Velocity uncorrelated

When a molecule gets into a fast flow channel, its higher-than-average velocity will persist for some average distance S along the flow axis. After it has gone distance 5, the molecule finds itself in a new channel and it therefore assumes a new velocity uncorrelated with its initial velocity. This change in velocity represents a random step. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

A number of simulation methods based on Equation (7.115) have been described. Thess differ in the assumptions that are made about the nature of frictional and random forces A common simplifying assumption is that the collision frequency 7 is independent o time and position. The random force R(f) is often assumed to be uncorrelated with th particle velocities, positions and the forces acting on them, and to obey a Gaussiar distribution with zero mean. The force F, is assumed to be constant over the time step o the integration. 

It has to be noted that the measurement values for range and velocity are not uncorrelated according to the LFMCW measurement described in section 8. As a consequence, the observed measurement errors ft, can also be considered as correlated random variables for a single sensor s data. For 24GHz pulse radar networks, developed also for automotive applications, a similar idea has been described by a range-to-track association scheme [12], because no velocity measurements are provided in such a radar network. 

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. 

From equation 1.41, the total shear stress varies linearly from a maximum fw at the wall to zero at the centre of the pipe. As the wall is approached, the turbulent component of the shear stress tends to zero, that is the whole of the shear stress is due to the viscous component at the wall. The turbulent contribution increases rapidly with distance from the wall and is the dominant component at all locations except in the wall region. Both components of the mean shear stress necessarily decline to zero at the centre-line. (The mean velocity gradient is zero at the centre so the mean viscous shear stress must be zero, but in addition the velocity fluctuations are uncorrelated so the turbulent component must be zero.)... 

At high Reynolds numbers, the energy-containing scales of one velocity component and the dissipative scales of another velocity component will be uncorrelated. We can then write... 

In locally isotropic turbulence, the fluctuating velocity gradient and scalar gradient will be uncorrelated, and sf will be null. Thus, at sufficiently high Reynolds number, the scalar-flux dissipation is negligible. 

At high Reynolds numbers, the scalar gradient and the velocity gradient will be uncorrelated due to local isotropy. Thus, Gf will be negligible. Likewise, the mean-scalar-curvature term Cf, defined by... 

III) The molecular mixing model must be uncorrelated with the velocity at high... 

III) Velocity and scalar gradients must be uncorrelated (III ) Local scalar isotropy must be correct... 

No fully satisfactory method is available for correlating the drag on irregular particles. Settling behavior has been correlated with most of the more widely used shape factors. Settling velocity may be entirely uncorrelated with the visual sphericity obtained from the particle outline alone (B8). General correlations for nonspherical particles are discussed in Chapter 6. 

We can determine the importance of dynamical pair correlations by comparing G(t) with Gs(t). Because velocities of different molecules and for different degrees of freedom are uncorrelated at t = 0,... 

The oldest and best known example of a Markov process in physics is the Brownian motion.510 A heavy particle is immersed in a fluid of light molecules, which collide with it in a random fashion. As a consequence the velocity of the heavy particle varies by a large number of small, and supposedly uncorrelated jumps. To facilitate the discussion we treat the motion as if it were one-dimensional. When the velocity has a certain value V, there will be on the average more collisions in front than from behind. Hence the probability for a certain change AV of the velocity in the next At depends on V, but not on earlier values of the velocity. Thus the velocity of the heavy particle is a Markov process. When the whole system is in equilibrium the process is stationary and its autocorrelation time is the time in which an initial velocity is damped out. This process is studied in detail in VIII.4. 

As was mentioned in Section II.A.6, in the Gross collision model the angular velocities, unlike the orientations, are considered to be uncorrelated at an instant of a strong collision. 

We start to follow the relative motion between two particles of mass mp separated by a specified initial distance. This initial distance is chosen to be large compared to the correlation length Xr (the average distance over which the relative velocity of the particle pair is correlated). Therefore, the initial relative motion of the particle pair, which is considered as the initial condition in the calculations, is uncorrelated and its initial relative velocity distribution is Maxwellian. The correlation length Xr is given by (1)... 

The Levy walk is physically more plausible than the Levy flight. How to derive the Levy walk from a Liouville approach of the kind described in Section III Here, we illustrate a path explored some years ago, to establish a connection between GME and this kind of superdiffusion [49,50]. We assume that there exists a waiting time distribution v /(x), prescribed, for instance, by the dynamic model illustrated in Section V. This function corresponds to a distribution of uncorrelated times. We can imagine the ideal experiment of creating the sequence x,, by drawing in succession the numbers of this distribution. Then we create the fluctuating velocity E,(f), according to the procedure illustrated in Section V. 

Eulerian equations for the dispersed phase may be derived by several means. A popular and simple way consists in volume filtering of the separate, local, instantaneous phase equations accounting for the inter-facial jump conditions [274]. Such an averaging approach may be restrictive, because particle sizes and particle distances have to be smaller than the smallest length scale of the turbulence. Besides, it does not account for the Random Uncorrelated Motion (RUM), which measures the deviation of particle velocities compared to the local mean velocity of the dispersed phase [280] (see section 10.1). In the present study, a statistical approach analogous to kinetic theory [265] is used to construct a probability density function (pdf) fp cp,Cp, which gives the local instantaneous probable num-... 

The averaging operation for the liquid droplet velocity described in the previous section introduces a particle velocity deviation from the mean (or correlated) velocity, noted as m" = Up — ui, and named the random uncorrelated velocity [280]. By definition, the statistical average (based on the particle probability density function) of this uncorrelated velocity is zero < u" >= 0. A conservation equation can be written for the associated kinetic energy 59i =< Up pip > /2 ... 

P. Fevrier, O. Simonin, and K. Squires. Partitioning of particle velocities in gas-solid turbulent flows into a continuous field and a spatially uncorrelated random distribution Theoretical formalism and numerical study. J. Fluid Mech., 533 1-46, 2005. 

That Eqs. (32) and (34) differ in the dependence of Z i2 on the molecular weights should not be altogether surprising, since in our simple treatment we have assumed that the velocities of molecules after collisions are uncorrelated with their velocities before collisions, whereas in fact such correlations in general exist and are functions of the ratio of the masses of the colliding particles. To take this and other remaining factors properly into account would require a treatment that is beyond the scope of this book. It may suffice to point out here that the square-root quantity in Eq. (34) is equivalent to y/lfpizNo, where... 

There is a quantity referred to in transport calculations called the velocity autocorrelation function (see Section 4.2.19). When applied to the velocity of particles in liquids, it refers to the time needed for a particle to be free of the influence of the previous movement of particles (i.e., uncorrelated). For KCl at 1045 K, the value calculated by Smedley and Woodcock by means of a simulation gave 3 x 10 s for the autocorrelation function—about one-tenth of the time for aj ump calculated by a hole model (see Table 5.33) for NaNO,. 

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