Vortice

If a fluid is placed between two concentric cylinders, and the inner cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagmed as a substantially enlranced diflfiisivity (typically with effective diflfiision coefficients several orders of magnitude above molecular difhision coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifiircation tlirough a complex sequence of front patterns, see figure A3.14.16. 

In writing the Lagrangean density of quantum mechanics in the modulus-phase representation, Eq. (140), one notices a striking similarity between this Lagrangean density and that of potential fluid dynamics (fluid dynamics without vorticity) as represented in the work of Seliger and Whitham [325]. We recall briefly some parts of their work that are relevant, and then discuss the connections with quantum mechanics. The connection between fluid dynamics and quantum mechanics of an electron was already discussed by Madelung [326] and in Holland s book [324]. However, the discussion by Madelung refers to the equations only and does not address the variational formalism which we discuss here. 

If a flow satisfies the condition of zero vorticity, that is, the velocity field v is such that V X V = 0, then there exists a function v such that v = Vv. In that case, one can describe the fluid mechanical system with the following Lagrangean density... 

Fig. 3. Flow past a circular cylinder for (a). Re < 5 where no separation is evident (b) 5 < Re < 40 and fixed vortices exist in a separation bubble or wake ...

As the Reynolds number rises above about 40, the wake begins to display periodic instabiUties, and the standing eddies themselves begin to oscillate laterally and to shed some rotating fluid every half cycle. These still laminar vortices are convected downstream as a vortex street. The frequency at which they are shed is normally expressed as a dimensionless Strouhal number which, for Reynolds numbers in excess of 300, is roughly constant ... 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

Here (0 is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irro-tational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects m be neglected. Such flows without viscous effec ts are called in viscid flows. 

When fluid flows past objects or through orifices or similar restrictions, vortices may periodically be shed downstream. Objects such as smokestacks, chemical-processing columns, suspended pipehnes, and electrical transmission lines can be sul ected to damaging vibrations and forces due to the vortices, especially if the shedding frequency is close to a natural vibration frequency of the objecl. The shedding can also produce sound. See Krzywoblocki (Appl. Mech. Rev., 6, 39 397 [1953]) and Marris (J. Basic Eng., 86, 185-196 [1964]). 

For flow past a cyhnder, the vortex street forms at Reynolds numbers above about 40. The vortices initially form in the wake, the point of formation moving closer to the cylinder as Re is increased. At a Reynolds number of 60 to 100, the vortices are formed from eddies attached to the cylinder surface. The vortices move at a velocity slightly less than V. The frequency of vortex shedding/is given in terms of the Strouhal number, which is approximately constant over a wide range of Reynolds numbers. 

For 40 < Re < 200 the vortices are laminar and the Strouhal number has a nearly constant value of 0.2 for flow past a cylinder. Between Re = 200 and 400 the Strouhal number is no longer constant and the wake becomes irregular. Above about Re = 400 the vortices become turbulent, the wake is once again stable, and the Strouhal number remains constant at about 0.2 up to a Reynolds number of about 10. ... 

Cavitation Loosely regarded as related to water hammer and hydrauhc transients because it may cause similar vibration and equipment damage, cavitation is the phenomenon of collapse of vapor bubbles in flowing liquid. These bubbles may be formed anywhere the local liquid pressure drops below the vapor pressure, or they may be injected into the hquid, as when steam is sparged into water. Local low-pressure zones may be produced by local velocity increases (in accordance with the Bernouhi equation see the preceding Conservation Equations subsection) as in eddies or vortices, or near bound-aiy contours by rapid vibration of a boundaiy by separation of liquid during water hammer or by an overaU reduction in static pressure, as due to pressure drop in the suction line of a pump. 

Vortex-Shedding Flowmeters These flowmeters take advantage of vortex shedding, which occurs when a fluid flows past a non-streamlined objec t (a Blunt body). The flow cannot follow the shape of the object and separates from it, forming turbulent vortices or eddies at the object s side surfaces. As the vortices move downstream, they grow in size and are eventually shed or detached from the objec t. 

Vapor-Liquid Separation This design problem may be important for a number of reasons. The most important is usually prevention of entrainment because of value or product lost, pollution, contamination of the condensed vapor, or fouling or corrosion of the surfaces on which the vapor is condensed. Vapor-liquid separation in the vapor head may also oe important when spray forms deposits on the w ls, when vortices increase head requirements of circulating pumps, and when shoiT circuiting allows vapor or unflashed liquid to be carried back to the circulating pump ana heating element. 

In a submerged-tube FC evaporator, all heat is imparted as sensible heat, resulting in a temperature rise of the circulating hquor that reduces the overall temperature difference available for heat transfer. Temperature rise, tube proportions, tube velocity, and head requirements on the circulating pump all influence the selec tion of circulation rate. Head requirements are frequently difficult to estimate since they consist not only of the usual friction, entrance and contraction, and elevation losses when the return to the flash chamber is above the liquid level but also of increased friction losses due to flashing in the return line and vortex losses in the flash chamber. Circulation is sometimes limited by vapor in the pump suction hne. This may be drawn in as a result of inadequate vapor-liquid separation or may come from vortices near the pump suction connection to the body or may be formed in the line itself by short circuiting from heater outlet to pump inlet of liquor that has not flashed completely to equilibrium at the pressure in the vapor head. 

When fluid flows around a curwe in a duct, or when fluid is confined between differentially rotating cylinders, secondary flows called Taylor vortices are generated. If a membrane is mounted on a rotating... 

The suction bell reduces entrance losses and helps to prevent vortices. If you use a basket strainer, the screen area should be four times the area of the entrance pipe. Avoid tight mesh. screens because they clog quickly (Figure 17-8). 

The lowering below the stack top of pieces of the plume by the vortices shed downwind of the stack is simulated by using a value h in place of the physical stack height h. This is somewhat less than the physical height when the stack gas exit velocity is less than 1.5 times the wind speed u,... 

Blade stall causes Karman vortices in the airfoil wake. Whenever the frequency of these vortices coincides with the natural frequency of the airfoil, flutter will occur. Stall flutter is a major cause of compressor blade failure. 

Rushton et al. [10] performed extensive measurements of the power requirements for geometrically similar systems and found that for baffled tanks, the Froude number plays no part in determining the power requirements, as vortices do not form in such systems. For unbaffled systems, the Froude number plays a part above N g of about... 

---