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

^{*}

^{*}Corresponding author

- DOI
- 10.1080/14029251.2018.1503447How to use a DOI?
- Keywords
- homogeneous isotropic turbulence; two-point correlation tensor; infinite-dimensional Lie algebra; minimal set of differential invariants
- Abstract
The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field

*B*(_{ij}*x*,*x*′,*t*) and isometries of the correlation space*K*^{3}equipped with a (pseudo-) Riemannian metrics*ds*^{2}(*t*) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies*ds*^{2}(*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 (*M*,^{t}*ds*^{2}(*t*)),*M*⊂^{t}*K*^{3}. Here the metric*ds*^{2}(*t*), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of*ds*^{2}(*t*) in*K*^{3}and present symmetries (or isometric motions in*K*^{3}) of the metric*ds*^{2}(*t*) which coincide with the sliding deformation of a surface arising under the geometric realization of*ds*^{2}(*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*λ*is a second-order differential invariant and show how_{g}*λ*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_{g}*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.- Copyright
- © 2018 The Authors. Published by Atlantis and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

## 1. Introduction

We deal with homogeneous isotropic turbulence and the 2-point velocity correlation tensor field *B _{ij}*(

*x*′,

*x*,

*t*) where

*x*′ and

*x*are the points of a three-dimensional space filled by turbulent fluid. The assumption of homogeneity and isotropy of turbulent flow (invariance with respect to rotation, reflection, and translation) implies that this tensor depends only on the length of the correlation vector

*B*is a symmetric tensor field and takes the diagonal form [11]. Therefore

_{ij}*B*is a metric tensor field given on the correlation space

_{ij}*K*

^{3}and

*K*

^{3}can be equipped with a Riemannian metric. The form of the components of

*B*specifies this metric as the so-called semi-reducible pseudo-Riemannian metric [8]. The components are the correlation functions that evolve due to the von Kármán-Howarth equation [7]. This is conceptually similar to the Ricci flow that acts directly on the metric under consideration. In order to avoid a redundant complexity of the exposition of material, we use only the elementary information about the structure of the two-point velocity correlation tensor of the velocity fluctuations for homogeneous isotropic flows. The modern theory of the properties and structure of second-order (Cartesian) correlation tensors is given in [15]. We only mention that in the course of this development, the authors examined carefully several important misleading or incorrect statements that have remained uncorrected in the literature of the theory of correlation tensors for homogeneous (or isotropic turbulence). Most of these problems arisen because of confusion over the circumstances in which the generating scalar functions can be, or must, pseudoscalar, see for details [15].

_{ij}The aim of this review is to present both geometric and isometry properties of the 2-point velocity correlation tensor field *B _{ij}*(

*x*′,

*x*,

*t*) for the case of homogeneous isotropic turbulence which were obtained in [3]–[6] and demonstrates their impact in turbulence. The review is organized as follows. Section 2 contains the results obtained (see [3] for more details) in a compressed form about the geometric realization of the two-point velocity correlation tensor. In Section 3, we present the Lagrangian system generated by

*ds*

^{2}(

*t*) and show that it is reduced to a Lagrangian system of the one-degree of freedom for each fixed time

*t*due to the second conservation law obtained. The first integrals of the equations of geodesic curves form the kinematic conservation laws. Section 4 is devoted to symmetries (isometric motions in

*K*

^{3}with the structure of a Riemannian manifold) of the metric

*ds*

^{2}(

*t*). In fact, the symmetry transformation looks like as a sliding motion of a surface (the geometric realization of the metric) along itself i.e. the form of this surface is invariant under the symmetry transformation. The properties of an infinite-dimensional Lie algebra obtained are discussed. The basis of differential invariants is constructed and a minimal generating set of differential invariants is derived. The well-known Taylor microscale

*λ*is a second-order differential invariant. We show that

_{g}*λ*can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. In Section 5 we consider the energy-momentum tensor associated with the metric

_{g}*ds*

^{2}and show that the components

*T*(

*z*) and

*T*(

*z*) and

*z*→ ∞ and

*B*for the large values of the correlation distances. The question about the asymptotic expansion of

_{NN}*B*in the physical space

_{NN}*B*which cannot be obtained directly from the von Kármán-Howarth equation in view of the equation in unclosed form and demands at least a suitable modeling. Here we are not concerned these questions.

