Thermodynamic theory of incompressible hydrodynamics
ABSTRACT The grand potential for open systems describes thermodynamics of fluid flows at low Mach numbers. A new system of reduced equations for the grand potential and the fluid momentum is derived from the compressible Navier-Stokes equations. The incompressible Navier-Stokes equations are the quasistationary solution to the new system. It is argued that the grand canonical ensemble is the unifying concept for the derivation of models and numerical methods for incompressible fluids, illustrated here with a simulation of a minimal Boltzmann model in a microflow setup.
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ABSTRACT: In flows through microdevices the continuum fluid mechanics description often breaks down and higher order corrections to the Navier-Stokes description arise both at the boundaries and in the bulk. The interaction between the flow geometry, rarefaction and compressibility is not completely understood for such flows. Recent advances in compu- tational kinetic theory, such as the entropic lattice Boltzmann method, provide a simple and realistic framework which enable the systematic study of such interactions. We consider a specific example of entropic lattice Boltzmann model and compare it with Grad's moment system. We show that for the model under consideration, the dispersion relation is closely related to that of Grad's ten-moment system. We perform a parametric study of the flow in a microcavity, which is a prototype problem, where the deviations from incompressible hydrodynamics can be studied conveniently. Simulation results obtained with the entropic lattice Boltzmann method are compared with those of the Direct Simulation Monte-Carlo method. Based on the parametric study, we discuss aspects of the interaction between rarefaction and compressibility.
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ABSTRACT: An alternative artificial compressibility (AC) scheme is proposed to allow the explicit simulation of the incompressible Navier-Stokes (INS) equations. Traditional AC schemes rely on an artificial equation of state that gives the pressure as a function of the density, which is known to enforce isentropic behavior. This behavior is nonideal, especially in viscously dominated flows. An alternative, the entropically damped artificial compressibility (EDAC) method, is proposed that employs a thermodynamic constraint to damp the pressure oscillations inherent to AC methods. The EDAC method converges to the INS in the low-Mach limit, and is consistent in both the low- and high-Reynolds-number limits, unlike standard AC schemes. The proposed EDAC method is discretized using a simple finite-difference scheme and is compared with traditional AC schemes as well as the lattice-Boltzmann method for steady lid-driven cavity flow and a transient traveling-wave problem. The EDAC method is shown to be beneficial in damping pressure and velocity-divergence oscillations when performing transient simulations. The EDAC method follows a similar derivation to the kinetically reduced local Navier-Stokes (KRLNS) method [Borok et al., Phys. Rev. E 76, 066704 (2007)]; however, the EDAC method does not rely on the grand potential as the thermodynamic variable, but instead uses the more common pressure-velocity system. Additionally, a term neglected in the KRLNS is identified that is important for accurately approximating the INS equations.Physical Review E 01/2013; 87(1-1):013309. DOI:10.1103/PhysRevE.87.013309 · 2.33 Impact Factor
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ABSTRACT: Concepts of the lattice Boltzmann method are discussed in detail for the one-dimensional kinetic model. Various techniques of constructing lattice Boltzmann models are discussed, and novel collision integrals are derived. Geometry of the ki- netic space and the role of the thermodynamic projector is elucidated.