Turbulence probability density functions

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

A model must be introduced to simulate fast chemical reactions, for example, flamelet, or turbulent mixer model (TMM), presumed mapping. Rodney Eox describes many proposed models in his book [23]. Many of these use a probability density function to describe the concentration variations. One model that gives reasonably good results for a wide range of non-premixed reactions is the TMM model by Baldyga and Bourne [24]. In this model, the variance of the concentration fluctuations is separated into three scales corresponding to large, intermediate, and small turbulent eddies. 

In a turbulent flow, the local value (i.e., at a point in space) of the mixture fraction will behave as a random variable. If we denote the probability density function (PDF) of by f - Q where 0 < ( < 1, the integer moments of the mixture fraction can be found by integration ... 

An attempt has been made by Tsouris and Tavlarides[5611 to improve previous models for breakup and coalescence of droplets in turbulent dispersions based on existing frameworks and recent advances. In both the breakup and coalescence models, two-step mecha-nisms were considered. A droplet breakup function was introduced as a product of droplet-eddy collision frequency and breakup efficiency that reflect the energetics of turbulent liquid-liquid dispersions. Similarly, a coalescencefunction was defined as a product of droplet-droplet collision frequency and coalescence efficiency. The existing coalescence efficiency model was modified to account for the effects of film drainage on droplets with partially mobile interfaces. A probability density function for secondary droplets was also proposed on the basis of the energy requirements for the formation of secondary droplets. These models eliminated several inconsistencies in previous studies, and are applicable to dense dispersions. 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

Calculations of subsonic and supersonic turbulent reacting mixing layers using probability density function methods. Physics of Fluids 10, 497 498. 

Dopazo, C. (1975). Probability density function approach for a turbulent axisymmetric heated jet. Centerline evolution. The Physics of Fluids 18, 397 404. 

Relaxation of initial probability density functions in the turbulent convection of scalar fields. The Physics of Fluids 22, 20-30. 

Drake, M. C., R. W. Pitz, and W. Shyy (1986). Conserved scalar probability density functions in a turbulent jet diffusion flame. Journal of Fluid Mechanics 171, 27-51. 

Dreeben, T. D. and S. B. Pope (1997a). Probability density function and Reynolds-stress modeling of near-wall turbulent flows. Physics of Fluids 9, 154—163. 

Gao, F. (1991). Mapping closure and non-Gaussianity of the scalar probability density function in isotropic turbulence. Physics of Fluids A Fluid Dynamics 3, 2438-2444. 

Haworth, D. C. and S. H. El Tahry (1991). Probability density function approach for multidimensional turbulent flow calculations with application to in-cylinder flows in reciprocating engines. AIAA Journal 29, 208-218. 

Janicka, J., W. Kolbe, and W. Kollmann (1979). Closure of the transport equation for the probability density function of turbulent scalar fields. Journal of Non-Equilibrium Thermodynamics 4, 47-66. 

Kollmann, W. and J. Janicka (1982). The probability density function of a passive scalar in turbulent shear flows. The Physics of Fluids 25, 1755-1769. 

Wall-boundary conditions in probability density function methods and application to a turbulent channel flow. Physics of Fluids 11, 2632-2644. 

The probability density function approach to reacting turbulent flows. In P. A. Libby and F. A. Williams (eds.), Turbulent Reacting Flows, pp. 185-203. Berlin Springer. 

Erratum Application of the velocity-dissipation probability density function model to inhomogeneous turbulent flows [Phys. Fluids A 3, 1947 (1991)]. Physics of Fluids A Fluid Dynamics 4, 1088. 

Raju, M. S. (1996). Application of scalar Monte Carlo probability density function method for turbulent spray flames. Numerical Heat Transfer, Part A 30, 753-777. 

Wall, C., B. J. Boersma, and R Moin (2000). An evaluation of the assumed beta probability density function subgrid-scale model for large eddy simulation of nonpremixed, turbulent combustion with heat release. Physics of Fluids 12, 2522-2529. 

The figure shows U >. S L in this region and Da is predominantly small. At the highest Reynolds numbers the region is entered only for very intense turbulence, U > SL. The region has been considered a distributed reaction zone in which reactants and products are somewhat uniformly dispersed throughout the flame front. Reactions are still fast everywhere, so that unbumed mixture near the burned gas side of the flame is completely burned before it leaves what would be considered the flame front. An instantaneous temperature measurement in this flame would yield a normal probability density function—more importantly, one that is not bimodal. 

Pozorski, J., and J. P. Minier. 1999. Probability density function modeling of dispersed two-phase turbulent flow. Phys. Rev. E 59 855-63. 

The present study is to elaborate on the computational approaches to explore flame stabilization techniques in subsonic ramjets, and to control combustion both passively and actively. The primary focus is on statistical models of turbulent combustion, in particular, the Presumed Probability Density Function (PPDF) method and the Pressure-Coupled Joint Velocity-Scalar Probability Density Function (PC JVS PDF) method [23, 24]. 

The Presumed Probability Density Function method is developed and implemented to study turbulent flame stabilization and combustion control in subsonic combustors with flame holders. The method considers turbulence-chemistry interaction, multiple thermo-chemical variables, variable pressure, near-wall effects, and provides the efficient research tool for studying flame stabilization and blow-off in practical ramjet burners. Nonreflecting multidimensional boundary conditions at open boundaries are derived, and implemented into the current research. The boundary conditions provide transparency to acoustic waves generated in bluff-body stabilized combustion zones, thus avoiding numerically induced oscillations and instabilities. It is shown that predicted flow patterns in a combustor are essentially affected by the boundary conditions. The derived nonreflecting boundary conditions provide the solutions corresponding to experimental findings. 

Figure 4. Probability density functions of temperature for Ht-air turbulent diffusion flame determined at various radial positions 134 mm downstream of the fuel line tip according to procedures indicated in Figure 3. The measurement positions are drawn schematically in the center of the figure to correspond to the radial positions r on the scale at the RHS.

Figure 5. Probability density function (pdf or histogram) for temperature X velocity for turbulent diffusion flame. These data correspond to a test zone along the axis, 50 fuel-tip diameters downstream from the fuel line tip.

Time- and space-resolved major component concentrations and temperature in a turbulent gas flow can be obtained by observation of Raman scattering from the gas. (1, 2) However, a continuous record of the fluctuations of these quantities is available only in those most favorable cases wherein high Raman scattering rate and/or slow rate of time variation of the gas allow many scattered photons (> 100) to be detected during a time resolution period which is sufficiently short to resolve the turbulent fluctuations. (2, 3 ) Fortunately, in other cases, time-resolved information still can be obtained in the forms of spectral densities, autocorrelation functions and probability density functions. (4 5j... 

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