Fluctuating element, velocity

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. 

The use of bold denotes that all quantities are vectors. The velocity V on the left-hand side is the instantaneous velocity, which is comprised of the sum of the mean velocity at the point (denoted by the overbar) and the fluctuating element of the velocity (denoted by the lowercase m). To permit the use of the Einstein summation convention, the component of the instantaneous velocity in direction i at a point is written as the sum of the mean and fluctuating components of the velocity in that direction. 

It is the fluctuating element of the velocity in a turbulent flow that drives the dispersion process. The foundation for determining the rate of dispersion was set out in papers by G. 1. Taylor, who first noted the ability of eddy motion in the atmosphere to diffuse matter in a manner analogous to molecular diffusion (though over much larger length scales) (Taylor 1915), and later identified the existence of a direct relation between the standard deviation in the displacement of a parcel of fluid (and thus any affinely transported particles) and the standard deviation of the velocity (which represents the root-mean-square value of the velocity fluctuations) (Taylor 1923). Roberts (1924) used the molecular diffusion analogy to derive concentration profiles... 

It is apparent from equations 3.2.4-3.2.7 that the determination of the concentration field is dependent on the values of the Gaussian dispersion parameters a, (or Oy in the fully coupled puff model). Drawing on the fundamental result provided by Taylor (1923), it would be expected that these parameters would relate directly to the statistics of the components of the fluctuating element of the flow velocity. In a neutral atmosphere, the factors affecting these components can be explored by considering the fundamental equations of fluid motion in an incompressible fluid (for airflows less than 70% of the speed of sound, airflows can reasonably be modeled as incompressible) when the temperature of the atmosphere varies with elevation, the fluid must be modeled as compressible (in other words, the density is treated as a variable). The set of equations governing the flow of an incompressible Newtonian fluid at any point at any instant is as follows ... 

In a turbulent flow, the pressure and velocity vary randomly in time and space. It can therefore be more meaningful to average the equations over time or space for example, if the equations hold at every instant, they must also hold on average over some finite time period. It may be recalled that the instantaneous velocity and pressure may be written as the sum of their mean and fluctuating elements as follows ... 

So far, some researchers have analyzed particle fluidization behaviors in a RFB, however, they have not well studied yet, since particle fluidization behaviors are very complicated. In this study, fundamental particle fluidization behaviors of Geldart s group B particle in a RFB were numerically analyzed by using a Discrete Element Method (DEM)- Computational Fluid Dynamics (CFD) coupling model [3]. First of all, visualization of particle fluidization behaviors in a RFB was conducted. Relationship between bed pressure drop and gas velocity was also investigated by the numerical simulation. In addition, fluctuations of bed pressure drop and particle mixing behaviors of radial direction were numerically analyzed. 

Velocity fluctuations can also cause extra apparent shear stress components. An element of fluid with a non-zero velocity component in the x-direction possesses an x-component of momentum. If this element of fluid also has a non-zero velocity component in the y-direction then as it moves in the y-direction it carries with it the x-component of momentum. The mass flow rate across a plane of area 8x8z normal to the y-coordinate direction is pvy8x8z and the x-component of momentum per unit mass is vx, so the rate of transfer of x-momentum in they-direction is given by the expression... 

It follows that if an element of fluid moves in they-direction in a region where the mean velocity gradient dvjdy is zero, a fluctuation v y gives rise, on average, to a zero fluctuation v x. The time-average product of the fluctuations (the Reynolds stress) is zero and the fluctuations are said to be uncorrelated. 

Turbulent mixing (i.e., the scalar flux) transports fluid elements in real space, but leaves the scalars unchanged in composition space. This implies that in the absence of molecular diffusion and chemistry the one-point composition PDF in homogeneous turbulence will remain unchanged for all time. Contrast this to the velocity field which quickly approaches a multi-variate Gaussian PDF due, mainly, to the fluctuating pressure term in (6.47). 

Since measurements of u v ), u w ), etc., usually indicate that the magnitudes of these covariances are much smaller than the mean square velocity fluctuations (m ), etc., it is generally assumed that the off-diagonal elements of P are negligible. Also, we use the notation, al = ((,1), cr, =... 

As the reactant elements pass through the shear layer, they pick up and convect turbulent fluctuations to the flame. There is no constraint that the flame speed and radial velocity be uniform along the axis however, since the flame... 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

To simplify the derivation, we may consider only the one-dimensional case. Denote s(to +1 ) as the distance traveled by a marked fluid element with starting time t0 during the time interval t. Let u be the instantaneous velocity and u be the fluctuating velocity. In the following derivation, the time-averaged fluid velocity is taken as zero. Thus, we have... 

Reynolds stress Nondiagonal element of the correlation dyad for fluctuation velocity components in a turbulent flow commonly interpreted as a shear component of the extra stress caused by the turbulence. 

This is the continuity equation for turbulent flow when the mean motion is two-dimensional. It will be noted that this equation has exactly the same form as the continuity equation for two-dimensional steady laminar flow with the mean values of the velocity components substituted in place of the steady values that apply in laminar flow. This result can, in fact, be deduced by intuitive reasoning and simply states that if an elemental control volume through which the fluid flows is considered, then over a sufficiently long period of time, the fluctuating components contribute nothing to the mass transfer through this control volume. 

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