Fluctuation velocity, definition

Note that (7.61) is second-order accurate in time. Also, by definition, the estimated mean fluctuating velocity should be null u X = 0. This condition will not be automatically satisfied due to numerical errors. Muradoglu et al. (2001) propose a simple correction algorithm that consists of subtracting the interpolated value of u X (e.g., the LOME found as in (7.45)) from uin) t + At). 

Smaller particles primarily see only the fluctuating velocity component. When the particle size is much less than 100 /zm, the turbulent properties of the fluid become important. This is the definition of the boundary size for microscale mixing. 

Only flows steady with respect to the time-mean properties will be considered here. The time-mean value of the fluctuation velocity is, by definition, equal to zero, w = 0. Correspondingly, pressure, temperature and concentration can also be split into mean and fluctuating values... 

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. 

Besides this mean value (here w) the mean values of the fluctuating velocities are important. The mean values of these fluctuating velocities are zero according to their definitions... 

With the assumption of isotropic turbulence (approximately valid in stirred vessels) the mean values of the fluctuating velocities are not a function of the direction. The definition of the effective value is... 

In Section 2-2, we used Figure 2-7 to illustrate the instantaneous velocity versus time signal and the mean velocity. A third velocity used widely for turbulent flows is the root-mean-square (RMS) velocity, or the standard deviation of the instantaneous velocity signal. Because the average fluctuation is zero by definition, the RMS velocity gives us an important measure of the amount or intensity of turbulence, but many different signals can return the same mean velocity and RMS fluctuating velocity, so more information is needed to characterize the turbulence. 

Upon evaluating this at L/2, using the definition of slip velocity (4.27), and Equation 4.29, the relationship between the boundary velocity, fluctuation velocity, and stress ratio may be written as... 

A very fine space resolution is required to measure the gradient of turbulent velocity fluctuations and calculate turbulent dissipation directly from the definition [5, 6]. 

Figure 1.25 shows the boundary layer that develops over a flat plate placed in, and aligned parallel to, the fluid having a uniform velocity upstream of the plate. Flow over the wall of a pipe or tube is similar but eventually the boundary layer reaches the centre-line. Although most of the change in the velocity component vx parallel to the wall takes place over a short distance from the wall, it does continue to rise and tends gradually to the value vx in the fluid distant from the wall (the free stream). Consequently, if a boundary layer thickness is to be defined it has to be done in some arbitrary but useful way. The normal definition of the boundary layer thickness is that it is the distance from the solid boundary to the location where vx has risen to 99 per cent of the free stream velocity v . The locus of such points is shown in Figure 1.25. It should be appreciated that this is a time averaged distance the thickness of the boundary layer fluctuates owing to the velocity fluctuations.

If a flow in the tank is turbulent, either because of high power levels or low viscosity, then a typical velocity pattern at a point would be illustrated by Fig. 3. The velocity fluctuation i can be changed into a root mean square value (RMS), which has great utility in estimating the intensity of turbulence at a point. So in addition to the definitions above, based on average velocity point, we also have the same quantities based on the root mean square fluctuations at a point. We re interested in this value at various rates of power dissipation, since energy dissipation is one of the major contributors to a particular value of RMS v. ... 

To understand atmospheric dispersion, therefore, it is necessary to understand the nature of turbulent flows and the structure of the atmosphere near the Earth s surface. Turbulent flows exhibit apparently random fluctuations in local velocity and pressure (Mathieu and Scott 2000). These fluctuations appear over a wide range of length and time scales. Reynolds (1895) drew an analogy between the behavior of the velocity in a turbulent flow and the velocity of the individual molecules in a gas, leading to the definition of the instantaneous velocity at a point as having a mean and a fluctuating component ... 

The velocity field is statistically homogeneous if all statistics are invariants under a shift in position. If the field is also statistically invariant under rotations and reflections of the coordinate system, then it is statistically isotropic. In chemical reaction engineering these mathematical definitions are usually somewhat relaxed, since turbulence is said to be isotropic if the individual velocity fluctuations are equal in all the three space dimensions. Otherwise it is said to be an-isotropic. Similarly, a flow field where turbulence levels do not change from one point to another is called homogeneous. 

The velocity fluctuation represents the flow that varies with periods shorter than the averaging time period. Recall that turbulence is a 3D phenomena. Therefore, we expect that fluctuations in the x-direction might be accompanied by fluctuations in the y- and z- directions. Turbuience, by definition, is a type of motion. Yet motions frequently cause variations in the temperature and concentration fields as well, if there is some mean gradient of that variable across the turbulent domain. Hence, it is common practice to portion each of these variables into mean and turbulent parts in the same manner as for the velocity. 

The positive-definite matrix Bfp will be determined by the structure of the fluid flow around the particle, and will more than likely be significantly anisotropic (Tenneti et al, 2012). Note that we have not included the fluid-velocity-fluctuation-dissipation model in Eq. (4.104) when writing the GPBE. Here, for clarity, we will focus exclusively on the interdependence of the fluid and particle velocities, for which it suffices to consider a ID velocity phase space for Vp and Vf. The GPBE for this case is given by... 

In flow of fluid through a dosed channel turbulence cannot exist permanently at the boundary between the solid and the flowing fluid. The velocity at the interface is zero because of the adherence of the fluid to the solid, and (except very-infrequently) velocity components normal to the wall do not exist. Within a thin volume immediately adjacent to the wall, the velocity gradient is essentially constant and the flow is viscous most of the time. This volume is called the viscous sublayer. Formerly it was assumed that this sublayer had a definite thickness and was always free from eddies, but measurements have shown velocity fluctuations in the sublayer caused by occasional eddies from the turbulent fluid moving into this region. Very close to the wall, eddies are infrequent, but there is no region that is completely free of eddies. Within the viscous sublayer only viscous shear is important, and eddy diffusion, if present at all, is minor. 

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