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We apply the Cameron—Martin—Wiener (formerly ‘Wiener—Hermite’) expansion of a random velocity field to the analytical study of turbulence. The kernels of this expansion contain all statistical information about the ensemble. Complete expressions are derived for constructing statistical quantities in terms of the kernels, and for the equations of motion of the kernels. We rigorously prove the Gaussian trend of the velocity field of the Navier—Stokes equation in the very late stage when the non-linear term is neglected. The n -dependence ( n is the order of derivative) of the flatness factor, minus three for derivatives of the velocity field, shows a rapid increase with n in this stage.
The late decay problem of the Burgers model of turbulence is studied analytically with a view to obtaining suggestive guidelines for fitting the non-linear aspects of the model turbulence. We can divide the energy spectrum density into two parts, the larger of which is a kind of steady solution, which we call the ‘equilibrium state’, which remains self-similar in time in terms of an appropriate variable. The deviation from this ‘equilibrium solution’ satisfies the Kármán—Howarth equation. As initial velocity field, we take two particular cases: ( a ) a pure Gaussian, and ( b ) a non-Gaussian velocity field. With these two cases a detailed spectral analysis has been obtained. The energy spectrum deviation from ‘equilibrium’ declines exponentially to zero for all wave-numbers. The Gaussian case shows that the flatness factor minus three increases rapidly with n , while the non-Gaussian case does not show any marked dependence on n .

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... So, developing approximate techniques through which approximate statistical moments can be obtained, is an important and necessary work. Since Meecham and his co-workers[2]developed a theory of turbulence involving a truncated Wiener-Hermite expansion (WHE) of the velocity field, many authors studied problems concerning turbulence[3][4][5][6][7][8]. A lot of general applications in fluid mechanics were also studied in[9][10][11]. ...

... S th ince Meecham and his co-workers eory of turbulence involving a truncated Wiener-Hermite expansion (WHE) of the velocity field, many authors studied problems concerning turbulence2627. A lot of general applications in fluid mechanics was also studied in [28]. Scattering problems attracted the WHE applications through many authors [29]. ...

In this paper, the cubic and quintic diffusion equation under stochastic non homogeneity is solved using Wiener-Hermite expansion and perturbation (WHEP) technique, Homotopy perturbation method (HPM) and Pickard approximation technique. The analytic solution of the linear case is obtained using Eigenfunction expansion .The Picard approximation method is used to introduce the first and second order approximate solution for the non linear case. The WHEP technique is also used to obtain approximate solution under dif-ferent orders and different corrections. The Homotopy perturbation method (HPM) is also used to obtain some approximation orders for mean and variance. Using mathematica-5, the methods of solution are illus-trated through figures, comparisons among different methods and some parametric studies.

... Since Meecham and his co-workers [2] developed a theory of turbulence involving a truncated Wiener-Hermite expansion (WHE) of the velocity field, many authors studied problems concerning turbulence345678 . The nonlinear oscillators were considered as an opened area for the applications of WHE as can be found in9101112131415 . ...

In this paper, nonlinear oscillators under cubic nonlinearity with stochastic inputs are considered. Two techniques are used to introduce approximate solutions, mainly; the Wiener‐Hermite expansion and perturbation (WHEP) technique and the homotopy perturbation method (HPM). Some statistical moments are computed using the two methods with the big aid of symbolic computations of mathematica‐5. Comparisons are illustrated through figures.

... Stochastic Navier-Stokes equations were studied theoretically by Wiener chaos in the recent work of Mikulevicius & Rozovskii (2004). However, in certain applications there are some serious limitations of Wiener chaos and several papers in the literature have focused on this (see Orszag & Bissonnette 1967;Crow & Canavan 1970;Canavan 1970;Kahng & Siegel 1970;Chorin 1974;Hogge & Meecham 1978). Generalized polynomial chaos, including Wiener chaos as a subset, converges to a secondorder random process exponentially (p-convergence) with respect to a certain PDF. ...