_{LL}## 2. Geometric Realization of the Two-Point Velocity Corellation Tensor

The results of this Section are given in details in [3].

First we recall the elementary information about the structure of the two-point velocity-correlation tensor of the velocity fluctuations for homogeneous isotropic flows. The modern theory of the properties and structure of second-order (Cartesian) correlation tensors is given in [15].

The statistical description of fluid turbulence employ the Reynolds decomposition to separate the fluid velocity

*x*,

*x*′ are the points of a three-dimensional space filled by turbulent fluid. Rewritten formula (2.1) as

*x*,

*x*′) where

*x*and

*x*′ are the starting point and endpoint correspondingly or

*K*

^{3}with the adjoined vector space of the correlation vectors

*t*i.e.

Moreover for isotropic turbulence,
*B _{ij}* can be expressed by using only the longitudinal correlational function

*B*takes the diagonal form with the components

_{ij}*B*

_{11}=

*B*and

_{LL}*B*

_{22}≡

*B*

_{33}=

*B*in a suitable system of the coordinates of the adjoined vector space. Further instead of directly employing the correlation function

_{NN}*B*and

_{LL}*B*, we use their normalized representations

_{NN}*f*and

*g*where

*dl*

^{2}(

*t*) is an indefinite quadratic form in view of the properties

*g*satisfies the relation (taken from the continuity) [2]

The property that *f* decays faster that

Hence
*f* and *g* are given on Figure 1.

The data presented we use to determine the qualitative behaviors of *f* and *g*, in particular, the algebraic properties of these correlation functions. Thus, we will assume that *f* is a positive everywhere function, *g* changes sign only in interval
*g* is a positive function on
*g* < 0 outside of
*g* means that the quadratic forms *dl*^{2}(*t*) have a variable signature. The normalized longitudinal correlational function

*h* is the normalized triple-correlation function and
*f*, *h* with the turbulence intensity

If we consider in the correlation space *K*^{3} an infinite cylindrical domain. Then the metric (induced by the quadratic form *B _{ij}*) of the surface which bounds this domain takes the form

*ρ*denotes the Euclidean radius of the cross-section {

*a*} ×

*S*

^{1}(

*ρ*) of the surface

*a*∈

*R*. We can account that

*ρ*= 1 and identify this manifold with

*f*,

*g*are non-dimensional with

*f*(0,

*t*) =

*g*(0,

*t*) = 1 and physically

*f*is a positive function such that

*f*→ 0 (

*g*→ 0) as

*f*and

*g*are bounded even functions such that

*f*≤ 1, |

*g*| ≤ 1 and

*f*goes faster to zero than

*f*is acceptable [2] and the map

**L**(

*t*) is determined by the formulas

Now we rewrite the metric *ds*^{2}(*t*) in the frame of the variable *q*

The metric (2.8) admits an one-parametric group of (isometric) motion

The scalar product of the generator *X* equals

*t*

*A point p*_{0} *is called the pole* [8] *of a (pseudo-)Riemannian manifold M if p*_{0} *is a fixed point of a group of diffeomorphisms*
*which acts on M*.

