Fluctuating velocity , defined

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 ,/ . 

Fig. 6 shows the FFT spectrum for calculated bed pressure drop fluctuations at various centrifugal accelerations. The excess gas velocity, defined by (Uo-U ,, was set at 0.5 m/s. Here, 1 G means numerical result of particle fluidization behavior in a conventional fluidized bed. In Fig. 6, the power spectrum density function has typical peak in each centrifugal acceleration. However, as centrifugal acceleration increased, typical peak shifted to high frequency region. Therefore, it is considered that periods of bubble generation and eruption are shorter, and bubble velocity is faster at hi er centrifugal acceleration. 

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... 

All of these relations contains terms involving statistical correlations among various products of fluctuating velocity, pressure, and stress terms. This renders them considerably more complex than their laminar flow counterparts. Reynolds succeeded in partially sol ving this dilemma by the expedient of introducing the turbulent stress tensor f, defined by... 

Alternatively, the fluctuating velocity field can be characterized by the energy spectrum defined as the Fourier transform of the two-point velocity correlation integrated over a spherical shell of wavenumbers with magnitude k ... 

The ultimate descriptio we would like to have of turbulent flow would be an explicit expression for V, 1/, and Vy, and as functions of time and position. Then we could predict the average and the fluctuating velocities at any point and any time. Currently it seems impossible to make such a description the problem Is much too complex. The next best thing is a statistical description of the flow, i.e., what fraction of the time V, v, Vy, etc., have certain values. So far most of the experimental and theoretical work done on turbulence has been directed at these statistical properties of the,flow. Below we give a set of definitions which are widely used in the turbulence literature to describe such statistical properties of the flow and some experimental values of the quantities so defined. 

About 10000 data points (N) are collected with LDV at a single point in the vessel with a data rate of around 200 Hz. From these data points the mean velocity is calculated (as given in Figs. 3 to 5) but also the fluctuating velocity V as defined by... 

The normalized spatial correlation of the fluctuating velocity component u/ with the fluctuating velocity component u- is defined as... 

In this class, the fluctuations in the velocity are truly random such that averaged values of turbulence parameters are independent of the position in space. Thus, the fluctuations do not vary with the axis of translation (Brodkey 1967). This type of turbulence can be generated with a well-defined equipment geometry. However, it can only be sustained over relatively short distances. An important parameter that can be used to represent turbulence is the root-mean-square velocity defined earher. For homogeneous turbulence, the values need not be equal but they should not vary over the field under consideration. 

Turbulent flow is described by conservation equations of continuity and momentum, known as the Reynolds-averaged Navier-Stokes (RANS) equations. Laminar velocity terms in conservation equations are replaced by the steady-state mean components and time-dependent fluctuating components defined by Equation 6.100. 

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. 

In turbulent flow, we expect both velocity and concentration to fluctuate. For the smoke plume, the velocity fluctuations are the wind gusts, and the concentration fluctuations can be reflected as sudden changes in odor. To rewrite this equation to include these fluctuations, we define... 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

Time Averaging In turbulent flows it is useful to define time-averaged and fluctuation values of flow variables such as velocity com-... 

The ratio to z depends only on (gag-, zjx, = 2/3 tga.g, and the ratio of x, /Xq has a constant value equal to 0.578. To clarify the trajectory equation of inclined jets for the cases of air supply through different types of nozzles and grills, a series of experiments were conducted. The trajectory coordinates were defined as the points where the mean values of the temperatures and velocities reached their maximum in the vertical cross-sections of the jet. It is important to mention that, in such experiments, one meets with a number of problems, such as deformation of temperature and velocity profiles and fluctuation of the air jet trajectory, which reduce the accuracy in the results. The mean value of the coefficient E obtained from experimental data (Fig. 7.25) is 0.47 0.06. Thus the trajectory of the nonisothermal jet supplied through different types of outlets can be calculated from... 

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

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