Pathline

Fig. 14. CFD predictions of turbulent flow (meshed REU geometry, air pathlines with spheres hidden for clarity, contours of turbulent kinetic energy k) for configuration B2. Copyright 2001 From Turbulent Resistance of Complex Bed Structures by J. Tobis. Reproduced by permission of Taylor and Francis, Inc., http //www.taylorandfrancis.com.

The advective control model presented in this chapter is an extension of the formulation presented by Mulligan and Ahlfeld (1999a) that uses both contaminant pathline and capture zone simulation to constrain plume capture designs. In this chapter, additional constraints and algorithmic procedures... 

Advective transport is simulated in the advective control model by particle tracking after the velocity field is determined. Following solution of equations (1) and (2), a particle can be tracked as it travels through any part of the domain (Shafer 1987). A particle pathline is the trace of particle position over time, which is determined as... 

In the first stage of the solution process, the advective control model seeks a pumping scheme in which the capture zone fully encompasses all control points representing the contaminant plume. The capture zone is simulated by tracking particles from extraction wells backwards through the velocity field. To represent the plume capture constraints numerically, a distance measure is used in which the minimum distance between each plume control point and all particles (see Figure 1) is constrained. When the distance between a control point and particle pathline equals zero then the plume control point lies within the capture zone. To ensure capture of the entire plume, the constraint function must equal zero for all control points. The reverse tracking formulation is stated as... 

Figure I. In reverse tracking, pathlines represent the capture zone and the constraint function measures the minimum distance between a plume control point and all pathlines. Pathlines are shown as dashed curves and the gray area represents the contaminant plume to be contained.

Shafer, J. M. (1987). Reverse pathline calculation of time-related capture zones in nonuniform flow. Ground Water, 25(3), 283-289. 

If the flow is steady in time, we speak about the x,y,z space as the phase space and the integral curves as trajectories, or in this case, the pathlines. If the fluid is also... 

Results demonstrate that when agitators are switched the slope of the pathline becomes discontinuous. We will see later in this chapter how this mechanism may produce an essentially stochastic response in the Lagrangian sense. Aref termed this chaotic advection, which he suggested to be a new intermediate regime between turbulent and laminar advection. The chaos has a kinematic origin, it is temporal—that is, along trajectories associated with the motion of individual fluid particles. Chaos is used in the sense of sensitivity of the motion to the initial position of the particle, and exponential divergence of adjacent trajectories. 

Pathline is the trajectory a fluid particle describes in the phase space streamline is a line that is tangent to the direction of flow at any point mathematically, streamlines are calculated via dx/dt = d[Pg.334]

A simplified particle-trajectory analysis has been used to explain the essential features of US-assisted filtration. Acoustic forces can cause particle trajectories to deviate from hydrodynamic pathlines towards particle collectors, thereby enhancing their collection efficiency in comparison to pure hydrodynamic interception. 

The total operating pressure drop of the filter can be approximated by equation 7. Knowing that the pressure drop across the two pathways is equal to the total pressure drop, the flowrate through each pathline can be calculated as follows ... 

Fig. 4 An iso-surface of axial vorticity in a cyclone, colored by velocity magnitude, is used to show the central vortex pathlines are used to show the swirling flow. (View this art in color at www.dekker.com.)...

Other types of flow lines also can be used. Oil-flow lines are pathlines that are constrained to a given boundary surface. When calculating the pathline, velocity components that are tangent to the given boundary surface are included and normal velocity components are ignored. This is useful for visualization... 

To determine the flow status inside the volume, so called Internal Points are calculated in the BEMflow software. From these points on, streamlines are calculated for a number of specified timesteps. For steady state flows, streamlines and pathlines are identical, while for transient flow channels as in the extruders the pathlines have to be calculated from the results for every single timestep respectively every relative screw position. This leads to an additional task for the flow analysis [145]. 

Figure 2-20. A sketch of the problem of Young el al. of a gas bubble that is held at a fixed position in a gravitational field because of the action of a surface-tension gradient at its surface (caused by a vertical gradient of temperature in the tank). The temperature decreases in the vertical direction. This means that the interfacial tension is higher at the top of the bubble and lower at the bottom. As a result, the interface is pulled from the bottom toward the top, dragging fluid with it, as shown by the sketch of fluid pathlines in the vicinity of the bubble. This causes the bubble to swim downward against the upward buoyant force produced by gravity, much as a swimmer pulling himself downward by the upward motion of his arms.

Equation (3-66) is the balance between centrilugal forces and the radial pressure gradient that is responsible for the fact that ur = 0. Thus, in the Couette flow problem the acceleration associated with curved fluid pathlines does not necessarily lead to a radial flow, but may be balanced by a radial pressure gradient. 

Thus lines of constant t/r are everywhere tangent to the local velocity that is, curves in space corresponding to constant values of

[Pg.447]

This is the velocity field for a stationary, nonrotating sphere in a simple shear flow. A sketch of the fluid pathlines in the xix2 plane for this case is reproduced in Fig. 8 2(a). If the sphere is allowed to rotate with some angular velocity 12, the corresponding velocity field... 

Figure 8-2. Fluid pathlines in the x x2 plane (see Fig. 8-1) for simple shear flow past (a) nonrotating and (b) rotating spheres. The nonrotating case was obtained from Eq. (8-51), and the rotating case was obtained by adding (8-54) to (8-51).

The composite velocity field for a sphere rotating with the angular velocity (8-52) in a simple shear flow, Eq. (8 46), is thus the sum of (8-51) and (8-54). The fluid pathlines in the X X2 plane for this case are shown in Fig. 8-2(b). The corresponding torque on the sphere is... 

Unfortunately, however, there are a large number of different types of flow conditions for which the boundary-layer form of the heat transfer correlation (9-255) is not applicable. This applies, basically, to any flow configuration in which the body is completely surrounded by a region of closed streamlines (or pathlines, if the flow is not 2D or axisymmetric). We will discuss high-Peclet-number heat transfer in such cases in Section L. Here, we consider... 

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