Computational fluid dynamics transport equations

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

A drawback of the Lagrangean artificial-viscosity method is that, if sufficient artificial viscosity is added to produce an oscillation-free distribution, the solution becomes fairly inaccurate because wave amplitudes are damped, and sharp discontinuities are smeared over an increasing number of grid points during computation. To overcome these deficiencies a variety of new methods have been developed since 1970. Flux-corrected transport (FCT) is a popular exponent in this area of development in computational fluid dynamics. FCT is generally applicable to finite difference schemes to solve continuity equations, and, according to Boris and Book (1976), its principles may be represented as follows. 

Much effort has been expended in the last 5 years upon development of numerical models with increasingly less restrictive assumptions and more physical complexities. Current development in PEFC modeling is in the direction of applying computational fluid dynamics (CFD) to solve the complete set of transport equations governing mass, momentum, species, energy, and charge conservation. 

A commercial Computational Fluid Dynamics package (FIDAP Version 7.6, Fluid Dynamics International, Evanston, IL) based on the finite element method was used to solve the governing continuity, momentum and heat transport equations. A mesh was defined with more nodes near the wall and the entrance of the tubular heat exchanger to resolve the larger variations of temperature and velocities near the wall and the entrance. 

Computational fluid dynamics is based on the principle of solving conservation equations for all relevant variables. The conservation equations include the transport of the variable throughout the domain, as well as either its creation or its destruction. Conserved variables include ... 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

As previously mentioned, the analysis of microfluidic systems can be rather difficult for a variety of reasons. The direct implementation of the Navier-Stokes equations toward surface-directed microfluidic systems requires careful attention when considering the advection of the free surface and the associated curvature of this surface. Consequently, sophisticated computational fluid dynamics software packages are required for a comprehensive three-dimensional analysis of the fluid transport within surface-directed microfluidic devices. However, a time-consuming comprehensive analysis may be beyond the requirements of designing and manufacturing functional surface-directed microfluidic platforms. Consequently, empirical approximations and scaling arguments are commonly used in the characterization of microfluidic physics. 

The integration of the transport equation, (12.4.2-2), for the one-point joint velocity-composition micro-PDF, f, is tedious. Also, comparison with measured data is usually only possible at the level of certain moments of f. An approach often tried to reduce the computation, is to derive and solve the transport equations for a limited number of moments of f. In what follows, /will be simply referred to as the micro-PDF. Micro-PDF moment methods are also called Computational Fluid Dynamics (CFD) methods and have been extensively developed over the last decades [Anderson, 1995 Hirsch, 2007]. 

Classical computational fluid dynamics (CFD) deals with a well-established system of equations. Typically, transport parameters and kinetic constants (if any) are also well defined. Scientific CFD problems are complicated by the presence of turbulence in the system. The spectrum of turbulence covers many orders of magnitude, which requires exhausting computing resources to resolve all the space scale and timescales. 

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