On flux terms in volume averaging

This note examines the modeling of non-convective fluxes (e.g., stress, heat flux and others) as they appear in the general, unclosed form of the volume-averaged equations of multiphase flows. By appealing to the difference between slowly and rapidly varying quantities, it is shown that the natural closure of these terms leads to the use of a single, slowly-varying combined average flux, common to both phases, plus rapidly-varying local contributions for each phase. The result is general and only rests on the hypothesis that the spatial variation of the combined average flux is adequately described by a linear function of position within the averaging volume. No further hypotheses on the nature of the flow (e.g., about specific flow regimes) prove necessary. The result agrees with earlier ones obtained by ensemble averaging, is illustrated with the example of disperse flows and discussed in the light of some earlier and current literature. A very concise derivation of the general averaged balance equation is also given.     

On the theorems for local volume averaging of multiphase systems

Proofs of the theorems of local volume averaging which relate averages of derivatives to derivatives of averages are presented. A distribution function whose derivatives are proportional to the Dirac delta function is used in the development and the proofs demonstrated are intended to be simpler than those found in the literature.     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

On higher-order averaging of time-periodic systems: reconciliation of two averaging techniques

Nonlinear Dynamics - In this paper we show how higher-order averaging can be used to remedy serious technical issues with the direct application of the averaging theorem. While doing so, we...     

A rigorous derivation of the bioheat equation for local tissue heat transfer based on a volume averaging theory

A general three-dimensional bioheat equation for local tissue heat transfer has been derived with less assumptions, exploiting a volume averaging theory commonly used in fluid-saturated porous media. The volume averaged energy equations obtained for the arterial blood, venous blood and tissue were combined together to form a single energy equation in terms of the tissue temperature alone. The resulting energy equation turns out to be remarkably simple as we define the effective thermal conductivity tensor, which accounts not only for the countercurrent heat exchange mechanism but also for the thermal dispersion mechanism. The present equation for local tissue heat transfer naturally reduces to the Weinbaum-Jiji equation for the unidirectional case.     

Techniques of ray averaging

Ray methods are used in coastal and harbour wave disturbance investigations where the area to be modelled is large compared to the wavelength. The interpretation of forward-plotted ray diagrams, once obtained, has always been a difficult problem. The technique described in this paper calculates wave amplitudes during the ray plotting process and requires only minor modifications to existing ray plotting programs. The idea is to superimpose a grid of square elements over the entire sea area under study, and to perform a spatial averaging of the rays crossing each square element. This ‘square-averaging’ technique has a number of advantages. It smooths the rapid amplitude variations near caustics, calculates the interference of several wave trains, and generates amplitudes automatically in a square array covering the whole studied sea area. Two types of sensitivity tests are carried out. These tests are designed to determine the accuracy of the predicted wave amplitudes with respect to: (1) the square size per wavelength, and (2) the ray density. These two factors largely determine the computing storage, time and cost of a ray model. An upper limit on the square size per wavelength and a lower limit on the ray density are obtained.     

Turbulence averaging within spark ignition engines

 This paper examines velocity averaging within Spark-Ignition (SI) engines, a non-stationary system. Comparison is made between the mean and turbulence velocities found from (a) Ensemble, (b) Cyclic and (c) Wavelet-based averaging. The various methods of extracting turbulence within this flow system result in qualitatively similar average velocities; however, there are significant differences in the turbulence velocities and spectral content of the flow field based on the definition used. The differing interpretation of turbulence results in a subjectivity to the physical understanding of the flows. The experience in extracting coherent structures in stationary turbulence suggests that wavelet analysis offers a unique insight that has applicability for engine studies. Received: 25 February 1998 / Accepted: 12 August 1998     

随机平均规范形方法

计算随机规范形系数是应用随机规范形方法的关键．提出一种应用随机平均计算随机规范形系数的方法．为了说明方法的有效性,对白噪声激励的Duffing系统,经过变换,对于相应的平均方程,比较了精确解、规范形方法解和能量包络方法解的稳态概率密度,结果表明,当非线性项系数较小时,三者完全一致．当非线性项系数较大时,规范形方法所得解与精确解相差不大．     

Stochastic averaging of energy harvesting systems

A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.     