We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien–Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter $E$ for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of $E$ on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of $E$, which are skewed with their peaks increasing towards larger values of $E$ as the Reynolds number increases. We then study the effect of multiple frequencies described by a periodically correlated random process. We find that the mean value of $E$ is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing ‘smart’ PDFs of the stochastic input that can result in improved heat transfer enhancement rates.

The most fundamental change in Physics this century has been in the role of the observer. One consequence of the latter is symmetry, i. e., that two observers must perceive the same physical reality structure, as in relativity, and a second consequence is Quantum Mechanics. What we propose here is that an underlying nonlinear partial differential equation (PDE) can describe high-energy physics phenomena.

Wiener-Hermite expansion linked with perturbation technique (WHEP) is used to solve the stochastic tow-dimensions non-linear Navier-Stokes equations. An approximate formula for the ensemble average, variance and some higher statistical moments of the stochastic solution process are obtained using WHEP technique and some cases study are considered to illustrate the method of analysis.

Turbulence studies were initiated to solve the problems of heat and mass transfer from the earth's surface. This chapter discusses the importance of the statistical approach in wall turbulence experimental studies. The existing experimental techniques are reviewed from this standpoint. Turbulence is essentially a statistical phenomenon. All the quantities involved are therefore, random variables characterized by the corresponding probability distributions. The complete solution of the turbulence problem would consist of the determination of time evolution of probability distribution of the hydrodynamic, enthalpy, and other fields starting from a known set of distribution functions at the initial moment and employing the relevant physical laws. The physical and mathematical models of the wall turbulence rely on experimental evidence. Integral prediction methods and simple differential procedures, such as the mixing length concept, are based on the experimental data on the characteristics of the mean motion, such as wall shear stress and mean velocity distributions. More elaborate models require experimental evidence of second or higher order statistical moments of the relevant quantities.

The literature relating to the one-dimensional Burgers equation is surveyed. About thirty-five distinct solutions of this equation are classified in tabular form. The physically interesting cases are illustrated by means of isochronal graphs.

A statistical framework of weak turbulence is applied to investigate the maintenance of atmospheric turbulence during a long period, even after the external energy supply from solar radiations has stopped. Thus, the problem of hydrodynamical turbulence without any mean motion is dealt with. Main attention is drawn to one-dimensional Burgers equation, together with a discussion devoted mainly to Millionshtchikov's hypothesis, which may be applied as a consequence of the assumed weak turbulence. Remarkably, this leads to an explicit proof of the existence of the ‘cascade process’ in turbulence. The results are illustrated by numerical calculations.

In this paper, a stochastic perturbed nonlinear diffusion equation is studied under a stochastic nonlinear nonhomogeneity. The Pickard approximation method is used to introduce a reference first order approximate solution. Under different correction levels, the WHEP technique is used to obtain approximate solutions. Using Mathematica-5, the solution algorithm is operated and several comparisons among correction levels together with error curves have been demonstrated. The method of solution is illustrated through case studies and figures.

In this paper, nonlinear oscillators under quadratic nonlinearity with stochastic inputs are considered. Different methods are used to obtain first order approximations, namely; the WHEP technique, the perturbation method, the Pickard approximations, the Adomian decompositions and the homotopy perturbation method (HPM). Some statistical moments are computed for the different methods using mathematica 5. Comparisons are illustrated through figures for different case-studies.

The stochastic solution to the Burgers' equation evolving from a random initial condition is obtained numerically by means of a conservative difference scheme. The performance of the scheme is checked by means of a number of mandatory numerical tests involving different mesh sizes. Various multi-point statistical characteristics, e.g. three-point third-order correlation and two-point third- and fourth-order correlations, are calculated from the random solution and shown graphically. The results for the correlation function and normalized spectrum are compared with earlier findings from numerical simulations of other authors performed with different numerical techniques and the agreement is quantitatively very good. A kind of stochastic self-similarity is observed for large times. In the end the usefulness of the data gathered is demonstrated through comparing the results for the statistical properties from the direct numerical simulation with the predictions of so-called ‘random-point approximation’ and the agreement turns out to be fully satisfactory.

