Hopf Functional Equation Research Articles

Study of the Hopf functional equation for turbulence: Duhamel principle and dynamical scaling.

We consider a formulation for the Hopf functional differential equation which governs statistical solutions of the Navier-Stokes equations. By introducing an exponential operator with a functional derivative, we recast the Hopf equation as an integro-differential functional equation by the Duhamel principle. On this basis we introduce a successive approximation to the Hopf equation. As an illustration we take the Burgers equation and carry out the approximations to the leading order. Scale invariance of the statistical Navier-Stokes equations in d dimensions is formulated and contrasted with that of the deterministic Navier-Stokes equations. For the statistical Navier-Stokes equations, critical scale invariance is achieved for the characteristic functional of the dth derivative of the vector potential in d dimensions. The deterministic equations corresponding to this choice of the dependent variable acquire the linear Fokker-Planck operator under dynamic scaling. In three dimensions it is the vorticity gradient that behaves like a fundamental solution (more precisely, source-type solution) of deterministic Navier-Stokes equations in the long-time limit. Physical applications of these ideas include study of a self-similar decaying profile of fluid flows. Moreover, we reveal typical physical properties in the late-stage evolution by combining statistical scale invariance and the source-type solution. This yields an asymptotic form of the Hopf functional in the long-time limit, improving the well-known Hopf-Titt solution. In particular, we present analyses for the Burgers equations to illustrate the main ideas and indicate a similar analysis for the Navier-Stokes equations.

Symmetry analysis and invariant solutions of the multipoint infinite systems describing turbulence

The present work concerns multipoint description of turbulence in terms of the probability density functions (pdf's) and the characteristic Hopf functional. Lie symmetries of infinite systems of equations for the pdf's and the Hopf functional equation are discussed. Based on symmetries, invariant solutions for turbulence statistics are calculated.

Analysis of symmetric and skew-symmetric weight functions for a semi-infinite interfacial crack in transversely isotropic piezoelectric bimaterials

This paper presents an analysis of symmetric and skew-symmetric weight functions for in-plane interfacial crack problems in piezoelectric bimaterials. The symmetric weight function matrix is obtained by means of the solution of a Wiener–Hopf functional equation and the skew-symmetric weight function matrix is developed by the construction of a corresponding full field solution. The explicit expressions for the symmetric and skew-symmetric weight functions are given and applied to in-plane deformation problems for a semi-infinite crack between two dissimilar piezoelectric materials. The validity of the present method in the determination of the field intensity factors is demonstrated by illustrative examples, in which both piezoelectric bimaterials and piezoelectric/elastic bimaterials are considered. It is shown that, for the present interfacial crack problem, both symmetric and skew-symmetric weight function matrices are necessary in the general integral formula for the evaluation of field intensity factors, and that the contribution of the skew-symmetric element of the applied load is not negligible in fracture analysis.

Stroh formalism in analysis of skew-symmetric and symmetric weight functions for interfacial cracks

The focus of the paper is on the analysis of skew-symmetric weight functions for interfacial cracks in two-dimensional anisotropic solids. It is shown that the Stroh formalism proves to be an efficient tool for this challenging task. Conventionally, the weight functions, both symmetric and skew-symmetric, can be identified as non-trivial singular solutions of a homogeneous boundary-value problem for a solid with a crack. For a semi-infinite crack, the problem can be reduced to solving a matrix Wiener–Hopf functional equation. Instead, the Stroh matrix representation of displacements and tractions, combined with a Riemann–Hilbert formulation, is used to obtain an algebraic eigenvalue problem, which is solved in a closed form. The proposed general method is applied to the case of a quasi-static semi-infinite crack propagating between two dissimilar orthotropic media: explicit expressions for the weight functions are evaluated and then used in the computation of the complex stress intensity factor corresponding to a general distribution of forces acting on the crack faces.

