From: Bruce Boghosian (bmb@Think.COM)
Date: Thu May 27 1993 - 22:57:23 UTC
Date: 27 May 1993 15:41:58 -0400 From: butterwo@cs.unc.edu (Jeff Butterworth) In article <1993May27.071230.22564@crs4.it> ralf@grappa.crs4.it (Ralph Santos) writes: >> I am programming a fluid flow model using a hexagonal mesh. Its a 2-D >> channel flow with a solid boundary at the 'top', a solid boundary at the >> 'bottom' and the flow from 'left' to 'right'. The solid boundaries are non- >> slip. >> >> I would appreciate hints on how to do the left and right boundary conditions. >> The two methods mentioned so far are (1) using periodic boundary conditions, and then driving the system with an artificial gravity, and (2) introducing particles with different probabilities at the left and right boundaries. Indeed, both methods can be effective. With regard to the first, it has been noted that the artificial gravity can be a volumetric force, set up by introducing occasional non-momentum-conserving collisions at random sites throughout the medium. The driving force per unit volume is then equal to the amount of momentum introduced per unit volume per unit time. Note that there has to be some obstacle when you do this -- you have to have drag past some object -- otherwise you have a source for momentum and no sink, so the momentum keeps ramping up... Also note that when taking quantitative measurements you sometimes have to be careful that the wake doesn't wrap around and distort the incoming flow. With regard to the second method, constant velocity boundary conditions can be set up by sampling the incoming particles at the enterance and exit from the Fermi-Dirac distribution that is the equilibrium for the lattice gas (assuming that you are using a lattice gas that satisfies semi-detailed balance). Outgoing particles at either the enterance or exit are just allowed to leave the system. You have to ensure that the incoming flux is equal to the outgoing flux in order to achieve a steady state. For details on the Fermi-Dirac equilibrium, see one of the introductory texts on lattice gases that have been quoted on this list in the recent past. I think that you can implement constant pressure boundary conditions in this way too, though I have never tried it. Note that the two methods correspond to different mathematical formulations of the fluid: The first method solves the Navier-Stokes equations with an external force term and periodic boundary conditions, while the second method solves it without the external force term but with constant velocity boundary conditions. Finally, note that either method introduces a probabilistic component to the lattice gas rule. -- Bruce M. Boghosian | Internet: bmb@think.com Thinking Machines Corporation | Bitnet: bmb%think.com@mitvma.bitnet 245 First Street | Phone: (617) 234-2140 (work) Cambridge, MA 02142-1264 USA | FAX: (617)-234-4444
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