Implementation of Knudsen Layer Phenomena in Rarefied High-Speed Gas Flows
Publication: Journal of Aerospace Engineering
Volume 32, Issue 6
Abstract
A numerical investigation of Knudsen layer effects in high-speed flows in a rarefied flow regime has been carried out. The conventional compressible flow computational fluid dynamics (CFD) solver is reformed based on the effective mean free path model to augment the validity of the Navier-Stokes-Fourier equations in the slip-transition flow regime. The mean free path is expressed as a function of the distance from the surface and local flow gradients, and has been used to modify linear constitutive relations, as well as slip and jump boundary conditions, to incorporate Knudsen layer correction. The improved solver has been validated against direct simulation Monte Carlo (DSMC) data of benchmark test cases of hypersonic flow ( and 12.7) over a flat plate in the slip-flow regime (), and Mach 10 flow over a circular cylinder in the transition flow regime (). The results show that the accuracy of the conventional CFD solver has been significantly improved due to the implementation of the Knudsen layer approach in the near-wall region and the bulk-flow region. Crucial nonlinear trends are captured in the nonequilibrium regions of high-speed flows, such as shock waves and the Knudsen layer. The location of an oblique shock around the flat plate is accurately captured and overall, the Knudsen layer incorporation has exhibited good agreement with the DSMC with deviations within 4% in the upstream region of the cylinder.
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Data Availability Statement
Some of the codes generated or used during the study are available from the corresponding author by request:
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Boundary condition code of Maxwell, Smoluchwoski, and Le temperature jump.
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Postprocessing utility to compute gradient-based Knudsen number.
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Code to calculate normalized MFP over a flat plate.
Acknowledgments
This work is supported by the Ministry of Human Resource Development (MHRD) fellowship. The authors thank Dr. V. K. Sarswat for comments that greatly improved the manuscript.
References
Anderson, J. D. 2000. Hypersonic and high-temperature gas dynamics. Reston, VA: American Institute of Aeronautics and Astronautics.
Bahukudumbi, P. 2003. “A unified engineering model for steady and quasi-steady shear-driven gas microflows.” Microscale Thermophys. Eng. 7 (4): 291–315. https://doi.org/10.1080/10893950390243581.
Bahukudumbi, P., and A. Beskok. 2003. “A phenomenological lubrication model for the entire Knudsen regime.” J. Micromech. Microeng. 13 (6): 873. https://doi.org/10.1088/0960-1317/13/6/310.
Becker, M., and D. E. Boylan. 1966. Flow field and surface pressure measurements in the fully merged and transition flow regimes on a cooled sharp flat plate. Tullahoma, TN: Von Karman Gas Dynamics Facility, Arnold Engineering Development Center, Air Force Systems Command, Arnold Air Force Station.
Bhagat, A., H. Gijare, and N. Dongari. 2019. “Modeling of Knudsen layer effects in the micro-scale backward-facing step in the slip flow regime.” Micromachines 10 (2): 118. https://doi.org/10.3390/mi10020118.
Bird, G. A. 1976. Vol. 76 of Molecular gas dynamics. Oxford: Clarendon Press.
Bird, R. B., W. E. Stewart, and E. N. Lightfoot. 2007. Transport phenomena. New York: Wiley.
Blazek, J. 2015. Computational fluid dynamics: Principles and applications. Amsterdam, Netherlands: Butterworth-Heinemann, Elsevier.
Boyd, I. D., G. Chen, and G. V. Candler. 1995. “Predicting failure of the continuum fluid equations in transitional hypersonic flows.” Phys. Fluids 7 (1): 210–219. https://doi.org/10.1063/1.868720.
Burnett, D. 1936. “The distribution of molecular velocities and the mean motion in a non-uniform gas.” Proc. London Math. Soc. 2 (1): 382–435.
Casseau, V., D. E. Espinoza, T. J. Scanlon, and R. E. Brown. 2016a. “A two-temperature open-source CFD model for hypersonic reacting flows, part two: Multi-dimensional analysis.” Aerospace 3 (4): 45. https://doi.org/10.3390/aerospace3040045.