We note that if *p* = *p*_{0} (*p*_{0} is the pole of *g** _{τ}*) then

*X*

^{2}(

*p*

_{0}) = 0 and due to (2.9)

*p*

_{0}coincides with the roots of the equation

*G*(

*q*, ⋅) = 0. Therefore the points

*q*

^{*}∈[−

**L**

^{*}(

*t*),

**L**

^{*}(

*t*)] wherein

*G*vanishes are the poles of

**g**

*. In view of our assumption on*

_{τ}*g*(|

*r*|,

*t*), the equation

*G*(

*q*, ⋅) = 0 has only 4 roots

*i*= 1, …, 4 such that

*t*. This metric determines for

*q*∈

*I*

_{1}the element of length of the surface of revolution in

Therefore the model manifold defined by (2.8) for *q* ∈ *I*_{1} is a cylindrical-type surface
*q*} × *S*^{1}(1) equals *G*^{1/2}(*q*, *t*). For *q* ∈ *I _{i}*,

*i*= 2,3 the function

*G*is negative. The metric

*ds*

^{2}(

*t*) is of a fixed sign. Here the rotation presents the motion along the pseudo-circle of the radius |

*G*(

*q*, ⋅)|

^{1/2},

*q*∈

*I*. Indeed, let us fix the point

_{i}*p*= (

_{a}*q*,

_{a}*ϕ*) on the cross-section {

_{a}*q*} ×

_{a}*S*

^{1}(1) and consider the action of the group

**g**

*on*

_{τ}*p*i.e. the orbit

_{a}*q*} ×

_{a}*S*

^{1}(1) and if

*p*does not coincide with the poles

_{a}**g**

*then*

_{τ}*t*which coincides with the so-called pseudo-circle under the embedding

*q*} ×

_{a}*S*

^{1}(1) of

**L**

^{*}(

*t*) (respectively

*−*

**L**

^{*}(

*t*) and

**g**

*on the point*

_{τ}*p*is a motion along these piecewise linear isotropic curves when

*G*(

*q*,

*t*) with the length of the velocity vector

*p*(or the length of arch), with respect to the vector field generated by

**g**

*(*

_{τ}*p*), is determined by the formula

The constant *χ* can be fixed by normalizing the velocity vector

The Gaussian curvature *K*_{++} of the manifold
*G* vanishes for
*K*_{++} is a singular function at
*t*. If zero is of infinite order then *G*(*q*, *t*) ≡ 0 in a neighborhood of
*G* is an analytical function. The same argument we can apply to investigation of the behavior of the Gaussian curvature of the manifold
*f* and therefore *g* have to go faster to zero than
*f* (and *G*) as
*K*| have to go faster to infinity than

We indicate a connection between the Gaussian curvature *K*_{++} of
*λ _{g}* (see, e.g. [11]) which is defined by

*q*|. Moreover, the formula (2.11) gives the connection between the geometry of

*η*is the Kolmogorov length scale and

The Kolmogorov scale *η* varies with the viscosity *v* and the dissipation of turbulent energy *ε* according to *η* = (*ν*^{3} / *ε*)^{1/4}. In the limit of infinite Reynolds numbers or vanishing the viscosity *ν*, *ν* decreases and the Gaussian curvature *K*_{++} restricted on the cross-section {0} × *S*^{1} grows infinitely. It means that
*q* = 0 which forms the so-called break circle where the manifold loses smoothness.

The peculiarity of the metric presented consists in arising the singularity of “the shrinking cylinder type” due to alternative sign of the transversal correlation function *B _{NN}*. It means that we can describe “shrinking phenomenon” for a singled out cylinder in terms of singularity points of the metric (2.8).