Symmetric and skew-symmetric weight functions in 2D perturbation models for semi-infinite interfacial cracks

In this paper we address the vector problem of a 2D half-plane interfacial crack loaded by a general asymmetric distribution of forces acting on its faces. It is shown that the general integral formula for the evaluation of stress intensity factors, as well as high-order terms, requires both symmetric and skew-symmetric weight function matrices. The symmetric weight function matrix is obtained via the solution of a Wiener–Hopf functional equation, whereas the derivation of the skew-symmetric weight function matrix requires the construction of the corresponding full field singular solution. The weight function matrices are then used in the perturbation analysis of a crack advancing quasi-statically along the interface between two dissimilar media. A general and rigorous asymptotic procedure is developed to compute the perturbations of stress intensity factors as well as high-order terms.

Diffraction of SH-waves on interface cracks under the action of concentrated forces

In the mathematically rigorous statement, we solve the problem of diffraction of elastic SH-waves on a finite (semiinfinite) crack located on the plane boundary of two perfectly joined materials. As a sounding field, we consider the field of a point source. The problem is reduced to the Wiener – Hopf functional equation, which is solved by the method of factorization. An approximate solution of this equation is obtained. We numerically study the distribution of the field in the radiation zone and the distinctive features of its formation depending on the location of the source of radiation.

On a Fourier space master equation for Navier-Stokes turbulence

Three-dimensional homogeneous isotropic turbulence is formulated in terms of a discrete stochastic process in Fourier space. The time-dependent joint probability distribution of the stochastic Fourier modes is governed by a multivariate master equation. It is demonstrated that the characteristic functional of the stochastic process obeys the Hopf functional equation. As a first application of the method stochastic simulations of the Burgers's turbulence model are performed and shown to yield the expected energy spectrum.

On the Solution of Wiener--Hopf Problems Involving Noncommutative Matrix Kernel Decompositions

Many problems in physics and engineering with semi-infinite boundaries or interfaces are exactly solvable by the Wiener--Hopf technique. It has been used successfully in a multitude of different disciplines when the Wiener--Hopf functional equation contains a single scalar kernel. For complex boundary value problems, however, the procedure often leads to coupled equations which therefore have a kernel of matrix form. The key step in the technique is to decompose the kernel into a product of two functions, one analytic in an upper region of a complex (transform) plane and the other analytic in an overlapping lower half-plane. This is straightforward for scalar kernels but no method has yet been proposed for general matrices. In this article a new procedure is introduced whereby Padé approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel. As well as being simple to apply, the use of approximants allows the accuracy of the factorization to be increased almost indefinitely. The method is demonstrated by way of example. Scattering of acoustic waves by a semi-infinite screen at the interface between two compressible media with different physical properties is examined in detail. Numerical evaluation of the approximate factorizations together with results on an upper bound of the absolute error reveal that convergence with increasing Padé number is extremely rapid. The procedure is computationally efficient and of simple analytic form and offers high accuracy (error $ < 10^{-7}\%$ typically). The method is applicable to a wide range of initial or boundary value problems.

A master equation representation of two-dimensional turbulence

A stochastic representation of the turbulent dynamics of the two-dimensional Navier-Stokes equation is developed. This stochastic formulation interprets the fluctuating velocity field as a discrete, multivariate stochastic process which is governed by a master equation. By deriving the Hopf functional equation of statistical fluid mechanics it is shown that this approach yields a complete description of the stochastic properties of the turbulent velocity field. On the basis of the multivariate master equation a stochastic simulation method for the two-dimensional Navier-Stokes equation is constructed. This method is applied to the stochastic simulation of high-Reynolds-number turbulent flows.

Burgers's turbulence model as a stochastic dynamical system: Master equation and simulation.

By means of Burgers's equation a stochastic description of turbulent fluid flows is explained which is based on a discrete master equation. The latter governs the dynamics of a discrete multivariate stochastic process representing the random velocity field of the fluid. From the characteristic function corresponding to this stochastic process, the Hopf functional equation of turbulence is obtained. This implies that the infinite hierarchy of correlation functions can be derived from the master equation. The master-equation description naturally leads to a simple stochastic simulation algorithm which is well suited for numerical implementation. Stochastic simulations of the Burgers model of turbulence are performed and are shown to yield very accurate results.