Casseau, V., R. C. Palharini, T. J. Scanlon, and R. E. Brown. 2016b. “A two-temperature open-source CFD model for hypersonic reacting flows, part one: Zero-dimensional analysis.” Aerospace 3 (4): 34. https://doi.org/10.3390/aerospace3040034.
Cercignani, C. 1969. Mathematical methods in kinetic theory. New York: Springer.
Cercignani, C., A. Frangi, S. Lorenzani, and B. Vigna. 2007. “Bem approaches and simplified kinetic models for the analysis of damping in deformable mems.” Eng. Anal. Boundary Elem. 31 (5): 451–457. https://doi.org/10.1016/j.enganabound.2006.11.010.
Chapman, S., T. G. Cowling, and D. Burnett. 1970. The mathematical theory of non-uniform gases: An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge: Cambridge Univ. Press.
Chen, X.-X., Z.-H. Wang, and Y.-L. Yu. 2014. “Nonlinear shear and heat transfer in hypersonic rarefied flows past flat plates.” AIAA J. 53 (2): 413–419. https://doi.org/10.2514/1.J053168.
Dongari, N. 2012. “Micro gas flows: Modelling the dynamics of Knudsen layers.” Ph.D. thesis, Dept. of Mechanical and Aerospace Engineering, Univ. of Strathclyde.
Dongari, N., R. W. Barber, D. R. Emerson, S. K. Stefanov, Y. Zhang, and J. M. Reese. 2013a. “The effect of Knudsen layers on rarefied cylindrical Couette gas flows.” Microfluid. Nanofluid. 14 (1–2): 31–43. https://doi.org/10.1007/s10404-012-1019-2.
Dongari, N., F. Durst, and S. Chakraborty. 2010. “Predicting microscale gas flows and rarefaction effects through extended Navier–Stokes–Fourier equations from phoretic transport considerations.” Microfluid. Nanofluid. 9 (4–5): 831–846. https://doi.org/10.1007/s10404-010-0604-5.
Dongari, N., C. White, T. J. Scanlon, Y. Zhang, and J. M. Reese. 2013b. “Effects of curvature on rarefied gas flows between rotating concentric cylinders.” Phys. Fluids 25 (5): 052003. https://doi.org/10.1063/1.4807072.
Dongari, N., Y. Zhang, and J. M. Reese. 2011a. “Modeling of Knudsen layer effects in micro/nanoscale gas flows.” J. Fluids Eng. 133 (7): 071101. https://doi.org/10.1115/1.4004364.
Dongari, N., Y. Zhang, and J. M. Reese. 2011b. “Molecular free path distribution in rarefied gases.” J. Phys. D: Appl. Phys. 44 (12): 125502. https://doi.org/10.1088/0022-3727/44/12/125502.
Fichman, M., and G. Hetsroni. 2005. “Viscosity and slip velocity in gas flow in microchannels.” Phys. Fluids 17 (12): 123102. https://doi.org/10.1063/1.2141960.
Gijare, H., A. Assam, and N. Dongari. 2015. “Aero-thermodynamics optimization of re-entry capsule in the slip flow regime.” In Proc., 23rd National Heat and Mass Transfer Conf. and 1st Int. ISHMT-ASTFE Heat and Mass Transfer Conf., 1–8. Trivandrum: Liquid Propulsion Systems Centre, Indian Space Research Organisation.
Gijare, H., A. Bhagat, and N. Dongari. 2017. “Effect of chemical non-equilibrium on flow parameters in the intermediate hypersonic regime.” In Proc., 24th National and 2nd Int. ISHMT-ASTFE Heat and Mass Transfer Conf., 535–541. Hyderabad, India: BITS-Pilani.
Gijare, H., A. Bhagat, and N. Dongari. 2019. “Effect of Knudsen layer on the heat transfer in hypersonic rarefied gas flows.” Int. J. Therm. Sci. 142 (Aug): 134–141. https://doi.org/10.1016/j.ijthermalsci.2019.04.016.
Grad, H. 1949. “Note on n-dimensional hermite polynomials.” Commun. Pure Appl. Math. 2 (4): 325–330. https://doi.org/10.1002/cpa.3160020402.
