The classical Navier-Stokes (NS) model is inappropriate for the high-speed rarefied gas flow simulations due to the linear NS constitutive relations. A modified NS model has been proposed by incorporating the molecular diffusivity terms in the NS constitutive relations and gave good agreement with the experimental data for simulating the one-dimensional shock wave structure. The current work comprehensively investigates enhancing this modified NS model for two-dimensional high-speed rarefied gas flows and determining its proper slip and jump boundary conditions. A new temperature jump condition is also proposed for this modified NS model. This modified NS model and the new temperature jump condition are implemented in OpenFOAM by modifying the solver rhoCentralFoam, which solves the classical NS equations. Two high-speed gas flow cases are selected to validate the modified NS model with the new temperature jump condition as the sharp leading-edge wedge and truncated leading-edge flat plate with argon as the working fluid. Combining the modified NS equations and the new temperature jump with the wall-function type to correct the viscosity in the Knudsen layer predicts the computational results approaching the DSMC data.
GoshayeshiBRoohiEStefanovS. A novel simplified Bernoulli trials collision scheme in the direct simulation Monte Carlo with intelligence over particle distances. Phys Fluids2015; 27: 107104.
2.
Shoja-SaniARoohiEKahromM, et al. Investigation of aerodynamic characteristics of rarefied flow around NACA 0012 airfoil using DSMC and NS solvers. Eur J Mech B Fluids2014; 48: 59–74.
3.
BhattacharyaDKEuBC. Nonlinear transport processes and fluid dynamics: effects of thermoviscous coupling and nonlinear transport coefficients on plane Couette flow of Lennard-Jones fluids. Phys Rev A1987; 35(2): 821–836.
4.
OuJChenJ. Hypersonic aerodynamics of blunt plates in near-continuum regime by improved Navier–Stokes model. AIAA J2020; 58: 4037–4046.
5.
GreenshieldsCJReeseJM. The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion. In: Proceedings of the 2nd European conference for AeroSpace Sciences, Brussels, Belgium EUCASS2007, 2007.
6.
SambasivamR. Erweiterte Navier-Stokes-Gleichungen: Ableitungen und Anwendungen auf Strömungsprobleme. PhD Thesis, Friedrich Alexander-Universität Erlangen-Nürnberg, 2012.
7.
LeNTPXiaoHMyongRS. A triangular discontinuous Galerkin method for non-Newtonian implicit constitutive models of rarefied and microscale gases. J Comput Phys2014; 273: 160–184.
8.
MyongRS. A computational method for Eu’s generalized hydrodynamic equations of rarefied and microscale gas dynamics. J Comput Phys2001; 168: 47–72.
9.
YuanZZhaoWJiangZ, et al. Modified nonlinear coupled constitutive relations model for hypersonic nonequilibrium flows. J Thermophys Heat Transf2020; 34(1): 1–12.
10.
GreenshieldsCJReeseJM. The structure of shock waves as a test of Brenner’s modifications to the Navier-Stokes equations. J Fluid Mech2007; 580: 407–429.
11.
BrennerH. Kinematics of volume transport. Physica A2005; 349: 11–59.
ChristouCKokou DadzieS. An investigation of heat transfer in a cavity flow in the noncontinuum regime. J Heat Transf2017; 139: 092002.
14.
DadzieSKChristouC. Bi-velocity gas dynamics of a micro lid-driven cavity heat transfer subject to forced convection. Int Commun Heat Mass Transf2016; 78: 175–181.
15.
WallsPLLAbedianB. Bivelocity gas dynamics of micro-channel couette flow. Int J Eng Sci2014; 79: 21–29.
16.
ReddyMHLDadzieSK. Accurate constitutive relations for shock wave structures in gases. In: Edited By: HernándezS., University of A Corua, Spain, and VorobieffP., University of New Mexico, USA Advances in fluid dynamics with emphasis on multiphase and complex flow, 2021, vol. 132, pp.165–176.
17.
ReddyMHLDadzieSK. Effects of molecular diffusivity on shock-wave structures in monatomic gases. Phys Rev E2021; 104: 035111.
18.
VaradeVDuryodhanVSAgrawalA, et al. Low Mach number slip flow through diverging microchannel. Comput Fluids2015; 111: 46–61.
19.
AkhlaghiHRoohiEStefanovS. A comprehensive review on micro- and nano-scale gas flow effects: slip-jump phenomena, Knudsen paradox, thermally-driven flows, and Knudsen pumps. Phys Rep2023; 997: 1–60.
20.
MaxwellJC. On stresses in rarefied gases arising from inequalities of temperature. Phil Trans R Soc1879; 170: 231–256.
21.
LeNTPTranNHTranTN. Modified Patterson temperature jump condition considering viscous heat generation. Int J Heat Mass Transf2018; 126: 1267–1274.
22.
MaslenSH. On heat transfer in slip flow. J Aerosp Sci1958; 25: 400–401.
23.
LeNTPVuNALocLT. Effect of the sliding friction on heat transfer in high-speed rarefied gas flow simulations in CFD. Int J Therm Sci2016; 109: 334–341.
24.
LockerbyDAReeseJMGallisMA. A wall-function approach to incorporating Knudsen-layer effects in gas microflow simulations. In: Proceedings of the 24th international symposium on rarefied gas dynamics, Bari, Italy 2004, pp.731–736.
25.
LeNTPWhiteCReeseJM, et al. Langmuir-Maxwell and Langmuir-Smoluchowski boundary conditions for thermal gas flow simulations in hypersonic aerodynamics. Int J Heat Mass Transf2012; 55: 5032–5043.
26.
RanaATorrilhonMStruchtrupH. A robust numerical method for the R13 equations of rarefied gas dynamics: Application to lid driven cavity. J Comput Phys2013; 236: 169–186.
27.
GradH. On the kinetic theory of rarefied gases. Commun Pure Appl Math1949; 2: 331–407.
28.
DadzieSK. A thermo-mechanically consistent Burnett regime continuum flow equation without Chapman–Enskog expansion. J Fluid Mech2013; 716: R6.
