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Corrected Navier-Stokes equations for compressible flows

Jinglei XuDong MaPengxin LiuLin BiXianxu YuanLongfei Chen
Dec 2022
摘要
For gas flows, the Navier-Stokes (NS) equations are established bymathematically expressing conservations of mass, momentum and energy. Theadvantage of the NS equations over the Euler equations is that the NS equationshave taken into account the viscous stress caused by the thermal motion ofmolecules. The viscous stress arises from applying Isaac Newton's second law tofluid motion, together with the assumption that the stress is proportional tothe gradient of velocity1. Thus, the assumption is the only empirical elementin the NS equations, and this is actually the reason why the NS equationsperform poorly under special circumstances. For example, the NS equationscannot describe rarefied gas flows and shock structure. This work proposed acorrection to the NS equations with an argument that the viscous stress isproportional to the gradient of momentum when the flow is under compression,with zero additional empirical parameters. For the first time, the NS equationshave been capable of accurately solving shock structure and rarefied gas flows.In addition, even for perfect gas, the accuracy of the prediction of heat fluxrate is greatly improved. The corrected NS equations can readily be used toimprove the accuracy in the computation of flows with density variations whichis common in nature.
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