A closed equation for the joint probability density function of the magnitudes of fluctuations of a turbulent scalar reacting field and its gradient

Flow Turbulence Combust

Stavrov

2008

A joint probability distribution function of a conservative scalar (mixture fraction) and its gradient is predicted numerically. Statistical moments of this function are compared to their approximations, direct numerical simulation data, and also to the results obtained by simplified models for a conditional rate of scalar dissipation, the surface density function, and the one-point PDF of scalar fluctuation under homogeneous isotropic turbulence. The results allow to evaluate the performance of existing statistical micromixing models.

Section: The Closed Equation For the Joint Pdf Of A Scalar And Its Grmentioning

confidence: 98%

Section: The Closed Equation For the Joint Pdf Of A Scalar And Its Grmentioning

confidence: 99%

Section: The Closed Equation For the Joint Pdf Of A Scalar And Its Grmentioning

confidence: 99%

“…Equation (4) has been closed in [12] with the assumption of homogeneous and isotropic turbulent fields. Some improvements of the closure were done in [17].…”

Section: The Closed Equation For the Joint Pdf Of A Scalar And Its Grmentioning

confidence: 99%

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Modeling of the Joint PDF of a Scalar and Its Gradient in Turbulent Mixing

Flow Turbulence Combust

Stavrov

2008

“…Furthermore, of great interest for turbulent reacting flows are the distributions of length and time scales in a concentration space, since it is precisely they that determine the properties of a turbulent field in the zones of localization of the reaction. Therefore, we derive the equation for the PDF of length and time scales using the equation obtained earlier for the joint PDF of the concentration field and its gradient [11].…”

mentioning

confidence: 99%

Length-scale distributions of the concentration pulsations of a passive impurity in a homogeneous turbulent flow

Чорный

2005

J Eng Phys Thermophys

Self Cite

A relation for calculating the probability density function f t λ (ϕ) of the length scales of a passive concentration field in homogeneous turbulence has been obtained by consideration of the joint statistics of the concentration field and its gradient. The closed equation derived for f t λ (ϕ) has been solved numerically using the data of direct numerical modeling of homogeneous turbulence for the mean characteristics involved in the equation as the coefficients. The results obtained for different values of the Schmidt number have been compared.Introduction. Turbulence considerably intensifies the mixing and transfer of substances and heat in a flow. Many processes in the environment and in technical devices that are associated with flames, chemical reactions, and propagation of impurities would be completely impossible without efficient mixing. Although the history of study of turbulent mixing is rather long, its nature remains to be completely understood. We know of the equations using which one describes turbulent flows, but their numerical solution involves many difficulties.Direct numerical modeling of combined turbulent flows, which develops in connection with the progress made in computer technologies, cannot be used for problems of practical significance [1]. Therefore, one continues to develop statistical approaches with one method of averaging or another. One of them is the method of the probability density function (PDF) of different hydro-and thermodynamic parameters [2], which has been fruitfully used in studying reacting media.The advantages of the PDF method are primarily a simple and accurate representation of the influence of chemical sources on changes in the quantities sought and a lower consumption of computer time. Its main problem is that of modeling of the contribution of the process of fine-structure mixing of a scalar field of concentration (micromixing) to the general pattern of mixing. Micromixing is determined by the mechanism using which the scalar field brought to the state of "rough homogeneity" by the averaged flow and large vortices in the flow is broken by a statistically homogeneous turbulent velocity field down to the smallest turbulence scales due to the small-scale motion of the medium and next to the molecular level through molecular diffusion.The presence of all kinds of values of the concentrations and the existence of the entire range of scales create constantly interacting categories of turbulent mixing: the evolution of the length-scale range leads to a constant change in the boundary conditions for the action of molecular diffusion, thus influencing the rate of change in the concentration pulsations. The structure of the pulsation field is dependent on the small scales directly influencing the rate of dissipation of concentration pulsations more strongly than on large scales.Depending on the formulation of the problem, one selects the normalized temperature, the concentrations of the reactants or the inert impurity, the Schwab-Zel'dovich variables for turbulent flows ...

Calculation of the statistical characteristics of a turbulent flow under the action of external forces

Zhukova

2006

J Eng Phys Thermophys

Self Cite

A study is made of the statistical characteristics of isotropic velocity and scalar fields in supply of the kinetic energy of turbulence from the energy of the average flow. Two models based on the distributions of the kinetic turbulence energy and the intensity of scalar-field pulsations by wave numbers and length scales are used for calculation of the statistical characteristics of turbulent velocity and scalar fields. The calculation results are compared to the data of the direct numerical modeling performed under the same initial conditions. Introduction.A great deal of chemical reactions realized in industrial devices are "rapid" (i.e., the time of chemical reaction is much shorter than the time of turbulent mixing). In this case, the chemical-reaction rate is determined by the rate of mixing of the reagents to the molecular level. Turbulent mixing seeks to make the scalar field homogeneous, in so doing considerably intensifying this process due to the formation of large scalar gradients in the flow. The joint action of turbulent transfer and diffusion makes the mixing (from a large-scale one to that at the molecular level) efficient.The formalism of the probability density function of the turbulent pulsations is used to statistically describe turbulent mixing [1]. Models based on the joint probability density function (JPDF) are used to study the turbulent reacting scalar field. Knowledge of a JPDF makes it possible to correctly average strongly nonlinear terms describing chemical interaction in transfer equations for the concentrations of the mixture components and the temperature. Methods of construction of the equations for different JPDFs in turbulent flows have already been developed at present [1][2][3][4][5]. Open equations for a number of JPDFs have been presented in [2]. Data of experiments and direct numerical modeling as well as the assumptions of the Gaussian or another existing form of conditional distribution functions are used to close them.The conservative-scalar method where a special variable (conservative scalar), the transfer equations for which contain no term describing chemical interaction, is constructed are widely used for description of flows with chemical reactions. The conditional moments from the JPDF of the conservative scalar and its gradient (conditional scalar-dissipation function, flame-surface density, etc.) are then used in the equations for nonconservative scalars. Also, the equation for the JPDF of the scalar and its gradient is used with allowance for the terms characterizing chemical interaction, e.g., when it is necessary to calculate very low concentrations of harmful reaction products [6].Closed equations for the JPDF of the scalar and its gradient have been presented in [6][7][8]. The models presented contain the unknown coefficients to determine which one must use experimental results and data of direct numerical modeling or construct additional models.Since the models mentioned in [6][7][8] for JPDFs are single-point ones from the viewpoint of the statistics use...