Closure of the hierarchy of fluid equations by means of the polytropic-coefficient function (PCF)
- Association EURATOM-OAeW, Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck (Austria)
- LECAD Laboratory, University of Ljubljana, SI-1000 Ljubljana (Slovenia)
The continuity and momentum equations of a fluid plasma component may be viewed as four scalar evolution equations for the four scalar fluid variables n(x-vector,t)(density) and u(x-vector,t)(fluid velocity), which are zeroth- and first order velocity moments of the velocity distribution function (VDF). However, the momentum equation in addition contains the gradient of the pressure p(x-vector,t), which is a second-order velocity moment for which another equation, the 'closure equation', is needed. In the present work, closure by means of the polytropic-coefficient function (PCF) is discussed which, by analogy with the well-known polytropic coefficient (also called the 'polytropic index' or 'polytropic exponent') in macroscopic thermodynamic systems, is formally defined by {gamma}(x-vector,t) = (nDp/Dt)(pDn/Dt) = (n/p)(Dp/Dn), with D/Dt = {partial_derivative}/{partial_derivative}t+u-vector{center_dot}{partial_derivative}/{partial_derivative}x-vector, which amounts to the closure equation if {gamma}(x-vector,t) is known. In fluid problems, however, the PCF is usually unknown and hence must be assumed or guessed, but in kinetic problems it can be calculated exactly. These general concepts are first developed and then applied specifically to the basic Tonks-Langmuir (TL) model [L. Tonks and I. Langmuir, Phys. Rev. 34, 876, 1929]. It is shown for the first time that results obtained from the fluid equations closed with the correct PCF coincide with the corresponding results calculated on the basis of the exact kinetic solution [K.-U. Riemann, Phys. Plasmas 13, 063508 (2006)], but differ visibly from those obtained from the approximate fluid equations closed with the zero-pressure approximation [Riemann et al., Plasma Phys. Control. Fusion 47, 1949 (2005)]. Also, it is again confirmed that the correct PCF may be a strongly varying function of position, so that the simple constant values of {gamma} usually assumed [K.-U. Riemann, XXVIII International Conference on Phenomena in Ionized Gases, 479 (2007)] may lead to markedly erroneous results especially near material walls. All of these findings lead us to conclude that better approximations to the PCF are needed for closing fluid equations in an appropriate manner.
- OSTI ID:
- 21506954
- Journal Information:
- AIP Conference Proceedings, Vol. 1306, Issue 1; Conference: Workshop on new frontiers in advanced plasma physics, Trieste (Italy), 5-16 Jul 2010; Other Information: DOI: 10.1063/1.3533190; (c) 2010 American Institute of Physics; ISSN 0094-243X
- Country of Publication:
- United States
- Language:
- English
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