Turbulence spectra

This length is representative of the dimension at which dissipation occurs and defines a cut-off of the turbulence spectrum. For large A) there is a large spread of the two extreme lengths characterizing turbulence. This spread is reduced with the increasing temperature found in combustion of the consequent increase in v(). 

In other cases the application of this concept has been further extended simulating faster turbulent fluctuations that are within the turbulence spectrum. For such dynamic simulations, using Reynolds averaged models, the Ic-quantity represents the turbulent kinetic energy accumulated on the fraction of the spectrum that is represented by the modeled scales. Therefore, to compare the simulated results obtained with this type of models with experimental data, that is averaged over a sufficient time period to give steady-state data (representing the whole spectrum of turbulence), both the modeled and the resolved scales have to be considered [68]. 

For steady-state simulations, considering the turbulence spectrum in terms of eddy size (length scale), the scales predicted by the k-e model is thus much larger than the particle size. The inclusion of turbulence production due to the bubbles relative motion is therefore based on the assumption of an inverse cascade of turbulence. 

This means that the aggregate size distribution mirrors the turbulence spectrum of the reactor. Variables that determine mixing are reactor design, number and design of baffles, impeller design, power input, feed concentration, feed rate, location and number of inlet tubes, and so on. The position of the inlet tube(s) and the conditions near these feed points are also important and generally the solutions should be introduced close to the agitator [23, 24]. 

The size of drops encountered in stirred dispersions is comparable to the size of energy transmitting eddies. By consideration of stresses exerted on a drop in this range of the turbulence spectrum, Hinze (12) characterized the maximum drop size by a critical Weber number... 

Penetration theory can also be applied to turbulent conditions by assuming the turbulence spectrum to consist of large eddies, capable of surface renewal, and small eddies responsible for the presence of eddy diffusivity The small eddies are damped when an element of liquid reaches the interface so that, during its residence time there, mass transfer occurs in accordance with the assumptions of the penetration theory If all the eddies stay at the interface for the same interval of time we talk about penetration theory with regular surface renewal or the Higbie model If there is random distribution of residence times with an age-independent fractional rate of surface renewal, s, the term penetration theory with random surface renewal, or the Danckwerts nK)del, is employed In the case of the Higbie model, the mass transfer coefficient is the same as that given by eqn (18). For the Danckwerts model it takes the form... 

Here is the root-mean-square (rms) relative velocity between two points in the fluid separated by a distance d. For very large Reynolds numbers of the main stream (much larger than the Re value required for assumption of universal equilibrium), Kolmogorov s theory proposes that the turbulence spectrum be divided into two subranges. The inertial subrange is that part of the spectrum in which viscous dissipation is unimportant and... 

Following [10] the flame response to a turbulent flow can be considered as the sum of its responses to the vortices composing the turbulence spectrum in a first approximation. The principle regimes of interaction were identified by [10] as a function of two main parameters the ratio of the maximum rotational speed of the vortices to laminar flame speed U0/si and of the vortex core diameter to laminar flame thickness d/si While most qualitative results were confirmed by experimental investigations [11,12], the simulations did not reproduce flame extinction observed in the experiments. Complex chemical kinetics phenomena are a possible explanation for flame extinction due to excessive strain as encountered during flame-vortex interaction. 

While the fully isotropic assumption is not a good match to physical reality, the implications of isotropy are profound for turbulence modeling and measurements. Isotropy allows the entire turbulent spectrum to be defined from one component of fluctuating velocity, because the flow is perfectly without directional preference. It allows simplification of the equations to include only the normal stresses. It also allows one to make spectral arguments to simplify the measurement of the dissipation. This assumption is so powerful that it is often invoked in the hope that it will be good enough for a flrst approximation, despite the fact that it is a poor match for the full physical reality. 

Probably the most advanced in-situ high-temperature measurements have been reported in refs. [32, 61]. The authors were able to measure the temperature and temperature fluctuations within the silicon melt during industrial crystal growth using thermocouples and optical sensors. The superiority of the latter has been established, since signals from thermocouples do not adequately reproduce the high-frequency part of the turbulence spectrum. 

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