Random field scalar

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... 

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

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... 

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 ... 

For example, all information is lost concerning the relative spatial locations of two random samples. As discussed in Chapter 2, this fact implies that all information concerning the spatial derivatives of the scalar fields is lost when the scalar field is described by its one-point joint PDF. 

Figure 3.7. Two random scalar fields (x, t ) as a function of x = xi with fixed t = t. The scalar fields were extracted from DNS of isotropic turbulence (R>. = 140, (U) = 0) with collinear uniform mean scalar gradients. Dashed line Sc = 1/8 solid line Sc = 1. The corresponding velocity field is shown in Fig. 2.1. (Courtesy of P. K. Yeung.)... 

In the LEM, turbulence is modeled by a random rearrangement process that compresses the scalar field locally to simulate the reduction in length scales that results from turbulent mixing. For example, with the triplet map, defined schematically in Fig. 4.2, a random length scale / is selected at a random point in the computational domain, and the scalar field is then compressed by a factor of three.14 The PDF for /,... 

During the time intervals between random eddy events, (4.37) is solved numerically using the scalar fields that result from the random rearrangement process as initial conditions. A standard one-dimensional parabolic equation solver with periodic boundary conditions (BCs) is employed for this step. The computational domain is illustrated in Fig. 4.3. For a homogeneous scalar field, the evolution of t) will depend on the characteristic length... 

The transported PDF models discussed so far in this chapter involve the velocity and/or compositions as random variables. In order to include additional physics, other random variables such as acceleration, turbulence dissipation, scalar dissipation, etc., can be added. Examples of higher-order models developed to describe the turbulent velocity field can be found in Pope (2000), Pope (2002a), and Pope (2003). Here, we will limit our discussion to higher-order models that affect the scalar fields. 

Convection of a passive scalar by a quasi-uniform random straining field. Journal of Fluid Mechanics 64, 737-762. 

The nuclear hyperfine interaction splits the paramagnetic states of an electron when it is close to a nucleus with a magnetic moment. For a random orientation of spins and nuclei, the tensor quantities in Eq. (4.11) are replaced by scalar distributions, and the resonance magnetic field is shifted from the Zeeman field // by... 

In these wave packet simulations, the molecular axis of the FHF system is assumed to be aligned along the space-fixed axis Z electric field vector. This assumption involves a maximum interaction of the IR and UV laser pulses with the system. Recalling that the time-dependent interaction potential is given by the scalar product of the electric field vector and the dipole vector, i.e. (t) /j, cos 9, it is clear that for field polarizations perpendicular to the molecular axis [9 = 90°) the interaction of the IR laser pulse with the anion vanishes, and for any molecular orientation different from 0= 0° or 180° the interaction is less efficient. Consider now an ensemble of randomly oriented FHF molecules, as in Fig. 4.13(c). Since the UV pulse is tuned to match the energy gap between anion and neutral... 

This Appendix supplements Section 2.4. The problem is to find the mean concentration field a(x, t) for an arbitrary scalar entity, given a turbulent velocity field u(x, f), a specified source density homogeneous initial and boundary conditions a = 0 on the outer boundary Sq of a region V. Hence, all scalar is introduced into the flow by the sources (p within V. We consider an ensemble of realizations of the turbulent flow, denoted by a superscript co, so that a" , u" and (p " =

Peclet number limit, the relationship between a ", u" and (p is given by Eq. (14), here rewritten as... 

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