Navier-Stokes equation, conserved order

Dissipative particle dynamics (DPD) is a technique for simulating the motion of mesoscale beads. The technique is superficially similar to a Brownian dynamics simulation in that it incorporates equations of motion, a dissipative (random) force, and a viscous drag between moving beads. However, the simulation uses a modified velocity Verlet algorithm to ensure that total momentum and force symmetries are conserved. This results in a simulation that obeys the Navier-Stokes equations and can thus predict flow. In order to set up these equations, there must be parameters to describe the interaction between beads, dissipative force, and drag. 

The Navier-Stokes equations are recovered by substituting this first-order expression for the pressure tensor into the conservation theorem with n — mvi (i.e, into equation 9.55). 

The general heat-conduction equation, along with the familiar diffusion equation, are both consequences of energy conservation and, like we have just seen for the Navier-Stokes equation, require a first-order approximation to the solution of Boltz-man s equation. 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

Computational fluid dynamics (CFD) emerged in the 1980s as a significant tool for fluid dynamics both in research and in practice, enabled by rapid development in computer hardware and software. Commercial CFD software is widely available. Computational fluid dynamics is the numerical solution of the equations of continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conservation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. 

The second-order expansion of Eq. [11] leads to a macroscopic momentum conservation equation that differs from Navier-Stokes equations only in irrelevant terms of higher order provided that the mean velocity u is small. Thus lattice-gases may be used as models for fluids. 

It can be seen from Fig. 1 that gas flows in micron size channels are typically relevant to the slip flow regime, at any rate for usual pressure and temperature conditions. For lower sizes, i.e., for Knudsen numbers higher than 10 , the slip flow regime could remain valid, provided that classical velocity slip and temperature jump boundary conditions are modified (taking into account higher-order terms as explained below) and/or that Navier—Stokes equations are extended to more general sets of conservation equatiOTis, such as the quasi-gasodynamic (QGD), the quasi-hydrodynamic (QHD), or the Burnett equatiOTis [3]. 

In order to simulate temperature-dependent flow systems, in addition to the continuity equation and the Navier- Stokes equations, the conservation of energy is introduced as an additional descriptive fundamental equation of the flow problem. 

A physical model solves essentially Navier-Stokes equation with the forcing (the wind stress, Coriolis force, and buoyancy force) under adequate approximations. In order to include the density effects, conservation equations for temperature and salinity are also solved. In order to obtain a reahstic prediction for vertical stratification, turbulent closure model is also employed. As a result, the gravitational, wind-driven, and topographically induced flows can be reproduced within... 

Substitution of the first order dense gas flux approximations (2.687) and (2.691) into the conservation equations results in the set of non-equilibrium gas dynamics equations (i.e., sometimes named the Navier-Stokes equations) for a mono-atomic dense gas. If the zero order dense gas flux approximations were employed instead, the outcome is the set of equilibrium Euler equations. 

Pontaza JP, Reddy JN (2003) Spectral/hp least-squares finite element formulation for the incompressible Navier-Stokes equation. J Comput Phys 190 523-549 Pontaza JP, Reddy JN (2004) Space-time coupled spectral/hp least squares finite element formulation for the incompressible Navier-Stokes equation. J Comput Phys 190 418-459 Pontaza JP (2006) A least-squares finite element formulation for unsteady incompressible flows with improved velocity-pressure coupling. J Comput Phys 217 563-588 Pontaza JP (2006) A new consistent splitting scheme for incompressible Navier-Stokes flows a least-squares spectral element implementation. J Comput Phys 225 1590-1602 Pontaza JP, Reddy JN (2006) Least-squares finite element formulations for viscous incompressible and compressible fluid flows. Appl Mech Eng 195 2454-2494 Post D, Kendall R (2003) Software project management and quality engineering practices for complex, coupled multiphysics, massively parallel computational simulations Lessons learned from ASCI. Int J High Perform Comput Appl 18(4) 399-416 Prather MJ (1986) Numerical advection by conservation of second order moments. J Geophys Res 91(D6) 6671-6681... 

The recovery of the Navier-Stokes also follows from momentum conservation but requires that a first-order approximation be made to the full solution to the Boltzman-equation. We sketch the main steps of the recovery below (see [huangk63]). 

In order to determine the distributions of pressure, velocity, and temperature the principles of conservation of mass, conservation of momentum (Newton s Law) and conservation of energy (first law of Thermodynamics) are applied. These conservation principles represent empirical models of the behavior of the physical world. They do not, of course, always apply, e.g., there can be a conversion of mass into energy in some circumstances, but they are adequate for the analysis of the vast majority of engineering problems. These conservation principles lead to the so-called Continuity, Navier-Stokes and Energy equations respectively. These equations involve, beside the basic variables mentioned above, certain fluid properties, e.g., density, p viscosity, p conductivity, k and specific heat, cp. Therefore, to obtain the solution to the equations, the relations between these properties and the pressure and temperature have to be known. (Non-Newtonian fluids in which p depends on the velocity field are not considered here.) As discussed in the previous chapter, there are, however, many practical problems in which the variation of these properties across the flow field can be ignored, i.e., in which the fluid properties can be assumed to be constant in obtaining fire solution. Such solutions are termed constant... 

This is the Qiapman-Enskog normal solution of the Boltzmann equation. When the solution is inserted into the expressions for P and Jr in the conservation laws, it leads to the Navier-Stokes hydrodynamic equations, which involve the first and second spatial derivatives of the functions /i, u, and T. If we use the order /u,, . .. terms in Eq. (119), we are led to the Burnett,... 

Once one has decided to formulate a dynamical theory for Fourier components conserved variables at long times and small k, extra simplifications occur. Note that the right-hand side of Mori s equation (10) may be a quite complicated function of k for arbitrary k. However, for small A , it is common to expand K t) /ft, and in a power series in k. For the conserved-fluid variables, schemes that expand the equations of motion to order A , k, k j and k, respectively, are the Euler, Navier-Stokes, Burnett, and super-Burnett equations. Since = /k j, and since ift contains A once, K contains A twice, while x does not contain A at all [Eqs. (5), (6), and (11)], the leading term in / > = is 0 ik), while the leading term in Kx Ms 0(k ). The leading term in i(o is imaginary while the leading term in F is real. The Euler equations are reversible, i.e., show no dissipation, while the introduction of the k term in K at the Navier-Stokes level causes irreversibility. 

Momentum conservation requires that an equal and opposite force be applied to the fluid. Both discrete and continuous degrees of freedom are subject to Langevin noise in order to balance the frictional and viscous losses, and thereby keep the temperature constant. The algorithm can be applied to any Navier-Stokes solver, not just to LB models. For this reason, we will discuss the coupling within a (continuum) Navier-Stokes framework, with a general equation of state p p). We use the abbreviations for the viscosity tensor (46), and... 

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