Navier numerical technique

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 strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

The Lattice Boltzmann Method (LBM), including the method Cellular Automaton (AC), present a powerful alternative to standard apvproaches known like "of up toward down" and "of down toward up". The first approximation study a continuous description of macroscopic phenomenon given for a partial differential equation (an example of this, is the Navier-Stokes equation used for flow of incompressible fluids) some numerical techniques like finite difference and the finite element, they are used for the transformation of continuous description to discreet it permits solve numerically equations in the compniter. 

Fort the solution of complex problems of dynamics of fluids, exist traditionally two kinds of points of view the first is macroscopic, which is considered continuous, with an ap>proach of differential equations in p>artial derivatives, for example of Navier-Stokes equations used for flow of incompressible fluids and numerical techniques for its solution. The second pwint of view is microscopic it has its basis in kinetics theory of gases and statistical mechanics. 

Eree surface flows and interfaces between two or more immiscible fluids or phases are observed in many natural and industrial processes at macro- and microscales. Different numerical techniques are developed to simulate these flows. However, due to the corrplexity of the problem, each technique is tailored to a particular category of flows. Einite element (EE), finite volume (EV) and finite difference (ED) methods are all potentially applicable to generalized Navier-Stokes equations. However, they have to be coupled with a technique to track moving fluid boundaries and interfaces. The difficulty in tackling interfacial flows is inherently related to the corrplexity of interface topology and the fact that the interface location is unknown. 

Mader, C. L. 2004. Numerical Modeling of Water Waves, 2nd ed. Boca Raton, FL CRC Press. This manual covers all aspects of this topic from basic fluid dynamics and the basic models to the most complex, including the compressible Navier Stokes techniques to model waves generated in various ways. 

The volume of fluid (VOF) method represents a category of numerical techniques used to trace the free surface of the fluid or the interface of two types of adjacent fluids. The fluids or mixture are described with a mesh grid, which is either stationary or moves with the flow front or interfaces in a prescribed manner. The interfaces of the different components in each mesh grid are then calculated for each step. Hence, the VOF technique is an advection method which describes only the flow front and must be adapted to other constitutive equations (e.g., Navier-Stokes) to describe the physics in the motion of the flow. 

S-3.2.3 Alternative Numerical Techniques. As mentioned earlier, other methods for solving the Navier-Stokes equations exist. Two of these are described briefly below. 

In the two-fluid formulation, the velocity field of each of these two continuous phases is described by its own continuity equation and Navier—Stokes equation (e.g., Anderson and Jackson, 1967 Rietema and Van den Akker, 1983 Sokolichin et al, 2004 Tabib et al, 2008). Each of these Navier-Stokes equations comprises a mutual phase interaction force. Only in the size, or numerical value, of this phase interaction force, particle size may come out. However, in selecting the computational grid or the numerical technique for solving the flow fields of the two continuous phases, particle size is completely irrelevant—due to the definition of mutually interpenetrating continua. 

Taylor, C. and Hood, P., 1973. A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73-100. 

Lee, R. L., Gresho, P. M. and Sani, R. L., 1979. Smoothing techniques for certain primitive variable solutions of the Navier-Stokes equations. Int. J. Numer. Methods Eng. 14, 1785-1804. 

Alternative methods of analysis have been examined and evaluated. Shokoohi and Elrod[533] solved the Navier-Stokes equations numerically in the axisymmetric form. Bogy15271 used the Cosserat theory developed by Green.[534] Ibrahim and Linl535 conducted a weakly nonlinear instability analysis. The method of strained coordinates was also examined. In spite of the mathematical or computational elegance, all of these methods suffer from inherent complexity. Lee15361 developed a 1 -D, nonlinear direct-simulation technique that proved to be a simple and practical method for investigating the nonlinear instability of a liquid j et. Lee s direct-simulation approach formed the... 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

Here we consider three theoretical approaches. As for rigid spheres, numerical solutions of the complete Navier-Stokes and transfer equations provide useful quantitative and qualitative information at intermediate Reynolds numbers (typically Re < 300). More limited success has been achieved with approximate techniques based on Galerkin s method. Boundary layer solutions have also been devised for Re > 50. Numerical solutions give the most complete and... 

As far as convective heat transfer is concerned, liquid and gaseous flows musf be considered separately. Liquid flow has been investigated experimentally, whereas analytical, numerical and molecular simulation techniques have been applied to understand the characteristics of gaseous flow and heat transfer. While the Navier-Stokes equations can still be applied, due to the small size of microchannels, some deviations from the conventionally sized applications have been observed. Flow regime boundaries are significantly different, as well as flow and heat transfer characteristics. 

In this work, microscale evaporation heat transfer and capillary phenomena for ultra thin liquid film area are presented. The interface shapes of curved liquid film in rectangular minichannel and in vicinity of liquid-vapor-solid contact line are determined by a numerical solution of simplified models as derived from Navier-Stokes equations. The local heat transfer is analyzed in term of conduction through liquid layer. The data of numerical calculation of local heat transfer in rectangular channel and for rivulet evaporation are presented. The experimental techniques are described which were used to measure the local heat transfer coefficients in rectangular minichannel and thermal contact angle for rivulet evaporation. A satisfactory agreement between the theory and experiments is obtained. 

D time-dependent solution of the Navicr -Stokes equations. The main reason we do not discuss these flows here is that the analytical solution techniques that we develop have had relatively little impact on the analysis or understanding of turbulent flows. The most powerfifl theoretical tools for turbulence research and for the prediction of turbulent flows are currently direct numerical solutions (DNS) of the Navier Stokes equation, typically by use of spectral techniques for discretization. Again, the interested reader will find many texts and references to modem work on turbulent flows.2... 

The low Reynolds number approximation of the Navier-Stokes equations (also known as Stokes equations) is an acceptable model for a number of interfacial flow problems. For instance, the typical example of drop coalescence belongs to this case. A BI method [3] arises from a reformulation of the Stokes equations in terms of BI expressions and the subsequent numerical solution of the integral equations. This technique is further described in chapter 18. 

In order to obtain a correlation, the outflow of the effervescent spray was simulated by a numerical model based on the Navier-Stokes equations and the particle tracking method. The external gas flow was considered turbulent. In droplet phase modeling, Lagrangian approach was followed. Droplet primary and secondary breakup were considered in their model. Secondary breakup consisted of cascade atomization, droplet collision, and coalescence. The droplet mean diameter under different operating conditions and liquid properties were calculated for the spray SMD using the curve fitting technique [43] ... 

Electrokinetic transport phenomena in porous media have been studied much in the past decade both theoretically and numerically however, most results were actually based on linearization approximations of the Poisson-Boltzmann equation [5] so that reliable applications were limited to cases where the EDL length was very thin or very thick compared to the channel size. To our knowledge, there are very few publications that present successful numerical simulations for the electrokinetic flows in charged porous media by solving the nonlinear Poisson-Boltzmann equation and the Navier-Stokes equations by the classical CFD techniques [3, 5]. Because of easy implementation and high efficiency for fluid-solid boundary conditions, the LPBM great advantage in simulations of such flows. 

A single set of conservation eqnations valid for both porous electrodes and the free electrolyte region is derived and nnmerically solved using a computational fluid dynamics technique. This numerical methodology is capable of simulating a two-dimensional cell with the fluid flow taken into consideration. The motion of the liquid electrolyte is governed by the Navier-Stokes equation with the Boussinesq approximation and the continuity equation as follows ... 

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