Azimuthal flow

Another set of experiments, by Paoletti et al. (2006), investigated the synchronization of stirred oscillatory BZ reaction over distances larger than the characteristic lengthscale of the flow. In this experiment the flow was composed of an annular ring of counter-rotating vortices with a superimposed additional oscillatory azimuthal flow. A simplified model of the corresponding velocity field can be written as... 

Since wind is circular, it is frequently easier to interpret and visualize the frequency of wind flow subjectively by displaying a wind rose, that is, wind frequencies for each direction oriented according to the azimuth for that direction. Figure 21-8 is a wind rose showing both directional frequencies and wind speed frequencies by six classes from 3-hourly observations for a 5-year period (1965-1969) for O Hare Airport, Chicago. The highest frequencies are from the south and west, the lowest from the southeast and east. 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

The transition from laminar to turbulent flow on a rotating sphere occurs approximately at Re = 1.5 4.0 x 104. Experimental work by Kohama and Kobayashi [39] revealed that at a suitable rotational speed, the laminar, transitional, and turbulent flow conditions can simultaneously exist on the spherical surface. The regime near the pole of rotation is laminar whereas that near the equator is turbulent. Between the laminar and turbulent flow regimes is a transition regime, where spiral vortices stationary relative to the surface have been observed. The direction of these spiral vortices is about 4 14° from the negative direction of the azimuthal angle,. The phenomenon is similar to the flow transition on a rotating disk [19]. 

For turbulent flow on a rotating sphere or hemisphere, Sawatzki [53] and Chin [22] have analyzed the governing equations using the Karman-Pohlhausen momentum integral method. The turbulent boundary layer was assumed to originate at the pole of rotation, and the meridional and azimuthal velocity profiles were approximated with the one-seventh power law. Their results can be summarized by the... 

The Chilton-Colburn analogy can be also used to estimate the local mass transfer rate in laminar flow where the wall shear stress is related to the azimuthal velocity gradient by... 

With a view to any simulation, a few important items have to be addressed. First of all, it has to be decided whether the flow to be simulated is 2-D, 21/2-D, or 3-D. When the flow is, e.g., axis-symmetrical and steady, a 2-D simulation may suffice. For a flow field in which all variables, including the azimuthal velocity component, may not depend on the azimuthal coordinate, a 21/2-D simulation may be most appropriate. Most other cases may require a full 3-D simulation. It is tempting to reduce the computational job by casting the 3-D flow field into a 2-D mode. The experience, however, is that in 2-D simulations the turbulent viscosity tends to be overestimated in this way, the flow... 

A second choice to be made relates to the size of the flow domain. It may be worthwhile to limit the calculational job by reducing the size of the flow domain, e.g., by identifying an axis or plane of symmetry, or, in a cylindrical vessel with baffles mounted on the wall, due to periodicity in the azimuthal direction. Commercial software accomplishes these choices by means of symmetry cells and cyclic cells, respectively although such choices reduce the size of the simulation, they may eliminate the possibility of finding the real (asymmetric, unstable, or transient) 3-D flow field. The presence of feed pipes or drain or withdrawal pipes may also make the use of symmetry or cyclic cells impossible. Again, this issue only plays a role in RANS-type simulations. 

Venneker et al. (2002) used as many as 20 bubble size classes in the bubble size range from 0.25 to some 20 mm. Just like GHOST , their in-house code named DA WN builds upon a liquid-only velocity field obtained with FLUENT, now with an anisotropic Reynolds Stress Model (RSM) for the turbulent momentum transport. To allow for the drastic increase in computational burden associated with using 20 population balance equations, the 3-D FLUENT flow field is averaged azimuthally into a 2-D flow field (Venneker, 1999, used a less elegant simplification )... 

Along an axial line at r = 0.026666667 m, graph the azimuthal vorticity. Qualitatively explain the observed behavior in terms of the flow field and the signs of the vorticity. Where is the flow rotating clockwise and counterclockwise Provide a physical interpretation for the observed fluid rotation. 

Transitions from steady-state to time-dependent surface-tension-driven motions are well known also and are important in meniscus-defined crystal growth systems. For example, the experiments of Preisser et al. (51) indicate the development of an azimuthal traveling wave on the axisymmetric base flow in a small-scale floating zone. 

In their milestone work, Melander and Hussain found that the method of complex helical wave decomposition was instrumental in modeling both laminar as well as turbulent shear flows associated with coherent vortical structures, and revealed much new important data about this phenomenon than had ever been known before through standard statistical procedures. In particular, this approach plays a crucial role in the description of the resulting intermittent fine-scale structures that accompany the core vortex. Specifically, the large-scale coherent central structure is responsible for organizing nearby fine-scale turbulence into a family of highly polarized vortex threads spun azimuthally around the coherent structure. 

The only quantum number that flows naturally from the Bohr approach is the principal quantum number, n the azimuthal quantum number Z (a modified k), the spin quantum number ms and the magnetic quantum number mm are all ad hoc, improvised to meet an experimental reality. Why should electrons move in elliptical orbits that depend on the principal quantum number n Why should electrons spin, with only two values for this spin Why should the orbital plane of the electron take up with respect to an external magnetic field only certain orientations, which depend on the azimuthal quantum number All four quantum numbers should follow naturally from a satisfying theory of the behaviour of electrons in atoms. 

It is instructive to present the solution to equation (7.106) for the case of simple shear flow. For a spheroid oriented with its symmetry axis defined by the polar angle 9 relative to the z axis, and azimuthal axis <(> measured in the (x, y) plane relative to the x axis, equation (7.106) produces the following two equations for a simple shear flow of the form, v = G (0, x, 0) ... 

Dispersion coefficients were obtained by assuming that all dispersion occurs in the azimuthal direction, perpendicular to the flow, and by assuming that the annulus is a thin, infinitely wide slab. This is not unreasonable since the bed is thin compared with its radius, and the tracer is typically distributed over one quadrant. The steady state equation is... 

When an isolated sphere is held stationary in a creeping flow field containing suspended particles (see Figure 2), the equation of continuity in spherical coordinates, allowing azimuthal symmetry, is given by... 

The sample is subjected to compression by moving the crosshead downwards at a constant speed. The sample is extruded from between the two discs, undergoing elongational or biaxial flow the sample is stretched radially and azimuthally as it flows outwards between the approaching discs. Lubrication ensures that shear flow cannot occur. Elongational viscosity is calculated directly from the measured force-distance data, disc radius and crosshead speed no rheological model is required (Campanella and Peleg, 2002). 

Write the simplified heat-conduction equation for (a) steady one-dimensional heat flow in cylindrical coordinates in the azimuth (< ) direction and (b) steady onedimensional heat flow in spherical coordinates in the azimuth (0) direction. 

We now wish to examine the applications of Fourier s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification. 

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