Velocity component, random

Random motion is ubiquitous. At the molecular level, the thermal motions of atoms and molecules are random. Further, motions in macroscopic systems are often described by random processes. For example, the motion of stirred coffee is a turbulent flow that can be characterized by random velocity components. Randomness means that the movement of an individual portion of the medium (i.e., a molecule, a water parcel, etc.) cannot be described deterministically. However, if we analyze the average effect of many individual random motions, we often end up with a simple macroscopic law that depicts the mean motion of the random system (see Box 18.1). 

The average random force over the time step is taken from a Gaussian with a varianc 2mk T y(St). Xj is one of the 3N coordinates at time step i E and R are the relevan components of the frictional and random forces at that time n, is the velocity component. 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

Wind speed has velocity components in all directions so that there are vertical motions as well as horizontal ones. These random motions of widely different scales and periods are essentially responsible for the movement and diffusion of pollutants about the mean downwind path. These motions can be considered atmospheric turbulence. If the scale of a turbulent motion (i.e., the size of an eddy) is larger than the size of the pollutant plume in its vicinity, the eddy will move that portion of the plume. If an eddy is smaller than the plume, its effect will be to difhise or spread out the plume. This diffusion caused by eddy motion is widely variable in the atmosphere, blit even when the effect of this diffusion is least, it is in the vicinity of three orders of magnitude greater than diffusion by molecular action alone. 

In turbulent flow there is a complex interconnected series of circulating or eddy currents in the fluid, generally increasing in scale and intensity with increase of distance from any boundary surface. If, for steady-state turbulent flow, the velocity is measured at any fixed point in the fluid, both its magnitude and direction will be found to vary in a random manner with time. This is because a random velocity component, attributable to the circulation of the fluid in the eddies, is superimposed on the steady state mean velocity. No net motion arises from the eddies and therefore their time average in any direction must be zero. The instantaneous magnitude and direction of velocity at any point is therefore the vector sum of the steady and fluctuating components. 

If the magnitude of the fluctuating velocity component is the same in each of the three principal directions, the flow is termed isotropic. If they are different the flow is said to be anisotropic. Thus, if the root mean square values of the random velocity components... 

Fig. 2.9.10 Maps of the temperature and of the experimental data. The right-hand column convection flow velocity in a convection cell in refers to numerical simulations and is marked Rayleigh-Benard configuration (compare with with an index 2. The plots in the first row, (al) Figure 2.9.9). The medium consisted of a and (a2), are temperature maps. All other random-site percolation object of porosity maps refer to flow velocities induced by p = 0.7 filled with ethylene glycol (temperature thermal convection velocity components vx maps) or silicon oil (velocity maps). The left- (bl) and (b2) and vy (cl) and (c2), and the hand column marked with an index 1 represents velocity magnitude (dl) and (d2).

As we saw in Chapter 1, the one-point joint velocity, composition PDF contains random variables representing the three velocity components and all chemical species at a particular spatial location. The restriction to a one-point description implies the following. 

Handley fluidised soda glass particles using methyl benzoate, and obtained data on the flow pattern of the solids and the distribution of vertical velocity components of the particles. It was found that a bulk circulation of solids was superimposed on their random movement. Particles normally tended to move upwards in the centre of the bed and downwards at the walls, following a circulation pattern which was less marked in regions remote from the distributor. 

The key problem in using Eq. (3.1) is the specification of p. We ask whether we can derive an expression for p. The velocity components u, v, and w, although random, are related through conservation of mass and momentum for the flow, that is, they are governed by the stochastic Navier-Stokes and continuity equations. In general, as we have noted, an exact solution for u, v, and w is unobtainable. We can, however, consider an idealized situation in which the statistical properties of u, v, and w are specified a priori. Then, in so doing, we wish to see if we can obatin an exact solution of Eq. (2.4) from which p can be obtained through Eq. (2.6). 

The mean velocity components are expressed as //, v, and ii>. We assume that the velocity components are stationary Gaussian random processes, so that, based on the preceding discussion, the autocovariances of u, V, and w can be written as (Papoulis, 1965, p. 397)... 

