Data-driven nonlinear constitutive relations for rarefied flow computations

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 coarsegraining model could make the DNCR method to be an effective tool in the rarefied gas community,especially for hypersonic engineering applications.

MixedMultiscale Finite Volume Methods for Elliptic Problems in Two-Phase Flow Simulations

We develop a framework for constructing mixed multiscale finite volume methods for elliptic equations with multiple scales arising from flows in porous media.Some of the methods developed using the framework are already known[20];others are new.New insight is gained for the known methods and extra flexibility is provided by the new methods.We give as an example a mixed MsFV on uniform mesh in 2-D.This method uses novel multiscale velocity basis functions that are suited for using global information,which is often needed to improve the accuracy of the multiscale simulations in the case of continuum scales with strong non-local features.The method efficiently captures the small effects on a coarse grid.We analyze the new mixed MsFV and apply it to solve two-phase flow equations in heterogeneous porous media.Numerical examples demonstrate the accuracy and efficiency of the proposed method for modeling the flows in porous media with non-separable and separable scales.