Models/modeling Navier-Stokes equations

Each of these different types of flows is governed by a set of equations having special features. It is essential to understand these features to select an appropriate numerical method for each of these types of equations. It must be remembered that the results of the CFD simulations can only be as good as the underlying mathematical model. Navier-Stokes equations rigorously represent the behavior of an incompressible Newtonian fluid as long as the continuum assumption is valid. As the complexity increases (such as turbulence or the existence of additional phases), the number of phenomena in a flow problem and the possible number of interactions between them increases at least quadratically. Each of these interactions needs to be represented and resolved numerically, which may put strain on (or may exceed) the available computational resources. One way to deal with the resolution limits and... 

A similar analysis can be performed for the shocks in the model Navier-Stokes equation l(. It is noticeable that the value of the parameter B remains the same, but the Navlei> -Stokes profile coincides with the Boltzmann shock profile for weak shocks only. 

Quian Y H, D Humieres D and Lallemand P 1992 Lattice BGK models for Navier-Stokes equation Euro. Phys. Lett. 17 479... 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

Dynamic meteorological models, much like air pollution models, strive to describe the physics and thermodynamics of atmospheric motions as accurately as is feasible. Besides being used in conjunction with air quaHty models, they ate also used for weather forecasting. Like air quaHty models, dynamic meteorological models solve a set of partial differential equations (also called primitive equations). This set of equations, which ate fundamental to the fluid mechanics of the atmosphere, ate referred to as the Navier-Stokes equations, and describe the conservation of mass and momentum. They ate combined with equations describing energy conservation and thermodynamics in a moving fluid (72) ... 

For most applications, the engineer must instead resort to turbulence models along with time-averaged Navier-Stokes equations. Unformnately, most available turbulence models obscure physical phenomena that are present, such as eddies and high-vorticity regions. In some cases, this deficiency may partially offset the inherent attractiveness of CFD noted earlier. 

The dimensionless numbers are important elements in the performance of model experiments, and they are determined by the normalizing procedure ot the independent variables. If, for example, free convection is considered in a room without ventilation, it is not possible to normalize the velocities by a supply velocity Uq. The normalized velocity can be defined by m u f po //ao where f, is the height of a cold or a hot surface. The Grashof number, Gr, will then appear in the buoyancy term in the Navier-Stokes equation (AT is the temperature difference between the hot and the cold surface) ... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

The basic model equations for a description of hydrodynamical flow are the Navier-Stokes equations, representing momentum conservation in the fluid... 

While there are mairy variants of the basic, model, one can show that there is a well-defined minimal set of niles that define a lattice-gas system whose macroscopic behavior reproduces that predicted by the Navier-Stokes equations exactly. In other words, there is critical threshold of rule size and type that must be met before the continuum fluid l)cliavior is matched, and onec that threshold is reached the efficacy of the rule-set is no loner appreciably altered by additional rules respecting the required conservation laws and symmetries. 

Chapter 9 provides an introductory discussion of a research area that is rapidly growing in importance lattice gases. Lattice gases, which are discretized models of continuous fluids, represent an early success of CA modeling techniques. The chapter begins with a short primer on continuum fluid dynamics and proceeds with a discussion of CA lattice gas models. One of the most important results is the observation that, under certain constraints, the macroscopic behavior of CA models exactly reproduces that predicted by the Navier-Stokes equations. 

This chapter is organized into two main parts. To give the reader an appreciation of real fluids, and the kinds of behaviors that it is hoped can be captured by CA models, the first part provides a mostly physical discussion of continuum fluid dynamics. The basic equations of fluid dynamics, the so-called Navier-Stokes equations, are derived, the Reynolds Number is defined and the different routes to turbulence are described. Part I also includes an important discussion of the role that conservation laws play in the kinetic theory approach to fluid dynamics, a role that will be exploited by the CA models introduced in Part II. 

These conditions show us immediately that in the case of the four-neighbor HPP lattice (V = 4) f is noni.sotropic, and the macroscopic equations therefore cannot yield a Navier-Stokes equation. For the hexagonal FHP lattice, on the other hand, we have V = 6 and P[. is isotropic through order Wolfram [wolf86c] predicts what models are conducive to f lavier-Stokes-like dynamics by using group theory to analyze the symmetry of tensor structures for polygons and polyhedra in d-dimensions. 

In Spite of the existence of numerous experimental and theoretical investigations, a number of principal problems related to micro-fluid hydrodynamics are not well-studied. There are contradictory data on the drag in micro-channels, transition from laminar to turbulent flow, etc. That leads to difficulties in understanding the essence of this phenomenon and is a basis for questionable discoveries of special microeffects (Duncan and Peterson 1994 Ho and Tai 1998 Plam 2000 Herwig 2000 Herwig and Hausner 2003 Gad-el-Hak 2003). The latter were revealed by comparison of experimental data with predictions of a conventional theory based on the Navier-Stokes equations. The discrepancy between these data was interpreted as a display of new effects of flow in micro-channels. It should be noted that actual conditions of several experiments were often not identical to conditions that were used in the theoretical models. For this reason, the analysis of sources of disparity between the theory and experiment is of significance. 

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. 

We use computational solution of the steady Navier-Stokes equations in cylindrical coordinates to determine the optimal operating conditions.Fortunately in most CVD processes the active gases that lead to deposition are present in only trace amounts in a carrier gas. Since the active gases are present in such small amounts, their presence has a negligible effect on the flow of the carrier. Thus, for the purposes of determining the effects of buoyancy and confinement, the simulations can model the carrier gas alone (or with simplified chemical reaction models) - an enormous reduction in the problem size. This approach to CVD modeling has been used extensively by Jensen and his coworkers (cf. Houtman, et al.) ... 

An Eulerian-Eulerian (EE) approach was adopted to simulate the dispersed gas-liquid flow. The EE approach treats both the primary liquid phase and the dispersed gas phase as interpenetrating continua, and solves a set of Navier-Stokes equations for each phase. Velocity inlet and outlet boundary conditions were employed in the liquid phase, whilst the gas phase conditions consisted of a velocity inlet and pressure outlet. Turbulence within the system was account for with the Standard k-e model, implemented on a per-phase basis, similar to the recent work of Bertola et. al.[4]. A more detailed description of the computational setup of the EE method can be found in Pareek et. al.[5]. 

A standard approach to modeling transport phenomena in the field of chemical engineering is based on convection-diffusion equations. Equations of that type describe the transport of a certain field quantity, for example momentum or enthalpy, as the sum of a convective and a diffusive term. A well-known example is the Navier-Stokes equation, which in the case of compressible media is given as... 

For applications in the field of micro reaction engineering, the conclusion may be drawn that the Navier-Stokes equation and other continuum models are valid in many cases, as Knudsen numbers greater than 10 are rarely obtained. However, it might be necessary to use slip boimdaty conditions. The first theoretical investigations on slip flow of gases were carried out in the 19th century by Maxwell and von Smoluchowski. The basic concept relies on a so-called slip length L, which relates the local shear strain to the relative flow velocity at the wall ... 

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