Random velocity

By Eq. (1-55), we have px — m Jujaf dv. Since mut is the random momentum in the -direction (. e., the momentum associated with the -component of the random velocity), the (i,j) component of the pressure tensor is the average of the random flow in the -direction of the -directed momentum. From the definition of the temperature, Eq. (1-45), the hydrostatic pressure, defined as one-third of the trace of the pressure tensor, is... 

This is Hie random flow of the energy associated with the random velocity of the particles. The third term has parts of the form ... 

The term mass transfer is used to denote the transference of a component in a mixture from a region where its concentration is high to a region where the concentration is lower. Mass transfer process can take place in a gas or vapour or in a liquid, and it can result from the random velocities of the molecules (molecular diffusion) or from the circulating or eddy currents present in a turbulent fluid (eddy diffusion). 

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... 

Brunk, B. K., Koch, D. L., and Lion, L. W., Hydrodynamic pair diffusion in isotropic random velocity fields with application to turbulent coagulation. Phys. Fluids 9,2670-2691 (1997). 

Tang, D.H., F.W. Schwartz and L. Smith (1982). Stochastic modeling of mass transport in a random velocity field. Water Resources Research 18(2), pp. 231-244. 

Unlike simple random variables that have no space or time dependence, the statistics of the random velocity field in homogeneous turbulence can be described at many different levels of complexity. For example, a probabilistic theory could be formulated in terms of the set of functions U(x, t) (x, t) e R3 x R However, from a CFD modeling perspective, such a theory would be of little practical use. Thus, we will consider only one-point and two-point formulations that describe a homogeneous turbulent flow by the velocity statistics at one or two fixed points in space and/or time. 

For a fixed point in space x and a given instant t, the random velocity field Ui(x, t) can be characterized by a one-point probability density function (PDF) fufiVi x, t) defined by4... 

Despite its widespread use in the statistical description of turbulent reacting flows, the one-point joint velocity PDF does not describe the random velocity field in sufficient detail to understand the physics completely. For example, the one-point description tells us nothing about the statistics of velocity gradients, e.g.,... 

Because the random velocity field U(x, t) appears in (1.28), p. 16, a passive scalar field in a turbulent flow will be a random field that depends strongly on the velocity field (Warhaft 2000). Thus, turbulent scalar mixing can be described by a one-point joint velocity, composition PDF /u,< (V, i/r,x, t) defined by... 

Equation (2.2) can be considered as the fundamental governing equation for the concentration of an inert constituent in a turbulent flow. Because the flow in the atmosphere is turbulent, the velocity vector u is a random function of location and time. Consequently, the concentration c is also a random fimction of location and time. Thus, the dispersion of a pollutant (or tracer) in the atmosphere essentiaUy involves the propagation of the species molecules through a random medium. Even if the strength and spatial distribution of the source 5 are assumed to be known precisely, the concentration of tracer resulting from that source is a random quantity. The instantaneous, random concentration, c(x, y, z, t), of an inert tracer in a turbulent fluid with random velocity field u( c, y, z, t) resulting from a source distribution S x, y, z, t) is described by Eq. (2.2). 

Figure 10 show the effect of the number of particles used in the Monte Carlo calculation The calculations were carried out with 250, 1000, and 2000 particles using randomized velocities, case c of Fig. 12. Although as the number of particles is increased from 250 to 1000, a noticeable reduction in fluctuations occurs, from 1000 to 2000 particles the change is not as pronounced.

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). 

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 ... 

The random velocities of atoms and molecules are described by velocity distribution functions which can often be approximated by a Maxwellian distribution (as in Eq. 2.10). If radiating atoms have such a distribution, the resulting line profile is a Gaussian,... 

To observe a 7s — 9 transition requires that there be a 9p admixture in the 9 state. For odd this admixture is provided by the diamagnetic interaction alone, which couples states of and 2, as described in Chapter 9. For even states the diamagnetic coupling spreads the 9p state to all the odd 9( states and the motional Stark effect mixes states of even and odd (. Due to the random velocities of the He atoms, the motional Stark effect and the Doppler effect also broaden the transitions. Together these two effects produce asymmetric lines for the transitions to the odd 9t states, and double peaked lines for the transitions to even 9( states. The difference between the lineshapes of transitions to the even and odd 9i states comes from the fact that the motional Stark shift enters the transitions to the odd 9( states once, in the frequency shift. However, it enters the transitions to the even 9( states twice, once in the frequency shift and once in the transition matrix element. Although peculiar, the line shapes of the observed transitions can be analyzed well enough to determine the energies of the 9( states of >2 quite accurately.25... 

The dark halo hypothesis is based on the assumption that companions are physical if they are not then they do not measure the mass of the main galaxy, but characterise mean random velocities of galaxies ... 

CvetkovicVD,Dagan G (1994) Transport ofkinetically sorbing solute by steady random velocity in heterogeneous porous formation. J Fluid Mech 265 189-215... 

V in the Maxwellian and in the arguments of the expansion polynomials, due to the fact that V C t = j2Tfm in many fusion plasma applications usually. Hence fa can equally well be expanded in polynomials in either v or in the random velocity v. ... 

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