## 3. Lagrangian system

We write the metric *ds*^{2}(*t*) in the conformal form at first for *q* ∈ *I*_{1}. Let us consider on the interval *I*_{1} a new measure *dζ* with the density
*ds*^{2}(*t*) as

*q*∈

*I*,

_{i}*i*= 2,3 this metric takes the form

Here

While for the metric (3.2) given on

Here we use the coordinates (*ζ*, *θ*) for each chart of the manifold *M ^{t}*. Integrability of these systems of equations is established by the usual way. The systems of equations (3.3) and (3.4) admit the following first integrals

correspondingly where **M** and **N** depend on the time *t* in general. Let us consider now the vector
*g** _{τ}*. Here [⋅, ⋅]

_{dl2}denotes the vector product with respect to the metric

*dζ*

^{2}±

*dϕ*

^{2}. The equality

*F*(

*ζ*,

*t*)

_{c}*ϕ*≡

_{θ}**M**= const means that the momentum vector

*t*. On the plane (

*dζ*,

*dϕ*) the momentum vector

*ds*

^{2}(

*t*) with respect to

**g**

*), we can write*

_{τ}*dζ*,

*dϕ*]

_{dl2}equals –1 (1) for the signature (+

*−*)((++)). Using the above-mentioned first integrals or the compatible differential constraints to (3.3),(3.4), the reduction of (3.3) on invariant manifolds defined by these constraints leads to the following equation

*F*(

*ζ*,

*t*)

*ϕ*=

_{θ}**M**for the different signature. As a result, we obtain

*t*as a parameter. Equation (3.5) coincides with the well-known equation of the motion of a unit mass point in the potential field with the effective potential energy

*V*(the terminology of Newtonian mechanics is used). Then the Lagrangian

Here
*t*. Therefore the couple

*dθ*leads to the following equation for

*ζ*=

*ζ*(

*ϕ*)

Geometrically, the parameter *θ* is the length of arch of a non-isotropic geodesic curve with the initial position at the corresponding pole. If this length is changed then in the course of during this process the “total mechanical” energy and momentum are preserved quantities along geodesic curves. The angle *α* between the vector

The case of cos *α* = 0 corresponds to the motion along the orbit
*F* ≡ const) and this orbit presents unclosed curve in the case of
*i* = 2,3. Notice that in the sharp contrast to the previous consideration about the dynamics of a fluid particle on the surface of revolution
*ds*^{2}(*t*) = 0) which is given by *ϕ* = *ζ*−*ζ*_{0} (or *ϕ* = −(*ζ*−*ζ*_{0})). This curve coincides with the pseudo-circle of zero radius. The canonical parameter *θ* (the length of the corresponding isotropic curve) is defined by the formula

Here (*ζ*_{0}, 0) is the coordinate of the corresponding pole of
*V*.

The above-mentioned conservation laws are standard consequence of invariance of the flow with respect to the translation and rotation groups. Note that the quantity

## 4. Isometries

We consider the functional of simple action in (*M ^{t}*,

*ds*

^{2}) and present isometric motions in

*K*

^{3}. The peculiarity of these transformations consists in: we look for their in the class of equivalence transformations that preserve the length of tangent vectors

## 4.1. Equivalence transformation

Instead of the variables (*ρ*, *ϕ*), we consider

Consider the set of piecewise smooth curves
*t*. Then the formula

*τ*is the so-called natural parameter i.e

Therefore

*ξ*,

*η*). Consider an infinitesimal transformation of the variables

*ξ*and

*η*

*ξ*,

*η*), we consider the co-vector (

*τ*,

_{ξ}*τ*) defined by the formulas (for brevity we omitted the index

_{η}*τ*)

Then (4.1) is transformed into

This equation is the eikonal-type one (the relativistic eikonal equation for the signature (+−)). Therefore in order to find isometries of
*τ* invariant. The restatement of the variational symmetry in the terms of symmetry of partial differential equations (4.5) enables us to extend the class of isometry transformations admitted by the functional
*ξ*, *η*, *u*^{1}, *u*^{2}) space where *u*^{1} = *τ*, *u*^{2} = Λ^{2}. Infinitesimally, we look for an operator in the following form [12], [13]

Here *Y*_{1} denotes the first prolongation of *Y*. In the case of the signature (++) the coefficients of the operator *Y* have been calculated in [12]. For the relativistic eikonal equation i.e. for the signature (+−) these calculations are given in [5]. The infinitesimal operator *Y* reads:

