Wiener-Hermite Expansion Research Articles

The Wiener-Hermite expansion applied to decaying isotropic turbulence using a renormalized time-dependent base

The problem of decaying isotropic turbulence has been studied using a Wiener-Hermite expansion with a renormalized time-dependent base. The theory is largely deductive and uses no modelling approximations. It has been found that many properties of large Reynolds number turbulence can be calculated (at least for moderate time) using the moving-base expansion alone. Such properties found are the spectrum shape in the dissipation range, the Kolmogorov constant, and the energy cascade in the inertial subrange. Furthermore, by using a renormalization scheme, it is possible to extend the calculation to larger times and to initial conditions significantly different from the equilibrium form. If the initial spectrum is the Kolmogorov spectrum perturbed with a spike or dip in the inertial subrange, the process proceeds to eliminate the perturbation and relax to the preferred spectrum shape. The turbulence decays with the proper dissipation rate and several other properties are found to agree with measured data. The theory is also used to calculate the energy transfer and the flatness factor of turbulence.

Nonlinear Stochastic Differential Equations in Statistical Physics and Integral Representation

Integral representation (IR), or the so-called ·wiener-Hermite expansion, is proposed as a new method of solving a class of nonlinear stochastic differential equations (NLSDE's) that appear in the theory of Brownian motion of anharmonic oscillators, semi-classical treatment of laser oscillation or in the Kraichnan-Wyld formulation of turbulence. The basic procedure of application of IR is established and simple NLSDE's are analyzed. Results for power spectral density agree well with numerical experiment data even in strongly nonlinear cases. It is also shown that IR is a completed form of renormalized perturbation expansion, and provides a systematic way to improve various approximation schemes such as fourth cumulant discard approximation or equivalent linearization procedure.

An Operator Formalism to the Path Integral Method

An operator formalism to the path integral method is developed. In this formalism the operator characterizing the trajectories of the particle is introduced and the usual path integral is expressed in terms of expectation value of a functional of that operator. We also investigate the Wiener-Hermite expansion on the basis of our operator formalism. Feynman's path integral is justified as a sort of analytic continuation of the path integral in the Wiener process.

Quasi-Probability for Three-Level System and Superradiance

A quasi-probability distribution function based on the coherent state for U(3) is intro­ duced, and its properties are then investigated. With the aid of this function it is shown that the co-operative spontaneous emission from a system of the three-level atoms can be described as a diffusion-like process on C 2 • In particular, the limit of validity of the semi­ classical approach is examined. The case in which the semiclassical approach is not applicable can also be treated approximately by means of the Wiener-Hermite expansion method.

Burgers’ model with a renormalized Wiener–Hermite representation

The use of the Wiener–Hermite expansion for the turbulence problem is reviewed. The expansion is known to give good results for lower Reynolds’ number flows, up to a fluctuation Reynolds’ number of 20 using more recent time-dependent bases. The use and meaning of these bases is discussed. A new development in Wiener–Hermite expansion is used to calculate Burgers’ model: It is known that these expansions are not unique; by taking advantage of this arbitrariness, a renormalization is presented and used to improve the convergence of the expansion. The procedure involves calculation for turbulence using a time-dependent base for a short time. The calculation is then stopped and the resulting functions are adjusted in such a way as to minimize the non-Gaussian part of the energy, at the same time preserving the last value of the transfer function. Following this, the calculation is resumed using these new, adjusted values for the functions; then the process is repeated. For the first time it was possible to obtain the equilibrium form k−2, of the energy spectrum for Burgers’ model of turbulence. The calculation proceeds, holding the non-Gaussian part of the energy to but a few percent. Good results are obtained up to fluctuation Reynolds’ numbers of 100.

Wiener-Hermite expansion and the inertial subrange of a homogeneous isotropic turbulence

The two-term truncated three-dimensional Wiener-Hermite expansion is applied to the study of the inertial subrange energy spectrum of a homogeneous isotropic turbulence. It is found that this expansion, up to the second term, will predict Kolmogoroff's inverse five-thirds law if the relaxation time is modified according to Kolmogoroff's second hypothesis. It is found that in the inertial subrange the energy spectrum of the non-Gaussian part is much smaller than that of the Gaussian part.

Gaussian fields and random flow

The high-frequency component of the random solution of a model problem is shown to be statistically orthogonal to the Gaussian component. This is shown to be a consequence of the existence of an equilibrium range. It is concluded that random flow fields can be viewed as being approximately Gaussian only in a very special sense and, in particular, that Wiener–Hermite expansions can provide a useful description only of large-scale hydrodynamical phenomena.