In this paper, nonlinear oscillators under mixed quadratic and cubic nonlinearities with stochastic inputs are considered. Different methods are used to obtain second order approximations, namely; the Wiener–Hermite and perturbation (WHEP) technique and the homotopy perturbation method (HPM). Some statistical moments are computed for the different methods using mathematica 5. Comparisons are illustrated through figures for different case-studies.

In this paper, a nonlinear diffusion equation is studied under stochastic nonhomogeneity through homogeneous boundary conditions.
The analytical solution for the linear case is obtained using the eigenfunction expansion. The Pickard approximation method
is used to introduce a first order approximate solution for the nonlinear case. The WHEP technique is also used to obtain
approximate solution under different orders and different corrections. Using Mathematica-5, the solution algorithm is operated
through first order approximation. The method of solution is illustrated through case studies and figures.

The Wiener-Hermite expansion linked with perturbation technique (WHEP) was used to solve perturbed non-linear stochastic differential
equations. In this article, the homotopy perturbation method is used instead of the conventional perturbation methods which
generalizes the WHEP technique such that it can be applied on non-linear stochastic differential equations without the necessity
of the presence of the small parameter. The technique is called homotopy WHEP and is demonstrated through many non-linear
problems.

DOI:https://doi.org/10.1103/RevModPhys.17.323

The symmetry properties of the Cameron-Martin-Wiener (Wiener-Hermite) kernels are investigated in the isotropic velocity field as required for the invariance of the velocity distribution of the field under rigid rotation and reflection. As a sufficient condition for the invariance, it is found that the even kernels must be isotropic true tensors, and the odd kernels (i) isotropic true tensors or (ii) isotropic pseudotensors. In incompressible turbulent flow, it is shown that the odd kernels must be isotropic pseudotensors. Assuming that the nonlinear term of Burgers' equation can be treated as a perturbation in the very late decay stage, it is shown that the energy spectrum density must vanish at zero wavenumber in order to get a valid physical model of turbulence. Applying the above conclusion, it is seen that the odd kernels in the Burgers' model must be odd functions, which suggests that the model thus obtained is the one-dimensional counterpart of turbulent flow which is incompressible.

A Wiener-Hermite functional expansion is used to treat a random initial value process involving the Burgers model equation. The nonlinear model equation has many of the characteristics of the Navier-Stokes equation. It is found that the functional expansion converges better the larger the separation variable in the correlation function (the nearer to joint normal is the distribution). To the present order, the treatment is similar to a quasinormal assumption. The computations show that the correlation function quickly approaches an equilibrium form for quite different initial values. The power spectrum function approaches an equilibrium form also, where it falls off like the inverse second power of the wavenumber.

Higher‐order time‐correlations and the associated skewnesses and flatnesses were measured in a turbulent field downstream of a grid using high‐speed computing techniques. The results were obtained using samples of 160 020 digitized data recorded at time intervals of 1∕12 800 sec during time periods of approximately 12.5 sec. Comparison is made between the measured correlations and the higher‐order correlation curves corresponding to a Gaussian probability density distribution of turbulent velocities. The departures from Gaussianity are shown, and non‐Gaussian probability distributions are proposed which correspond considerably better to experimental reality. Several relations between correlation coefficients of different orders are obtained for the non‐Gaussian probability distributions and confirmed by comparison with the measured correlations, skewnesses, and flatnesses.

The Wiener‐Hermite functional expansion, which is the expansion of a random function about a Gaussian function, is here substituted into the Burgers one‐dimensional model equation of turbulence. The result is a hierarchy of equations which (along with initial conditions) determine the kernel functions which play the role of expansion coefficients in the series. Initial conditions are postulated, based on physical reasoning, criteria of simplicity, and the assumption that the series is to represent the late decay stage (in which the Gaussian correction is small and also decreasing with time). These are shown to justify an iterative solution to the equations. The first correction to the Gaussian approximation is calculated. This is then tested by evaluating the correction to the flatness factor, which for an exactly Gaussian function has the value 3, but which has been found by experiment (in real three‐dimensional fluids, of course) to have a value which deviates from the Gaussian value increasingly rapidly with the order of the derivative. We utilize this effect as a test of the inherent ability of the Wiener‐Hermite expansion to bring to realization the physical properties implicit in the Navier‐Stokes or Burgers equations. The various contributions to the flatness‐factor deviation, when computed, do show a potential capability of providing a theoretical basis for the effect.

