Fluctuating velocity

The shear rate between the average velocities is called macroscale shear present in eddies of 500 p.m or larger in size. The shear rate between the fluctuating velocities, present in smaller than 100 p.m eddies, is called microscale shear. A mixing tank, therefore, has several types of shear, ie, macroscale and microscale, maximum and minimum, average in the impeller zone and in the entire tank. 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-... 

Most of the energy dissipation occurs on a length scale about 5 times the Kolmogorov eddy size. The characteristic fluctuating velocity for these energy-dissipating eddies is about 1.7 times the Kolmogorov velocity. 

In the present discussion only the problem of steady flow will be considered in which the time average velocity in the main stream direction X is constant and equal to ux. in laminar flow, the instantaneous velocity at any point then has a steady value of ux and does not fluctuate. In turbulent flow the instantaneous velocity at a point will vary about the mean value of ux. It is convenient to consider the components of the eddy velocities in two directions—one along the main stream direction X and the other at right angles to the stream flow Y. Since the net flow in the X-direction is steady, the instantaneous velocity w, may be imagined as being made up of a steady velocity ux and a fluctuating velocity ut, . so that ... 

If the magnitude of the fluctuating velocity component is the same in each of the three principal directions, the flow is termed isotropic. If they are different the flow is said to be anisotropic. Thus, if the root mean square values of the random velocity components... 

The parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

At lower Reynolds numbers, the axial velocity profile will not be flat and it might seem that another correction must be added to Equation (9.14). It turns out, however, that Equation (9.14) remains a good model for real turbulent reactors (and even some laminar ones) given suitable values for D. The model lumps the combined effects of fluctuating velocity components, nonflat velocity profiles, and molecular diffusion into the single parameter D. 

Baldi, S. and Yianneskis, M. (2004) On the quantification of energy dissipation in the impeller stream of a stirred vessel from fluctuating velocity gradient measurements. Chem. Eng. Sci., 59 (13), 2659-2671. 

In the velocity field of the determining eddies, which is characterized by the turbulent fluctuation velocity the particles experience a dynamic stress according to the Reynolds stress Eq. (2) ... 

In order to use Eqs. (3) and (4) or the data given in Fig. 1, for the calculation of maximum turbulent fluctuation velocity the maximum energy dissipation e , must be known. With fully developed turbulence and defined reactor geometry, this is a fixed value and directly proportional to the mean mass-related power input = P/pV, so that the ratio ,/ can be described as an exclusive function of reactor geometry. In the following, therefore details will be provided on the calculation of power P and where available the geometric function ,/ . 

In 1965, Van der Grinten4 introduced the interesting concept of measureable-ness a as qualification of an analytical method in relation to the process to be regulated, with its specific fluctuation velocity. In this connection he also defined... 

In order to simplify the notation, we will first denote the fluctuating velocity gradient by... 

In locally isotropic turbulence, the fluctuating velocity gradient and scalar gradient will be uncorrelated, and sf will be null. Thus, at sufficiently high Reynolds number, the scalar-flux dissipation is negligible. 

However, the mean velocity field will depend on conditional moments of the fluctuating velocity through the mean velocity transport equation,... 

In a velocity, composition PDF code, each notional particle is described by its position Xw(f), fluctuating velocity uw(f), composition and weight w(n)(t). The treatment... 

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