General and exact pressure evolution equation (original) (raw)

Exact Pressure Evolution Equation for Incompressible Fluids

2008

An important aspect of computational fluid dynamics is related to the determination of the fluid pressure in isothermal incompressible fluids. In particular this concerns the construction of an exact evolution equation for the fluid pressure which replaces the Poisson equation and yields an algorithm which is a Poisson solver, i.e., it permits to time-advance exactly the same fluid pressure \textit{without solving the Poisson equation}% . In fact, the incompressible Navier-Stokes equations represent a mixture of hyperbolic and elliptic pde's, which are extremely hard to study both analytically and numerically. In this paper we intend to show that an exact solution to this problem can be achieved adopting the approach based on inverse kinetic theory (IKT) recently developed for incompressible fluids by Ellero and Tessarotto (2004-2007). In particular we intend to prove that the evolution of the fluid fields can be achieved by means of a suitable dynamical system, to be identified with the so-called Navier-Stokes (N-S) dynamical system. As a consequence it is found that the fluid pressure obeys a well-defined evolution equation. The result appears relevant for the construction of Lagrangian approaches to fluid dynamics.

An exact pressure evolution equation for the incompressible Navier-Stokes equations

2006

In this paper the issue of the determination of the fluid pressure in incompressible fluids is addressed, with particular reference to the search of algorithms which permit to advance in time the fluid pressure without actually solving numerically the Poisson equation. Based on an inverse kinetic approach recently proposed for the incompressible Navier-Stokes equations we intend to prove that an exact evolution equation can be obtained which advances in time self-consistently the fluid pressure. The new equation is susceptible of numerical implementation in Lagrangian CFD simulation codes.

A Survey of the Compressible Navier-Stokes Equations

Taiwanese Journal of Mathematics

This paper presents mathematical properties of solutions to the Navier-Stokes equations for compressible fluids. We first review existence results for the Cauchy problem, and describe some regularity properties of solutions in the presence of possibly vanishing densities. Finally, we address the problem of the low Mach number limit leading to incompressible models.

Thermodynamic Theory of Incompressible Hydrodynamics

Physical Review Letters, 2005

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.

Lattice Boltzmann Inverse Kinetic Approach for Classical Incompressible Fluids

AIP Conference Proceedings, 2008

In spite of the large number of papers appeared in the past which are devoted to the lattice Boltzmann (LB) methods, basic aspects of the theory still remain unchallenged. An unsolved theoretical issue is related to the construction of a discrete kinetic theory which yields exactly the fluid equations, i.e., is non-asymptotic (here denoted as LB inverse kinetic theory). The purpose of this paper is theoretical and aims at developing an inverse kinetic approach of this type. In principle infinite solutions exist to this problem but the freedom can be exploited in order to meet important requirements. In particular, the discrete kinetic theory can be defined so that it yields exactly the fluid equation also for arbitrary non-equilibrium (but suitably smooth) kinetic distribution functions and arbitrarily close to the boundary of the fluid domain. This includes the specification of the kinetic initial and boundary conditions which are consistent with the initial and boundary conditions prescribed for the fluid fields. Other basic features are the arbitrariness of the "equilibrium" distribution function and the condition of positivity imposed on the kinetic distribution function. The latter can be achieved by imposing a suitable entropic principle, realized by means of a constant H-theorem. Unlike previous entropic LB methods the theorem can be obtained without functional constraints on the class of the initial distribution functions. As a basic consequence, the choice of the the entropy functional remains essentially arbitrary so that it can be identified with the Gibbs-Shannon entropy. Remarkably, this property is not affected by the particular choice of the kinetic equilibrium (to be assumed in all cases strictly positive). Hence, it applies also in the case of polynomial equilibria, usually adopted in customary LB approaches. We provide different possible realizations of the theory and asymptotic approximations which permit to determine the fluid equations with prescribed accuracy. As a result, asymptotic accuracy estimates of customary LB approaches and comparisons with the Chorin artificial compressibility method are discussed.

Lattice Boltzmann modeling and simulation of compressible flows

Frontiers of Physics, 2012

In this mini-review we summarize the progress of Lattice Boltzmann(LB) modeling and simulating compressible flows in our group in recent years. Main contents include (i) Single-Relaxation-Time(SRT) LB model supplemented by additional viscosity, (ii) Multiple-Relaxation-Time(MRT) LB model, and (iii) LB study on hydrodynamic instabilities. The former two belong to improvements of physical modeling and the third belongs to simulation or application. The SRT-LB model supplemented by additional viscosity keeps the original framework of Lattice Bhatnagar-Gross-Krook (LBGK). So, it is easier and more convenient for previous SRT-LB users. The MRT-LB is a completely new framework for physical modeling. It significantly extends the range of LB applications. The cost is longer computational time. The developed SRT-LB and MRT-LB are complementary from the sides of convenience and applicability. Only after the physical model is fixed can the corresponding numerical method be established. Clearly, the first kind of view starts LB research from the second step, numerical method design. It does not consider the improvements of the physical modeling. In other words, it assumes that the original hydrodynamic equations are sufficiently exact for modeling the problem under consideration. The second kind of view puts LB research on the more fundamental step, physical modeling. For this view the numerical method is the second important issue. It accepts any reasonable numerical methods no matter they are new or traditional. The second kind of view aims at physical problems. The point of the second view is that, compared with the traditional hydrodynamic equations, the LB framework contains

A new formulation of equations of compressible fluids by analogy with Maxwell's equations

Fluid Dynamics Research, 2010

A compressible ideal fluid is governed by Euler's equation of motion and equations of continuity, entropy and vorticity. This system can be reformulated in a form analogous to that of electromagnetism governed by Maxwell's equations with source terms. The vorticity plays the role of magnetic field, while the velocity field plays the part of a vector potential and the enthalpy (of isentropic flows) plays the part of a scalar potential in electromagnetism. The evolution of source terms of fluid Maxwell equations is determined by solving the equations of motion and continuity. The equation of sound waves can be derived from this formulation, where time evolution of the sound source is determined by the equation of motion. The theory of vortex sound of aeroacoustics is included in this formulation. It is remarkable that the forces acting on a point mass moving in a velocity field of an inviscid fluid are analogous in their form to the electric force and Lorentz force in electromagnetism. The significance of the reformulation is interpreted by examples taken from fluid mechanics. This formulation can be extended to viscous fluids without difficulty. The Maxwell-type equations are unchanged by the viscosity effect, although the source terms have additional terms due to viscosities.