Turbulent flow means that, superimposed on the large-scale flow field (e.g., the Gulf Stream), we find random velocity components along the flow (longitudinal turbulence) as well as perpendicular to the flow (transversal turbulence). The effect of the turbulent velocity component on the transport of a dissolved substance can be described by an expression which has the same form as Fick s first law (Eq. 18-6), where the molecular diffusion coefficient is replaced by the so-called turbulent or eddy diffusion coefficient, E. For instance, for transport along the x-axis ... 

For the motion of a gas-solid suspension in the riser, both the gas and particle velocities have local averaged and random components. Thus, it is desirable to develop a mechanistic model which incorporates a variety of interactive effects due to both the gas and particle velocity components (see Chapter 5) as given in the following [Sinclair and Jackson, 1989] ... 

As a result of random collisions of gaseous molecules, the molecular velocities keep on changing. Consider a gas molecule of mass m having a velocity component u. Then, the kinetic energy, e, associated with this velocity component is 1/2 mu2. The probability that this molecule has its velocity component between u and u + du is given by p(u) du. In the 19th century, Boltzmann had shown that the probability for a molecule to have an energy e was proportional to er /KT. It is apparent that we can equate this probability with p(u) du. Thus,... 

The previous section has shown that turbulence combined with a different domain decomposition (i.e. a different number of processors for the following) is sufEcient to lead to totally different instantaneous flow realizations. It is expected that a perturbation in initial conditions will have the same effect as domain decomposition. This is verified in runs TC3 and TC4 which are run on one processor only, thereby eliminating issues linked to parallel implementation. The only difference between TC3 and TC4 is that in TC4, the initial solution is identical to TC3 except at one random point where a 10 perturbation is applied to the streamwise velocity component. Simulations with different locations of the perturbation were run to ensure that their position did not affect results. 

Let us now select a three-dimensional system of cartesian axes fixed with respect to the vessel. Our assumption of complete randomness of molecular motion tells us that we shall expect to find as many molecules moving with velocity components of a given range along the x axis as along the y and z axes. That is, the motion is isotropic. If we define three distribution functions P(y ), P vy), and P(vz) such that P Vx) dvx represents the fraction of all the N molecules which have x velocity components between Vx and Vx + dVx and the other two functions have similar relations with Vy and Va, the assumption of randomness tells us that the three functions are the same for the different components. The assumption of independent motion further tells us that the fraction of all molecules with... 

Any particular ion starts off after a collision with a velocity that may be in any direction this is the randomness in its walk. The initial velocity can be ignored precisely because it can take place in any direction and therefore does not contribute to the drift (preferred motion) of the ion. But the ion is all the time under the influence of the applied-force field. This force imparts a component to the velocity of the ion, an extra velocity component in the same dkection as the force vector F. It is this additional velocity component due to the force F that is called the drift velocity Vj. What is its average value ... 

Having defined the atomic coordinates, initial velocities must next be assigned. The atomic velocity components may be chosen randomly from either a Gaussian distribution at the desired temperature or from a uniform distribution in the interval fmax)> where can be chosen to be equal to the... 

Equations (16.14), (16.18), and (16.22) govern the fluid velocity and temperature in the lower atmosphere. Although these equations are at all times valid, their solution is impeded by the fact that atmospheric flow is turbulent (as opposed to laminar). It is difficult to define turbulence instead we can cite a number of the characteristics of turbulent flows. Turbulent flows are irregular and random, so that the velocity components at any location vary randomly with time. Since the velocities are random variables, their exact values can never be predicted precisely. Thus (16.14), (16.18), and (16.22) become partial differential equations whose dependent variables are random functions. We cannot therefore expect to solve any of these equations exactly rather, we must be content to determine some convenient statistical properties of the velocities and temperature. The random fluctuations in the velocities result in rates of momentum, heat, and mass transfer in turbulence that are many orders of magnitude greater than the corresponding rates due to pure molecular transport. 

The type of flow encountered when a highly underexpanded nozzle exhausts into a vacuum differs significantly from the random free-molecule flow characterized by the Max-well-Boltzmann distribution function, which is normally encountered in vacuum practice. The difference is illustrated in Fig. 1 and is primarily due to the large directional velocity component given to the gas molecules while still in a dense continuum flow condition in the nozzle. The velocity distribution function for the exhaust gas thus differs from that of the random flow in that it contains both the random thermal velocity components and the large directional velocity component. 

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