Stochastic averaging of quasi-generalized Hamiltonian systems

A stochastic averaging method for generalized Hamiltonian systems (GHS) subject to light dampings and weak stochastic excitations is proposed. First, the GHS are briefly reviewed and classified into five classes, i.e., non-integrable GHS, completely integrable and non-resonant GHS, completely integrable and resonant GHS, partially integrable and non-resonant GHS and partially integrable and resonant GHS. Then, the averaged and FPK equations and the drift and diffusion coefficients for the five classes of quasi-GHS are derived. Finally, the stochastic averaging for a nine-dimensional quasi-partially integrable GHS is given to illustrate the application of the proposed procedure, and the results are confirmed by using those from Monte Carlo simulation.     

Application of stochastic averaging to non-linear ecosystems

Stochastic models for the prey–predator type ecosystems in a random environment are proposed and investigated. They are the stochastic versions of the classical Lotka–Volterra model, the predator-saturation model, and the time-delay model. Variations in the birth rate of the preys and the death rate of the predators are modeled as random processes. For the three stochastic models, the stochastic averaging procedure of Stratonovich and Khasminskii is applied to obtain the probability distributions of the prey and predator populations at the state of statistical stationarity. Effects of different system parameters on the ecosystem behaviors are evaluated. The analytical results are substantiated by results obtained from Monte Carlo simulations.     

Optimisation of temporal averaging processes in PIV

A hybrid of correlation and vector averaging is introduced to capitalise on the advantages of each process. An extensive series of Monte Carlo simulations have been conducted to investigate hybrid averaging and evaluate it against both vector and correlation averaging. The simulations show that hybrid averaging improves the measurement accuracy over both correlation and vector averaging over a wide range of imaging conditions. The simulations are validated by applying hybrid averaging to experimental micro- and macro-flows. In pulsatile conditions, correlation averaging yields an averaged correlation function that is multi-modal, which can result in unpredictable measurements. A Monte Carlo simulation shows the benefits of hybrid averaging over correlation averaging in such conditions. This has been experimentally validated on the unsteady wake behind a shedding circular cylinder at Re = 98.     

Solving three-dimensional problems of two-phase filtration using averaging

The present work examines a method of solving three-dimensional problems of two-phase filtration based on averaging the equations and introducing functions reflecting the layering of the flow in collectors inhomogeneous over the thickness. These functions are constructed from the calculation of the two-dimensional flow in the plane of the vertical cross section of the bed. This approach is generalized to the case of the displacement of petroleum by aqueous solutions of chemical reagents. Inhomogeneous problems of multiphase filtration may only be solved numerically. However, in the case of three-dimensional flow, even the use of an effective difference scheme is beset with considerable difficulties, dueprimarily to the increased requirements for memory and speed of operation of the computer. In [1] a principle of approximate integration of the filtration equations in a thin inclined bed was proposed. Assuming a hydrostatic law of vertical pressure distribution, the equations were averaged, and successive solution of problems of ever smaller dimensionality was carried out, not with the initial curves of phase permeability, but with curves of averaged phase permeability. Subsequently, there was further development of this principle of capillary-gravitational equilibrium, and others, to deal with the case of flooding of an inhomogeneously laminar bed [2, 3]. A significantly different approach to the determination of the auxiliary functions, not involving assumptions to as the relations between the viscous drag and the capillary and gravitational forces, is to use the solution of the two-dimensional problem of the replacement of petroleum by water in the plane of the vertical cross section of the bed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 71–75, May–June, 1977.     

Probability distribution and conditional averaging in turbulent flows

The results of an experimental investigation of the turbulence characteristics in the plane mixing layer and in the wake behind a cylinder are given. Measurements are made of the distribution of the velocity and temperature probabilities, the intermittency coefficient, and the conditionally averaged values of the square of the velocity and temperature derivatives.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 31–37, November–December, 1977.     