Greenshields, C. J., and J. M. Reese. 2012. “Rarefied hypersonic flow simulations using the Navier–Stokes equations with non-equilibrium boundary conditions.” Prog. Aerosp. Sci. 52 (Jul): 80–87. https://doi.org/10.1016/j.paerosci.2011.08.001.
Guo, Z., J. Qin, and C. Zheng. 2014. “Generalized second-order slip boundary condition for nonequilibrium gas flows.” Phys. Rev. E 89 (1): 013021. https://doi.org/10.1103/PhysRevE.89.013021.
Guo, Z., B. Shi, and C. G. Zheng. 2007. “An extended Navier-Stokes formulation for gas flows in the Knudsen layer near a wall.” Europhys. Lett. 80 (2): 24001. https://doi.org/10.1209/0295-5075/80/24001.
Hirschfelder, J., R. B. Bird, and C. F. Curtiss. 1964. Molecular theory of gases and liquids. New York: Wiley.
Jaiswal, S., and N. Dongari. 2015. “Implementation of Knudsen layer effects in open source CFD solver for effective modeling of microscale gas flows.” In Proc., 23rd National Heat and Mass Transfer Conf. and 1st Int. ISHMT-ASTFE Heat and Mass Transfer Conf., 1–8. Trivandrum: Liquid Propulsion Systems Centre, Indian Space Research Organisation.
Jasak, H. 2010. “Openfoam: A year in review.” In Proc., 5th OpenFOAM Workshop. Bracknell, UK: ESI-Open CFD.
Karniadakis, G., A. Beskok, and N. Aluru. 2006. Vol. 29 of Microflows and nanoflows: Fundamentals and simulation. New York: Springer.
Kennard, E. H. 1938. Kinetic theory of gases, with an introduction to statistical mechanics. London: McGraw-Hill.
Kogan, M. 1969. Rarefied gas dynamics. New York: Plenum Press.
Kurganov, A., S. Noelle, and G. Petrova. 2001. “Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations.” SIAM J. Sci. Comput. 23 (3): 707–740. https://doi.org/10.1137/S1064827500373413.
Kurganov, A., and E. Tadmor. 2000. “New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations.” J. Comput. Phys. 160 (1): 241–282. https://doi.org/10.1006/jcph.2000.6459.
Le, N. T., and E. Roohi. 2015. “A new form of the second-order temperature jump boundary condition for the low-speed nanoscale and hypersonic rarefied gas flow simulations.” Int. J. Therm. Sci. 98 (Dec): 51–59. https://doi.org/10.1016/j.ijthermalsci.2015.06.017.
Le, N. T., N. A. Vu, and L. T. Loc. 2016. “New type of Smoluchowski temperature jump condition considering the viscous heat generation.” AIAA J. 55 (2): 474–483. https://doi.org/10.2514/1.J055058.
Le, N. T., C. White, J. M. Reese, and R. S. Myong. 2012. “Langmuir–Maxwell and Langmuir–Smoluchowski boundary conditions for thermal gas flow simulations in hypersonic aerodynamics.” Int. J. Heat Mass Transfer 55 (19–20): 5032–5043. https://doi.org/10.1016/j.ijheatmasstransfer.2012.04.050.
Li, J.-M., B.-X. Wang, and X.-F. Peng. 2000. “Wall-adjacent layer analysis for developed-flow laminar heat transfer of gases in microchannels.” Int. J. Heat Mass Transfer 43 (5): 839–847. https://doi.org/10.1016/S0017-9310(99)00109-X.
Lilley, C. R., and J. E. Sader. 2007. “Velocity gradient singularity and structure of the velocity profile in the Knudsen layer according to the Boltzmann equation.” Phys. Rev. E 76 (2): 026315. https://doi.org/10.1103/PhysRevE.76.026315.
Lockerby, D. A., and J. M. Reese. 2008. “On the modelling of isothermal gas flows at the microscale.” J. Fluid Mech. 604: 235–261. https://doi.org/10.1017/S0022112008001158.
Lockerby, D. A., J. M. Reese, and M. A. Gallis. 2005. “Capturing the Knudsen layer in continuum-fluid models of nonequilibrium gas flows.” AIAA J. 43 (6): 1391–1393. https://doi.org/10.2514/1.13530.
