Abstract
To overcome the defects of traditional rarefied numerical methods such as the Direct Simulation Monte Carlo (DSMC) method and unified Boltzmann equation schemes and extend the covering range of macroscopic equations in high Knudsen number flows, data-driven nonlinear constitutive relations (DNCR) are proposed first through the machine learning method. Based on the training data from both Navier-Stokes (NS) solver and unified gas kinetic scheme (UGKS) solver, the map between responses of stress tensors and heat flux and feature vectors is established after the training phase. Through the obtained off-line training model, new test cases excluded from training data set could be predicated rapidly and accurately by solving conventional equations with modified stress tensor and heat flux. Finally, conventional one-dimensional shock wave cases and two-dimensional hypersonic flows around a blunt circular cylinder are presented to assess the capability of the developed method through various comparisons between DNCR, NS, UGKS, DSMC and experimental results. The improvement of the predictive capability of the coarse-graining model could make the DNCR method to be an effective tool in the rarefied gas community, especially for hypersonic engineering applications.
Highlights
Due to non-equilibrium effects inside the shock wave with large macroscopic gradients on a scale of several mean free paths, traditional hydrodynamic methods such as Navier-Stokes equations are no longer valid
In order to get the solutions of these approximations, discrete velocity and ordinate methods (DVM) such as gaskinetic unified algorithms (GKUA) [5] are widely employed and a new framework named unified gas kinetic scheme (UGKS) [6] proposed by Xu and collaborators a few decades ago is regarded as a multiscale scheme for both hydrodynamic and rarefied regimes
Except Direct Simulation Monte Carlo (DSMC), Boltzmann equation and moment equation solvers, many coupling models have been proposed recently such as traditional CFD-DSMC hybrid method [16], unified gaskinetic wave-particle method (UGKWP) [17], general synthetic iterative scheme (GSIS) [18], unified stochastic particle method based on the BGK model (USP-BGK) [19] and simplified unified wave-particle method (SUWP) [20]
Summary
Due to non-equilibrium effects inside the shock wave with large macroscopic gradients on a scale of several mean free paths, traditional hydrodynamic methods such as Navier-Stokes equations are no longer valid. Except DSMC, Boltzmann equation and moment equation solvers, many coupling models have been proposed recently such as traditional CFD-DSMC hybrid method [16], unified gaskinetic wave-particle method (UGKWP) [17], general synthetic iterative scheme (GSIS) [18], unified stochastic particle method based on the BGK model (USP-BGK) [19] and simplified unified wave-particle method (SUWP) [20] These methods try to couple hydrodynamic and microscopic descriptions of the rarefied flow to construct an efficient and accurate hybrid scheme. Data-driven nonlinear constitutive relations (DNCR) will be developed by machine learning method to provide a predictive approach for rarefied nonequilibrium flows Both NS equations solver and UGKS solver provide training data during training phase to map feature vectors and responses of stress tensor, heat flux and flow parameter derivatives between NS and UGKS data. Conclusions and future work are summarized in the last section
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