Lagrangian fluctuating velocity

The crosscorrelation of Lagrangian fluctuating velocities, needed in equation (5b), is obtained from CARPT experiments by... 

Computations of the Lagrangian integral time scale for different experiments yielded values in the range of 0.4 to 0.8 seconds for different compartments. These time scales are not strongly dependent on gas superficial velocity or column diameter. This implies that the Lagrangian fluctuating velocity is statistically independent of itself beyond about 0.6 seconds i in this interval the particle moves from one eddy (correlated region of flow) to another. 

The chaotic motion of the solids in gas fluidized beds necessitates the measurement of the fluctuating and mean velocities of the solids for thorough understanding of their dynamic behavior. The statistical information of the fluctuating velocity may be obtained from the Lagrangian autocorrelations. 

For an arbitrary value of the Lagrangian timescale, Tj =0.001 s and p = 1240kg/m Harriott estimated that the slip velocity for 50 and 300 pm particles should be approximately between 2 and 14% of the fluctuating velocity, respectively. 

Basset term in Tchen s original equation. Lee (1981) derived the values of the rms particle fluctuating velocity, Mp, and fluid-particle relative turbulent velocity, Mr, based on the assumption that the particle experiences Lagrangian fluid energy spectrum for particle size up to 1000 pm. The experimental data of Snyder and Lumley... 

We shall see that a conditional acceleration model in the form of (6.48) is equivalent to a stochastic Lagrangian model for the velocity fluctuations whose characteristic correlation time is proportional to e/k. As discussed below, this implies that the scalar flux (u,

joint velocity, composition PDF level, and thus that a consistent scalar-flux transport equation can be derived from the PDF transport equation. 

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 Lagrangian velocity autocorrelations can be used to evaluate several important quantities that characterize the fluctuating motion of solids,... 

Because Newton s equations of motion are conservative, the natural ensemble is NVE (micro-canonical), Aat is one in which the internal energy rather than the temperature is held constant. This is inconvenient if one wishes to compare with experiment where it is the temperature that is generally controlled. In order to perform molecular dynamics in the canonical ensemble, a thermostat must be applied to the system. This is accomplished by constructing a pseudo-Lagrangian. Many forms for temperature-conserving Lagrangians have been proposed, most of which can be written in a form that adds a frictional (velocity-dependent) term to the equations of motion (Allen and Tildesley 1989). Physically, the thermostat can be thought of as a heat bath to which the system is coupled. In the NPT ensemble, in which the pressure is held constant, the cell size and shape fluctuates. The choice of dynamical variables is critical. If the lattice parameters are chosen as in the method of Parrinello and Rahman (1981), the time evolution may depend on the chosen size or shape of the supercell. This difficulty is... 

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. 

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