Artificial neural network approach for turbulence models: A local framework
aa r X i v : . [ phy s i c s . f l u - dyn ] J a n Artificial neural network approach for turbulence models: A localframework
Chenyue Xie , ∗ Xiangming Xiong , and Jianchun Wang † Program in Applied and Computational Mathematics,Princeton University,Princeton, NJ 08544, USAand Department of Mechanics and Aerospace Engineering,Southern University of Science and Technology,Shenzhen 518055, People’s Republic of China (Dated: January 27, 2021)
Abstract
A local artificial neural network (LANN) framework is developed for turbulence modeling. TheReynolds-averaged Navier-Stokes (RANS) unclosed terms are reconstructed by artificial neuralnetwork (ANN) based on the local coordinate system which is orthogonal to the curved walls. Weverify the proposed model for the flows over periodic hills. The correlation coefficients of the RANSunclosed terms predicted by the LANN model can be made larger than 0.96 in an a priori analysis,and the relative error of the unclosed terms can be made smaller than 18%. In an a posteriori analysis, detailed comparisons are made on the results of RANS simulations using the LANN andSpalart-Allmaras (SA) models. It is shown that the LANN model performs better than the SAmodel in the prediction of the average velocity, wall-shear stress and average pressure, which givesthe results that are essentially indistinguishable from the direct numerical simulation (DNS) data.The LANN model trained in low Reynolds number Re = 2800 can be directly applied in the casesof high Reynolds numbers Re = 5600, 10595, 19000, 37000 with accurate predictions. Furthermore,the LANN model is verified for flows over periodic hills with varying slopes. These results suggestthat the LANN framework has a great potential to be applied to complex turbulent flows withcurved walls. ∗ Corresponding author. Email: [email protected] † Email: [email protected] . INTRODUCTION The Reynolds-averaged Navier-Stokes (RANS) simulation has been widely applied tostudy complex turbulent flows in industrial applications, combustion, astrophysics, and en-gineering problems for its low computing requirements[1–3], which can be derived by timeaveraging of the Navier-Stokes equations[2]. Since the pioneering works of Reynolds by de-composing the instantaneous quantity into its time-averaged and fluctuating quantities[4],a series of model-driven approaches have been proposed to develop RANS models. Theseinclude the eddy viscosity models[5, 6], the Spalart-Allmaras model[7], the k − ǫ model[8–11],the k − ω model[12–14], the Reynolds stress model(RSM)[15], etc[2].Recently, data-driven techniques have been incorporated into turbulence models [16–46]. The discrepancies in the Reynolds stress anisotropy tensor are reconstructed by thesupervised learning[16]. Duraisamy et al. proposed a data-driven approach to the mod-eling of turbulence with enforcing consistency between the data and the model[17, 18].Ling et al. developed the neural network architectures embedded invariance propertiesin RANS simulations[22]. A physics-informed Bayesian framework for quantifying andreducing model-form uncertainties in RANS simulations has been proposed by Xiao etal. [24]. Reynolds stresses modeling discrepancies can be reconstructed by a physics-informedmachine-learning approach[27]. A physics-based implicit treatment was proposed to modelReynolds stress by using machine learning techniques[33]. Wu et al. proposed a metric toquantitatively assess the conditioning of RANS equations with data-driven Reynolds stressclosures[36]. Furthermore, the recent progresses on data-driven turbulence models have beensummarized