Vladimir N. Grebenev

GEOMETRIC REALIZATION OF THE TWO-POINT VELOCITY CORRELATION TENSOR FOR ISOTROPIC TURBULENCE

A new geometric view of homogeneous isotropic turbulence is contemplated employing the two-point velocity correlation tensor of the velocity fluctuations. We show that this correlation tensor generates a family of pseudo-Riemannian metrics. This enables us to specify the geometry of a singled out Eulerian fluid volume in a statistical sense. We expose the relationship of some geometric constructions with statistical quantities arising in turbulence.

A Chorin-Type Formula for Solutions to a Closure Model for the von Kármán–Howarth Equation

Abstract The article is devoted to studying the Millionshtchikov closure model (a particular case of a model introduced by Oberlack [14]) for isotropic turbulence dynamics which appears in the context of the theory of the von Kármán-Howarth equation. We write the model in an abstract form that enables us to apply the theory of contractive semigroups and then to present a solution to the initial-boundary value problem by Chorin-type formula.

Approximate Lie symmetries of the Navier-Stokes equations

Abstract In the framework of the theory of approximate transformation groups proposed by Baikov, Gaziziv and Ibragimov [1], the first-order approximate symmetry operator is calculated for the Navier-Stokes equations. The symmetries of the coupled system obtained by expanding the dependent variables of the Navier-Stokes equations in the perturbation series with respect to a small parameter (viscosity) are used to derive approximate symmetries in the sense by Baikov et al.

Explicit series solution of a closure model for the von Kármán-Howarth equation

The homotopy analysis method (HAM) is applied to a nonlinear ordinary differential equation (ODE) emerging from a closure model of the von Kármán–Howarth equation which models the decay of isotropic turbulence. In the infinite Reynolds number limit, the von Kármán–Howarth equation admits a symmetry reduction leading to the aforementioned one-parameter ODE. Though the latter equation is not fully integrable, it can be integrated once for two particular parameter values and, for one of these values, the relevant boundary conditions can also be satisfied. The key result of this paper is that for the generic case, HAM is employed such that solutions for arbitrary parameter values are derived. We obtain explicit analytical solutions by recursive formulas with constant coefficients, using some transformations of variables in order to express the solutions in polynomial form. We also prove that the Loitsyansky invariant is a conservation law for the asymptotic form of the original equation. 2000 Mathematics subject classification: primary 34B15; secondary 22E70.

Lie algebra methods for the applications to the statistical theory of turbulence

Abstract Approximate Lie symmetries of the Navier-Stokes equations are used for the applications to scaling phenomenon arising in turbulence. In particular, we show that the Lie symmetries of the Euler equations are inherited by the Navier-Stokes equations in the form of approximate symmetries that allows to involve the Reynolds number dependence into scaling laws. Moreover, the optimal systems of all finite-dimensional Lie subalgebras of the approximate symmetry transformations of the Navier-Stokes are constructed. We show how the scaling groups obtained can be used to introduce the Reynolds number dependence into scaling laws explicitly for stationary parallel turbulent shear flows. This is demonstrated in the framework of a new approach to derive scaling laws based on symmetry analysis [11]-[13].

Interfacial phenomenon for one-dimensional equation of forward-backward parabolic type

SummaryAn interfacial phenomenon for a class of the solutions of a nonlinear forward-backward parabolic equation in R × (0,T) is investigated. In general, short time-period of interfaces is considered. This inner analysis allows to construct on some time interval a solution of the Cauchy problem for certain initial data.

Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor

The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field Bij (x,x′,t) and isometries of the correlation space K3 equipped with a (pseudo-) Riemannian metrics ds2(t) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds2(t) as the semi-reducible pseudo-Riemannian metric. This construction presents the template for the application of methods of Riemannian geometry in turbulence to observe, in particular, the deformation of length scales of turbulent motion localized within a singled out fluid volume of the flow in time. This also allows to use common concepts and technics of Lagrangian mechanics for a Lagrangian system (Mt, ds2(t)), Mt ⊂ K3. Here the metric ds2(t), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of ds2(t) in K3 and present symmetries (or isometric motions in K3) of the metric ds2(t) which coincide with the sliding deformation of a surface arising under the geometric realization of ds2(t). We expose the fine structure of a Lie algebra associated with this symmetry transformation and construct the basis of differential invariants. Minimal generating set of differential invariants is derived. We demonstrate that the well-known Taylor microscale λg is a second-order differential invariant and show how λg can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. Finally, we establish that there exists a nontrivial central extension of the infinite-dimensional Lie algebra constructed wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For turbulence, we give the asymptotic expansion of the transversal correlation function for the geometry generated by a quadratic form.

Symmetry transformations of an ideal steady fluid flow determined by a potential function

First, we consider a transformation Ξ of 3D trajectories (fluid particle paths) of inviscid steady flows using the dual stream function approach for the (local) representation of velocity fields u→(x,y,z)=∇ λ×∇ μ. This enables to derive the equation governing the deformation of trajectories by the gradient field ξ→=∇ μ along the surface λ(x, y, z) = λ0. In fact, Ξ is a symmetry transformation and it looks formally like the filament motion which preserves the curvature. Then, we investigate in detail a fine structure of a Lie algebra associated with an extension of the transformation Ξ which creates a visual appearance of sliding stream surfaces λ(x, y, z) = λ0 along itself. The minimal set of generating differential invariants is found. This set consists of a single invariant which coincides with a Hamiltonian function.