear from Fig. 7(a) that the ANN-based and gradient models can predict the inverse energy cascade in turbulence, with the ANN-based model performing better than the gradient model, as already found above. The DNS result for *ɛ*_{SGS-back} in Fig. 7(b) shows that there exist a few backward energy transfer events in the stage of excessive energy dissipation, although the average transfer of energy during this stage is in the forward direction.^{36} The ANN-based model captures these backward energy transfer events accurately, which the Smagorinsky and gradient models cannot do. This means that the ANN-based model has great potential for modeling SGS stress taking account of the inverse energy cascade. We have also trained another ANN model (hereinafter referred to as the ANN-1 model) that does not consider the input features $SijSij$ and *abc*. To exclude the influence of the training method on the model, the ANN and ANN-1 models in the present study are trained by the same method, as described in Sec. II. Figure 8 shows the *x*-direction evolution of *ɛ*_{SGS-back} from the ANN and ANN-1 models and the FDNS data at *t* = 1.5. Clearly, the performance of the ANN-1 model in capturing the few backward energy transfer events is poorer than that of the ANN model, which indicates that $SijSij$ and *abc* are important input features in training an ANN-based SGS model that takes account of the inverse energy cascade.

## IV. CONCLUSION

This study has investigated the closure of the SGS stress tensor with account taken of the turbulent inverse energy cascade in the framework of an ANN. It has considered 14 input features, among which the rms rate-of-strain tensor and the product of eigenvalues of the rate-of-strain tensor have been employed to describe the backward energy transfer process in turbulence. The correlation coefficients between the SGS stress tensor calculated by the ANN-based model and that obtained from DNS data are larger than 0.95, which indicates that the ANN-based model accurately predicts the SGS stress tensor in the stage of backward energy transfer. The performance of the gradient model is slightly poorer than that of the ANN-based model, while the Smagorinsky model performs unsatisfactorily. In an *a priori* test, we have compared the SGS stress tensor, SGS dissipation, and SGS backscatter from the ANN-based model with those from the DNS data, Smagorinsky model, and gradient model. In comparison with the gradient model, the ANN-based model can even predict the few backward energy transfer events in the stage of excessive energy dissipation. In addition, compared with the diagonal component of the SGS stress tensor, the off-diagonal component may be intimately linked to the inverse energy cascade. This study is a preliminary attempt at the design of a new SGS model that takes account of the inverse energy cascade in a machine learning framework. Future studies are required with regard to the following aspects: an *a posteriori* test is needed to further verify the ANN model’s performance; the input features and the structure of the ANN need to be further optimized, since the ANN-based model presented here is complex and time-consuming; and the application of the ANN-based model to complex flows involving an inverse energy cascade should be investigated. In addition, generalization of the ANN-based model is very important. In the future work, we plan to verify a generalization of the present model through DNS of, for example, a set of isotropic turbulence cases with higher Reynolds numbers, transitional channel flows, and transitional boundary-layer flows, all involving a backward energy transfer process.

## ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (Grant No. 12002318), the Fundamental Research Program of Shanxi Province (Grant No. 202303021221118), and the China Postdoctoral Science Foundation (Grant No. 2022M721630).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Feng Liu**: Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Supervision (equal); Writing – original draft (lead); Writing – review & editing (equal). **Zhuangzhuang Wu**: Data curation (lead); Investigation (lead); Software (lead); Validation (lead); Visualization (lead). **Pengfei Lv**: Methodology (equal); Project administration (equal); Visualization (equal). **Wei Yang**: Formal analysis (equal); Methodology (equal); Software (equal). **Congcong Chen**: Formal analysis (equal); Writing – review & editing (equal). **Junfeng Xu**: Methodology (equal); Resources (equal); Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

*Turbulent Flows*(