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.     

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．     

A modified method of averaging for solving a class of nonlinear equations

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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.     

On some composite failure criteria based on the averaging method

The construction of an anisotropic criterion for composite failure is based on the use of the averaging method and micromechanical criteria for each of the components. The averaging method permits calculating not only the effective properties of a composite using the solutions of a local problem on the periodicity cell but also the stresses and strains in each composite. Using the superposition method and studying the composite brittle failure, we can calculate the stress distribution for an arbitrary radial (simple) macroloading as the linear combination of six basic problems on the periodicity cell. The limit load for an effective material is obtained from the condition that some of the composite components starts to fail. For given trajectories, the limit loads depend on the geometry of elastic properties and mainly on the ratio of ultimate strengths for different components of the composite.     

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     

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.     

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     

Theory of averaging of elastic fields in media with a monoclinic structure

Conclusions The proposed relations of averaging theory, together with complex Kolosov-Muskhelishvili potentials for isotropic matrices and Lekhnitskii potentials for rectilinearly anisotropic matrices with prismatic fillers, constitute a closed system of equations in the problem of determining the internal fields and the complete set of effective elastic constants of composite media with uniform external static stresses.By combining relations of the averaging theory and well-known solutions of boundary-value problems on the stress-state of an infinite medium with an individual inclusion, we can directly construct the solution of the problem of determining the macroscopic parameters of a composite system with an arbitrary structure.Conformal mapping of the external boundary of the determining element onto a unit circle is an efficient method of calculating contour integrals in averaging theory with a high degree of accuracy.When the initial terms are retained in an expansion of the complex potentials in degrees of inclusion interaction, it is possible to obtain approximate analytic formulas for all of the effective constants. In special cases, these formulas coincide with the asymptotic formulas found from the exact solutions.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 23, No. 1, pp. 3–18, January, 1987.     

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.     

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.     

Two-phase flow in heterogeneous porous media: The method of large-scale averaging

The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.     

Conditional averaging of PIV plane wake data using a cross-correlation approach

A new approach is investigated for conditional averaging of a series of random snapshots of periodic velocity fields obtained from PIV measurements. This approach departs from conventional conditional averaging approaches that require a reference signal for extracting phase information and uses the cross-correlation between the velocity fields as a phase identifier. The methodology is applied on the natural and forced turbulent wake of a circular cylinder, the latter serving also as a validation/verification case. The results show that the vortex formation and shedding processes in the wake are drastically affected by inflow oscillations and reveal the underlying vortex dynamics. Additionally, detailed velocity measurements are reported for the two cases investigated.     

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.     

On the spatial-temporal averaging method for modeling transport in porous media

The spatial-temporal averaging procedure is considered with a nonhomogeneous distribution of elementary domains in the spatial-temporal space and the probabilistic interpretation of the ST-averaging is also given. Several averaging theorems and corollaries about the averages of spatial and temporal derivatives are presented and rigorously proved which allow elementary domain to vary in space and time. The macroscopic transport equation in the most general condition and the simplified macroscopic equation under the special form of distributions are developed which may be reduced to the classical macroscopic transport equation as the spatial-temporal average degenerates into the volume average.     

Ultraconvergence for averaging discontinuous finite elements and its applications in Hamiltonian system

This paper discusses the k-degree averaging discontinuous finite element solution for the initial value problem of ordinary differential equations.When k is even,the averaging numerical flux (the average of left and right limits for the discontinuous finite element at nodes) has the optimal-order ultraconvergence 2k + 2.For nonlinear Hamiltonian systems (e.g.,Schro¨dinger equation and Kepler system) with momentum conservation,the discontinuous finite element methods preserve momentum at nodes.These properti...     

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.     

Estimation of the relative error for the averaging method

A quasilinear ordinary differential system is considered. It and the corresponding averaged system have the same equilibrium point. With some assumptions of regularity and stability, it is shown that the relative error due to the averaging method, tends to zero when the independent variable tends to infinite. The capital point lies in the choice of the two solutions which are compared.     

Improved averaging method for turbulent flow simulation. Part II: Calculations and verification

This is the second of two articles intended to develop, apply and verify a new method for averaging the momentum and mass transport equations for turbulence. Part I presented the theoretical development of a new space-time filter (STF) averaging procedure. The new method, as well as all existing averaging procedures, are applied to the one-dimensional transient equations of momentum and scalar transport in a Burgers' flow field. Dense-grid ‘exact’ results from the unaveraged equations are presented to depict the dynamic behaviour of the flow field and serve as a basis for verifying the coarse-grid STF predictions. In this paper, a finite difference procedure is used to numerically solve the new STF averaged equations, as well as the other forms of the averaged equations derived in Part I. All averaged equations are solved on the same coarse grid. The velocity and scalar fields, predicted from each equation form, are intercompared according to a verification procedure based on the statistical and spectral properties of the results. It is found that the new STF procedure improves coarse-grid dynamic predictions over the existing methods of averaging.