Its Lie (infinite-dimensional) subalgebra is of the form

*τ*is a scalar invariant of

*X*. Therefore

*X*is a symmetry operator admitted by the functional

*ξ*,

*η*) and Ψ(

*ξ*,

*η*) satisfy the Cauchy-Riemann differential equations Φ

*= Ψ*

_{ξ}*and Φ*

_{η}*= −Ψ*

_{η}*in the case of the eikonal equation and Φ*

_{ξ}*= Ψ*

_{ξ}*, Φ*

_{η}*= Ψ*

_{η}*(the so-called*

_{ξ}*h*-conjugate functions [10]) for the relativistic eikonal equation. To expose a fine structure of the equivalence transformation generated by

*X*, we consider the case of the signature (++) and the complex coordinates

*z*=

*ξ*+

*iη*and

The operator *X* takes the form

Here
*F*. The tangent space is spanned by

For small perturbations
*X*_{1}. Infinitesimal holomorphic transformations of the variables *z* and

Using the Laurent series

Therefore the basis of the operator *X*_{1}

The factor Λ^{2} is transformed into

*z*↦

*z*

^{*}and

The linear hull of the direct sum
**W**. The direct calculation gives the following commutations relations

The last relation follows from the formulas

The algebra **W** is isomorphic two copies of the Witt algebra *W*.

Notice that {*k*_{0}, *k*_{±1}} form a subalgebra of **W** and its projection {*l*_{0}, *l*_{±1}} isomorphic to *sl*(2, *C*). The transformations of the variables *z* and
*z* ↦ *z*^{*} consists in the set Mb of Möbius transformations *φ*[17]:

For the transformations
*C*). Mb is isomorphic to the group Aut(P) of all biholomorphic maps of the Riemann sphere *P*. It is used the compactification

Consider the infinitesimal operator (4.10) for the signature (+−) of the metric. It leads to the following relationships for the coordinates of the operator *X*_{1}

It gives the transformation

*u*=

_{ξ}*v*,

_{η}*u*=

_{η}*v*and generates the Lie group of orientation-preserving conformal diffeomorphisms of the Minkowski plane

_{ξ}**F**:

**F**and all its derivatives on compact subsets

*S*

^{1}is the unit circle. The conformal group Conf

_{+}(

*S*

^{1}) × Diff

_{+}(

*S*

^{1}). Diff

_{+}(

*S*

^{1}) turns out to be a Lie group with models in the Fréchet space of smooth real-valued functions

**F**:

*S*

^{1}↦ R endowed with the uniform convergence on

*S*

^{1}of

**F**and all its derivatives. The corresponding Lie algebra Lie(Diff

_{+}(

*S*

^{1})) is the Lie algebra of smooth vector space fields Vect(

*S*

^{1}) (see, e.g. [17]). A finite dimensional counterpart of

*Y*∈ Vect(

*S*

^{1}) has the form

*Y*=

*y*(θ) ∂ / ∂

*θ*, where

*z*of

*S*

^{1}are represented as

*z*=

*e*. The representation of

^{iθ}*y*(

*θ*) by a convergent Fourier series

*S*

^{1}):

Consider the complexification of Vect(*S*^{1})

*S*

^{1}. Define

The liner hull of the
*W* is a part of the complexified Lie algebra

^{2}. The commutation relations for

*k*.

_{n}and k_{m}## 4.2. Invariant differentiations

Now we define the operators of invariant differentiations. Below we use the symbols ++ and +− which corresponds to the consideration of **G**_{++} and **G**_{+}_{−}. In the variables *ξ* and *η*, the operators of invariant differentiations are determined by the formula [14]

**c ^{i}** are chosen to be

**c**

^{1}=(0,1,0),

**c**

^{2}=(0,1,1).

