Joint Probability Density Function Of Velocity Research Articles

A posteriori subgrid-scale model tests based on the conditional means of subgrid-scale stress and its production rate

Traditional a posteriori tests of subgrid-scale (SGS) models often compare large eddy simulation (LES) profiles of various statistics with measurements. In this study we propose and employ a new a posteriori test to study SGS model performance. We compare the conditional means of the LES-generated SGS stress and stress production rate conditional on the resolvable-scale velocity with measurements. These statistics must be reproduced by the SGS model for LES to correctly predict the one-point resolvable-scale velocity joint probability density function. Our tests using data obtained in convective atmospheric boundary layers show that the a posteriori results are consistent with our a priori tests based on the same conditional statistics. The strengths and deficiencies of the models observed here were also identified in our a priori tests. The remarkable consistency between the two types of tests suggests that statistical model tests based on the conditional SGS stress and its production rate are a highly capable approach for identifying specific model deficiencies and for evaluating SGS model performance in simulations.

Experimental investigation of a maximum entropy assumption for acceleration terms within a poly‐disperse moment framework

AbstractMoment transport methods are being developed to model poly‐dispersed multiphase flows by transporting statistical moments of the particle size–velocity joint probability density function (JPDF). A common feature of these methods is the requirement to reproduce or approximate the form of the JPDF from the transported moments for calculation of body force terms and other source terms. This paper examines the application of a maximum entropy technique against phase Doppler anemometry data sets from an electrostatically charged kerosene spray and also an automotive pressure swirl atomizer. An assessment of which moments are required to reproduce the JPDFs using a maximum entropy assumption to a sufficient level of accuracy is made. It is found that it is possible to reproduce the JPDFs to a high level of accuracy using a large number of moments; however, this incurs large computational overheads. If the moments to be transported are chosen on the basis of physical reasoning (such as the relationship between size and velocity due to drag) it is possible to reduce the number of moments to those which would be conserved via balance equations. This permits an approximation to the JPDF commensurate with the closure level of the moment transport method and thus the closure model method is naturally scalable with the degree of information from available conservation equations. Copyright © 2008 John Wiley & Sons, Ltd.

Probability density function modeling of scalar mixing from concentrated sources in turbulent channel flow

Dispersion of a passive scalar from concentrated sources in fully developed turbulent channel flow is studied with the probability density function (PDF) method. The joint PDF of velocity, turbulent frequency and scalar concentration is represented by a large number of Lagrangian particles. A stochastic near-wall PDF model combines the generalized Langevin model of Haworth and Pope [Phys. Fluids 29, 387 (1986)] with Durbin's [J. Fluid Mech. 249, 465 (1993)] method of elliptic relaxation to provide a mathematically exact treatment of convective and viscous transport with a nonlocal representation of the near-wall Reynolds stress anisotropy. The presence of walls is incorporated through the imposition of no-slip and impermeability conditions on particles without the use of damping or wall-functions. Information on the turbulent time scale is supplied by the gamma-distribution model of van Slooten et al. [Phys. Fluids 10, 246 (1998)]. Two different micromixing models are compared that incorporate the effect of small scale mixing on the transported scalar: the widely used interaction by exchange with the mean and the interaction by exchange with the conditional mean model. Single-point velocity and concentration statistics are compared to direct numerical simulation and experimental data at Reτ=1080 based on the friction velocity and the channel half width. The joint model accurately reproduces a wide variety of conditional and unconditional statistics in both physical and composition space.

Galilean invariance of velocity probability density function transport equation

Model equations for turbulent flows generally must have the same Galilean transformation properties as the exact transport equations. We examine the transport equation of one-point velocity joint probability density function (JPDF) under Galilean transformations. The JPDF and its transport equation as well as the unclosed terms in the equation are shown to be Galilean invariant. For constant density flows they are also invariant under extended Galilean transformations. It is also shown that conditional statistics of any Galilean invariant variables conditional on velocity are invariant. The present work provides a basis and justification for using such conditional statistics in studying turbulent flows.

Application of PDF modeling to swirling and nonswirling turbulent jets

The turbulence modeling in probability density function (PDF) methods is studied through applications to turbulent swirling and nonswirling co-axial jets and to the temporal shear layer. The PDF models are formulated at the level of either the joint PDF of velocity and turbulent frequency or the joint PDF of velocity, wave vector, and turbulent frequency. The methodology of wave vector models (WVMs) is based on an exact representation of rapidly distorted homogeneous turbulence, and several models are constructed in a previous paper [1]. A revision to a previously presented conditional-mean turbulent frequency model [2] is constructed to improve the numerical implementation of the model for inhomogeneous turbulent flows. A pressure transport model is also implemented in conjunction with several velocity models. The complete model yields good comparisons with available experimental data for a low swirl case. The individual models are also assessed in terms of their significance to an accurate solution of the co-axial jets, and a comparison is made to a similar assessment for the temporal shear layer. The crucial factor in determining the quality of the co-axial jet simulations is demonstrated to be the proper specification of a parameter ratio in the modeled source of turbulent frequency. The parameter specification is also shown to be significant in the temporal shear layer.

Structure and propagation of turbulent premixed flames astabilized in a stagnation flow

Turbulence statistics in premixed methane/air flames stabilized by a stagnation plate have been measured using the two-component laser Doppler velocimetry technique. Experimental conditions cover equivalence ratios of 0.67 to 1.16, incident turbulence intensities of 0.1 to 0.7 m/s, and global stretch rates of 100 to 230 sec−1. Results on turbulence statistics show that in crossing the flame the increase in fluctuation is more pronounced in the normal component while the Reynolds stress remains almost zero, and that the velocity joint probability density function is bimodal for rich flames and mono-modal for lean flames. It is also demonstrated that the present stagnation flame configuration can be usefully applied for the determination, of the turbulent flame speed ST. Values of ST determined as functions of the equivalence ratio and turbulence intensity agree well with those in the literture. It is suggested that the stagnation flame offers potential as a viable configuration for the study of planar, turbulent premixed flames uncomplicated by fluctuations generated by the stabilization regions associated with the conventional conical Bunsen flames and V-flames

A Lagrangian two-time probability density function equation for inhomogeneous turbulent flows

An exact equation for the Lagrangian two-time velocity joint probability density function (pdf) is derived from the Navier–Stokes equation. The pdf equation contains as an unknown the conditional expectation of the fluid acceleration. A linear Markov model is proposed which leads to a modeled equation that is consistent both with Kolmogorov’s theory in the inertial subrange and with Reynolds-stress models. The dissipation rate is obtained from the joint pdf in a way that is consistent with the modeled dissipation equation. A Monte Carlo method can be used to solve the modeled two-time pdf equation for inhomogeneous turbulent flows.