by Duraisamy et al. [41].In this paper, we propose a local artificial neural network (LANN) framework for recon-structing the RANS unclosed terms in the local coordinate system orthogonal to the curvedwall. We find that the Reynolds unclosed terms predicted by the LANN model exhibit highaccuracy in the a priori analysis for flows over periodic hills. We also study the accuracyof the proposed LANN model in the a posteriori tests by examining the average velocity,wall-shear stress and average pressure. These tests suggest that the LANN model is a veryattractive approach for developing models of RANS unclosed terms in complex turbulentflows with curved walls.This paper is organized as follows. Section II briefly describes the governing equations2nd computational method. Section III discusses the DNS database of compressible flowsover periodic hills. Section IV introduces the LANN model for the reconstruction of RANSunclosed terms from the averaged flow fields. Section V presents both a priori and a pos-teriori results of the LANN model. Some discussions on the proposed LANN models arepresented in Section VI. Conclusion are drawn in Section VII. II. GOVERNING EQUATIONS, AND NUMERICAL METHOD
The dimensionless Navier-Stokes equations for compressible turbulence of ideal gas in theconservation form are[47–51]: ∂ρ∂t + ∂ ( ρu j ) ∂x j = 0 , (1) ∂ ( ρu i ) ∂t + ∂ [ ρu i u j + pδ ij ] ∂x j = 1 Re ∂σ ij ∂x j , (2) ∂ E ∂t + ∂ [( E + p ) u j ] ∂x j = 1 α ∂∂x j ( κ ∂T∂x j ) + 1 Re ∂ ( σ ij u i ) ∂x j , (3) p = ρT / ( γM a ) , (4)where u i is the i -th velocity component ( i = 1 , , p is the pressure, ρ is the density,and T is the temperature. The viscous stress is defined by σ ij = 2 µS ij − µδ ij S kk , where S ij = ( ∂u i /∂x j + ∂u j /∂x i ) is the strain rate tensor. The molecular viscosity µ = T / S ) T + S ( S = 110 . K/T f ) is determined from Sutherland’s law [52], and the thermal conductivity κ isthen calculated from the molecular viscosity with the constant Prandtl number assumption.The total energy per unit volume E is defined by E = pγ − + ρ ( u j u j ).The hydrodynamic and thermodynamic variables in Eqs. (1-4) are normalized by a set ofreference variables: the reference velocity U f , temperature T f , length L f , density ρ f , energyper unit volume ρ f U f , viscosity µ f , thermal conductivity κ f and pressure p f = ρ f U f . Thereare three reference governing parameters: the reference Reynolds number Re ≡ ρ f U f L f /µ f ,the reference Mach number M a = U f /c f , and the reference Prandtl number P r ≡ µ f C p /κ f .The speed of sound is defined by c f ≡ p γRT f , where γ ≡ C p /C v is the ratio of specificheat at constant pressure C p to that at constant volume C v and is assumed to be equal to1.4. Moreover, R ≡ C p − C v is the specific gas constant. The parameter P r is assumed tobe equal to 0.7. The parameter α is given by α ≡ P rRe ( γ − M a .3he RANS equations governing the dynamics of the mean scales, which can be obtainedby projecting the physical variables into the time-averaged variables by a Reynolds operation¯ f ( x ) = lim T R →∞ T R R t + T R t f ( x , t ) dt , where ¯ f denotes a time averaged variable, T R is theintegration time[4]. Favre-filtering (mass-weighted filtering: ˜ f = ρf / ¯ ρ )[53] is used to avoidadditional RANS unclosed terms and simplify the treatments in the equation of conservationof mass in compressible flows. The Favre average obeys the following decomposition rules: f = ¯ f + f ′ and f = ˜ f + f ′′ .The dimensionless compressible governing equations for the time-averaged variables canbe expressed as follows: ∂ ¯ ρ∂t + ∂ ( ¯ ρ ˜ u j ) ∂x j = 0 , (5) ∂ ( ¯ ρ ˜ u i ) ∂t + ∂ ( ¯ ρ ˜ u i ˜ u j + ¯ pδ ij ) ∂x j − Re ∂ ˜ σ ij ∂x j = ∂τ ij ∂x j , (6) ∂ ˜ E ∂t + ∂ [( ˜ E + ¯ p )˜ u j ] ∂x j − Re ∂ (˜ σ ij ˜ u i ) ∂x j − α ∂∂x j (˜ κ ˜ Tx j ) = ∂C p Q i ∂x i + ∂J i ∂x i , (7)¯ p = ¯ ρ ˜ T / ( γM a ) , (8)where the time-averaged total energy ˜ E is defined by ˜ E = ¯ pγ − + ¯ ρ (˜ u j ˜ u j ), the time-averagedviscous stress is ˜ σ ij = 2˜ µ ˜ S ij − ˜ µδ ij ˜ S kk , where ˜ S ij = ( ∂ ˜ u i /∂x j + ∂ ˜ u j /∂x i ), and ˜ µ is calculatedfrom Sutherland’s law.The RANS unclosed terms appearing on the right hand sides of Eqs. (5-8) are defined as τ ij = − ¯ ρ ] u ′′ i u ′′ j , Q j = − ¯ ρ ] u ′′ j T ′′ , (9)where τ ij is the Reynolds stress, Q j is the turbulent heat flux, J i is the triple correlationterm J i = − ¯ ρ ( ^ u j u j u i − g u j u j ˜ u i ) ≈ τ ij ˜ u j [54, 55].In this paper, we model the Reynolds stress τ ij and turbulent heat flux Q j , and neglectother unclosed terms. Meanwhile, we assume that the kinematic viscosity satisfies thefollowing condition: σ ij = 2 ρν ( S ij − δ ij S kk ) = 2 ¯ ρν ( f S ij − δ ij f S kk ), where ν is the kinematicviscosity, and the term Re ∂σ ij − f σ ij ∂x j would not appear in the filtered momentum equation. III. DNS DATABASE OF COMPRESSIBLE TURBULENCE
The DNS data of the compressible flows over the periodic hills (the baseline geome-try of the periodic hill are depicted by piecewise cubic polynomials[56, 57]) are obtained4
IG. 1. The configuration of the flow over periodic hill: every fourth curvilinear grid line is shown. from the direct numerical simulation with a high-order finite difference Navier-Stokes solver“OpenCFD-SC”[48, 58]. A sixth-order compact finite difference scheme is used for spacediscretization and a third order Runge-Kutta scheme is used for time integration[48, 59].No-slip velocity and adiabatic conditions are imposed on the upper and lower walls for thevelocity and temperature, respectively. We implement periodic boundary conditions in thestreamwise x − and spanwise z − directions. A body-fitted curvilinear gird system is used inall of the simulations[60]. The flows are driven by a body force F ( t ) in the streamwise direc-tion, which is a function of time only and maintain the average mass-flux remains constantat every time step[50, 61]: ∂∂t R vol ρudv = 0.The periodic-hill channel flow configuration is shown in Fig. 1[50, 51, 57, 61–63]. Thelengths are non-dimensionalized by the height of the hill h . The dimensionless computationaldomain are: (0 , L x ) × , L y × (0 , L z ), where L x = 9 , L y = 3 . , L z = 4 .
5. The cross-sectionalReynolds number over the hill crest is defined as Re S = R S ( ρu ) | x =0 dSh/ ( R S dSµ wall ), where h is the hill height. The volumetric Reynolds number is Re v = R V ρu dV h/ ( R V dV µ wall ). Arelationship between Re S and Re v is Re v = 0 . Re S [55, 61]. The wall temperature is fixed at300 K . We present the DNS results at the Reynolds number Re ranging from 2800 to 10595and Mach number M a = 0 . M a = R S ( u ) | x =0 dS/ ( R S dSc wall )) with the grid resolution of256 × × z = 0 is shown in Fig. 2(a). The flow fieldis divided into two region: the reverse flow with u < u > IG. 2. Contours of the streamwise velocity field at Re = 2800: (a) the instantaneous streamwisevelocity field in the (x-y) plane at z = 0, (b) the mean streamwise velocity ˜ u and the streamlines.FIG. 3. The mean streamwise and normal velocities ˜ u and ˜ u profiles at different stations x = 0,0.5, 1, 2, 3, 4, 5, 6, 7, 8, 8.5 with Re = 2800, 5600, 10595: (a) ˜ u at Re = 2800, (b) ˜ u at Re = 2800,(c) ˜ u at Re = 5600, (d) ˜ u at Re = 10595. occur behind the first hill. The separation and reattachment points are x sep = 0 .
227 and x reatt = 5 .