*J*

^{++}and

*J*

^{+−}are found as solutions of the equations

_{λ}=λ

_{1}∂ / ∂λ

_{1}+λ

_{2}∂ / ∂λ

_{2}and correspondingly ∂

*=*

_{μ}*μ*

_{1}∂ / ∂

*μ*

_{1}+

*μ*

_{2}∂ / ∂

*μ*

_{2}. Here

*X*

_{1}for the signatures (++) and (+−) correspondingly. These operators read

*=Ψ*

_{ξ}*, Φ*

_{η}*=−Ψ*

_{η}*for the operator*

_{ξ}*h*-holomorphic conditions Φ

*= Ψ*

_{ξ}*, Φ*

_{η}*= Ψ*

_{η}*in the case of the operator*

_{ξ}What about equation (4.14), we have the following two functionally independent solutions

We give the following comments to these relationships obtained. Let us consider nontrivial solutions of the equation

As a result, we obtain the operators of invariant differentiations

The scalar invariants of the first-order of **G**_{++} and **G**_{+}_{−} are defined by

Splitting these equations, we can easily find the following functional-independent solutions

Therefore the universal first-order differential invariants
**G**_{++} and Lie group **G**_{+}_{−} are of the form

The actions of
*ξ*, *η*) to the geodesic curves of length equals *τ*. It means that the geodesic curves under the action of **G**_{++} and **G**_{+}_{−} transformed into the geodesic curves.

According to Tresse theorem [14], using the operators
**G**_{++} and **G**_{+}_{−} by functional operations and the invariant differentiation. In particular, the Gaussian curvature *K* reads

*K*

_{++},

*K*

_{+}

_{−}are the differential invariants of the second order of

**G**

_{++}and

**G**

_{+}

_{−}correspondingly. This fact is checked by direct calculations which show that

*X*for the signature (++) and (+−) correspondingly. These two differential invariants

*K*

_{++}and K

_{+}

_{−}can be expressed in the form

*K*

_{++}and the Taylor microscale

*λ*and the result above, we can claim that the Taylor microscale is a differential invariant of

_{g}**G**

_{++}.

## 4.3. Local shape structure of
M I i t

Consider the manifold
**G**_{++} that preserves the form of the metric and generates infinitesimal deformations of the surface

Therefore we have infinite number of transformations of the form

We establish that these transformations generates sliding

Here *H*, *K* are the average and Gaussian curvature, *L*, *M* and *N* are the coefficients of the second quadratic form. The conformal transformations conserve the form of metrics, it means that we have *h* ≡ 0. This factum also follows from the well-known result about the transformation of metrics to the canonical (conformal) form by a quasi-conformal mapping *ω* = *ω*(*z*), *z* = *ξ* + *iη* which satisfies the Beltrami equation

The conformal property of the transformation
*p* and therefore *h* = 0. Here Λ^{2}(*ξ*, *η*) is transformed to

This assertion we use to classify the shape of
*u _{α}*,

*α*= 1,2 and then

*w*=

*u*

_{1}+

*iu*

_{2}and define for the positive Gaussian curvature

*K*

_{++}> 0 the function

Here

The derivation of equation (4.20) from the kinematic system of equations (4.19) is given in [18] The direct calculations of the Cristoffel symbols in (4.21) show that

Therefore the function *W* is a holomorphic function and the complex function of displacement *w* = *w*(*z*) reads

*z*) is a holomorphic function. In the case when

### Theorem 4.1 ([18]).

*The condition B* ≡ 0 *is satisfied for the second-order algebraic surfaces of a positive Gaussian curvature and only for such surfaces*.

As a corollary of this theorem, we can claim that in each chart where
*K*_{++} < 0 there is no similar result to classify surfaces of negative Gaussian curvature. It is well-known that each second-order algebraic surface of revolution with positive Gauss curvature *K*_{++} > 0 can be transformed by an infinitesimal pressure along any parallel into the surface of variable curvature. If this infinitesimal pressure is not scaling then we get a surface which different from the second-order algebraic surfaces.

