Velocity PDF

The above definition can be extended to include an arbitrary number of random variables. For example, the one-point joint velocity PDF /u(V x, t) describes all three velocity... 

A semi-colon is used in the argument list to remind us that V is an independent (sample space) variable, while x and t are fixed parameters. Some authors refer to fyx (Vj x, f ) as the one-point, one-time velocity PDF. Here we use point to refer to a space-time point in the four-dimensional space (x, t). 

In homogeneous turbulence, the one-point joint velocity PDF can be written as /u(V t), and can be readily measured using hot-wire anemometry or laser Doppler velocimetry (LDV). 

In fully developed homogeneous turbulence,7 the one-point joint velocity PDF is nearly Gaussian (Pope 2000). A Gaussian joint PDF is uniquely defined by a vector of expected values (j, and a covariance matrix C ... 

The mean velocity can be computed directly from the joint velocity PDF 8... 

The last equality follows from the definition of the one-point marginal velocity PDF, e.g.,... 

The reader familiar with turbulence modeling will recognize the covariance matrix as the Reynolds stresses. Thus, for fully developed homogeneous turbulence, knowledge of the mean velocity and the Reynolds stresses completely determines the one-point joint velocity PDF. 

Note that homogeneity rules out wall-bounded turbulent flows (e.g., pipe flow) wherein the velocity PDF can be far from Gaussian. 

In general, the one-point joint velocity PDF can be used to evaluate the expected value of any arbitrary function / (U) of U using the definition... 

For Gaussian random variables, an extensive theory exists relating the joint, marginal, and conditional velocity PDFs (Pope 2000). For example, if the one-point joint velocity PDF is Gaussian, then it can be shown that the following properties hold ... 

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.,... 

The two-point description of homogeneous turbulence begins with the two-point joint velocity PDF /u,u (V, V x, x. t) defined by... 

The spatial correlation functions are computed from the two-point joint velocity PDF based on two points in space. Obviously, the same idea can be extended to cover two points in time. Indeed, the Eulerian9 two-time joint velocity PDF /Lf (J (V, V x, t, t ),... 

More precisely, the Fourier coefficients in (4.27) can be replaced by random variables with the following properties k. U = 0 and (U ) = 0 for all k such that k > kc. An energy-conserving scheme would also require that the expected value of the residual kinetic energy be the same for all choices of the random variable. The LES velocity PDF is a conditional PDF that can be defined in die usual manner by starting from die joint PDF for the discrete Fourier coefficients U. ... 

The velocity field is assumed to be statistically homogeneous. Thus, die LES velocity PDF does not depend explicitly on x, only implicitly through U (x, t). 

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. 

At this point, the next step is to decompose the velocity into its mean and fluctuating components, and to substitute the result into the left-hand side of (6.42). In doing so, the triple-correlation term (UiUjUk) will appear. Note that if the joint velocity PDF were known (i.e., by solving (6.19)), then the triple-correlation term could be computed exactly. This is not the case for the RANS turbulence models discussed in Section 4.4 where a model is required to close the triple-correlation term. 

Note that, since the joint velocity PDF will be known from the solution of (6.46), the model can be formulated in terms of V, the moments of U and their gradients, or any arbitrary function of V. However, as with Reynolds-stress models, in practice (Pope 2000) the usual choice of functional dependencies is limited to... 

A general modeling strategy that has been successfully employed to model the joint velocity PDF in a wide class of turbulent flows30 is to develop stochastic models which... 

Homogeneous, linear Fokker-Planck equations are known to admit a multi-variate Gaussian PDF as a solution.33 Thus, this closure scheme ensures that a joint Gaussian velocity PDF will result for statistically stationary, homogeneous turbulent flow. 

Transport by velocity fluctuations is treated exactly at the joint velocity PDF level. 

Since the mean velocity and Reynolds-stress fields are known given the joint velocity PDF /u(V x, t), the right-hand side of this expression is closed. Thus, in theory, a standard Poisson solver could be employed to find (p)(x, t). However, in practice, (U)(x, t) and (u,Uj)(x, t) must be estimated from a finite-sample Lagrangian particle simulation (Pope 2000), and therefore are subject to considerable statistical noise. The spatial derivatives on the right-hand side of (6.61) are consequently even noisier, and therefore are of no practical use when solving for the mean pressure field. The development of numerical methods to overcome this difficulty has been one of the key areas of research in the development of stand-alone transported PDF codes.38... 

The resolved velocity U would then be found from an LES simulation, and the LES velocity PDF (defined in Section 4.2) would be written in terms of the unresolved velocity uL Alternatively, the filtered density function (FDF) approach can be used with a variant... 

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

Choosing the drift coefficient to be linear in U and the diffusion matrix to be independent of U ensures that the Lagrangian velocity PDF will be Gaussian in homogeneous turbulence. Many other choices will yield a Gaussian PDF however, none have been studied to the same extent as the LGLM. 

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