34, respectively[50, 61, 62].The comparison of the mean streamwise and normal velocities ˜ u and ˜ u at eleven stations x = 0, 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 8.5 for Reynolds numbers Re = 2800, 5600, 10595 are shownin Fig. 3. ˜ u and ˜ u computed from the present simulations are in excellent agreementwith the DNS results of Breuer el al. [50, 62]. Figure 4 shows the Favre-averaged Reynoldsstresses ρu ′′ u ′′ and ρu ′′ u ′′ at Re = 2800. It can be seen that the solved ρu ′′ u ′′ and ρu ′′ u ′′ by present simulation agree well with the results of Breuer el al. [50, 62]. Furthermore, theaveraged pressure distribution ¯ p − ¯ p ave (¯ p ave = L x R L x ¯ pdx ) along the lower wall is shown inFig. 5. The pressure from the present simulation is close to the previous DNS[62]. Thesecomparisons validate the accuracy of the present direct numerical simulations.6 IG. 4. The Favre-averaged Reynolds stresses ρu ′′ u ′′ and ρu ′ u ′ profiles at different stations with Re = 2800.FIG. 5. The averaged pressure distribution ¯ p − ¯ p ave along the lower wall for Re = 2800. IG. 6. Schematic diagram of the ANN’s network structure.
IV. THE STRUCTURE OF THE LANN MODEL
A fully connected ANN is used to reconstruct the nonlinear relation between averageinput features and RANS unclosed terms τ ij and Q j . The network structure of the ANN isshown in Fig. 6[64–66]. Neurons in layer l of ANN receive signals X l − j from layer l − X li to neurons in layer l + 1[67–69]. The transfer function is calculated as X li = σ ( s li + b li ) , (10) s li = X j W lij X l − j , (11)where W lij , b li , σ are the weight, bias parameter, and activation function, respectively. Wetrain the ANN to update the weights and bias parameters so that the final output X O approximates well the RANS unclosed terms τ ij and Q j . Five ANNs are trained to predicteach independent component of τ ij and Q j separately.In this research, the fully-connected ANN contains four layers of neurons ( M : 64 : 32 : 1)between the set of inputs and targets. The input layer has M neurons, while the output layerconsists of a single neuron. Two hidden layers are activated by the Leaky-Relu activation8 IG. 7. Transformation between the global and local reference frames in the flow over periodichill. function: σ ( a ) = a, if a > , . a, if a ≤ . (12)Meanwhile, linear activation σ ( a ) = a is used to the output layer. The loss function is definedby the difference between the output X O and the RANS unclosed terms from DNS ( h ( X O − τ ij ) i or h ( X O − Q j ) i ), where hi represents the average over the entire domain[64]. The lossfunction is minimized by the back-propagation method with Adam optimizer (learning rateis 0.001)[70].The proper choice of input variables for flows over periodic hills with varying slopes isimportant for the present ANN architecture to model the Reynolds stress τ ij and the turbu-lent heat flux Q j accurately. As shown in the previous work[41], the first-order derivativesof averaged velocities have been used to establish a functional relation between { ∂ ˜ u i ∂x j , d } ( d is the nearest distance from the walls) and the RANS unclosed terms[41].In the RANS simulations of flows over periodic hills with varying slopes, the RANS un-closed terms have been predicted by machine learning models, where the input featurescontain the first-order derivatives of the mean velocity and temperature in the global ref-erence frame[41]. Due to that the angle between the local wall-normal direction and theglobal y direction varies with the spatial position, it is difficult to extend the trained ma-chine learning model to flows over periodic hills with varying slopes in the global referenceframe, which makes it useless for other flows with general boundaries. In order to optimize9 NN Inputs OutputsANN1 ∂ ˜ u ξ ∂ξ , ∂ ˜ u ξ ∂η , ∂ ˜ u η ∂ξ , ∂ ˜ u η ∂η , ¯ ρ, d, ¯ µ τ ξη ANN2 ∂ ˜ u ξ ∂ξ , ∂ ˜ u ξ ∂η , ∂ ˜ u η ∂ξ , ∂ ˜ u η ∂η , ∂ ˜ T∂ξ , ∂ ˜ T∂η , ¯ ρ, d, ¯ µ Q ξ TABLE I. Set of inputs and outputs for the ANNs. the input features while maintaining accuracy