Consider now the manifolds

This change of coordinates is simply a rotation on the angle π/2. Let us perform a Wick rotation *η* ↦ *iη*, *i*^{2} = −1 that results in

*i*

^{2}= −1,

*j*= 2, 3 a transformation of

## 5. Asymptotic Expansion of the Transversal Correlation Function

Consider in *K*^{3} an infinite cylindrical domain. Fix the cross-section {0} × *D*^{2} of this domain where *D*^{2} is a two-dimensional disk. Then the quadratic form *dl*^{2}(*t*) induces on the above-mentioned cross-section the metric

^{2},

*v*,

*w*) of piecewise smooth curves

*γ*:

*J*→

*D*

^{2}with fixed endpoints

*γ*(0) =

*v*and

*γ*(1) =

*w*and the functional of action of the trajectory

*γ*

We can rewrite it in the following form

The classical energy-momentum tensor is defined by

*τ*this tensor takes the form

*T _{ik}* is a traceless tensor due to the equality

*j*=

_{μ}*T*and has an automatically vanishing divergence

_{μν}δω^{ν}*T*. Since

_{μν}*T*is traceless, we have

_{ik}*T*(

*z*) =

*T*(

_{zz}*z*) = 1/4 (

*T*

_{11}

*−*2

*iT*

_{12}

*−T*

_{22}) and

*T*(

*z*) and

*n*−2 is chosen so that for the scale transformation

*z*↦

*λ*

^{−1}

*z*under which

*T*(

*z*) ↦

*λ*

^{2}

*T*(

*λ*

^{−1}

*z*) we have

*L*

_{−n}↦

*λ*

^{n}L_{−n}, and

Recall that *L _{n}* are defined on

It leads to the commutation relations [1]

Then from (5.7),(5.8) follow that *L _{n}* and

This algebra is the Lie algebra of the central extension of the group of diffeomorphisms of the circle whose basic elements {*L _{n}*},

*n*∈

*Z*. Here

*δ*= 1 for

_{k}*k*= 0 and

*δ*= 0 for

_{k}*k*≠ 0. The quantity

*c*is known as the central charge.

### Theorem 5.1 ([1]).

*Let*
*and*
*be the fields with* {*L _{n}*}

*satisfying the commutation relationship*(5.9).

*Then for large z and w it holds*

The proof of this Theorem is based on the direct calculations. The first term of the right-hand side of (5.9) equals

The left-hand side of (5.9) reads

This integral can be rewritten as

Indeed, integrating the last term of this integral by parts and combining with the first term we get the right-hand part of (5.12). The second term of (5.9) that is

Further transformations are based on using the well-known formula from the Complex Analysis

Since

Applying the same procedure to the first and the second terms of (5.14), we can see that this integral equals

Combining (5.17) and (5.18), we found that the integral

*R*(

*z, w*) on the variable

*z*–

*w*since this function are free of poles at

*z*=

*w*, hence

*R*(

*z, w*) does not contribute to the integral. Therefore comparing (5.19) with (5.13), it is derived that

This relationship is called *the operator product expansion* in CFT. The same assertion are hold for the
*c* and

### Corollary 5.1.

We apply (5.20) to find asymptotic expansion the transversal correlation function

*T*(

*z*) =

*T*= (∂

_{zz}*)*

_{z}τ^{2}and

Since

The sign “−” appears in view of the negative values of

We presented the asymptotic behavior of the transversal correlation function *B _{NN}* as

*K*

^{3}with

## References

### Cite this article

TY - JOUR AU - Vladimir N. Grebenev AU - Martin Oberlack PY - 2021 DA - 2021/01/06 TI - Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor JO - Journal of Nonlinear Mathematical Physics SP - 650 EP - 672 VL - 25 IS - 4 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2018.1503447 DO - 10.1080/14029251.2018.1503447 ID - Grebenev2021 ER -