Velocity field random

Radiation field, interaction with nega-ton-positon field, 642 Radtke, M. ( ., 408 Raiffa, Howard, 314 Random coding, 227 Random particle velocity, 19 Random processes, 99,102 Gaussian, 176 harmonic analysis of, 180... 

At high Reynolds number, the velocity U(x, t) is a random field, i.e., for fixed time t = t the function U(x, D varies randomly with respect to x. This behavior is illustrated in Fig. 2.1 for a homogeneous turbulent flow. Likewise, for fixed x = x lJ(x. t) is a random process with respect to t. This behavior is illustrated in Fig. 2.2. The meaning of random in the context of turbulent flows is simply that a variable may have a different value each time an experiment is repeated under the same set of flow conditions (Pope 2000). It does not imply, for example, that the velocity field evolves erratically in time and space in an unpredictable fashion. Indeed, due to the fact that it must satisfy the Navier-Stokes equation, (1.27), U(x, t) is differentiable in both time and space and thus is relatively smooth. ... 

Figure 2.1. Three components of the random field U(x, t ) as a function of x = xi with fixed t =t. The velocity was extracted from DNS of isotropic turbulence (Rk = 140) with (U> = 0. (Courtesy of P. K. Yeung.)...

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... 

Note that hv operates on the random field U(r, f) and (for fixed parameters V, x, and t) produces a real number. Thus, unlike the LES velocity PDF described above, the FDF is in fact a random variable (i.e., its value is different for each realization of the random field) defined on the ensemble of all realizations of the turbulent flow. In contrast, the LES velocity PDF is a true conditional PDF defined on the sub-ensemble of all realizations of the turbulent flow that have the same filtered velocity field. Hence, the filtering function enters into the definition of /u u(V U ) only through the specification of the members of the sub-ensemble. 

The joint velocity, composition PDF is defined in terms of the probability of observing the event where the velocity and composition random fields at point x and time t fall in the differential neighborhood of the fixed values V and ip ... 

We start by considering an arbitrary measurable10 one-point11 scalar function of the random fields U and 0 Q U, 0). Note that, based on this definition, Q is also a random field parameterized by x and t. For each realization of a turbulent flow, Q will be different, and we can define its expected value using the probability distribution for the ensemble of realizations.12 Nevertheless, the expected value of the convected derivative of Q can be expressed in terms of partial derivatives of the one-point joint velocity, composition PDF 13... 

The expected value on the left-hand side is taken with respect to the entire ensemble of random fields. However, as shown for the velocity derivative starting from (2.82) on p. 45, only two-point information is required to estimate a derivative.14 The first equality then follows from the fact that the expected value and derivative operators commute. In the two integrals after the second equality, only /u,[Pg.264]

In summary, due to the linear nature of the derivative operator, it is possible to express the expected value of a convected derivative of Q in terms of temporal and spatial derivatives of the one-point joint velocity, composition PDF. Two-point information about the random fields U and

expected value and derivative operators commute, and does not appear in the final expression (i.e., (6.9)). 

Note that A, and , will, in general, depend on multi-point information from the random fields U and 0. For example, they will depend on the velocity/scalar gradients and the velocity/scalar Laplacians. Since these quantities are not contained in the one-point formulation for U(x, t) and 0(x, f), we will lump them all into an unknown random vector Z(x, f).16 Denoting the one-point joint PDF of U, 0, and Z by /u,,z(V, ip, z x, t), we can express it in terms of an unknown conditional joint PDF and the known joint velocity, composition PDF ... 

Note that, even though Y and to are fixed, the initial velocity and composition will be random variables, since U(x, 0 and (. f) are random fields. 

Our next goal will be to incorporate into the theory some more detailed information regarding the statistical properties of turbulence (Puhl, Altares and Nicolis 1987). For this purpose we model the turbulent velocity field u as a random field with zero mean ( 0). u is further... 

Doppler broadening arises from the random thermal agitation of the active systems, each of which, in its own test frame, sees the appHed light field at a different frequency. When averaged over a Maxwellian velocity distribution, ie, assuming noninteracting species in thermal equilibrium, this yields a line width (fwhm) in cm C... 

If we were to forget that the flow of current is due to a random motion which was already present before the field was applied—if we were to disregard the random motion entirely and assume that each and every electron, in the uniform field X, moves with the same steady velocity, the distance traveled by each electron in unit time would be the distance v used in the construction of Fig. 16 this is the value which would lead to a current density j under these assumptions, since all electrons initially within a distance v of the plane AB on one side would cross AB in unit time, and no others would cross. Further, in a field of unit intensity, the uniform velocity ascribed to every electron would be the u of (34) this quantity is known as the mobility of the charged particle. (If the mobility is given in centimeters per second, the value will depend on whether electrostatic units or volts per centimeter are used for expressing the field strength.)... 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

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