and generality, we proposed the local artifi-cial neural network (LANN) model, which reconstructs the nonlinear function of the inputfeatures and RANS unclosed terms τ ij and Q j in the local coordinate system orthogonal tothe nearest wall as shown in Fig. 7. The LANN model guarantees that the nearest distancebetween the present point and the wall ( d ) can be measured along the η direction of thelocal coordinate system, which is general for flows over periodic hills with varying slopes. Aset of input variables and output variables of different ANNs are shown in Table I. As shownin Fig. 6, the input and output features X I ( x, y ) , X O ( x, y ) in the global reference frame aretransformed to X I ( ξ, η ) , X O ( ξ, η ) in local reference frame for flows over periodic hills: ∂ ˜ u ξ ∂ξ ∂ ˜ u ξ ∂η∂ ˜ u η ∂ξ ∂ ˜ u η ∂η = A ∂ ˜ u x ∂x ∂ ˜ u x ∂y∂ ˜ u y ∂x ∂ ˜ u y ∂y A T (13) (cid:16) ∂ ˜ T∂ξ ∂ ˜ T∂η (cid:17) = (cid:16) ∂ ˜ T∂x ∂ ˜ T∂y (cid:17) A T (14) τ ξξ τ ξη τ ηξ τ ηη = A τ xx τ xy τ yx τ yy A T (15) (cid:16) Q ξT Q ηT (cid:17) = (cid:16) Q xT Q yT (cid:17) A T (16)where A = sin ( θ ) − cos ( θ ) cos ( θ ) sin ( θ ) , cos ( θ ) = ∆ xr , sin ( θ ) = ∆ yr , r = p ∆ x + ∆ y .In order to increase the robustness of the ANN training, the first-order derivatives ofthe mean velocity and temperature in X I are normalized by their root mean square (rms)values, which is similar to the previous data-driven strategies[20, 24, 27, 30, 66, 69]: Z I = X I /X rmsI . (17)Besides, we suppress overfitting with a cross-validation. The performance of the modelis estimated by the data which have not been used for training. In this research, the inputsand outputs of the LANN model are the mean flow features and the RANS unclosed terms10 IG. 8. Learning curves of the proposed LANN model of the unclosed Reynolds stress τ ξξ . τ ij , Q j , respectively, which are obtained from the DNS data. The three-dimensional (3D)DNS data are generated using 256 × ×
128 degrees of freedom while the two-dimensional(2D) RANS is performed at the grid resolutions of 256 ×
129 and 128 ×
65. Finally, thenetwork is trained by the Adam algorithm[70] with early stopping (if validation errors did notdecrease improve for 10 epochs, the training would exit with the best model correspondingto the lowest validation loss until then)[65]. The learning rate and batch size of the ANNare 0.001 and 1000, respectively. The total data for ANN training are 32639 grid points at Re = 2800. 70% of data is for training, and the left 30% is for testing. The training andtesting losses show similar behavior and correlate closely after 100 global iterations as shownin Fig. 8, which implies that the learning process is reasonable. V. TEST RESULTS OF THE LANN MODEL
In this section, we conduct both a priori and a posteriori tests to evaluate the per-formance of the LANN model for flows over periodic hills. The LANN model trained at Re = 2800 is used to produce reliable and repeatable predictions at Re = 2800, 5600, 10595,19000, 37000. We calculated the correlation coefficients and relative errors of the predictedRANS unclosed terms τ ij and Q j in the a priori test. In the a posteriori test, results ofthe RANS simulations with the LANN model are compared with the SA model and theDNS database. It is shown that the RANS simulations with the proposed LANN model canpredict the statistics of the averaged DNS data with high accuracy.11 τ ξξ τ ξη τ ηη Q ξ Q η train 0.967 0.992 0.993 0.980 0.986test 0.966 0.992 0.993 0.979 0.987 E r τ ξξ τ ξη τ ηη Q ξT Q ηT train 0.140 0.111 0.082 0.177 0.164test 0.141 0.110 0.084 0.181 0.162TABLE II. Correlation coefficient (C) and relative error ( E r ) of τ ξξ , τ ξη , τ ηη , Q ξT , and Q ηT for theLANN model in the local reference frame at Re = 2800. A. A priori tests
