Vorticity diffusion

In this parabolic equation we see that vorticity diffuses from the boundaries into the interior, with an effective diffusion coefficient that is the kinematic viscosity v = p/p. 

The analysis can be significantly simplified by reahzing that the rate with which the vorticity diffuses inwards, and hence establishes the fluid motion, is represented by the kinematic viscosity coeflBcient, which is of the order of 10 cmVsec and is at least one order of magnitude greater than the droplet surface regression rate. Hence quasi-steadiness for both the gas and liquid motion, with a stationary droplet surface and constant interfacial heat and mass flux, can be assumed. Once the fluid mechanical aspect of the problem is solved, the transient liquid-phase heat and mass transfer analyses, with a regressing droplet surface, can be performed. 

The Vorticity Diffusion of A-Vortices in Drag Reducing Solutions... 

The vorticity diffusion in a-vortex attached to the wall in a complex eigenvalue critical point is investigated. It is shown that extensional viscosity is produced near the critical point. The internal strainfield of the vortex rods is damped and an accelerated vorticity diffusion occurs due to polymer additives. 

This description is based on the assumption that the overall behaviour can be studied in a Newtonian fluid, and that a locally higher viscosity can be introduced to evaluate the vorticity diffusion. Since the main mechanism is an intertial one and the restoring time of the molecules rather small this simplification appears to be permissible. 

The aim of this investigation was to make a step towards the explanation of drag reduction by means of an altered vorticity diffusion behaviour in the ejections. We were able to show that we presumably will have a damping of the strain and an acceleration of the vorticity diffusion in the ejection type of events. 

An other common explanation for drag reduction is that the extensional viscosity stabilizes the flow of the viscous sublayer or the organisation of the different features in a grand structure. It would also be possible to incorporate the enhanced vorticity diffusion discussed here into such an explanation. Without knowing the grand structure in Newtonian fluids this is naturally highly speculative. 

In this model the momentum transport in the ejections will be reduced by damping the axial flow component. Moreover the dissipation is reduced due to a reduction of the internal shear by a vorticiy diffusion process. Finally the vortex core increases in width and the axial momentum is smeered over larger areas. Therefore it is thinkable that the competition between vorticity diffusion and the cascade process, which both are not dissipative processes, would result in a reduction of the energy transport since the diffusion transports the energy in the opposite direction than the cascade process This means towards vortices of larger scale. 

With these remarks the limits of such models of drag reduction are given. It seems therefore reasonable to investigate this flow behaviour in much more detail before conclusive statements on the drag reducing mechanism can be made. However most of the main feature of the drag reduction can be explained by this vorticity diffusion mechanism. 

It is not straightforward to determine the transient solution that leads to the solid-body rotation of the fluid starting from rest The solution of the Navier-Stokes equations in the form ur=Q,ug r,z,t),u =0), for which the azimuthal component ug evolves in time nnder the effect of a process of vorticity diffusion from the lateral circular wall toward the tank axis, does not portray reality. A secondaiy flow Ur 0 and 0) actually occurs, which carries vorticity from the boundary layer on the horizontal bottom wall into the water layer. Boundary layers play a key part in rotating flows they allow steady-state flows to be established more rapidly. ... 

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. 

A little bit of physical intuition as to how the vortices form in the first place may help in explaining the behavior as TZ is increased still further. We know that u = 0 at the cylinder s surface. We also know that the velocity increases rapidly as we get further from that surface. Therefore vortices are due to this rapid local velocity variation. If the variation is small enough, there is enough time for the vorticity to diffuse out of the region just outside the cylinder s surface and create a large von Karman vortex street of vorticity down stream [feyn64]. 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

As the fluid flows over the forward part of the sphere, the velocity increases because the available flow area decreases, and the pressure decreases as a result of the conservation of energy. Conversely, as the fluid flows around the back side of the body, the velocity decreases and the pressure increases. This is not unlike the flow in a diffuser or a converging-diverging duct. The flow behind the sphere into an adverse pressure gradient is inherently unstable, so as the velocity (and lVRe) increase it becomes more difficult for the streamlines to follow the contour of the body, and they eventually break away from the surface. This condition is called separation, although it is the smooth streamline that is separating from the surface, not the fluid itself. When separation occurs eddies or vortices form behind the body as illustrated in Fig. 11-1 and form a wake behind the sphere. 

In PF, the transport of material through a vessel is by convective or bulk flow. All elements of fluid, at a particular axial position in the direction of flow, have the same concentration and axial velocity (no radial variation). We can imagine this ideal flow being blurred or dispersed by backmixing of material as a result of local disturbances (eddies, vortices, etc.). This can be treated as a diffusive flow superimposed on the convective flow. If the disturbances are essentially axial in direction and not radial, we refer to this as axial dispersion, and the flow as dispersed plug flow (DPF). (Radial dispersion may also be significant, but we consider only axial dispersion here.)... 

