Fluid Flow in Two and Three Dimensions

Two methods are used in commercial computational fluid dynamics (CFD) codes the finite volume method and the finite element method. To a beginner, it probably makes little difference which method is used. The author has used finite element methods for fluid flow for over 30 years, and that is the method used in FEMLAB , which is the CFD program illustrated here. 

What does matter, though, is the difference between laminar and turbulent flow. A common misconception is that if you see vortices (swirling motion), the flow is turbulent. That is not necessarily true. In turbulent flow, there are fine-scale oscillations on time scales of 0.1 ms, and spatial fluctuations on the scale of 30 pm (30 x 10 m, or 3% of 1 mm). To model turbulent flow, one option is direct numerical simulation (DNS). In DNS, the Navier-Stokes equations are integrated in time on a very small spatial scale. 

Introduction to Chemical Engineering Computing, by Bruce A. Finlayson Copyright 2006 John Wiley Sons, Inc. 

This chapter focuses on fluid flow, leaving the combination of fluid flow, heat transfer, and diffusion to Chapter 11. Examples of fluid flow include entry flow into a pipe, flow in a microfluidic T-sensor, turbulent flow in a pipe, time-dependent start-up of pipe flow, flow in an orifice, and flow in a serpentine mixer. The examples demonstrate many of the techniques that are useful in the program FEMLAB. 

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Fluid Flow in Two and Three Dimensions 11 Convective Diffusion Equation in... 

The form of the friction-loss term is very dependent on the geometry of the system. The problem is much simpler if the flow is all in one direction, as in a pipe, rather than in two or three dimensions, as is flow around an airplane. Therefore, first we consider fluid friction in long, constant-diameter pipes in steady flow. This case is of great, practical significance and is the easiest case to treat mathematically. The starting and stopping of flow in pipes is discussed in Sec. 7.4. I... 

Computer flow-analysis programs used throughout the plastics industry worldwide utilize two- and three-dimension models in conjunction with rheology equations. Models range from a simple Poiseuille s equation for fluid flow to more complex mathematical models involving differential calculus. These models are only approximations. Their relational techniques, coupled with the user s assumptions, determine whether the findings of the flow analysis have any real validity. What actually happens is determined after processing the plastic. See flow, Poi-seuille. 

Chemical engineering processes involve the transport and transfer of momentum, energy, and mass. Momentum transfer is another word for fluid flow, and most chemical processes involve pumps and compressors, and perhaps centrifuges and cyclone separators. Energy transfer is used to heat reacting streams, cool products, and run distillation columns. Mass transfer involves the separation of a mixture of chemicals into separate streams, possibly nearly pure streams of one component. These subjects were unified in 1960 in the first edition of the classic book. Transport Phenomena (Bird et al., 2002). This chapter shows how to solve transport problems that are one-dimensional that is, the solution is a function of one spatial dimension. Chapters 10 and 11 treat two- and three-dimensional problems. The one-dimensional problems lead to differential equations, which are solved using the computer. 

In Section 5-2, an introdnction to the field of computational fluid dynanfics is given, with an emphasis on the fundamental equations that are used to describe processes that are conunon in nfixing applications. An overview of the numerical methods used to solve these equations is presented in Section 5-3. Numerical simulations of stirred tanks are normally done in either two or three dimensions. In two dimensional (2D) simnlations, the geometry and flow field are assumed to... 

A flow field is best cliaraclerizcd by the velocity distribution, and thus a flow is said to be one-, two-, or three-dimensional if Ihe flow velocity varies in one, two, or three primary dimensions, respectively. A typical fluid flow involves a three-dimensional geometry, and the velocity may vary in all three dimensions, rendering the flow three-dimensional [V (.r. y, z) in rectangular or V (r, 0, z) in cylindrical coordinates]. However, the variation of velocity in certain directions can be small relative to the variation in oUicr directions and can be ignored with negligible error, In such cases, the flow can be modeled conveniently as being one- or two-dimensional, which is easier to analyze. 

The three-dimensional simulation domain is described Figure 1 for a two-compartment filter-press electrolyser. The cathode and anode compartments are separated by a rigid impermeable membrane, to prevent S02 crossover. The conductivity of a CMX Neocepta membrane is assumed for simulation. Both Pt electrodes are flat and parallel. Counter-current fluid flow is maintained under steady-state conditions in the plane-parallel compartments. The dimensions are those of the FM01-LC model, manufactured by ICI Chemical Polymer Company, and used in the Westinghouse pilot test facility implemented in the Marcoule Laboratory [9],... 

Fluid flow may be steady or unsteady, uniform or nonuniform, and it can also be laminar or turbulent, as well as one-, two-, or three-dimensional, and rotational or irrotational. One-dimensional flow of incompressible fluid in food systems occurs when the direction and magnitude of the velocity at all points are identical. In this case, flow analysis is based on the single dimension taken along the central streamline of the flow, and velocities and accelerations normal to the streamline are negligible. In such cases, average values of velocity, pressure, and elevation are considered to represent the flow as a whole. Two-dimensional flow occurs when the fluid particles of food systems move in planes or parallel planes and the streamline patterns are identical in each plane. For an ideal fluid there is no shear stress and no torque additionally, no rotational motion of fluid particles about their own mass centers exists. 

In order to connect the above expressions to Onsager s theory, it is necessary to extend equations such as (6.3.1) to three dimensions. This equation shows that a force in one direction leads to velocity changes in the other two spatial directions. A general treatment of fluid flow requires that one identify the components of the pressure tensor associated with the force which leads to fluid flow. The force vector F has, in general, a component in each of the three Cartesian directions. The component in the x-direction, F, gives rise to three pressure components, one in the same direction, P x, and two shear components, Pxy and Pxz- Six more pressure components are obtained from the force components in the y- and z-directions, Fy and F. As a result, there is a second-rank pressure tensor Py with nine components. Analysis on the basis of classical dynamics leads to the conclusion that the off-diagonal elements, Py and Py,-, are equal. As a result, six distinct elements of this tensor must be determined to define it. The second important step is to write an equation of continuity in terms of momentum. Defining the momentum density If as... 

In the previous sections we considered flows with a smooth spatial structure in which the relative dispersion of fluid trajectories is exponential in time and can be characterized by a single timescale, the inverse of the Lyapunov exponent. This is also valid for two-dimensional turbulent flows that have a smooth velocity field in the small-scale enstrophy cascade range (Bennett, 1984). A similar behavior occurs in any dimension at scales below the Kolmogorov scale (the so-called Batchelor or viscous-convective range, see below). In the inertial range of fully developed three-dimensional turbulence, however, the velocity field has a broad range of timescales and they all contribute to the relative dispersion of particle trajectories and affect the transport properties of the flow. 

Chaos does not occur as long as the torus attraaor is stable. As a parameter of the system is varied, however, this attractor may go through a sequence of transformations that eventually render it unstable and lead to the possibility of chaotic behavior. An early suggestion for how this happens arose in the context of turbulent fluid flow and involved a cascade of Hopf bifurcations, each of which generate additional independent frequencies. Each additional frequency corresponds to an additional dimension in phase space the associated attractors are correspondingly higher dimensional tori so that, for example, two independent frequencies correspond to a two-dimensional torus (7 ), whereas three independent frequencies would correspond to a three-dimensional torus (T ). The Landau theory suggested that a cascade of Hopf bifurcations eventually accumulates at a particular value of the bifurcation parameter, at which point an infinity of modes becomes available to the system this would then correspond to chaos (i.e., turbulence). 

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