We evaluate the performance of the LANN model by calculating the correlation coefficient C ( R ) and the relative error E r ( R ) of τ ij and Q j . C ( R ) and E r ( R ) are defined, respectively,by C ( R ) = h ( R − h R i )( R model − h R model i ) i ( h ( R − h R i ) ih ( R model − h R model i ) i ) / , (18) E r ( R ) = p h ( R − R model ) i p h R i , (19)where h·i denotes averaging over the volume.Table II shows the correlation coefficients and relative errors of τ ξη and Q ξ in the localreference frame for LANN model in both training and testing sets at Re = 2800. Thedifference between the results of training and testing sets is small, which implies that thetraining process of ANN is not overfitting. The correlation coefficients are larger than 0.96,and the relative errors are less than 0.18 for the LANN model. B. A posteriori tests
We evaluate the performance of the LANN model for flows over periodic hills with varyingslopes at Re = 2800, 5600, 10595,19000, 37000. Furthermore, in order to show that theLANN model can be applied to flows over periodic hills with varying slopes, the a posteriori studies of the LANN model applied to flow over periodic hill with the total horizontal lengthof the domain L x = 3 . α + 5 . α = 1 . α controls thewidth of the hill, and the length of the flat section between the hills is 5.142, which is keptconstant. The two dimensional Reynolds-averaged Navier-Stokes equations are solved witha finite volume solver “OpenCFD-EC” developed by Li et al. [72, 73]. The spatial gradientsare calculated with a second-order accurate discretization. The temporal advancement of12he equations is achieved using an implicit LU-SGS method. The flow is set to be periodicin the streamwise direction. No-slip condition and adiabatic condition are set at walls forthe velocity and temperature, respectively.Eddy viscosity turbulence models have been widely used for aeronautical, meteorolog-ical, and other applications[2]. The Boussinesq hypothesis is applied to establish the re-lation between RANS unclosed terms and the first-order derivatives of mean velocity andtemperature[5]. The traceless part of the Reynolds stress τ ij is proportional to the productof the mean strain rate tensor ˜ S ij and the eddy viscosity µ t . The Reynolds stress τ ij andturbulent heat flux Q j are[7]: τ ij = − µ t (2 ˜ S ij − ∂ ˜ u k ∂x k ) + ¯ ρkδ ij (where the last term is gen-erally ignored for one-equation models because k is not readily available), Q j = − C p µ t P r T ∂ ˜ T∂x j .The eddy viscosity is given by µ t = ρν t f ν . (20)The Spalart-Allmaras (SA) model solves a transport equation for ν t . The governing equationfor the intermediate variable ν t is[7]: ∂ν t ∂t + ∂∂x j ( ν t ˜ u j ) = c b S t ν t + 1 σ [ ∂∂x j ((˜ ν + ν t ) ∂ν t ∂x j ) + c b ∂ν t ∂x j ∂ν t ∂x j ] − c w f w ( ν t d ) , (21)where S t = Ω + ν t k d f ν , f ν = χ χ + c ν , f ν = 1 − χ χ/f ν , χ = ρν t µ , f w = g [ C w g + C w ] / , g = r + C w ( r − r ), r = ν t S t k d , Ω is the magnitude of the vorticity vector, and d is thenearest distance from the walls. Meanwhile, the model coefficients in the SA model are σ = 2 / C b = 0 . C b = 0 . κ = 0 . C w = c b κ + (1 + C b ) /σ , C w = 0 . C w = 2, C ν = 7 .
1, and C w = C b /k + (1 + C b ) /σ . The periodic boundary condition is appliedin the streamwise x-direction for ν t . ν wallt = 0 is imposed on the upper and lower wallboundaries.The performances of the LANN model are evaluated by calculating the average velocity,wall-shear stress and average pressure. In the a posteriori tests, the initial conditions forthe LANN model are generated from the steady-state flow fields calculated by the RANSsimulation with the SA model. The variations of the grid spacing in wall units in the x and y directions along the bottom wall for RANS simulations with the SA and LANN modelsare shown in Fig. 9. The maximum grid spacings in the x and y directions are located at thedownstream wall due to a large increase in the friction velocity in this region. Figure 10 showsthe mean streamwise velocity contours and the streamlines from RANS simulations with theSA and LANN models. The separation and reattachment points are at x sep = 0 . , . IG. 9. Grid spacing in wall units along the bottom wall of RANS simulations with the SA andLANN models at Re = 2800 with a grid resolution of 256 ×
129 and L x = 9 .
