Streamlines

From the preceding it is clear that the ideas of perfect-fluid flow and of the boundary layer are intimately tied together. Both are generally needed for completely describing physically interesting flows, although sometimes one alone is sufficient. We consider perfect-fluid flows in this chapter and the boundary layer in the next. First we must introduce the idea of streamlines. 

In one-dimensional flow, the direction of flow at every point in the flow is the same, although the velocity may not be the same at every point (e.g., laminar flow in a tube). In two- and three-dimensional flows, the velocity and direction both change from place to place. For unsteady (i.e., time-varying) flows, they also change from one instant to the next. For steady flow we can map out the velocity and direction at any point see Fig. 10.2, in which the velocity at any point is represented by an arrow showing the relative velocity and direction of the flow. 

If we follow the history of a fluid particle starting at A, we see that it moves, not in a straight line, but rather along a curve, whose direction at any point is tangent to the flow direction. Such a curve, showing the path of any fluid particle in steady flow, is called a streamline. Obviously, there is a streamline passing through every point in the flow so if all the streamlines 

Point values of the flow velocity and direction for steady, two-dimensional flow. 

If we use the alternative view of a streamline—a line across which there is no flow—then it is clear that the boundaries of solid objects immersed in the flow must be streamlines. For real fluid flows, the fluid adjacent to the boundary of a solid body does not move relative to the body it clings to the wall. Thus, in real fluids the wall is a streamline of zero velocity. In the theory of perfect-fluid flow, the imaginary perfect fluid has no tendency to cling to walls, because it has no viscosity. Thus, the streamline adjacent to a solid body in perfect-fluid flow is one with finite velocity. This leads to the idea that we may divide a perfect-fluid flow along a streamline and substitute a solid body for the flow on one side of the streamline without changing the mathematical character of the flow on the other side of the streamline. Thus, to compute the flow around some solid body in perfect-fluid theory, we need only find the flow which has a streamline with the same shape as the solid body and then conceptually substitute the solid body for that part of the flow this does not affect the rest of the flow. Several examples of this procedure will be shown. 

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Using CD s streamlines the automatic comparison of data. Because of their large storage capacity and their reduced dimensions CD s provide a complete historical database for all steam generator tubes from a mobile inspection platform. 

Development of weighted residual finite element schemes that can yield stable solutions for hyperbolic partial differential equations has been the subject of a considerable amount of research. The most successful outcome of these attempts is the development of the streamline upwinding technique by Brooks and Hughes (1982). The basic concept in the streamline upwinding is to modify the weighting function in the Galerkin scheme as... 

In the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection tenns (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called selective or inconsistent upwinding. Selective upwinding can be interpreted as the introduction of an artificial diffusion in addition to the physical diffusion to the weighted residual statement of the differential equation. This improves the stability of the scheme but the accuracy of the solution declines. 

Brooks, A. N, and Hughes, T. J.R., 1982. Streamline-upwind/Petrov Galerldn formulations for convection dominated hows with particular emphasis on the incompressible Navier -Stokes equations. Cornput. Methods Appl Meek Eng. 32, 199-259. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

Therefore the viscoelastic extra stress acting on a fluid particle is found via an integral in terms of velocities and velocity gradients evalua ted upstream along the streamline passing through its current position. This expression is used by Papanastasiou et al. (1987) to develop a finite element scheme for viscoelastic flow modelling. 

In the consistent streamline upwind Petrov-Galerkin (SUPG) scheme all of the terms in Equation (3.52) are weighted using the function defined by Equation (3.53) and hence Wjj = Wj. 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

Luo, X. L, and Tanner, R. L, 1989. A decoupled finite element streamline-upwind scheme for viscoelastic flow problems. J. Non-Newtonian Fluid Mech. 31, 143-162. 

Working equations of the streamline upwind (SU) scheme for the steady-state energy equation in Cartesian, polar and axisymmetric coordinate systems... 

Following the procedure described in in Chapter 3, Section 3 the streamlined-upwind weighted residual statement of the energy equation is formulated as... 

Least-square.s and streamline upwind Petrov-Galerkin (SUPG) schemes... 

The inconsistent streamline upwind scheme described in the last section is fonuulated in an ad hoc manner and does not correspond to a weighted residual statement in a strict sense. In tins seetion we consider the development of weighted residual schemes for the finite element solution of the energy equation. Using vector notation for simplicity the energy equation is written as... 

Figure 9.1 Distortion of flow streamlines around a spherical particle of radius R. The relative velocity in the plane containing the center of the sphere equals v, as r ...

In the derivation of both Eqs. (9.4) and (9.9), the disturbance of the flow streamlines is assumed to be produced by a single particle. This is the origin of the limitation to dilute solutions in the Einstein theory, where the net effect of an array of spheres is treated as the sum of the individual nonoverlapping disturbances. When more than one sphere is involved, the same limitation applies to Stokes law also. In both cases contributions from the walls of the container are also assumed to be absent. 

The viscosity of a suspension of ellipsoids depends on the orientation of the particle with respect to the flow streamlines. The ellipsoidal particle causes more disruption of the flow when it is perpendicular to the streamlines than when it is aligned with them the viscosity in the former case is greater than in the latter. For small particles the randomizing effect of Brownian motion is assumed to override any tendency to assume a preferred orientation in the flow. 

Equations (9.42) and (9.46) reveal that the range of a values in the Mark-Houwink equation is traceable to differences in the permeability of the coil to the flow streamlines. It is apparent that the extremes of the nondraining and free-draining polymer molecule bracket the range of intermediate permeabilities for the coil. In the next section we examine how these ideas can be refined still further. 

These results arise from considering the same polymer molecule under different conditions of permeability to the streamlines of solvent flow. 

Rather than discuss the penetration of the flow streamlines into the molecular domain of a polymer in terms of viscosity, we shall do this for the overall friction factor of the molecule instead. The latter is a similar but somewhat simpler situation to examine. For a free-draining polymer molecule, the net friction factor f is related to the segmental friction factor by... 

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