Abstract
On the basis of the kinetic model, the steady efflux of monatomic gas from a high pressure camera (the Knudsen number \({\text{Kn}} \ll 1\)) through a long channel between two parallel plates into a vacuum camera under a constant temperature on the bounding surfaces is studied. Using asymptotic estimates for relatively long channels, the flow domain is divided into three subdomains: (1) a neighborhood of the channel entry, (2) the main part of the flow in the channel that occupies almost all channel length, and (3) a neighborhood of the channel exit. The flow in subdomain (1) is not considered due to its low speed. In the main subdomain (2), the flow is slow and is driven by a low pressure gradient (the diffusion area). In subdomain (3), the flow gets faster, and the gas expands in the channel and in the vacuum camera. In subdomain (2), we have the continuum flow regime, and the well-known results of the linear one-dimensional theory of viscous gas flows in long channels (Poiseuille flow) are used. In the subdomain of fast flow, the full nonlinear kinetic equation (S-model) is used. The condition of asymptotic matching of solutions in two subdomains is replaced by the boundary condition of solution coupling in a certain section the position of which is chosen from the smoothness condition of the full solution of the problem. The kinetic equation is solved by the method of time marching to steady state using the conservative second-order scheme with respect to all variables implemented in Nesvetay software package. The proposed solution method can be considered as a hybrid one because the Navier–Stokes and kinetic equations are solved simultaneously.
Similar content being viewed by others
REFERENCES
F. Sharipov and V. Seleznev, “Data on internal rarefied gas flows,” J. Phys. Chem. Ref. Data. 27, 657–706 (1998).
F. M. Sharipov and V. D. Seleznev, Rarefied Gas Flow in Channels and Microchannels (Ural. Otd. Ross. Akad. Nauk, Ekaterinburg, 2008) [in Russian].
F. Sharipov and V. D. Seleznev, “Rarefied gas flow through a long tube at any pressure ratio,” J. Vac. Sci. Technol. A 12, 2933–2935 (1994).
I. Graur and F. Sharipov, “Non-isothermal flow of rarefied gas through a long pipe with elliptic cross section,” Microfluidics Nanofluidics 6 (2), 267–275 (2009).
S. Varoutis, C. Day, and F. Sharipov, “Rarefied gas flow through channels of finite length at various pressure ratios,” Vacuum 86, 1952–1959 (2012).
S. Pantazis, D. Valougeorgis, and F. Sharipov, “End corrections for rarefied gas flows through capillaries of finite length,” Vacuum 97, 26 – 29 (2013).
S. Pantazis, D. Valougeorgis, and F. Sharipov, “End corrections for rarefied gas flows through circular tubes of finite length,” Vacuum 101, 306 – 312 (2014).
D. Valougeorgis, N. Vasileiadis, and V. Titarev, “Validity range of linear kinetic modeling in rarefied pressure driven single gas flows through circular capillaries,” Eur. J. Mech. B. Fluids, Special Issue on Non-equilibrium Gas Flows. 64, 2–7 (2017).
V. A. Titarev, “Rarefied gas flow in a planar channel caused by arbitrary pressure and temperature drops,” Int. J. Heat Mass Transfer 55, 5916 – 5930 (2012).
V. A. Titarev and E. M. Shakhov, “Rarefied gas flow through a long circular pipe into vacuum,” Proc. 28th Int. Symp., AIP Conf. Proc. on Rarefied Gas Dynamics 1501, 2012, pp. 465–472.
V. A. Titarev and E. M. Shakhov, “End effects in rarefied gas outflow into vacuum through a long tube,” Fluid Dyn. 48, 697–707 (2013). № 5. C. 146–158.
E. M. Shakhov, “On a generalization of the relaxation kinetic Krook equation, Izv. Akad. Nauk SSSR, Ser. Mekh. Zhidkosti Gaza, No. 5, 142–145 (1968).
P. L. Bhatnagar, E. P. Gross, and M. Krook, “A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems,” Phys. Rev. 94, 511–525 (1954).
E. M. Shakhov, “Linearized two-dimensional problem of rarefied gas flow in a long channel,” Comput. Math. Math. Phys. 39, 1192–1200 (1999).
N. A. Konopel’ko, V. A. Titarev, and E. M. Shakhov, “Unsteady rarefied gas flow in a microchannel driven by a pressure difference,” Comput. Math. Math. Phys. 56, 470–482 (2016).
V. A. Titarev, “Software package Nesvetay-3D for the simulation of 3D flows of a monoatomic rarefied gas,” Nauka obraz. Electron. J., No. 6, 124–154 (2014).
V. A. Titarev, “Application of model kinetic equations to hypersonic rarefied gas flows,” Comput. & Fluids, Special issue “Nonlinear flow and transport” 169, 62–70 (2018).
V. A. Titarev, “Numerical methods for model kinetic equations and their application to external high-speed flows,” in Continuum Mech., Applied Mathematics and Scientific Computing: Godunov’s Legacy (Springer, 2020), pp. 353–358.
V. A. Titarev, “Rarefied gas flow in a circular pipe of finite length,” Vacuum 94, 92–103 (2013).
V. A. Titarev and E. M. Shakhov, “Rarefied gas flow into vacuum through a pipe composed of two circular sections of different radii,” Vacuum, Special Issue “Advances in Vacuum Gas Dynamics” 109, 236–245 (2014).
V. A. Titarev, E. M. Shakhov, and A. A. Frolova, “Shock wave reflection from a short orifice open to vacuum,” Vacuum 161, 232–241 (2019).
M. N. Petrov, A. A. Tambova, V. A. Titarev, S. V. Utyuzhnikov, and A. V. Chikitkin, “FlowModellium software package for calculating high-speed flows of compressible fluid,” Comput. Math. Math. Phys. 58, 1865–1886 (2018).
A. Chikitkin, M. Petrov, V. Titarev, and S. Utyuzhnikov, “Parallel versions of implicit LU-SGS method,” Special Iss. Lobachevskii J. Math. on Parallel Structure of Algorithms 39, 503–512 (2018).
V. A. Titarev, S. V. Utyuzhnikov, and A. V. Chikitkin, “OpenMP + MPI parallel implementation of a numerical method for solving a kinetic equation,” Comput. Math. Math. Phys. 56, 1919–1928 (2016).
J.C. Chai, H.S. Lee, and V. Patankar Suhas, “Ray effect and false scattering in the discrete ordinates method,” Num. Heat Transfer, Part B: Fundamentals 24, 373–389 (1993).
S. Brull and L. Mieussens, “Local discrete velocity grids for deterministic rarefied flow simulations,” J. Comput. Phys. 266, 22–46 (2014).
V. V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows (Kluwer, Dordrecht, 2001).
V. I. Kolobov, R. R. Arslanbekov, V. V. Aristov, A. A. Frolova, and S. A. Zabelok, “Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement,” J. Comput. Phys. 223, 589–608 (2007).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by A. Klimontovich
Rights and permissions
About this article
Cite this article
Titarev, V.A., Shakhov, E.M. A Hybrid Method for the Computation of a Rarefied Gas Jet Efflux through a Very Long Channel into Vacuum. Comput. Math. and Math. Phys. 60, 1936–1949 (2020). https://doi.org/10.1134/S0965542520110135
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542520110135