Stationary turbulent closure via the Hopf functional equation

Exact closed-form solutions are exhibited for the Hopf equation for stationary incompressible 3D Navier-Stokes flow, for the cases of homogeneous forced flow (including a solution with depleted nonlinearity) and inhomogeneous flow with arbitrary boundary conditions. This provides an exact method for computing two- and higher-point moments, given the mean flow.

Diffraction from the juncture of pressure release and locally reacting half-planes

Solutions to two coupled half-plane problems are derived. In the first, an exact solution to the canonical problem of acoustic plane-wave diffraction from a pressure release (zero impedance) surface coupled to a perfectly rigid (infinite impedance) surface is given using the Wiener–Hopf method. The solution is expressed in terms of an angular spectrum of plane waves. In the second, the rigid surface is replaced by one characterized by a locally reacting finite impedance. Here, an approximate kernel to the Wiener–Hopf functional equation is used which leads to a complete and readily interpretable solution. This solution is expressed in terms of the canonical angular spectrum for the first problem multiplied by a function that depends on material parameters of the locally reacting surface. Polar plots of the far-field amplitude level for the diffracted field are presented.

Functional calculus in strong plasma turbulence

The theory of electrostatic plasma turbulence is considered. The basic equations for the dynamics of the hierarchy of the moment equations are derived and the difficulty of the closure problem for strong plasma turbulence is discussed. The characteristic functional in phase space is introduced and its relations to the correlation functions are described. The Hopf functional equation for dynamics of the characteristic functional is derived, and its equivalence to the hierarchy of the moment equations is established. Similar formulations were carried out in velocity-wave vector space. characteristic functional are considered and their relationships are studied. An approximate solution for Hopf's equation for the nearly normal turbulence is obtained which is shown to predict diffusion of the mean distribution function in velocity space.

Elastic equilibrium of a wedge with a crack: PMM vol. 43, no. 1, 1979, pp 153–159

There is examined the plane homogeneous problem of elasticity theory on the equilibrium of an infinite wedge in whose bisector plane there is an inner rectilinear semi-infinite crack. The wedge faces and the crack edges are load-free. The stresses tend to zero at infinity but their principal vector and principal moment differ from zero and are given by a condition. The problem is reduced to solving a Wiener — Hopf functional equation by using a Mellin transformation. An exact solution of the equation is given and the stress intensity coefficient at the crack vertex is calculated.

Ensemble mechanics for the random-forced Navier-Stokes flow

It is shown that the so-called statistical theory of turbulence, which can be exactly incorporated by the Hopf functional equation, is imperfect in that it fails to ensure the irreversible approach to a unique ultimate steady state of turbulence (for a steady boundary condition) expected from observation; and that this imperfection is removed if a stochastic random-force term is added into the Navier-Stokes equation. The ensemble mechanics for the random-forced Navier-Stokes flow is formulated by taking into account the natural random force, which has usually been neglected in the Navier-Stokes equation.

An approximate method for solving Hopf's equation of Burger's model of turbulence

The Hopf functional equation of Burger's model of turbulence is considered and the Lewis-Kraichnan version of the functional formulation is discussed. An approximate method based on series expansion of natural logarithm of the characteristic functional is presented. The equations governing the hierarchy of cumulants are derived. Several possible methods of closing the system of equations are discussed.

Functional Calculus of a Transferable Scalar in a Turbulent Flow

Abstract The problem of dispersion of a transferable scalar in a turbulent fluid is considered. The Hopf functional equation for the characteristic functional of the velocity vector and transferable scalar is derived and discussed. The special case of heat conduction with random initial condition is studied in detail. It is shown that the Hopf equation accepts a Gaussian characteristic functional solution and several examples are discussed.