Velocity Field Information

Detailed hydrodynamic studies have been performed for pump-fed in-line mixers with rotors and stators comprised of teeth with slots between them. LeClair (1995) reported an early attempt for a Kady mixer. Although the simulation was quite simplistic and considered only a small section with assumed perfect symmetry, the results revealed a complex circulation pattern in the stator slot. 

For turbulent flow through rotor-stator devices with teeth, the aforementioned velocity field results indicate that flow stagnation on the leading edge of the downstream stator teeth provides a major energy field for emulsification and dispersion. It is not clear from these results what role is played by flow in the shear gap. The simulations indicate that the flow in the rotor-stator gap is not a simple shear flow but is more like a classical turbulent shear flow. Use of nominal shear rate may not be useful in scale-up. 

Simulations were performed and some LDA data were acquired for a similar device that has an enlarged shear gap of 4 mm rather than the standard 0.5 mm gap. The results indicate that there is much less stagnation on the stator teeth, so 

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The PIV techniques provide instantaneous velocity field information. The spatial and temporal resolution of PIV can be controlled with proper selection of lighting source, i.e., laser imaging system, i.e., camera and fi ame grabber, and optical arrangements/compo-nents. The continuous development of this hardware in the near future is expected to result in very high spatial and temporal resolution velocity measurements. Therefore, the p-PIV and p-holographic PIV techniques will develop as high-resolution indirect instantaneous shear stress measurement techniques in the near future. 

By making use of the spatial information, the velocity field of an extended, structured object can be obtained unambiguously without errors caused by uncertainty in the position of a feature within the slit. 

The filtering process used in LES results in a loss of information about the SGS velocity field. For homogeneous turbulence and the sharp-spectral filter, the residual velocity field5... 

There is no direct information on the velocity field, and thus a turbulence model is required to provide this information. 

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

Let us return for the moment to Eq. (2.2). In atmospheric problems it is impossible to solve the equations of motion analytically. Under these conditions information about the instantaneous velocity field u is available only from direct measurements or from numerical simulations of the fluid flow. In either case we are confronted with the problem of reconstructing the complete, continuous velocity field from observations at discrete points in space, namely the measuring sites or the grid points of the numerical model. The sampling theorem tells us that from a set of discrete values, only those features of the field with scales larger than the discretization interval can be reproduced in their entirety (Papoulis, 1%5). Therefore, we decompose the wind velocity in the form... 

The combination of the attraction of the electric field and the retarding effects leads to a maximum velocity for each ion. Measurement of these velocities gives information about the structure of the solution. Different cation and anion velocities give rise to a potential difference this is the liquid junction potential. It is interesting to know the magnitude of this potential, as it affects the measured potential of the whole electrochemical cell in other words, ion conductivities need to be measured. 

Given prior information in terms of a lower and upper bound, a prior bias, and constraints in terms of measured data, the MRE provides exact expressions for the posterior pdf and expected value of the inverse problem. The plume source is also characterized by a pdf The problem solved in their study is the same as Skaggs and Kabala s problem. For the noise-free data, MRE was able to reconstruct the plume evolution history indistinguishable from the true history. As for data with noise, the MRE method managed to recover the salient features of the source history. Another advantage using the MRE approach is that once the plume source history is reconstructed, future behavior of the plume can be easily predicted due to the probabilistic framework of MRE. Woodbury et al. [71] extended the MRE approach to reconstruct a 3D plume source within a ID constant velocity field and constant dispersivity system. 

The Prandtl number via has been found to be the parameter which relates the relative thicknesses of the hydrodynamic and thermal boundary layers. The kinematic viscosity of a fluid conveys information about the rate at which momentum may diffuse through the fluid because of molecular motion. The thermal diffusivity tells us the same thing in regard to the diffusion of heat in the fluid. Thus the ratio of these two quantities should express the relative magnitudes of diffusion of momentum and heat in the fluid. But these diffusion rates are precisely the quantities that determine how thick the boundary layers will be for a given external flow field large diffusivities mean that the viscous or temperature influence is felt farther out in the flow field. The Prandtl number is thus the connecting link between the velocity field and the temperature field. 

The last term is the rate of viscous energy dissipation to internal energy, E = jy dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function <1> for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such cases additional information, which may come from empirical correlations, is needed. 

The individual heat transfer coefficients for the medium and the liquor can be calculated from general correlations using the properties of the fluid and the velocity fields in the system.Additional information can be found in the continuing series of the American Society of Mechanical Engineers. Detailed studies and bibliographies on prediction of individual heat transfer coefficients for evaporating films on/in horizontal/vertical tubes, can be found in other publications. " ... 

Although there is no immediately useful information that we can glean from (2-56), we shall see that it provides a constraint on allowable constitutive relationships for T and q. In this sense, it plays a similar role to Newton s second law for angular momentum, which led to the constraint (2 41) that T be symmetric in the absence of body couples. In solving fluid mechanics problems, assuming that the fluid is isothermal, we will use the equation of continuity, (2-5) or (2-20), and the Cauchy equation of motion, (2-32), to determine the velocity field, but the angular momentum principle and the second law of thermodynamics will appear only indirectly as constraints on allowable constitutive forms for T. Similarly, for nonisothermal conditions, we will use (2-5) or (2-20), (2-32), and either (2-51) or... 

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