Abstract
The formulation of the half-range moment method (HRMM), well defined in steady rarefied gas flows, is extended to linear oscillatory rarefied gas flows, driven by oscillating boundaries. The oscillatory Stokes (also known as Stokes second problem) and the oscillatory Couette flows, as representative ones for harmonically oscillating half-space and finite-medium flow setups respectively, are solved. The moment equations are derived from the linearized time-dependent BGK kinetic equation, operating accordingly over the positive and negative halves of the molecular velocity space. Moreover, the boundary conditions of the “positive” and “negative” moment equations are accordingly constructed from the half-range moments of the boundary conditions of the outgoing distribution function, assuming purely diffuse reflection. The oscillatory Stokes flow is characterized by the oscillation parameter, while the oscillatory Couette flow by the oscillation and rarefaction parameters. HRMM results for the amplitude and phase of the velocity and shear stress in a wide range of the flow parameters are presented and compared with corresponding results, obtained by the discrete velocity method (DVM). In the oscillatory Stokes flow the so-called penetration depth is also computed. When the oscillation frequency is lower than the collision frequency excellent agreement is observed, while when it is about the same or larger some differences are present. Overall, it is demonstrated that the HRMM can be applied to linear oscillatory rarefied gas flows, providing accurate results in a very wide range of the involved flow parameters. Since the computational effort is negligible, it is worthwhile to consider the efficient implementation of the HRMM to stationary and transient multidimensional rarefied gas flows.
Highlights
Oscillatory boundary-driven rarefied gas flows have attracted, over the years, considerable attention [1,2,3,4,5,6,7,8,9,10,11,12,13], due to their presence in a variety of systems, such as resonating filters, sensors and actuators, where the computation of the damping forces is crucial in controlling and optimizing the resolution and sensitivity of the signal [14]
The research work has been extended to binary gas mixtures, examining the propagation of sound waves, due to mechanical and thermal excitation caused by moving boundaries [18,19,20,21,22,23]
The implemented numerical schemes are mainly based on the stochastic Direct Simulation Monte Carlo (DSMC) method [1,8,15,16,17] and the deterministic solution of the Boltzmann equation or kinetic model equations by the discrete velocity method (DVM) [2,3,4,21,22]
Summary
Oscillatory boundary-driven rarefied gas flows have attracted, over the years, considerable attention [1,2,3,4,5,6,7,8,9,10,11,12,13], due to their presence in a variety of systems, such as resonating filters, sensors and actuators, where the computation of the damping forces is crucial in controlling and optimizing the resolution and sensitivity of the signal [14]. When either of these restrictions is relaxed, the flow is classified as rarefied and may be in the transition or free molecular regimes depending on the time and space characteristic scales [2,3,4,5,24,25,26] In this context, in the present work two prototype oscillatory gas flows, namely the oscillating plane Couette and Stokes flows are tackled, in the whole range of gas rarefaction and oscillation frequency by the half-range Hermite moment method. The output quantities from the half-range moment method include the amplitudes and the phases of the velocity and shear stresses and they are systematically compared with corresponding results based on the kinetic solution via the DVM to investigate the range of its validity in solving finite medium (slab) and half-space oscillatory flow problems.
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