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Modelling low Knudsen number transition flows using a computationally efficient continuum-based methodology

Mizzi, Simon and Emerson, David and Barber, Robert W. and Reese, Jason (2006) Modelling low Knudsen number transition flows using a computationally efficient continuum-based methodology. In: Proceedings of the 4th International Conference on Nanochannels, Microchannels and Minichannels (ICNMM2006). ASME. ISBN 0791847608

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Abstract

This paper presents a new technique that combines Grad's 13-moment equations (G13) with a phenomenological approach. The combination of these approaches and the proposed solution technique manages to capture important non-equilibrium phenomena that start to appear in the early transition-flow regime. In contrast to the fully-coupled 13-moment equations, a significant advantage of the present solution technique is that it does not require extra boundary conditions. The solution method is similar in form to the Maxwellian iteration used in the kinetic theory of gases. In our approach, Grad's equations for viscous stress and heat flux are used as constitutive relations for the conservation equations instead of being solved as equations of transport. This novel technique manages to capture non-equilibrium effects and its relative computational cost is low in comparison to other methods such as fully-coupled solutions involving many moments or discrete methods. In this study, the proposed numerical procedure is applied to a planar Couette flow and the results are compared to predictions obtained from the direct simulation Monte Carlo method. In the transition regime, this test case highlights the presence of normal viscous stresses and tangential heat fluxes that arise from non-equilibrium phenomena. These effects cannot be captured by the Navier-Stokes-Fourier constitutive equations or phenomenological modifications thereof. Moreover, simply using the G13 equations, along with the decoupled solution method, does not capture the nonlinearities occurring in the proximity of a solid wall. However, combining phenomenological scaling functions and slip boundary conditions with the G13 equations provides a better representation of these important non-equilibrium phenomena but at a relatively low computational cost.