Stochastic averaging for nonlinear vibration energy harvesting system

A stochastic averaging technique for the nonlinear vibration energy harvesting system to Gaussian white noise excitation is developed to analytically evaluate the mean-square electric voltage and mean output power. By introducing the generalized harmonic transformation, the influence of the external circuit on the mechanical system is equivalent to a quasi-linear stiffness and a quasi-linear damping with energy-dependent coefficients, and then the equivalent nonlinear system with respect to the mechanical states is completely established. The Itô stochastic differential equation with respect to the mechanical energy of the equivalent nonlinear system is derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the stationary probability density of the mechanical states, and then the mean-square electric voltage and mean output power are analytically obtained through the approximate relation between the electric quantity and the mechanical states. The agreements between the analytical results and those from the moment method and from Monte Carlo simulations validate the effectiveness of the proposed technique.     

Volume averaging for the analysis of turbulent spray flows

Spray flow calculations are usually based upon equations that have been developed by averaging droplet properties locally throughout the flow field. Presently, standard procedure for LES (large-eddy simulations) is to average these averaged equations once again to filter the short-length-scale fluctuations. In this paper, the theoretical foundations for the averaged spray equations are examined; then the volume-averaging process for LES and the volume-averaging process for two-phase flows are unified for the analysis of turbulent, two-phase flows. Comments are provided on the relationship between the averaging volume and the computational-cell volume. This paper provides generality to the weighting-function choice in the averaging process and precision to the definition of the volume over which the averaging is performed. New flux terms that result from the averaging process and appear in the governing averaged partial differential equations are identified and their modelling is discussed. Situations are identified where sufficient stratification of properties on the scale smaller than the averaging volume leads to the significance of these quantities. Evolution equations for averaged entropy and averaged vorticity are developed. The relationship amongst the curl of the average gas-phase velocity, the average of the gas-phase-velocity curl, and the rotation of the discrete droplets or particles is established. The needs and challenges for sub-grid modelling to account for small-vortex/droplet interactions are presented. Applications to spray combustion are discussed.     

Structural averaging of flow processes in nonhomogeneous porous media

The problem of constructing macroscopic analogs for the equations describing processes in nonhomogeneous porous media is considered. The classical results of the theory relate to the case in which the averaging procedure leads to the smoothing of the coefficients describing the inhomogeneity without modifying the structure of the equations of the process. It is natural to call such averaging coefficient averaging. In this paper another approach — structural averaging, in which the type of the equations themselves or their qualitative structure is modified, is investigated. In the overwhelming majority of cases, in addition to a small scale of inhomogeneity, these systems also contain one or more small (large) parameters reflecting important differences in the properties of the individual components of the medium or the physical components of the transport process itself. A typical example of the structural averaging problems generated by processes in highly nonhomogeneous media and, moreover, processes with nonequivalent diffusion and convective transport is investigated. The methods of asymptotic averaging [1,2] are employed. Processes in highly nonhomogeneous media were investigated in [3–6]. Studies [4, 8, 9] are concerned with the averaging of convection-diffusion systems.Based on paper read at the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991. Presented by V. N. Nikolaevskii.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 103–116, November–December, 1992.     

Fourier averaging: a phase-averaging method for periodic flow

A new phase-averaging method, denoted as Fourier averaging, is presented for the investigation of periodic flows. In such flows, the moments of velocity, as estimated from a small number of samples, show fluctuations in their phasewise development. In previous methods these fluctuations are reduced by calculating moments from large phase intervals. Fourier averaging, in contrast, neglects high-frequency fluctuations and assumes that they are of no physical relevance. This method supplies additional information on amplitudes and phase angles of discrete frequencies, which may then be used for visualizations of flow fields at any desired phase increment. The Fourier averaging method was verified empirically by LDA measurements and compared to other methods. It is shown that the results obtained by Fourier averaging are more accurate than for previously known methods. Received: 15 June 1998/Accepted: 15 April 1999