Lofthouse, A. J., L. C. Scalabrin, and I. D. Boyd. 2008. “Velocity slip and temperature jump in hypersonic aerothermodynamics.” J. Thermophys. Heat Transfer 22 (1): 38–49. https://doi.org/10.2514/1.31280.
Maxwell, J. C. 1867. “On the dynamical theory of gases.” Philos. Trans. R. Soc. London 157: 49–88. https://doi.org/10.1098/rstl.1867.0004.
Myong, R. 2016. “Theoretical description of the gaseous Knudsen layer in Couette flow based on the second-order constitutive and slip-jump models.” Phys. Fluids 28 (1): 012002. https://doi.org/10.1063/1.4938240.
Norouzi, A., and J. A. Esfahani. 2015. “Two relaxation time lattice Boltzmann equation for high Knudsen number flows using wall function approach.” Microfluid. Nanofluid. 18 (2): 323–332. https://doi.org/10.1007/s10404-014-1435-6.
Ohwada, T., Y. Sone, and K. Aoki. 1989. “Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules.” Phys. Fluids A: Fluid Dyn. 1 (12): 2042–2049. https://doi.org/10.1063/1.857478.
Park, J. H., P. Bahukudumbi, and A. Beskok. 2004. “Rarefaction effects on shear driven oscillatory gas flows: A direct simulation Monte Carlo study in the entire Knudsen regime.” Phys. Fluids 16 (2): 317–330. https://doi.org/10.1063/1.1634563.
Reese, J. M., Y. Zheng, and D. A. Lockerby. 2007. “Computing the near-wall region in gas micro-and nanofluidics: Critical Knudsen layer phenomena.” J. Comput. Theor. Nanosci. 4 (4): 807–813. https://doi.org/10.1166/jctn.2007.2372.
Smoluchowski von Smolan, M. 1898. “Ueber wärmeleitung in verdünnten gasen.” [On heat conduction in diluted gases] Ann. Phys. 300 (1): 101–130. https://doi.org/10.1002/andp.18983000110.
Sone, Y. 2012. Kinetic theory and fluid dynamics. New York: Springer.
Sone, Y., S. Takata, and T. Ohwada. 1990. “Numerical analysis of the plane Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for hard-sphere molecules.” Eur. J. Mech. B Fluids 9 (3): 273–288.
To, Q. D., C. Léonard, and G. Lauriat. 2015. “Free-path distribution and Knudsen-layer modeling for gaseous flows in the transition regime.” Phys. Rev. E 91 (2): 023015. https://doi.org/10.1103/PhysRevE.91.023015.
Tu, C., L. Qian, F. Bao, and W. Yan. 2017. “Local effective viscosity of gas in nano-scale channels.” Eur. J. Mech. B. Fluids 64 (Jul–Aug): 55–59. https://doi.org/10.1016/j.euromechflu.2017.01.007.
Veijola, T., and M. Turowski. 2001. “Compact damping models for laterally moving microstructures with gas-rarefaction effects.” J. Microelectromech. Syst. 10 (2): 263–273. https://doi.org/10.1109/84.925777.
Vincenti, W. G., and C. H. Kruger. 1965. Vol. 246 of Introduction to physical gas dynamics. New York: Wiley.
Wang, W.-L. 2004. “A hybrid particle/continuum approach for nonequilibrium hypersonic flows.” Ph.D. thesis, Dept. of Aerospace Engineering, Univ. of Michigan.
Weller, H. G., G. Tabor, H. Jasak, and C. Fureby. 1998. “A tensorial approach to computational continuum mechanics using object-oriented techniques.” Comput. Phys. 12 (6): 620–631. https://doi.org/10.1063/1.168744.
Zhang, Y.-H., X.-J. Gu, R. W. Barber, and D. R. Emerson. 2006. “Capturing Knudsen layer phenomena using a lattice Boltzmann model.” Phys. Rev. E 74 (4): 046704. https://doi.org/10.1103/PhysRevE.74.046704.
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Published In
Journal of Aerospace Engineering
Volume 32 • Issue 6 • November 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Aug 7, 2018
Accepted: Jul 12, 2019
Published online: Sep 11, 2019
Published in print: Nov 1, 2019
Discussion open until: Feb 11, 2020
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