Quasi-Stable Energy Spectrum of Isotropic Turbulence

The Wiener-Hermite expansion with the time-dependent ideal random function is applied to the nearly Gaussian isotropic turbulence. The first two terms of the expansion are used for the numerical calculations of the initial Gaussian distributions whose energy spectra are not so different from the expected quasi-stable spectrum. The kinematic viscosity is set at 0.01. The characteristic wave number and velocity are considered to be of the order of 0.5 and 1 respectively in these initial conditions. It is shown that the variations of the shapes of the energy spectra, starting from these initial conditions, support Kolmogoroff's quasi-stable spectrum E ( k )∼ k -5/3 in the wave number range 1< k <2.8. E ( k )∼ k 2 in the range k ∼0 and E ( k )∼ k -σ (4<σ<5) in the range 8< k <11, are also suggested although the inclination is not so definite as in the above range.

Time-Dependent Wiener-Hermite Expansion for the Nearly Gaussian Turbulence in the Burgers' System

The proper choice of the time-dependent ideal random function improves the approximation of the truncated Wiener-Hermite expansion in the Burgers' system. The quasistable energy spectrum E(k)∼k−2 is supported.

Wiener-Hermite Expansion Applied to Passive Scalar Dispersion in a Non-Uniform Turbulent Flow

Abstract The dispersion of a passive scalar advected by a turbulent incompressible flow characterized by inhomogeneous and nonstationary statistics is investigated by expanding the velocity and scalar concentration fields in Wiener-Hermite functions. The time evolution equation for the ensemble mean concentration is determined in terms of Eulerian one-point, two-time and one-point, three-time correlation functions defined in the coordinate frame moving with the mean velocity. The example of a plane-parallel mean flow is studied in detail and eddy diffusivity tensors are extracted from the formalism.

Time-Dependent Wiener-Hermite Base for Turbulence

A Wiener-Hermite expansion with a time-dependent base for turbulence is tested on Burgers' equation. It is shown that the time-dependent base system improves the rate of series convergence and some useful results can be obtained. The moving-base system, developed by Bodner, allows the equipartition solution to be satisfied exactly with a two-term expansion, and is a modification of Bodner's moving base system for real fluid turbulence.

The Wiener-Hermite Expansion with Time-Dependent Ideal Random Function. II

The Wiener-Hermite Expansion with Time-Dependent Ideal Random Function. II

Relationship between a Wiener–Hermite expansion and an energy cascade

Meecham and his co-workers have developed a theory of turbulence involving a truncated Wiener–Hermite expansion of the velocity field. The randomness is taken up by a white-noise function associated, in the original version of the theory, with the initial state of the flow. The mechanical problem then reduces to a set of coupled integro-differential equations for deterministic kernels. We have solved numerically an analogous set for Burgers's model equation and have computed, for the sake of comparison, actual random solutions of the Burgers equation. We find that the theory based on the first two terms of the Wiener–Hermite expansion predicts an insufficient rate of energy decay for Reynolds numbers larger than two, because the equations for the kernels contain no convolution integrals in wave-number space and therefore permit no cascade of energy. An energy cascade in wave-number space corresponds to a cascade up through successive terms of the Wiener-Hermite expansion. Pictures of the Gaussian and non-Gaussian components of an actual solution of the Burgers equation show directly that only higher-order terms in the Wiener–Hermite expansion are capable of representing shocks, which dissipate the energy. Higher-order terms would be needed even for a nearly Gaussian field of evolving three-dimensional turbulence. ‘Gaussianity’, in the experimentalist's sense, has no bearing on the rate of convergence of a Wiener–Hermite expansion whose white-noise function is associated with the initial state. Such an expansion would converge only if the velocity field and its initial state were joint-normally distributed. The question whether a time-varying white-noise function can speed the convergence is treated in the paper following this one.

Some properties of a Lagrangian Wiener–Hermite expansion

Wiener proposed that the turbulent velocity be expanded in Hermite functionals of a Gaussian white noise random function advected by the fluid. This paper describes the mechanics of converting his suggestion into a computable model, and assesses its range of validity as an approximation for incompressible, homogeneous, and isotropic turbulence. The terms retained are a linear term, v1, representing an arbitrary Gaussian velocity, and a quadratic term, v2, representing a non-Gaussian contribution to the velocity needed for energy transfer. The requirement that advection by the dependent velocity v = v1 + v2 does not alter the statistics of the base necessitates a further truncation of the base to antisymmetric quadratic basis elements. Realizability of any statistics of v is common to all Wiener–Hermite expansions. The projected equations for the Lagrangian expansion conserve energy by non-linear interaction, preserve the inviscid Gaussian equipartition ensemble, and are invariant to random Galilean transformations. Numerical calculations with an approximate form of these equations reveal that irreversible relaxation to the inviscid equipartition solution is not a property of the Lagrangian model, and that the rapid convergence advanced as the original motivation for studying Wiener–Hermite expansions does not survive closure by truncation. The dynamics of the model is not inconsistent with the existence of an inertial range. A simple numerical search routine failed to produce a solution corresponding to such an equilibrium ensemble.