0: (a) SA model, (b)LANN model.FIG. 10. Hill flow contours of the mean streamwise velocity ˜ u and the streamlines at Re = 2800with a grid resolution of 256 ×
129 and L x = 9 .
0: (a) SA model, (b) LANN model. and x reatt = 7 . , .
80 for the SA and LANN models, respectively. The separation point at x sep = 0 .
316 predicted by the SA model is slightly closer to the DNS result ( x sep = 0 . x reatt = 4 .
80 predicted by the LANN model is much closer to theDNS result ( x reatt = 5 . τ w along the lowerwall at Re = 2800 (˜ τ w = Re ∂ ˜ u ξ ∂η , where ˜ u ξ is the flow velocity parallel to the wall and η isthe distance to the wall). ˜ τ w predicted by the LANN and SA models show similar behaviorsas the DNS data. The peak of the profile of ˜ τ w is recovered more accurately by the LANNmodel than the SA model. The averaged pressure along the lower wall at Re = 2800 is E r \ x E r ) of ˜ u for different models at Re = 2800 and y = 2 . × IG. 11. Profiles of the averaged wall shear stress ˜ τ w and the averaged pressure distribution ¯ p at Re = 2800 with a grid resolution of 256 ×
129 and L x = 9 .
0: (a) ˜ τ w , (b) ¯ p . shown in Fig. 11(b). The mean pressure predicted by the LANN model is closer to theDNS data than that predicted by the SA model in the range 1 ≤ x ≤
8. We compare themean streamwise velocity ˜ u at eleven locations in Fig. 12. Both the SA and LANN modelsaccurately predict ˜ u near the upper wall. The RANS simulation with the SA model do apoor job near the lower wall, especially behind the separation. In contrast, ˜ u predicted bythe RANS simulation with the LANN model are in good agreement with the DNS data atall locations, suggesting that the LANN model can predict the mean streamwise velocity˜ u of flows over periodic hill accurately. Furthermore, the performance of the LANN modeltrained at Re = 2800 are examined by predicting the mean streamwise velocity ˜ u profile forflows over periodic hills with higher Reynolds numbers Re = 5600 and 10595. We displaythe mean streamwise velocities ˜ u of DNS and RANS simulations with the SA and LANNmodels at Re = 5600 and 10595 in Fig. 12(b-c). One can see that some errors occur near15 IG. 12. Mean streamwise velocity ˜ u profiles with a grid resolution of 256 ×
129 and L x = 9 . Re = 2800, (b) Re = 5600, (c) Re = 10595. the lower and upper walls in the predicted ˜ u by the SA model. In contrast, the resultsof the LANN model are very close to those of the DNS. Table III shows relative errors of˜ u for different models at Re = 2800 and y = 2 .
5. We can see that the relative error ofthe LANN model is smaller than that of the SA model. Thus, the LANN model shows16
IG. 13. Mean streamwise velocity ˜ u profiles with a grid resolution of 128 ×
65 and L x = 9 .
0: (a) Re = 2800, (b) Re = 5600, (c) Re = 10595. significant advantage over the SA model on relative error in the a posteriori test.The generality of the LANN model are examined by plotting the mean streamwise velocity˜ u profiles for flows over periodic hills on coarser grids and in a different computationaldomain (the detailed information about the new L x can be given with L x = 3 . α + 5 . IG. 14. Mean streamwise velocity ˜ u profiles with a grid resolution of 256 ×
129 and L x = 9 . · · · · · ) experiments Rapp & Manhart[74]: a) Re = 19000, (b) Re = 37000. where α = 1 . u profiles for Re = 2800, 5600, 10595with a grid resolution of 128 ×
65 are shown in Fig. 13. ˜ u predicted by the LANN model ismuch closer to those of DNS, compared to those predicted by the SA model. Furthermore,we evaluated the performance of the LANN model against the experimental results by Rapp et al. at Re = 19000 and 37000[74]. As shown in Fig. 14, compared with the results fromthe experiments[74], the mean streamwise velocity ˜ u from the LANN model match betterthan those from the SA model. Finally, figure 15 shows the mean streamwise velocity ˜ u forflow over periodic hill with a total horizontal length of L x = 10 .