A new definition is given for the ``ideal random function'' (derivative of the Wiener function), which separates out infinite factors by fullest exploitation of the possibilities of the Dirac delta function. By allowing all integrals to be written formally as sums, this facilitates the definition and manipulation of the Wiener‐Hermite functionals, especially for vector random processes of multiple argument. Expansion of a random function in Wiener‐Hermite functionals is discussed. An expression is derived for the expectation value of the product of any number of Wiener‐Hermite functionals; this is all that is needed in principle to obtain full statistical information from the Wiener‐Hermite functional expansion of a random function. The method is illustrated by the calculation of the first correction to the flatness factor (measure of Gaussianity) of a nearly‐Gaussian random function.

The triple velocity correlation, in turbulence produced by inserting a square-mesh grid near the beginning of the working section of a wind tunnel, has been measured for mesh Reynolds numbers of RM = 5300, 21,200 and 42,400 (RM = UM/ν, where U is the mean wind speed in the working section of the tunnel and M is the centre to centre spacing of the rods making up the grid; ν is the kinematic viscosity of air). At the lowest Reynolds number the correlation has been measured at distances downstream of the grid varying from 20 to 120M. This range covers practically all of the initial period of the decay of turbulence, where the turbulent intensity varies as t−1.

By electrical analysis of the output from a hot-wire anemometer, it is possible to measure rapidly and accurately all the quantities appearing in the theoretical equations for the decay of isotropic turbulence. The technique for these measurements is described, and possible extensions and limitations of the method are discussed. Measurements have been made of the second and fourth derivatives of the longitudinal double-velocity correlation at the origin, the third derivative of the longitudinal triple-velocity correlation at the origin, the statistical distribution in time of the velocity fluctuations, and the integral scale of turbulence.(Received February 24 1947)

The turbulence problem is formulated using the Wiener stochastic expansion. The expansion is useful for processes which are in some sense nearly normal, and can be used for non-linear non-Gaussian processes such as many turbulent fluid flows. Here we present the general formulation for statistically inhomogeneous and anistropic processes.
The transfer term in the energy equation, or equivalently the third-order velocity correlation, forms a sensitive measure of the amount of non-Gaussianity present in real fluid flows. Experimental evidence shows that in many flows this component is small compared with the Gaussian part. It is shown that a homogeneous and isotropic flow which has but a small non-Gaussian part possesses a distribution at one point which is Gaussian to terms of second order. The experiments suggest that immediately behind a grid in a wind tunnel the flow is very nearly normal. The non-Gaussian part grows at a moderate rate, at least within the range of distance downstream (or decay time) available in the usual experiments. This growth is probably due to the relative increase in the amount of energy in the smallest eddies, which are non-normal.
A necessary criterion for the validity of the zero-fourth-cumulant approximation is suggested: the transfer term in dimensionless form should be small. It is shown that calculations using the zero-fourth-cumulant approximation have given negative energy spectra when this condition is violated, probably for the reason that the process is no longer nearly Gaussian. However, even when this condition is fulfilled, it is shown that that approximation must be corrected.
It is suggested that the present theory is valid for quite large times of decay if initial energy spectra are chosen which are not too far from the actual physical values for fluid in turbulent flow. Equations are given for the next-higher-order term in a nearly normal approximation. The expansion is also used in § 6 to describe turbulent mixing problems and is compared with the zero-fourth-cumulant approximation for these problems. Computational results are presented in § 7 and compared with experiments by Stewart and Townsend.

The Theory of Homogeneous Turbulence

- G K Batchelor

Air Force Office of Scientific Research (see also

- W-H Kahng
- A Siegel