Algorithms, for fluid flow

Generally, the design of modern solution algorithms for fluid flow problems is associated with the choice of primitive variables, the grid arrangement, and the solution approach [133]. In the class of pressure-based solution algorithms, both fully coupled and segregated approaches have been proposed. In the coupled approach, the discretized forms of the momentum and continuity... 

Moukalled F, Darwish M (2000) A Unified Formulation of the Segregated Class of Algorithms for Fluid Flow at All Speeds. Numerical Heat Transfer, Part B, 37 103-139... 

In the following, we use the term MFC to describe the generic class of particle-based algorithms for fluid flow which consist of successive free-streaming and multi-particle collision steps. The name SRD is reserved for the most widely used algorithm which was introduced by Malevanets and Kapral [18]. The name refers to the fact that the collisions consist of a random rotation of the relative velocities Svi = V, - u of the particles in a collision cell, where u is the mean velocity of all particles in a cell. There are a number of other MFC algorithms with different collision rules [31-33]. For example, one class of algorithms uses modified collision rules which provide a nontrivial coUisional contribution to the equation of state [33,34]. As a result, these models can be used to model non-ideal fluids or multi-component mixtures with a consolute point... 

Swarbrick, S. J. and Nassehi, V., 1992a. A new decoupled finite element algorithm for viscoelastic flow. Part 1 numerical algorithm and sample results. Int. J. Numer. Methods Fluids 14, 1367-1376,... 

Wanik, A. and Schnell, U. (1989), Some remarks on the PISO and SIMPLE algorithms for steady flow, Comput. Fluids, 17, 555. 

I.E. Barton, Comparison of SIMPLE- and PISO-type algorithms for transient flows. International Journal of Numerical Methods Fluids, 26 (1998) 459-483. 

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 eqnalions associated with a finite set of discrete points within the domain and specified by the user, directly or through an antomated algorithm. Regardless of the method, the numerical solution of the conservation equations for fluid flow is known as computational fluid dynamics (CFD). 

Townsend, P. and Webster, M. I- ., 1987. An algorithm for the three dimensional transient simulation of non-Newtonian fluid flow. In Pande, G. N. and Middleton, J. (eds). Transient Dynamic Analysis and Constitutive Laws for Engineering Materials Vul. 2, T12, Nijhoff-Holland, Swansea, pp. 1-11. 

Nichols, B. D., Hirt, C. W. and Hitchkiss, R. S., 1980. SOLA-VOF a solution algorithm for transient fluid flow with multiple free surface boundaries. Los Alamos Scientific Laboratories Report No. La-8355, Los Alamos, NM. 

Hunt and Kulmala have solved the full turbulent fluid flow for the Aaberg system using the k-e turbulent model or a variation of it as described in Chapter 13— the solution algorithm SIMPLE, the QUICK scheme, etc. Both commercial software and in-house-developed codes have been employed, and all the investigators have produced very similar findings. 

Cloutman, L. D., C. W. Hirt, and N. C. Romero. 1976. SOLA-ICE a numerical solution algorithm for transient compressible fluid flows. Los Alamos Scientific Laboratory report LA-6236. 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Since MPC dynamics yields the hydrodynamic equations on long distance and time scales, it provides a mesoscopic simulation algorithm for investigation of fluid flow that complements other mesoscopic methods. Since it is a particle-based scheme it incorporates fluctuations, which are essential in many applications. For macroscopic fluid flow averaging is required to obtain the deterministic flow fields. In spite of the additional averaging that is required the method has the advantage that it is numerically stable, does not suffer from lattice artifacts in the structure of the Navier-Stokes equations, and boundary conditions are easily implemented. 

Finally, we can also mention that laminar-flow systems with non-Newtonian fluids often require special numerical algorithms that are usually not available in CFD codes designed mainly for turbulent flows. 

Pfilzner, M., A. Mack, N. Brehm, A. Leonard, and I. Romaschov (1999). Implementation and validation of a PDF transport algorithm with adaptive number of particles in industrially relevant flows. In Computational Technologies for Fluid/Thermal/Structural/Chemical Systems with Industrial Applications, vol. 397-1, pp. 93-104. ASME. 

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

J<. Najib and D. Sandri, On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow, Num. Math., 72 (1995) 223-238. 

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