The decreased overall density of the mixing layer with combustion increases the dimensions of the large vortices and reduces the rate of entrainment of fluids into the mixing layer [13]. Thus it is appropriate to modify the simple phenomenological approach that led to Eq. (6.31) to account for turbulent diffusion by replacing the molecular diffusivity with a turbulent eddy dif-fusivity. Consequently, the turbulent form of Eq. (6.38) becomes... 

Figure 6.1 shows the apparatus diagram. The diffusion flame burner consisted of an air plenum with an exit diameter of 22 mm, forced at a Strouhal number of 0.73 (100 Hz) by a single acoustic driver, and a coaxial fuel injection ring of diameter 24 mm, fed by a plenum forced by two acoustic drivers at either 100 Hz (single-phase injection) or 200 Hz (dual-phase injection). The fuel was injected circumferentially directly into the shear layer and roll-up region for the air vortices. In addition, this fuel injection was sandwiched between the central air flow and the external air entrainment. Thus the fuel injection was a thin cylindrical flow acted upon from both sides by air flow. 

The soot formation and its control was studied in an annular diffusion flame using laser diagnostics and hot wire anemometry [17, 18]. Air and fuel were independently acoustically forced. The forcing altered the mean and turbulent flow field and introduced coherent vortices into the flow. This allowed complete control of fuel injection into the incipient vortex shedding process. The experiments showed that soot formation in the flame was controlled by changing the timing of fuel injection relative to air vortex roll-up. When fuel was injected into a fully developed vortex, islands of unmixed fuel inside the air-vortex core led to... 

The combustion efficiency of the benzene was beyond 99.999% even at an overall equivalence ratio of 1.0 (including the entrainment air which was 30% of the total air flow). With the controller off, the flame was extremely sooty and, in fact, sooting as the quartz tube would be blackened and the mass spectrometer sampling probe clogged in a matter of seconds, and soot was sucked into the scrubber system. This comparison between controller off and controller on conditions with benzene fuel is extremely dramatic and shows the efficacy of active combustion control in vortices to eliminating soot from diffusion flames. 

For CO 0, Eq. (11-7) reduces to the stream function for steady creeping flow past a rigid sphere, i.e., Eq. (3-7) with k = co. The parameter 3 may be regarded as a characteristic length scale for diffusion of vorticity generated at the particle surface into the surrounding fluid. When co is very large, 3 is small, and the flow can be considered irrotational except in the immediate vicinity of the particle. In the limit co go, Eq. (11-7) reduces to Eq. (1-29), the result for potential flow past a stationary sphere. 

The first term of Eq. (11-11) is the Stokes drag for steady motion at the instantaneous velocity. The second term is the added mass or virtual mass contribution which arises because acceleration of the particle requires acceleration of the fluid. The volume of the added mass of fluid is 0.5 F, the same as obtained from potential flow theory. In general, the instantaneous drag depends not only on the instantaneous velocities and accelerations, but also on conditions which prevailed during development of the flow. The final term in Eq. (11-11) includes the Basset history integral, in which past acceleration is included, weighted as t — 5) , where (t — s) is the time elapsed since the past acceleration. The form of the history integral results from diffusion of vorticity from the particle. 

Equation (11-11) depends on neglect of inertial terms in the Navier-Stokes equation. Neglect of inertia terms is often less serious for unsteady motion than for steady flow since the convective acceleration term is small both for Re 0 (Chapters 3 and 4), and for small amplitude motion or initial motion from rest. The second case explains why the error in Eq. (11-11) can remain small up to high Re, and why an empirical extension to Eq. (11-11) (see below) describes some kinds of high Re motion. Note also that the limited diffusion of vorticity from the particle at high cd or small t implies that the effects of a containing wall are less critical for accelerated motion than for steady flow at low Re. 

In this form one sees an analogy in the vorticity equation to the other transport equations— a substantial-derivative description of advective transport, a Laplacian describing the diffusive transport, and possibly a source term. It is interesting to observe that the vorticity equation does not involve the pressure. Since pressure always exerts a normal force that acts through the center of mass of a fluid packet (control volume), it cannot alter the rotation rate of the fluid. That is, pressure variations cannot cause a change in the vorticity of a flow field. 

The term p, V2u> reveals that vorticity (i.e., the strength of fluid rotation) can diffuse by molecular interactions throughout a flow field, with the viscosity being the diffusion coefficient. Quite often the source of vorticity is the fluid tumbling caused by the shearing action associated with a no-slip condition on a solid wall. This vorticity, once produced, is both convected and diffused throughout the flow. The relative strength of the convective and diffusive processes depends on the flow field and the viscosity. 

The pressure does not appear directly in the vorticity-transport equation. Thus, it is apparent that the convective and diffusive transport of vorticity throughout a flow cannot depend directly on the pressure field. Nevertheless, it is completely clear that pressure affects the velocity field, which, in turn, affects the vorticity. By taking the divergence of the incompressible, constant-viscosity Navier-Stokes equations, a relationship can be derived among the velocity, pressure, and vorticity fields. Beginning with the Navier-Stokes equations as... 

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