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Author: David Lee Davidson
Computational fluid dynamics has enormous potential for industry in the twenty-first century.
Continuum mechanics, one of our most successful physical theories, is readily applicable to the process industries. In continuum mechanics, the existence of molecules is ignored, and matter is treated as a continuous medium. The continuum hypothesis is valid, provided the equations of continuum mechanics are applied at sufficiently large length scales and time scales that the properties of individual molecules are not noticed. The mapping of the laws of mass, momentum, and energy conservation to the continuum results in field equations that describe the dynamics of the continuum. These field equations, variously known as the equations of motion, the equations of change, or simply the conservation equations, are nonlinear, partial differential equations that can be solved, in principle, when combined with the appropriate constitutive information1 and boundary conditions.
Continuum mechanics is the mechanical analog of classical electrodynamics, in which a set of field equations (Maxwell’s equations) describe the dynamics of the relevant variables of the electrical and magnetic fields. Whereas Maxwell’s equations are linear unless the constitutive behavior is nonlinear, the equations of continuum mechanics are nonlinear, regardless of the constitutive behavior of the materials of interest. The inherent nonlinearity of the conservation equations, which is due to convective transport of momentum, energy, and chemical species, is responsible for certain fluid mechanical phenomena, such as turbulence, that have no electrodynamic analog and that complicate solution of the conservation equations.
Analytical solutions (e.g., obtained by eigenfunction expansion, Fourier transform, similarity transform, perturbation methods, and the solution of ordinary differential equations for one-dimensional problems) to the conservation equations are of great interest, of course, but they can be obtained only under restricted conditions. When the equations can be rendered linear (e.g., when transport of the conserved quantities of interest is dominated by diffusion rather than convection) analytical solutions are often possible, provided the geometry of the domain and the boundary conditions are not too complicated. When the equations are nonlinear, analytical solutions are sometimes possible, again provided the boundary conditions and geometry are relatively simple. Even when the problem is dominated by diffusive transport and the geometry and boundary conditions are simple, nonlinear constitutive behavior can eliminate the possibility of analytical solution.
Consequently, numerical solution of the equations of change has been an important research topic for many decades, both in solid mechanics and in fluid mechanics. Solid mechanics is significantly simpler than fluid mechanics because of the absence of the nonlinear convection term, and the finite element method has become the standard method. In fluid mechanics, however, the finite element method is primarily used for laminar flows, and other methods, such as the finite difference and finite volume methods, are used for both laminar and turbulent flows. The recently developed lattice-Boltzmann method is also being used, primarily in academic circles. All of these methods involve the approximation of the field equations defined over a continuous domain by discrete equations associated with a finite set of discrete points within the domain and specified by the user, directly or through an automated algorithm. Regardless of the method, the numerical solution of the conservation equations for fluid flow is known as computational fluid dynamics (CFD).
CFD was initially done without automation because the need to solve these equations (e.g., in aircraft design) preceded the development of electronic computers by several decades. With the advent of electronic computers, more ambitious numerical calculations became possible. Initially, CFD codes were written for specific problems. It was natural to generalize these codes somewhat, and eventually, particularly as computational resources became more readily available, general-purpose CFD codes were developed. It was then recognized that a business could be built upon the development and licensing of these codes to industrial, academic, and government users. Today, many of the general-purpose commercial codes are quite sophisticated, cost a tiny fraction of their development cost, and are probably the mainstay of the industrial application of CFD.
Four steps are required to apply a general-purpose CFD code to an industrial problem. First, the domain must be defined. This amounts to constructing the geometry for the problem,2 which is typically done using a computer-assisted design (CAD)-like preprocessor.3 Within the preprocessor, relevant physics are defined, appropriate models are specified, boundary and initial conditions are applied, and solver parameters are specified. Because the conservation and constitutive equations must be discretized on the specified geometry, the domain discretization must be specified. This process, known as meshing or grid generation, is the second step in the application of a CFD code to an industrial problem. Meshing can be accomplished using two basic protocols: (1) structured meshing, which involves creating an assembly of regular, usually hexahedral (quadrilateral in two dimensions) elements or control volumes throughout the domain; and (2) unstructured meshing, which involves filling the geometry with control volumes, often tetrahedrons and prisms, in an irregular fashion. Unstructured mesh generators are usually simpler to use with complicated geometries and involve some degree of automation. For example, the user may specify one or more measures of surface grid density, and the mesh generator will fill the volume with elements according to some algorithm. In the third step, the equations are discretized over the specified grid, and the resulting nonlinear4 algebraic equations are solved. The development of solvers is still an active area of research, the goal being to improve the likelihood and rate of convergence. The fourth step, after satisfactory convergence is obtained, is to interrogate the solution to obtain the desired information. That information may be a single number extracted from the solution data set, an animation illustrating the transient macroscopic behavior of the entire flow field, or anything in between. Because the data sets can be quite large,5 robust tools for data set interrogation are often required. These are usually provided with the commercial CFD codes, but one leading commercial tool is a stand-alone CFD postprocessor (FIELDVIEW, 2002).
