Three-dimensional flow

Iterative solution methods are more effective for problems arising in solid mechanics and are not a common feature of the finite element modelling of polymer processes. However, under certain conditions they may provide better computer economy than direct methods. In particular, these methods have an inherent compatibility with algorithms used for parallel processing and hence are potentially more suitable for three-dimensional flow modelling. In this chapter we focus on the direct methods commonly used in flow simulation models. 

Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vaiy in all three coordinate directions, the concept of deformation previously introduced must be generahzed. The rate of deformation tensor Dy has nine components. In Cartesian coordinates. 

Many coating flows are subject to instabilities that lead to unacceptable coating defects. Three-dimensional flow instabilities lead to such problems as ribbing. Air entrainment is another common defect. 

The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD see the following subsection), is the A — model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio /cVe. 

CFD solutions, especially for complex three-dimensional flows,... 

Boyce, M.P., A Practical Three-Dimensional Flow Visualization Approach to the Complex Flow Characteristics in a Centrifugal Impeller, ASME Paper No. 66-GT-83, June 1983. 

Wu, C.H., A General Theory of Three-Dimensional Flow in Subsonic and Supersonic Turbomachines of Axial, Radial, and Mixed-Flow Type, NACA TN-2604, 1952. 

Holmquist, L.O., and Rannie, W.D., An Approximate Method of Calculating Three-Dimensional Flow in Axial Turbomachines (Paper) Meeting Inst. Aero. Sci., New York, January 24-28, 1955. 

Two-dimensional flow is in the x and y directions, while three-dimensional flow is in the x, y, and 2 directions. 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

H. B. Hadid, D. Henry. Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. 2. Three-dimensional flow. J Fluid Mech 333 57, 1997. 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

Two calculation procedures for steady, three-dimensional flows with recirculation, in Proceedings of the 3rd Int. Conf Numer. Methods Fluid Dyn., Paris (1972). 

Avalosse, Th., and Crochet, M. J., Finite element simulation of mixing 2. Three-dimensional flow through a Kenics mixer. AlChE J. 43, 588-597 (1997b). 

E. Allahyarov and G. Gompper, Mesoscopic solvent simulations multiparticle-collision dynamics of three-dimensional flows, Phys. Rev. E 66, 036702 (2002). 

The complex three-dimensional flow pattern within the cyclone is dominated by the radial (Fr) and tangential (V0) velocity components. The vertical component is also significant but plays only an indirect role in the separation. The tangential velocity in the vortex varies with the distance from the axis in a complex manner, which can be described by the equation... 

Lai F S (1980) Three-dimensional flow of air through nonuniform grain beds , Trans. ASAE, 23, 729-734. 

Computer aided design (CAD)/computer-aided manufacture (CAM) programs have been developed to carry out the necessary calculations. One current system known as Polyflow is offered by a Belgian organisation (Polyflow SA). Polyflow can be used to calculate three-dimensional flow and die swell, using finite element analysis. Other systems are available. 

Faust, C. R., Guswa, J. H., and Mercer, J. W., 1989, Simulation of Three Dimensional Flow of Immiscible Fluids within and below the Unsaturated Zone Water Resources Research, Vol. 25, No. 12, pp. 2449-2464. 

The flow patterns in the hydrocyclone are complex, and much development work has been necessary to determine the most effective geometry, as theoretical considerations alone will not allow the accurate prediction of the size cut which will be obtained. A mathematical model has been proposed by Rhodes et alP6), and predictions of streamlines from their work are shown in Figure 1.38. Salcudean and Gartshore137 have also carried out numerical simulations of the three-dimensional flow in a hydrocyclone and have used the results to predict cut sizes. Good agreement has been obtained with experimental measurements. 

There have been several studies in which the flow patterns within the body of the cyclone separator have been modelled using a Computational Fluid Dynamics (CFD) technique. A recent example is that of Slack et a/. 54 in which the computed three-dimensional flow fields have been plotted and compared with the results of experimental studies in which a backscatter laser Doppler anemometry system was used to measure flowfields. Agreement between the computed and experimental results was very good. When using very fine grid meshes, the existence of time-dependent vortices was identified. These had the potentiality of adversely affecting the separation efficiency, as well as leading to increased erosion at the walls. 

The flow problems considered in Volume 1 are unidirectional, with the fluid flowing along a pipe or channel, and the effect of an obstruction is discussed only in so far as it causes an alteration in the forward velocity of the fluid. In this chapter, the force exerted on a body as a result of the flow of fluid past it is considered and, as the fluid is generally diverted all round it, the resulting three-dimensional flow is more complex. The flow of fluid relative to an infinitely long cylinder, a spherical particle and a non-spherical particle is considered, followed by a discussion of the motion of particles in both gravitational and centrifugal fields. 

Chen, L. and Lindt, J. T., Three-Dimensional Flow Effects on Residence Time Distribution in Screw Extruders, SPE ANTEC Tech. Papers, 42, 170 (1996)... 

The performance of Double Wave screws was provided earlier by Kruder and others [20-22]. For these trials, the extrusion process was capable of higher rates and lower discharge temperatures than well designed conventional single-flighted screws. Three-dimensional flow simulations for a Double Wave and a Triple Wave section were performed by Fan [23] and Perdikoulias [24], These simulations showed the complex nature of the flows, including the acceleration of the flow as it... 

The conservation equations described in Section B.l show the mass, momentum, energy, and chemical species equations at a steady state in a one-dimensional flow field. Similarly, the conservation equations at a steady-state in two- or three-dimensional flow fields can be obtained. The results can be summarized in a vector form... 

---