Equilibrium characteristics of nearly normal turbulence

We discuss some consequences of assuming that two different non-linear model equations, and real turbulence are nearly Gaussian. It is supposed when necessary that the process is driven and it is supposed that the processes have become statistically stationary. These problems are discussed from the viewpoint of the Wiener–Hermite expansion for non-linear, nearly Gaussian processes. Expected equilibria forms are related to corresponding expressions obtained from the zero-fourth-cumulant assumption. The spectrum for Burgers’ model and for incompressible fluid flow problems is found from this viewpoint to beE∼k−2. The kinematical properties leading to such spectra are discussed. It is noted, as has been remarked earlier, that this spectrum is characteristic of flows with near discontinuities. A conjecture is offered concerning how these discontinuities are related to Gaussianity.

Application of the Wiener-Hermite Expansion to the Diffusion of a Passive Scalar in a Homogeneous Turbulent Flow

The stochastic expansion of Cameron, Martin, and Wiener is used for the velocity and concentration fields in a homogeneous turbulent flow convecting a passive scalar. The expansion is truncated and analytical solutions are obtained for the concentration field with the velocity distribution given and molecular diffusion neglected. For diffusion from a point source, the effective diffusivity is found in terms of Eulerian velocity correlations. For a statistically homogeneous scalar distribution, the spectrum function is determined in terms of the energy spectrum function. When the Prandtl number is large, the results obtained do not agree with the predictions of the universal equilibrium hypothesis.

The Wiener-Hermite Expansion with Time-Dependent Ideal Random Function

Extension of the Wiener-Hermite expansion to a new one with time-dependent ideal random function is proposed. Exact Gaussian solutions for an incompressible inviscid fluid, which are not expressed in terms of the truncated usual Wiener-Hermite expansion, are expressed by a single term of the new expansion. Application of this formulation to the shear flow turbulence is discussed and also the possibility of application to the nearly Gaussian phenomena is remarked.

Use of the Wiener—Hermite expansion for nearly normal turbulence

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.

Dynamical Properties of Truncated Wiener-Hermite Expansions

The use of truncated Wiener-Hermite functional expansions as a basis for the theory of turbulence is critically examined. An account is given of the application of such expansions to Burgers' model equation. The nature of Wiener-Hermite expansions is such that certain important consistency requirements, such as realizability, follow immediately. In order to gain insight into other aspects of the predicted turbulence dynamics, a simpler model problem in which there are only three interacting modes is first studied. It is shown that Wiener-Hermite closures do not faithfully represent the dynamics for this latter model. Then, the analysis is extended to Burgers' equation. It is shown that Wiener-Hermite closures do not preserve fundamental properties of the exact dynamics in equilibrium. Inviscid equipartition solutions do not survive in the closures. In addition, it is shown that these closures do not treat the effects of large scales on small scales properly. Numerical calculations be extended to the corresponding theory of Navier-Stokes turbulence. We conclude that truncated Wiener-Hermite expansions are unsuitable for the theory of high Reynolds-number turbulence.

Statistical Initial-Value Problem for Burgers' Model Equation of Turbulence

Some statistical characteristics of the Burgers' model equation are studied using a high-speed computer. With the initial turbulent velocity field given, the closed form solution is used to find the fluid velocity field at later time. The velocity correlation functions, energy spectra, and other statistical properties are examined. The energy spectrum falls off like the inverse second power of the wave number (agreeing with earlier Wiener-Hermite expansion results). Calculations based on the use of the quasi-normal assumption are also presented. It is found, as is known for the Navier-Stokes equation, that for general initial values of the energy spectrum function the function becomes (unacceptably) negative at later times. However, it is found that the calculation yields positive spectra (though not those of the ``experiment'' above) at least for moderate times, if the spectrum found from the numerical experiment is used at initial times. This suggests that for real fluids the quasi-normal assumption may at least be stable if the initial condition is not too far from the correct one.