929 ( α = 1 . IG. 15. Mean streamwise velocity ˜ u profiles at Re = 2800 with L x = 10 .
929 ( α = 1 . ×
65, (b) grid resolution of 256 × VI. DISCUSSION
The flow over periodic hill is one of the standard examples for developing new turbulencemodels of RANS[41], which includes the separation, recirculation, and reattachment. Oneimportant characteristic of RANS is that the RANS unclosed terms are very complex inturbulence near the boundary. It’s hard to reconstruct the RANS unclosed terms accuratelyand stably near a wall, which depend stronly on the distance to the walls[7, 41]. Due to theirregular and diverse nature of turbulence, it is difficult to explicitly derive the dependence ofRANS unclosed terms on the mean flow properties with analytical methods. The advantageof the LANN model in the local reference frame is that the η axis of the local coordinatesystem is orthogonal to the nearer wall and the nearest distance from the walls d canbe measured along the coordinate axis η , which is general and versatile for complex wall19onditions. In this research, it has been demonstrated that the LANN method is a powerfultool which can efficiently learn the high-dimensional and nonlinear relations between theRANS unclosed terms and the mean flow fields for flows over periodic hills with varyingslopes. The effects of more complex boundary conditions on the RANS simulations of wall-bounded turbulent flows will be modeled with the LANN framework in a follow-up study. VII. CONCLUSIONS
In this work, we proposed a framework of LANN for the RANS unclosed terms in RANSsimulations of compressible turbulence. The proposed LANN model depends on the localcoordinate system orthogonal to the wall for flows over periodic hills. In the a priori test,the correlation coefficients are larger than 0.96 and the relative errors are smaller than 18% for the LANN model. In an a posteriori analysis, we compare the performances of theLANN model with those of the SA model in the predictions of the average velocity, wall-shear stress and average pressure in flows over periodic hills with varying slopes α = 1 and1 .
5. There are non-negligible errors between the mean velocities predicted by the SA modeland the results of DNS near the walls, especially in the region right behind the separation.In contrast, the LANN model predicts the mean velocity accurately, and it also reconstructthe mean pressure closer to those of the DNS than those with the SA model at Reynoldsnumbers Re = 2800, 5600, 10595, 19000, 37000. In addition, the mean velocity in flows overperiodic hills with longer horizontal width L x = 10 .
929 ( α = 1 .
5) predicted by the LANNmodel is in well agreement with that of the DNS. The above comparison showed that theLANN model outperformed the SA model in the flows over periodic hills.The LANN model should also be very useful to wall-bounded turbulent flows with curvedwalls. There are several issues that need further exploration: the physical relationshipbetween averaged flow fields and the RANS unclosed terms, the hyperparameter space, thesymmetry and interpretation of the neural network models, the non-locality characteristicsof the RANS dynamics, and applications in more complex flows.20
CKNOWLEDGMENTS
We thank Weinan E and Chao Ma for helpful discussions. The authors are grateful toXinliang Li for providing the CFD codes OpenCFD-SC and OpenCFD-EC. The work ofChenyue Xie is supported in part by a gift to Princeton University from iFlytek. The workof Jianchun Wang is supported by the National Natural Science Foundation of China (NSFCGrants No. 91952104). [1] C. G. Speziale, Analytical methods for the development of Reynodls-stress closures in turbu-lence, Annu. Rev. Fluid Mech. , 107-157 (1991).[2] S. B. Pope, Turbulent Flows (Cambridge University Press, 2000).[3] P. A. Durbin, Some recent developments in turbulence closure modeling, Annu. Rev. FluidMech. , 77-103 (2018).[4] O. Reynolds, On the dynamical theory of incompressible viscous fluids and the determinationof the criterion, Philos. Trans. R. Soc. Lond. A , 123-164 (1895).[5] J. Boussinesq, Th´ e orie de l’´ e coulement tourbillant, Pr´ e sent´ e s par divers Savants Acad. Sci.Inst. Fr. , 46-50 (1877).[6] L. 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