Current Industrial Applications
CFD is routinely used today in a wide variety of disciplines and industries, including aerospace, automotive, power generation, chemical manufacturing, polymer processing, petroleum exploration, medical research, meteorology, and astrophysics. The use of CFD in the process industries has led to reductions in the cost of product and process development and optimization activities (by reducing down time), reduced the need for physical experimentation, shortened time to market, improved design reliability, increased conversions and yields, and facilitated the resolution of environmental, health, and right-to-operate issues. It follows that the economic benefit of using CFD has been substantial, although detailed economic analyses are rarely reported. A case study of the economic benefit of the application of CFD in one chemical and engineered-material company over a six-year period conservatively estimated that the application of CFD generated approximately a six-fold return on the total investment in CFD (Davidson, 2001a).
CFD has an enormous potential impact on industry because the solution of the equations of motion provides everything that is meaningful to know about the domain. For example, chemical engineers commonly make assumptions about the fluid mechanics in process units and piping that lead to great simplifications in the equations of motion. An agitated chemical reactor may be designed on the assumption that the material in the vessel is perfectly mixed, when, in reality, it is probably not perfectly mixed. Consequently, the fluid mechanics may limit the reaction rather than the reaction kinetics, and the design may be inadequate. CFD allows one to simulate the reactor without making any assumptions about the macroscopic flow pattern and thus to design the vessel properly the first time. Similarly, the geometrically complicated parts required for melt spinning can be designed with CFD rather than rules-of-thumb or experiments, resulting in "right the first time" designs (Davidson, 2001b). Commercial publications (e.g., CFX Update, Fluent News, and Applications from the Chemical Process Industry) are filled with case studies illustrating how CFD was applied to the design of a particular unit, the optimization of a particular process, or the analysis of a particular phenomenon with good results.
Areas of Research
There are, of course, limitations to the application of CFD, and active research is being done to overcome them. The primary limitation is in the area of turbulent flow. Turbulent flows are solutions to the equations of motion and can be computed directly, at least in principle. This approach, known as direct numerical simulation, requires a spatial grid fine enough to capture the smallest length scale of the turbulent fluid motion (the Kolmogorov scale) throughout the domain of interest and a correspondingly small time step. In typical problems of industrial interest, the ratio of the length scale of the domain to the Kolmogorov length scale is so large that the required grid is prohibitively large. Available computational resources are usually inadequate for this task except for relatively simple problems.
Consequently, industrial practitioners of CFD use turbulence models, usually by solving the Reynolds-averaged equations, that is, equations generated by averaging the equations of motion over a time scale that is much larger than the time scale of the turbulent fluctuations but much smaller than the smallest time scale of interest in the application. This procedure results in a set of equations that have the same form as the original equations of motion, but with time-averaged quantities in place of instantaneous quantities, plus one additional term that arises from the nonlinear convective terms in the original equations of motion. In the Reynolds-averaged momentum-conservation equation, for example, this additional term has the form of an additional stress, known as the Reynolds stress. This term is modeled based on the time-averaged quantities of the flow field. A variety of turbulence models are available (Wilcox, 1998), but the workhorse model of industrial CFD is the so-called k-epsilon model, which was introduced several decades ago (Casey and Wintergerste, 2000; Launder and Spalding, 1974). These turbulence models can lead to significant inaccuracies, and CFD practitioners must use them carefully.
Large eddy simulation (LES) is an alternative approach to turbulence modeling. Turbulent flows are characterized by an eddy cascade, in which large eddies transfer their kinetic energy to smaller eddies, which in turn transfer kinetic energy to even smaller eddies, and so on until, at the Kolmogorov scale, the kinetic energy is transformed into heat. LES attempts to solve for the larger eddies directly while modeling the smaller eddies. Although LES is more computationally intensive than other kinds of turbulence modeling, it has been applied to industrial-scale problems (Derksen, 2001).
