Eulerian fluctuating velocities

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

By extrapolation of the computed autocorrelation coefficient to zero time the square of the llagrangian fluctuating velocity can be obtained. The square root of this quantity can then be compared to the Eulerian rms velocity for the same column compartment. The two velocities should be equal in case of homogeneous turbulence. Devanathan (1991) shows that this is not the case in bubble columns but that homogeneous turbulence is approached at the highest gas velocities in the largest diameter column. 

In actual applications, the gas flow in a gravity settler is often nonuniform and turbulent the particles are polydispersed and the flow is beyond the Stokes regime. In this case, the particle settling behavior and hence the collection efficiency can be described by using the basic equations introduced in Chapter 5, which need to be solved numerically. One common approach is to use the Eulerian method to represent the gas flow and the Lagrangian method to characterize the particle trajectories. The random variations in the gas velocity due to turbulent fluctuations and the initial entering locations and sizes of the particles can be accounted for by using the Monte Carlo simulation. Examples of this approach were provided by Theodore and Buonicore (1976). 

The actual velocity field fluctuates wildly. Reynolds modeled it by a superposition of a Eulerian time mean value v defined by... 

In summary, the Eulerian two-fluid model is represented by Eqs. (5.112) and (5.113) in addition to a constitutive model for the fluid stress tensor Tf. As already mentioned, Eq. (5.112) was derived under the assumption that the particle-velocity distribution is very narrow (i.e. small particle Stokes number), and the particles must have the same internal coordinates. If these simplifications do not hold, for example under dense conditions when particle-particle collisions become important, then particle-velocity fluctuations must be taken into account, as discussed at the end of Chapter 4. 

Anonymous (1987). Corrsin, Stanley. Who s who in America 44 584. Marquis Chicago. Comte-Bellot, G., Corrsin, S. (1971). Simple Eulerian time correlation of fuU and narrow band velocity signals in grid-generated isotropic turbulence. J. FluidMech. 48(2) 271-337. Corrsin, S. (1951). The decay of isotropic temperature fluctuations in an isotropic turbulence. 

These equations complete the Lagrangian flamelet model. A transformation of coordinates different from that presented in Eqs. (5.75)-(5.77) results in the Eulerian flamelet model proposed by Pitsch [18]. In the Eulerian system, both velocity vector and scalar dissipation rate are functions of time, space, and the mixture fraction. The difference between these models appears to be the manner in which the fluctuations are taken into account. Because the differences are small, the Lagrangian flamelet model is more employed, because it is easier to implement and represents well the majority applications for diffusion flames. 

D Eulerian-Eulerian simulation of fluid catalytic cracking of gas oil. Snapshot of fluctuations around the statistically stationary flow field, (a) Solids volume fraction profile and (b) solids velocity field. 

D Eulerian-Eulerian simulation of fluid catalytic cracking of gas oil. Snapshot of fluctuations around the statistically stationary species field. Mass fraction in the gas phase (a) gasoil fraction of the feed, (b) gasoline fraction in the product, (c) Transient calculation of the meso-scale fluctuations around a statistically stationary state solids volume fraction and axial velocity at 0.1 m from the wall and 0.25 m from the bottom. 

In this method correlations between various velocity fluctuations are used to determine the turbulent energy spectrum, (fc), and several turbulence length scales. The correlations mentioned contain information about how velocities and other flow properties are statistically related in the turbulent flow. Turbulence measured at a fixed point can be described as a fluctuating waveform. If two instantaneous waveforms appear to have a corresponding behavior, they are said to be correlated. Equation (1.317) shows how velocity fluctuations at two points can be statistically correlated if the distance between the two points are small. The velocity fluctuations v-(f, r) and Vj t, r -I- x) are said to be correlated if Ryit, x) has a non-zero value, and uncorrelated if the Eulerian correlation tensor Ry t, x) is zero. The Ry (t, x) correlation can be normalized by introducing the rms velocity [8]. The Eulerian correlation function is defined by ... 

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