The second great limitation of CFD is dispersed, multiphase flows. Multiphase flows are common in industry, and consequently their simulation is of great interest. Like turbulent flows, multiphase flows (which may also be turbulent in one or more phases) are solutions to the equations of motion, and direct numerical simulation has been applied to them (Miller and Bellan, 2000). However, practical multiphase flow problems require a modeling approach. The models, however, tend to ignore or at best simplify many of the important details of the flow, such as droplet or particle shape and their impact on interphase mass, energy, and momentum transport, the impact of deformation rate on droplet breakup and coalescence, and the formation of macroscopic structures within the dispersed phase (Sundaresan et al., 1998).
Although the commercial CFD industry has greatly simplified the use of CFD codes by providing CAD-like preprocessors, automatic mesh generation, graphical user interfaces for all aspects of model definition, and on-line documentation, the industrial practice of CFD is still primarily in the hands of specialists. Regular use of a general-purpose code requires significant expertise in transport phenomena, an understanding of the capabilities and limitations of the modeling approaches used to handle turbulence and dispersed multiphase flows, an understanding of the relationship between mesh quality, convergence, and solution accuracy, and proficiency with the various means of interacting with the CFD code, including the graphical user interface, advanced command languages (when available), and user-accessible FORTRAN subroutines.6 For these reasons, attempts to train large numbers of engineers in the use of CFD have not been very successful (Davidson, 2001a). Nevertheless, the potential benefit of a much bro
ader CFD user base is very great. In our opinion, CFD should be accessible to every person in the enterprise who makes decisions the outcomes of which are governed by the laws of physics, from the CEO who makes strategic business decisions based on business goals to the operator who adjusts valve positions to meet process goals.
We believe that this can be achieved through the development of so-called "digital experts," stand-alone CFD (and other) applications that would be integrated into commercial CFD codes (as appropriate) and wrapped in interfaces that speak the language of the industrial application, not the language of CFD. Digital experts would automate geometry construction, mesh generation, solver selection, and other processes behind the scenes. In addition, they would contain all of the algorithms necessary to nurse the CFD codes to solution automatically, without having to ask the user to define satisfactory convergence, for example. Finally, they would extract the essential ingredients from the complete CFD solution and present them to the user in a convenient and familiar format, so the user would not have to be concerned with interrogation of the flow field by computation or visualization. A discussion of one digital expert that has been developed for melt-fiber spinning has been published (Davidson, 2001b).
The decision to develop an industrial digital expert is based on the relationship between development cost and benefit. Recently, a commercial product has become available with the potential to change that relationship significantly. EASATM (Enterprise Accessible Software Applications from AEA Technology), which was designed to help industrial practitioners develop digital experts, solves a number of problems for industrial developers, including construction of the graphical user interface, accessibility of the final product (the digital expert or EASAp) over the enterprise intranet, and the orchestration of computations on a heterogeneous computer network (Dewhurst, 2001). In essence, EASA allows an industrial CFD specialist to put bullet-proof digital experts in the hands of coworkers, with a consistent interface tailored to the user.
CFD is a powerful tool for solving a wide variety of industrial problems. Commercial general-purpose codes have the potential to solve a very broad spectrum of flow problems. Current research is concentrated on overcoming the principle weaknesses of CFD, namely how it deals with turbulence and dispersed multiphase flows. Development work on solver algorithms, meshing, and user interface generation are ongoing, with the objectives of improving accuracy, reducing solution time, and increasing accessibility. In spite of the limitations of CFD, the economic value of industrial applications has been demonstrated in a variety of industries, and its value as a research tool has been accepted in many areas, such as meteorology, med-icine, and astrophysics. In industry, CFD is presently primarily in the hands of specialists, but the development of digital experts and tools to facilitate the development of digital experts may revolutionize the way industry uses CFD by providing ready access throughout the enterpri
se. This would result in significant gains in productivity and profitability.
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Davidson, D.L. 2001b. SpinExpert: The Digital Expert for the Design and Analysis of Fiber Spinning Operations. Pp. 219-226 in Proceedings of the 3rd International ASME Symposium on Computational Technology (CFD) for Fluid/Thermal/Chemical/Stress Systems and Industrial Applications. Atlanta, Ga.: American Society of Mechanical Engineers.
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